Subgroups — C₇×#19756 #13196972

Order 1792.

808 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3358	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×D₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄×D₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3274	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1721	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1685	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1681	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1668	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1667	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1639	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1627	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1621	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1619	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1581	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₇×AGL(1,ℤ/8)	—	2	1	no	—
17	C₂²×C₇×SD₁₆	—	2	1	no	—
18	C₂²×C₇×D₈	—	2	1	no	—
19	C₇×#550	C₂²	1	1	yes	characteristic, complemented
20	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic, complemented
21	C₇×D₄²	C₂²	1	1	yes	characteristic
22	C₇×D₄²	C₂²	1	1	yes	characteristic
23	C₇×#493	C₂²	1	1	yes	characteristic
24	C₇×#493	C₂²	1	1	yes	characteristic, complemented
25	C₇×#493	C₂²	1	1	yes	characteristic, complemented
26	C₇×#492	C₂²	1	1	yes	characteristic, complemented
27	C₇×#486	—	2	1	no	—
28	C₇×#486	C₂²	1	1	yes	characteristic, complemented
29	C₇×#486	C₂²	1	1	yes	characteristic, complemented
30	C₇×#474	—	2	1	no	—
31	C₇×#474	C₂²	1	1	yes	characteristic, complemented
32	C₇×#469	—	2	1	no	—
33	C₇×#466	—	2	1	no	—
34	C₇×#465	—	2	1	no	—
35	C₇×#461	C₂²	1	1	yes	characteristic
36	C₇×#460	C₂²	1	1	yes	characteristic, complemented
37	C₇×#460	C₂²	1	1	yes	characteristic, complemented
38	C₇×#459	—	2	1	no	—
39	C₇×#459	C₂²	1	1	yes	characteristic
40	C₇×#459	C₂²	1	1	yes	characteristic
41	C₇×#459	C₂²	1	1	yes	characteristic
42	C₇×#459	C₂²	1	1	yes	characteristic, complemented
43	C₇×#450	—	2	1	no	—
44	C₇×#449	—	2	1	no	—
45	C₇×#448	—	2	1	no	—
46	C₇×#447	—	2	1	no	—
47	C₇×#447	—	2	1	no	—
48	C₇×#447	—	2	1	no	—
49	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
50	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
51	C₄×C₇×D₈	—	2	1	no	—
52	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
53	C₇×#436	C₂²	1	1	yes	characteristic, complemented
54	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
55	C₇×C₈×D₄	C₂²	1	1	yes	characteristic, complemented
56	C₇×#335	C₂²	1	1	yes	characteristic, complemented
57	C₇×#335	C₂²	1	1	yes	characteristic, complemented
58	C₇×#332	C₂²	1	1	yes	characteristic, complemented
59	C₇×#331	C₂²	1	1	yes	characteristic, complemented
60	C₇×#331	C₂²	1	1	yes	characteristic, complemented
61	C₇×#331	C₂²	1	1	yes	characteristic, complemented
62	C₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
64	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
65	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
66	C₇×#322	C₂²	1	1	yes	characteristic, complemented
67	#19756	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₇×#41	—	2	1	no	—
69	C₂²×C₇×Q₈	—	2	1	no	—
70	C₂²×C₇×D₄	—	2	1	no	—
71	C₂²×C₇×D₄	—	2	1	no	—
72	C₂²×C₇×D₄	—	2	1	no	—
73	C₂²×C₇×D₄	—	2	1	no	—
74	C₇×AGL(1,ℤ/8)	—	4	4	no	—
75	C₂×C₇×SD₁₆	—	2	1	no	—
76	C₂×C₇×SD₁₆	—	2	1	no	—
77	C₂×C₇×SD₁₆	—	2	1	no	—
78	C₂×C₇×SD₁₆	—	2	1	no	—
79	C₂×C₇×SD₁₆	—	2	1	no	—
80	C₂×C₇×SD₁₆	—	4	4	no	—
81	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
82	C₂×C₇×D₈	—	2	1	no	—
83	C₂×C₇×D₈	—	2	1	no	—
84	C₂×C₇×D₈	—	2	1	no	—
85	C₂×C₇×D₈	—	2	1	no	—
86	C₂×C₇×D₈	—	2	1	no	—
87	C₂×C₇×D₈	—	2	1	no	—
88	C₂×C₇×D₈	—	2	1	no	—
89	C₂×C₇×D₈	—	2	1	no	—
90	C₂×C₇×D₈	—	4	1	no	—
91	C₂×C₇×D₈	—	4	4	no	—
92	C₂×C₇×D₈	D₄	1	1	yes	characteristic
93	C₂×C₇×M₁₆	—	2	1	no	—
94	C₂²×C₇×C₈	—	2	1	no	—
95	C₂²×C₇×C₈	—	2	1	no	—
96	C₇×C₄⋊Q₈	—	2	1	no	—
97	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
98	C₇×#127	—	2	1	no	—
99	C₇×#127	C₂³	1	1	yes	characteristic
100	C₇×#127	C₂³	1	1	yes	characteristic
101	C₇×#127	C₂³	1	1	yes	characteristic
102	C₇×C₂²⋊Q₈	—	2	1	no	—
103	C₇×C₂²⋊Q₈	—	4	1	no	—
104	C₇×#121	—	2	1	no	—
105	C₇×#121	—	2	1	no	—
106	C₇×#121	—	2	1	no	—
107	C₇×#121	—	2	1	no	—
108	C₇×#121	—	2	1	no	—
109	C₇×#121	—	4	1	no	—
110	C₇×#120	—	4	1	no	—
111	C₇×#120	—	4	1	no	—
112	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
113	C₄×C₇×D₄	—	2	1	no	—
114	C₄×C₇×D₄	—	2	1	no	—
115	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
116	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
117	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
118	C₇×#107	—	2	1	no	—
119	C₇×#107	D₄	1	1	yes	characteristic
120	C₇×#106	D₄	1	1	yes	characteristic
121	C₇×C₄⋊C₈	—	2	1	no	—
122	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
123	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
124	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
125	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
126	C₇×#103	—	2	1	no	—
127	C₇×#103	D₄	1	1	yes	characteristic
128	C₇×#102	—	2	1	no	—
129	C₇×#102	—	2	1	no	—
130	C₇×#102	—	2	1	no	—
131	C₇×#102	—	2	1	no	—
132	C₇×#102	—	2	1	no	—
133	C₇×#102	—	2	1	no	—
134	C₇×#102	—	2	1	no	—
135	C₇×#102	—	2	1	no	—
136	C₇×#102	—	2	1	no	—
137	C₇×#102	—	4	1	no	—
138	C₇×#102	D₄	1	1	yes	characteristic
139	C₇×#102	D₄	1	1	yes	characteristic
140	C₇×#102	D₄	1	1	yes	characteristic
141	C₇×C₂²⋊C₈	—	2	1	no	—
142	C₇×C₂²⋊C₈	—	2	1	no	—
143	C₇×C₂²⋊C₈	—	2	1	no	—
144	C₄×C₇×C₈	—	2	1	no	—
145	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
146	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
147	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
148	#3358	C₂×C₇	1	1	yes	characteristic, complemented
149	D₄×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
150	D₄×D₈	C₂×C₇	1	1	yes	characteristic, complemented
151	#3274	C₂×C₇	1	1	yes	characteristic, complemented
152	#1721	C₂×C₇	1	1	yes	characteristic, complemented
153	#1685	C₂×C₇	1	1	yes	characteristic, complemented
154	#1681	C₂×C₇	1	1	yes	characteristic, complemented
155	#1668	C₂×C₇	1	1	yes	characteristic, complemented
156	#1667	C₂×C₇	1	1	yes	characteristic, complemented
157	#1639	C₂×C₇	1	1	yes	characteristic, complemented
158	#1627	C₂×C₇	1	1	yes	characteristic, complemented
159	#1621	C₂×C₇	1	1	yes	characteristic, complemented
160	#1619	C₂×C₇	1	1	yes	characteristic, complemented
161	#1581	C₂×C₇	1	1	yes	characteristic, complemented
162	SD₁₆⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
163	C₂⁴×C₇	—	4	1	no	—
164	C₂⁴×C₇	—	4	1	no	—
165	C₇×#41	—	4	2	no	—
166	C₇×#41	—	8	1	no	—
167	C₂×C₇×Q₈	—	2	1	no	—
168	C₂×C₇×Q₈	—	4	1	no	—
169	C₂×C₇×Q₈	—	4	2	no	—
170	C₂×C₇×Q₈	—	8	1	no	—
171	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
172	C₂×C₇×D₄	—	2	1	no	—
173	C₂×C₇×D₄	—	2	1	no	—
174	C₂×C₇×D₄	—	2	1	no	—
175	C₂×C₇×D₄	—	2	1	no	—
176	C₂×C₇×D₄	—	2	1	no	—
177	C₂×C₇×D₄	—	2	1	no	—
178	C₂×C₇×D₄	—	2	1	no	—
179	C₂×C₇×D₄	—	2	1	no	—
180	C₂×C₇×D₄	—	2	1	no	—
181	C₂×C₇×D₄	—	4	1	no	—
182	C₂×C₇×D₄	—	4	1	no	—
183	C₂×C₇×D₄	—	4	1	no	—
184	C₂×C₇×D₄	—	4	1	no	—
185	C₂×C₇×D₄	—	4	1	no	—
186	C₂×C₇×D₄	—	4	1	no	—
187	C₂×C₇×D₄	—	4	2	no	—
188	C₂×C₇×D₄	—	4	2	no	—
189	C₂×C₇×D₄	—	4	2	no	—
190	C₂×C₇×D₄	—	4	2	no	—
191	C₂×C₇×D₄	—	8	1	no	—
192	C₂×C₇×D₄	—	8	1	no	—
193	C₂×C₇×D₄	—	8	1	no	—
194	C₂×C₇×D₄	—	8	1	no	—
195	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
196	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
197	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₇	—	2	1	no	—
199	C₂²×C₄×C₇	—	2	1	no	—
200	C₂²×C₄×C₇	—	2	1	no	—
201	C₂²×C₄×C₇	—	4	1	no	—
202	C₂²×C₄×C₇	—	4	1	no	—
203	C₇×SD₁₆	—	4	2	no	—
204	C₇×SD₁₆	—	4	2	no	—
205	C₇×SD₁₆	—	4	2	no	—
206	C₇×SD₁₆	—	4	2	no	—
207	C₇×SD₁₆	—	8	1	no	—
208	C₇×SD₁₆	D₈	1	4	yes	complemented
209	C₇×D₈	—	2	2	no	—
210	C₇×D₈	—	4	2	no	—
211	C₇×D₈	—	4	2	no	—
212	C₇×D₈	—	4	2	no	—
213	C₇×D₈	—	4	2	no	—
214	C₇×D₈	—	4	2	no	—
215	C₇×D₈	—	4	2	no	—
216	C₇×D₈	—	8	1	no	—
217	C₇×D₈	—	8	1	no	—
218	C₇×D₈	—	8	2	no	—
219	C₇×M₁₆	—	4	2	no	—
220	C₂×C₇×C₈	—	2	1	no	—
221	C₂×C₇×C₈	—	2	1	no	—
222	C₂×C₇×C₈	—	2	1	no	—
223	C₂×C₇×C₈	—	2	1	no	—
224	C₂×C₇×C₈	—	2	1	no	—
225	C₂×C₇×C₈	—	4	2	no	—
226	C₂×C₇×C₈	—	4	2	no	—
227	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
228	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
229	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
230	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
231	C₇×C₄⋊C₄	—	2	1	no	—
232	C₇×C₄⋊C₄	—	2	1	no	—
233	C₇×C₄⋊C₄	—	2	1	no	—
234	C₇×C₄⋊C₄	—	4	1	no	—
235	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
236	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
237	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
238	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
239	C₇×C₂²⋊C₄	—	2	1	no	—
240	C₇×C₂²⋊C₄	—	2	1	no	—
241	C₇×C₂²⋊C₄	—	2	1	no	—
242	C₇×C₂²⋊C₄	—	4	1	no	—
243	C₇×C₂²⋊C₄	—	4	1	no	—
244	C₇×C₂²⋊C₄	—	4	1	no	—
245	C₇×C₂²⋊C₄	—	4	1	no	—
246	C₄²×C₇	—	2	1	no	—
247	C₄²×C₇	C₂⁴	1	1	yes	characteristic
248	C₂×AGL(1,ℤ/8)	—	2	1	no	—
249	C₂²×SD₁₆	—	2	1	no	—
250	C₂²×D₈	—	2	1	no	—
251	#550	C₂²×C₇	1	1	yes	characteristic, complemented
252	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
253	D₄²	C₂²×C₇	1	1	yes	characteristic
254	D₄²	C₂²×C₇	1	1	yes	characteristic
255	#493	C₂²×C₇	1	1	yes	characteristic
256	#493	C₂²×C₇	1	1	yes	characteristic, complemented
257	#493	C₂²×C₇	1	1	yes	characteristic, complemented
258	#492	C₂²×C₇	1	1	yes	characteristic, complemented
259	#486	—	2	1	no	—
260	#486	C₂²×C₇	1	1	yes	characteristic, complemented
261	#486	C₂²×C₇	1	1	yes	characteristic, complemented
262	#474	—	2	1	no	—
263	#474	C₂²×C₇	1	1	yes	characteristic, complemented
264	#469	—	2	1	no	—
265	#466	—	2	1	no	—
266	#465	—	2	1	no	—
267	#461	C₂²×C₇	1	1	yes	characteristic
268	#460	C₂²×C₇	1	1	yes	characteristic, complemented
269	#460	C₂²×C₇	1	1	yes	characteristic, complemented
270	#459	—	2	1	no	—
271	#459	C₂²×C₇	1	1	yes	characteristic
272	#459	C₂²×C₇	1	1	yes	characteristic
273	#459	C₂²×C₇	1	1	yes	characteristic
274	#459	C₂²×C₇	1	1	yes	characteristic, complemented
275	#450	—	2	1	no	—
276	#449	—	2	1	no	—
277	#448	—	2	1	no	—
278	#447	—	2	1	no	—
279	#447	—	2	1	no	—
280	#447	—	2	1	no	—
281	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
282	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
283	C₄×D₈	—	2	1	no	—
284	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
285	#436	C₂²×C₇	1	1	yes	characteristic, complemented
286	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
287	C₈×D₄	C₂²×C₇	1	1	yes	characteristic, complemented
288	#335	C₂²×C₇	1	1	yes	characteristic, complemented
289	#335	C₂²×C₇	1	1	yes	characteristic, complemented
290	#332	C₂²×C₇	1	1	yes	characteristic, complemented
291	#331	C₂²×C₇	1	1	yes	characteristic, complemented
292	#331	C₂²×C₇	1	1	yes	characteristic, complemented
293	#331	C₂²×C₇	1	1	yes	characteristic, complemented
294	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
295	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
296	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
297	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
298	#322	C₂²×C₇	1	1	yes	characteristic, complemented
299	C₂³×C₇	—	2	1	no	—
300	C₂³×C₇	—	2	1	no	—
301	C₂³×C₇	—	2	1	no	—
302	C₂³×C₇	—	4	1	no	—
303	C₂³×C₇	—	4	1	no	—
304	C₂³×C₇	—	4	1	no	—
305	C₂³×C₇	—	8	2	no	—
306	C₂³×C₇	—	8	2	no	—
307	C₂³×C₇	—	8	2	no	—
308	C₂³×C₇	—	8	2	no	—
309	C₂³×C₇	—	16	1	no	—
310	C₂³×C₇	—	16	1	no	—
311	C₇×Q₈	—	2	1	no	—
312	C₇×Q₈	—	4	1	no	—
313	C₇×Q₈	—	4	1	no	—
314	C₇×Q₈	—	8	2	no	—
315	C₇×Q₈	C₂×D₈	1	2	yes	complemented
316	C₇×D₄	—	2	1	no	—
317	C₇×D₄	—	2	1	no	—
318	C₇×D₄	—	2	1	no	—
319	C₇×D₄	—	2	1	no	—
320	C₇×D₄	—	4	1	no	—
321	C₇×D₄	—	4	1	no	—
322	C₇×D₄	—	4	1	no	—
323	C₇×D₄	—	4	1	no	—
324	C₇×D₄	—	4	1	no	—
325	C₇×D₄	—	4	1	no	—
326	C₇×D₄	—	4	1	no	—
327	C₇×D₄	—	4	1	no	—
328	C₇×D₄	—	4	1	no	—
329	C₇×D₄	—	4	1	no	—
330	C₇×D₄	—	4	1	no	—
331	C₇×D₄	—	4	1	no	—
332	C₇×D₄	—	8	2	no	—
333	C₇×D₄	—	8	1	no	—
334	C₇×D₄	—	8	1	no	—
335	C₇×D₄	—	8	1	no	—
336	C₇×D₄	—	8	2	no	—
337	C₇×D₄	—	8	2	no	—
338	C₇×D₄	—	8	2	no	—
339	C₇×D₄	—	8	2	no	—
340	C₇×D₄	—	16	1	no	—
341	C₇×D₄	C₂×D₈	1	2	yes	—
342	C₇×D₄	AGL(1,ℤ/8)	1	2	yes	complemented
343	C₂×C₄×C₇	—	2	1	no	—
344	C₂×C₄×C₇	—	2	1	no	—
345	C₂×C₄×C₇	—	2	1	no	—
346	C₂×C₄×C₇	—	2	1	no	—
347	C₂×C₄×C₇	—	2	1	no	—
348	C₂×C₄×C₇	—	4	1	no	—
349	C₂×C₄×C₇	—	4	1	no	—
350	C₂×C₄×C₇	—	4	1	no	—
351	C₂×C₄×C₇	—	4	1	no	—
352	C₂×C₄×C₇	—	4	1	no	—
353	C₂×C₄×C₇	—	4	1	no	—
354	C₂×C₄×C₇	—	4	1	no	—
355	C₂×C₄×C₇	—	8	2	no	—
356	C₂×C₄×C₇	—	8	2	no	—
357	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
358	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
359	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
360	C₇×C₈	—	2	1	no	—
361	C₇×C₈	—	2	1	no	—
362	C₇×C₈	—	2	1	no	—
363	C₇×C₈	—	4	1	no	—
364	C₇×C₈	—	4	1	no	—
365	C₇×C₈	—	4	1	no	—
366	C₇×C₈	—	4	1	no	—
367	C₇×C₈	—	4	1	no	—
368	C₇×C₈	C₂×D₈	1	2	yes	complemented
369	C₂×#41	—	2	1	no	—
370	C₂²×Q₈	—	2	1	no	—
371	C₂²×D₄	—	2	1	no	—
372	C₂²×D₄	—	2	1	no	—
373	C₂²×D₄	—	2	1	no	—
374	C₂²×D₄	—	2	1	no	—
375	AGL(1,ℤ/8)	—	4	4	no	—
376	C₂×SD₁₆	—	2	1	no	—
377	C₂×SD₁₆	—	2	1	no	—
378	C₂×SD₁₆	—	2	1	no	—
379	C₂×SD₁₆	—	2	1	no	—
380	C₂×SD₁₆	—	2	1	no	—
381	C₂×SD₁₆	—	4	4	no	—
382	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
383	C₂×D₈	—	2	1	no	—
384	C₂×D₈	—	2	1	no	—
385	C₂×D₈	—	2	1	no	—
386	C₂×D₈	—	2	1	no	—
387	C₂×D₈	—	2	1	no	—
388	C₂×D₈	—	2	1	no	—
389	C₂×D₈	—	2	1	no	—
390	C₂×D₈	—	2	1	no	—
391	C₂×D₈	—	4	1	no	—
392	C₂×D₈	—	4	4	no	—
393	C₂×D₈	C₇×D₄	1	1	yes	characteristic
394	C₂×M₁₆	—	2	1	no	—
395	C₂²×C₈	—	2	1	no	—
396	C₂²×C₈	—	2	1	no	—
397	C₄⋊Q₈	—	2	1	no	—
398	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
399	#127	—	2	1	no	—
400	#127	C₂³×C₇	1	1	yes	characteristic
401	#127	C₂³×C₇	1	1	yes	characteristic
402	#127	C₂³×C₇	1	1	yes	characteristic
403	C₂²⋊Q₈	—	2	1	no	—
404	C₂²⋊Q₈	—	4	1	no	—
405	#121	—	2	1	no	—
406	#121	—	2	1	no	—
407	#121	—	2	1	no	—
408	#121	—	2	1	no	—
409	#121	—	2	1	no	—
410	#121	—	4	1	no	—
411	#120	—	4	1	no	—
412	#120	—	4	1	no	—
413	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
417	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
418	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
419	#107	—	2	1	no	—
420	#107	C₇×D₄	1	1	yes	characteristic
421	#106	C₇×D₄	1	1	yes	characteristic
422	C₄⋊C₈	—	2	1	no	—
423	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
424	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
425	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
426	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
427	#103	—	2	1	no	—
428	#103	C₇×D₄	1	1	yes	characteristic
429	#102	—	2	1	no	—
430	#102	—	2	1	no	—
431	#102	—	2	1	no	—
432	#102	—	2	1	no	—
433	#102	—	2	1	no	—
434	#102	—	2	1	no	—
435	#102	—	2	1	no	—
436	#102	—	2	1	no	—
437	#102	—	2	1	no	—
438	#102	—	4	1	no	—
439	#102	C₇×D₄	1	1	yes	characteristic
440	#102	C₇×D₄	1	1	yes	characteristic
441	#102	C₇×D₄	1	1	yes	characteristic
442	C₂²⋊C₈	—	2	1	no	—
443	C₂²⋊C₈	—	2	1	no	—
444	C₂²⋊C₈	—	2	1	no	—
445	C₄×C₈	—	2	1	no	—
446	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
447	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
449	C₂²×C₇	—	2	2	no	—
450	C₂²×C₇	—	2	2	no	—
451	C₂²×C₇	—	4	1	no	—
452	C₂²×C₇	—	4	1	no	—
453	C₂²×C₇	—	4	1	no	—
454	C₂²×C₇	—	4	1	no	—
455	C₂²×C₇	—	4	1	no	—
456	C₂²×C₇	—	4	1	no	—
457	C₂²×C₇	—	4	1	no	—
458	C₂²×C₇	—	8	1	no	—
459	C₂²×C₇	—	8	1	no	—
460	C₂²×C₇	—	8	1	no	—
461	C₂²×C₇	—	8	1	no	—
462	C₂²×C₇	—	8	1	no	—
463	C₂²×C₇	—	8	1	no	—
464	C₂²×C₇	—	8	1	no	—
465	C₂²×C₇	—	8	1	no	—
466	C₂²×C₇	—	8	1	no	—
467	C₂²×C₇	—	16	4	no	—
468	C₂²×C₇	—	16	4	no	—
469	C₂²×C₇	D₄²	1	1	yes	characteristic
470	C₄×C₇	—	2	2	no	—
471	C₄×C₇	—	2	1	no	—
472	C₄×C₇	—	4	1	no	—
473	C₄×C₇	—	4	1	no	—
474	C₄×C₇	—	4	1	no	—
475	C₄×C₇	—	4	1	no	—
476	C₄×C₇	—	8	1	no	—
477	C₄×C₇	C₂²×D₈	1	1	yes	characteristic
478	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
479	C₄×C₇	#576	1	1	yes	characteristic
480	C₄×C₇	#577	1	1	yes	characteristic
481	C₂⁴	—	4	1	no	—
482	C₂⁴	—	4	1	no	—
483	#41	—	4	2	no	—
484	#41	—	8	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	4	1	no	—
487	C₂×Q₈	—	4	2	no	—
488	C₂×Q₈	—	8	1	no	—
489	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	—	2	1	no	—
495	C₂×D₄	—	2	1	no	—
496	C₂×D₄	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	4	1	no	—
500	C₂×D₄	—	4	1	no	—
501	C₂×D₄	—	4	1	no	—
502	C₂×D₄	—	4	1	no	—
503	C₂×D₄	—	4	1	no	—
504	C₂×D₄	—	4	1	no	—
505	C₂×D₄	—	4	2	no	—
506	C₂×D₄	—	4	2	no	—
507	C₂×D₄	—	4	2	no	—
508	C₂×D₄	—	4	2	no	—
509	C₂×D₄	—	8	1	no	—
510	C₂×D₄	—	8	1	no	—
511	C₂×D₄	—	8	1	no	—
512	C₂×D₄	—	8	1	no	—
513	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
514	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
515	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	1	no	—
518	C₂²×C₄	—	2	1	no	—
519	C₂²×C₄	—	4	1	no	—
520	C₂²×C₄	—	4	1	no	—
521	SD₁₆	—	4	2	no	—
522	SD₁₆	—	4	2	no	—
523	SD₁₆	—	4	2	no	—
524	SD₁₆	—	4	2	no	—
525	SD₁₆	—	8	1	no	—
526	SD₁₆	C₇×D₈	1	4	yes	complemented
527	D₈	—	2	2	no	—
528	D₈	—	4	2	no	—
529	D₈	—	4	2	no	—
530	D₈	—	4	2	no	—
531	D₈	—	4	2	no	—
532	D₈	—	4	2	no	—
533	D₈	—	4	2	no	—
534	D₈	—	8	1	no	—
535	D₈	—	8	1	no	—
536	D₈	—	8	2	no	—
537	M₁₆	—	4	2	no	—
538	C₂×C₈	—	2	1	no	—
539	C₂×C₈	—	2	1	no	—
540	C₂×C₈	—	2	1	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	4	2	no	—
544	C₂×C₈	—	4	2	no	—
545	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
546	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
547	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
548	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	1	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	4	1	no	—
553	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
554	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
555	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
556	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
557	C₂²⋊C₄	—	2	1	no	—
558	C₂²⋊C₄	—	2	1	no	—
559	C₂²⋊C₄	—	2	1	no	—
560	C₂²⋊C₄	—	4	1	no	—
561	C₂²⋊C₄	—	4	1	no	—
562	C₂²⋊C₄	—	4	1	no	—
563	C₂²⋊C₄	—	4	1	no	—
564	C₄²	—	2	1	no	—
565	C₄²	C₂⁴×C₇	1	1	yes	characteristic
566	C₂×C₇	—	4	2	no	—
567	C₂×C₇	—	4	2	no	—
568	C₂×C₇	—	8	1	no	—
569	C₂×C₇	—	16	1	no	—
570	C₂×C₇	—	16	1	no	—
571	C₂×C₇	—	16	1	no	—
572	C₂×C₇	#3265	1	1	yes	characteristic
573	C₂×C₇	D₄×D₈	1	1	yes	characteristic
574	C₂×C₇	#3288	1	1	yes	characteristic
575	C₂³	—	2	1	no	—
576	C₂³	—	2	1	no	—
577	C₂³	—	2	1	no	—
578	C₂³	—	4	1	no	—
579	C₂³	—	4	1	no	—
580	C₂³	—	4	1	no	—
581	C₂³	—	8	2	no	—
582	C₂³	—	8	2	no	—
583	C₂³	—	8	2	no	—
584	C₂³	—	8	2	no	—
585	C₂³	—	16	1	no	—
586	C₂³	—	16	1	no	—
587	Q₈	—	2	1	no	—
588	Q₈	—	4	1	no	—
589	Q₈	—	4	1	no	—
590	Q₈	—	8	2	no	—
591	Q₈	C₂×C₇×D₈	1	2	yes	complemented
592	D₄	—	2	1	no	—
593	D₄	—	2	1	no	—
594	D₄	—	2	1	no	—
595	D₄	—	2	1	no	—
596	D₄	—	4	1	no	—
597	D₄	—	4	1	no	—
598	D₄	—	4	1	no	—
599	D₄	—	4	1	no	—
600	D₄	—	4	1	no	—
601	D₄	—	4	1	no	—
602	D₄	—	4	1	no	—
603	D₄	—	4	1	no	—
604	D₄	—	4	1	no	—
605	D₄	—	4	1	no	—
606	D₄	—	4	1	no	—
607	D₄	—	4	1	no	—
608	D₄	—	8	2	no	—
609	D₄	—	8	1	no	—
610	D₄	—	8	1	no	—
611	D₄	—	8	1	no	—
612	D₄	—	8	2	no	—
613	D₄	—	8	2	no	—
614	D₄	—	8	2	no	—
615	D₄	—	8	2	no	—
616	D₄	—	16	1	no	—
617	D₄	C₂×C₇×D₈	1	2	yes	—
618	D₄	C₇×AGL(1,ℤ/8)	1	2	yes	complemented
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	8	2	no	—
632	C₂×C₄	—	8	2	no	—
633	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
634	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
635	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
636	C₈	—	2	1	no	—
637	C₈	—	2	1	no	—
638	C₈	—	2	1	no	—
639	C₈	—	4	1	no	—
640	C₈	—	4	1	no	—
641	C₈	—	4	1	no	—
642	C₈	—	4	1	no	—
643	C₈	—	4	1	no	—
644	C₈	C₂×C₇×D₈	1	2	yes	complemented
645	C₇	#19756	1	1	yes	characteristic, direct product factor, Sylow
646	C₂²	—	2	2	no	—
647	C₂²	—	2	2	no	—
648	C₂²	—	4	1	no	—
649	C₂²	—	4	1	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	1	no	—
653	C₂²	—	4	1	no	—
654	C₂²	—	4	1	no	—
655	C₂²	—	8	1	no	—
656	C₂²	—	8	1	no	—
657	C₂²	—	8	1	no	—
658	C₂²	—	8	1	no	—
659	C₂²	—	8	1	no	—
660	C₂²	—	8	1	no	—
661	C₂²	—	8	1	no	—
662	C₂²	—	8	1	no	—
663	C₂²	—	8	1	no	—
664	C₂²	—	16	4	no	—
665	C₂²	—	16	4	no	—
666	C₂²	C₇×D₄²	1	1	yes	characteristic
667	C₄	—	2	2	no	—
668	C₄	—	2	1	no	—
669	C₄	—	4	1	no	—
670	C₄	—	4	1	no	—
671	C₄	—	4	1	no	—
672	C₄	—	4	1	no	—
673	C₄	—	8	1	no	—
674	C₄	C₂²×C₇×D₈	1	1	yes	characteristic
675	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
676	C₄	C₇×#576	1	1	yes	characteristic
677	C₄	C₇×#577	1	1	yes	characteristic
678	C₂	—	4	2	no	—
679	C₂	—	4	2	no	—
680	C₂	—	8	1	no	—
681	C₂	—	16	1	no	—
682	C₂	—	16	1	no	—
683	C₂	—	16	1	no	—
684	C₂	C₇×#3265	1	1	yes	characteristic
685	C₂	C₇×D₄×D₈	1	1	yes	characteristic
686	C₂	C₇×#3288	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄²×C₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₇×#3358	C₂	semidirect
2. C₇×D₄×SD₁₆	C₂	semidirect
3. C₇×D₄×D₈	C₂	semidirect
4. C₇×#3274	C₂	semidirect
5. C₇×#1721	C₂	semidirect
6. C₇×#1685	C₂	semidirect
7. C₇×#1681	C₂	semidirect
8. C₇×#1668	C₂	semidirect
9. C₇×#1667	C₂	semidirect
10. C₇×#1639	C₂	semidirect
11. C₇×#1627	C₂	semidirect
12. C₇×#1621	C₂	semidirect
13. C₇×#1619	C₂	semidirect
14. C₇×#1581	C₂	semidirect
15. C₇×SD₁₆⋊C₈	C₂	semidirect
19. C₇×#550	C₂²	semidirect
20. C₇×D₄×Q₈	C₂²	semidirect
24. C₇×#493	C₂²	semidirect
25. C₇×#493	C₂²	semidirect
26. C₇×#492	C₂²	semidirect
28. C₇×#486	C₂²	semidirect
29. C₇×#486	C₂²	semidirect
31. C₇×#474	C₂²	semidirect
36. C₇×#460	C₂²	semidirect
37. C₇×#460	C₂²	semidirect
42. C₇×#459	C₂²	semidirect
49. C₇×C₈×Q₈	C₂²	semidirect
53. C₇×#436	C₂²	semidirect
55. C₇×C₈×D₄	C₂²	semidirect
56. C₇×#335	C₂²	semidirect
57. C₇×#335	C₂²	semidirect
58. C₇×#332	C₂²	semidirect
59. C₇×#331	C₂²	semidirect
60. C₇×#331	C₂²	semidirect
61. C₇×#331	C₂²	semidirect
62. C₇×Q₈⋊C₈	C₂²	semidirect
64. C₇×D₄⋊C₈	C₂²	semidirect
65. C₇×D₄⋊C₈	C₂²	semidirect
66. C₇×#322	C₂²	semidirect
67. #19756	C₇	direct (N × H)
148. #3358	C₂×C₇	semidirect
149. D₄×SD₁₆	C₂×C₇	semidirect
150. D₄×D₈	C₂×C₇	semidirect
151. #3274	C₂×C₇	semidirect
152. #1721	C₂×C₇	semidirect
153. #1685	C₂×C₇	semidirect
154. #1681	C₂×C₇	semidirect
155. #1668	C₂×C₇	semidirect
156. #1667	C₂×C₇	semidirect
157. #1639	C₂×C₇	semidirect
158. #1627	C₂×C₇	semidirect
159. #1621	C₂×C₇	semidirect
160. #1619	C₂×C₇	semidirect
161. #1581	C₂×C₇	semidirect
162. SD₁₆⋊C₈	C₂×C₇	semidirect
208. C₇×SD₁₆	D₈	semidirect
251. #550	C₂²×C₇	semidirect
252. D₄×Q₈	C₂²×C₇	semidirect
256. #493	C₂²×C₇	semidirect
257. #493	C₂²×C₇	semidirect
258. #492	C₂²×C₇	semidirect
260. #486	C₂²×C₇	semidirect
261. #486	C₂²×C₇	semidirect
263. #474	C₂²×C₇	semidirect
268. #460	C₂²×C₇	semidirect
269. #460	C₂²×C₇	semidirect
274. #459	C₂²×C₇	semidirect
281. C₈×Q₈	C₂²×C₇	semidirect
285. #436	C₂²×C₇	semidirect
287. C₈×D₄	C₂²×C₇	semidirect
288. #335	C₂²×C₇	semidirect
289. #335	C₂²×C₇	semidirect
290. #332	C₂²×C₇	semidirect
291. #331	C₂²×C₇	semidirect
292. #331	C₂²×C₇	semidirect
293. #331	C₂²×C₇	semidirect
294. Q₈⋊C₈	C₂²×C₇	semidirect
296. D₄⋊C₈	C₂²×C₇	semidirect
297. D₄⋊C₈	C₂²×C₇	semidirect
298. #322	C₂²×C₇	semidirect
315. C₇×Q₈	C₂×D₈	semidirect
342. C₇×D₄	AGL(1,ℤ/8)	semidirect
368. C₇×C₈	C₂×D₈	semidirect
526. SD₁₆	C₇×D₈	semidirect
591. Q₈	C₂×C₇×D₈	semidirect
618. D₄	C₇×AGL(1,ℤ/8)	semidirect
644. C₈	C₂×C₇×D₈	semidirect
645. C₇	#19756	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇×#3358	2
C₇×D₄×SD₁₆	2
C₇×D₄×D₈	2
C₇×#3274	2
C₇×#1721	2
C₇×#1685	2
C₇×#1681	2
C₇×#1668	2
C₇×#1667	2
C₇×#1639	2
C₇×#1627	2
C₇×#1621	2
C₇×#1619	2
C₇×#1581	2
C₇×SD₁₆⋊C₈	2
#19756	7

Subgroups — C7×#19756 #13196972

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19756 #13196972