Subgroups — C₇×#19803 #13197019

Order 1792.

728 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇×#3298	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×D₄×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×#3272	C₂	1	1	yes	maximal, characteristic
5	C₇×#1714	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1693	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#1679	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#1666	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇×#1665	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1654	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1651	C₂	1	1	yes	maximal, characteristic, complemented
12	C₇×#1639	C₂	1	1	yes	maximal, characteristic, complemented
13	C₇×#1627	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#1583	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇×#1580	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	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic
19	C₇×D₄×Q₈	C₂²	1	1	yes	characteristic, complemented
20	C₇×#547	C₂²	1	1	yes	characteristic
21	C₇×D₄²	C₂²	1	1	yes	characteristic
22	C₇×#500	—	2	1	no	—
23	C₇×#500	C₂²	1	1	yes	characteristic, complemented
24	C₇×#498	C₂²	1	1	yes	characteristic, complemented
25	C₇×#492	C₂²	1	1	yes	characteristic
26	C₇×#492	C₂²	1	1	yes	characteristic
27	C₇×#487	C₂²	1	1	yes	characteristic, complemented
28	C₇×#486	C₂²	1	1	yes	characteristic
29	C₇×#483	—	2	1	no	—
30	C₇×#475	C₂²	1	1	yes	characteristic, complemented
31	C₇×#474	—	2	1	no	—
32	C₇×#474	—	2	1	no	—
33	C₇×#474	C₂²	1	1	yes	characteristic
34	C₇×#474	C₂²	1	1	yes	characteristic, complemented
35	C₇×#468	—	2	1	no	—
36	C₇×#465	—	2	1	no	—
37	C₇×#465	—	2	1	no	—
38	C₇×#461	C₂²	1	1	yes	characteristic
39	C₇×#461	C₂²	1	1	yes	characteristic
40	C₇×#461	C₂²	1	1	yes	characteristic
41	C₇×#460	C₂²	1	1	yes	characteristic
42	C₇×#459	C₂²	1	1	yes	characteristic
43	C₇×#450	—	2	1	no	—
44	C₇×#450	—	2	1	no	—
45	C₇×#449	—	2	1	no	—
46	C₇×#448	—	2	1	no	—
47	C₇×#448	—	2	1	no	—
48	C₇×C₈×Q₈	C₂²	1	1	yes	characteristic, complemented
49	C₄×C₇×SD₁₆	C₂²	1	1	yes	characteristic
50	C₄×C₇×D₈	—	2	1	no	—
51	C₄×C₇×D₈	C₂²	1	1	yes	characteristic
52	C₇×#436	C₂²	1	1	yes	characteristic
53	C₇×C₈×D₄	C₂²	1	1	yes	characteristic
54	C₇×C₈×D₄	C₂²	1	1	yes	characteristic, complemented
55	C₂×C₇×#103	—	2	1	no	—
56	C₇×#334	C₂²	1	1	yes	characteristic, complemented
57	C₇×#334	C₂²	1	1	yes	characteristic, complemented
58	C₇×#332	C₂²	1	1	yes	characteristic, complemented
59	C₇×#332	C₂²	1	1	yes	characteristic, complemented
60	C₇×#332	C₂²	1	1	yes	characteristic, complemented
61	C₇×#331	C₂²	1	1	yes	characteristic
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
65	C₇×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
66	C₇×#322	C₂²	1	1	yes	characteristic, complemented
67	#19803	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×Q₈	—	2	1	no	—
71	C₂²×C₇×D₄	—	2	1	no	—
72	C₂²×C₇×D₄	—	2	1	no	—
73	C₇×AGL(1,ℤ/8)	—	4	4	no	—
74	C₂×C₇×SD₁₆	—	2	1	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	—	2	1	no	—
81	C₂×C₇×SD₁₆	—	2	1	no	—
82	C₂×C₇×SD₁₆	—	4	4	no	—
83	C₂×C₇×SD₁₆	D₄	1	1	yes	characteristic
84	C₂×C₇×D₈	—	2	1	no	—
85	C₂×C₇×D₈	D₄	1	1	yes	characteristic
86	C₂×C₇×M₁₆	—	2	1	no	—
87	C₂²×C₇×C₈	—	2	1	no	—
88	C₂²×C₇×C₈	—	2	1	no	—
89	C₇×C₄⋊Q₈	—	2	1	no	—
90	C₇×C₄⋊Q₈	—	2	1	no	—
91	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
92	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
93	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
94	C₇×#127	C₂³	1	1	yes	characteristic
95	C₇×#123	—	4	1	no	—
96	C₇×C₂²⋊Q₈	—	2	1	no	—
97	C₇×C₂²⋊Q₈	—	2	1	no	—
98	C₇×C₂²⋊Q₈	—	2	1	no	—
99	C₇×C₂²⋊Q₈	—	4	1	no	—
100	C₇×C₂²⋊Q₈	—	4	1	no	—
101	C₇×#121	—	2	1	no	—
102	C₇×#121	—	2	1	no	—
103	C₇×#121	—	2	1	no	—
104	C₇×#120	—	4	1	no	—
105	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
106	C₄×C₇×D₄	—	2	1	no	—
107	C₄×C₇×D₄	—	2	1	no	—
108	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
109	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
110	C₄×C₇×D₄	C₂³	1	1	yes	characteristic
111	C₂×C₇×C₄⋊C₄	—	2	1	no	—
112	C₇×#107	—	2	1	no	—
113	C₇×#107	—	2	1	no	—
114	C₇×#107	—	2	1	no	—
115	C₇×#107	—	2	1	no	—
116	C₇×#107	D₄	1	1	yes	characteristic
117	C₇×#106	—	2	1	no	—
118	C₇×#106	—	4	1	no	—
119	C₇×#106	D₄	1	1	yes	characteristic
120	C₇×C₄⋊C₈	—	2	1	no	—
121	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×#103	—	2	1	no	—
126	C₇×#103	—	2	1	no	—
127	C₇×#103	—	2	1	no	—
128	C₇×#103	—	2	1	no	—
129	C₇×#103	—	2	1	no	—
130	C₇×#103	—	2	1	no	—
131	C₇×#103	D₄	1	1	yes	characteristic
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	—	4	1	no	—
137	C₇×#102	D₄	1	1	yes	characteristic
138	C₇×#102	D₄	1	1	yes	characteristic
139	C₇×#102	D₄	1	1	yes	characteristic
140	C₇×C₂²⋊C₈	—	2	1	no	—
141	C₇×C₂²⋊C₈	—	2	1	no	—
142	C₇×C₂²⋊C₈	—	2	1	no	—
143	C₄×C₇×C₈	—	2	1	no	—
144	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
145	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
146	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
147	Q₈×D₈	C₂×C₇	1	1	yes	characteristic, complemented
148	#3298	C₂×C₇	1	1	yes	characteristic, complemented
149	D₄×SD₁₆	C₂×C₇	1	1	yes	characteristic, complemented
150	#3272	C₂×C₇	1	1	yes	characteristic
151	#1714	C₂×C₇	1	1	yes	characteristic, complemented
152	#1693	C₂×C₇	1	1	yes	characteristic, complemented
153	#1679	C₂×C₇	1	1	yes	characteristic, complemented
154	#1666	C₂×C₇	1	1	yes	characteristic, complemented
155	#1665	C₂×C₇	1	1	yes	characteristic, complemented
156	#1654	C₂×C₇	1	1	yes	characteristic, complemented
157	#1651	C₂×C₇	1	1	yes	characteristic, complemented
158	#1639	C₂×C₇	1	1	yes	characteristic, complemented
159	#1627	C₂×C₇	1	1	yes	characteristic, complemented
160	#1583	C₂×C₇	1	1	yes	characteristic, complemented
161	#1580	C₂×C₇	1	1	yes	characteristic, complemented
162	C₂⁴×C₇	—	4	1	no	—
163	C₇×#41	—	4	2	no	—
164	C₇×#41	—	8	1	no	—
165	C₂×C₇×Q₈	—	2	1	no	—
166	C₂×C₇×Q₈	—	2	1	no	—
167	C₂×C₇×Q₈	—	2	1	no	—
168	C₂×C₇×Q₈	—	4	1	no	—
169	C₂×C₇×Q₈	—	4	1	no	—
170	C₂×C₇×Q₈	—	4	2	no	—
171	C₂×C₇×Q₈	—	4	2	no	—
172	C₂×C₇×Q₈	—	8	1	no	—
173	C₂×C₇×Q₈	—	8	1	no	—
174	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
179	C₂×C₇×D₄	—	4	1	no	—
180	C₂×C₇×D₄	—	4	2	no	—
181	C₂×C₇×D₄	—	4	2	no	—
182	C₂×C₇×D₄	—	4	1	no	—
183	C₂×C₇×D₄	—	8	1	no	—
184	C₂×C₇×D₄	—	8	1	no	—
185	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
186	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
187	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₇	—	2	1	no	—
189	C₂²×C₄×C₇	—	2	1	no	—
190	C₂²×C₄×C₇	—	2	1	no	—
191	C₂²×C₄×C₇	—	4	1	no	—
192	C₂²×C₄×C₇	—	4	1	no	—
193	C₂²×C₄×C₇	—	4	1	no	—
194	C₇×SD₁₆	—	2	2	no	—
195	C₇×SD₁₆	—	4	2	no	—
196	C₇×SD₁₆	—	4	2	no	—
197	C₇×SD₁₆	—	4	2	no	—
198	C₇×SD₁₆	—	4	2	no	—
199	C₇×SD₁₆	—	4	2	no	—
200	C₇×SD₁₆	—	8	1	no	—
201	C₇×SD₁₆	—	8	1	no	—
202	C₇×SD₁₆	—	8	1	no	—
203	C₇×D₈	—	4	2	no	—
204	C₇×D₈	SD₁₆	1	4	yes	complemented
205	C₇×M₁₆	—	4	2	no	—
206	C₂×C₇×C₈	—	2	1	no	—
207	C₂×C₇×C₈	—	2	1	no	—
208	C₂×C₇×C₈	—	2	1	no	—
209	C₂×C₇×C₈	—	2	1	no	—
210	C₂×C₇×C₈	—	2	1	no	—
211	C₂×C₇×C₈	—	4	2	no	—
212	C₂×C₇×C₈	—	8	1	no	—
213	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
214	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
215	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
216	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
217	C₇×C₄⋊C₄	—	2	1	no	—
218	C₇×C₄⋊C₄	—	2	1	no	—
219	C₇×C₄⋊C₄	—	2	1	no	—
220	C₇×C₄⋊C₄	—	2	1	no	—
221	C₇×C₄⋊C₄	—	2	1	no	—
222	C₇×C₄⋊C₄	—	2	1	no	—
223	C₇×C₄⋊C₄	—	2	1	no	—
224	C₇×C₄⋊C₄	—	4	1	no	—
225	C₇×C₄⋊C₄	—	4	1	no	—
226	C₇×C₄⋊C₄	—	4	1	no	—
227	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
228	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
229	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
230	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
231	C₇×C₂²⋊C₄	—	2	1	no	—
232	C₇×C₂²⋊C₄	—	2	1	no	—
233	C₇×C₂²⋊C₄	—	2	1	no	—
234	C₇×C₂²⋊C₄	—	4	1	no	—
235	C₇×C₂²⋊C₄	—	4	1	no	—
236	C₇×C₂²⋊C₄	—	4	1	no	—
237	C₇×C₂²⋊C₄	—	4	1	no	—
238	C₄²×C₇	—	2	1	no	—
239	C₄²×C₇	C₂⁴	1	1	yes	characteristic
240	C₂×AGL(1,ℤ/8)	—	2	1	no	—
241	C₂²×SD₁₆	—	2	1	no	—
242	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic
243	D₄×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
244	#547	C₂²×C₇	1	1	yes	characteristic
245	D₄²	C₂²×C₇	1	1	yes	characteristic
246	#500	—	2	1	no	—
247	#500	C₂²×C₇	1	1	yes	characteristic, complemented
248	#498	C₂²×C₇	1	1	yes	characteristic, complemented
249	#492	C₂²×C₇	1	1	yes	characteristic
250	#492	C₂²×C₇	1	1	yes	characteristic
251	#487	C₂²×C₇	1	1	yes	characteristic, complemented
252	#486	C₂²×C₇	1	1	yes	characteristic
253	#483	—	2	1	no	—
254	#475	C₂²×C₇	1	1	yes	characteristic, complemented
255	#474	—	2	1	no	—
256	#474	—	2	1	no	—
257	#474	C₂²×C₇	1	1	yes	characteristic
258	#474	C₂²×C₇	1	1	yes	characteristic, complemented
259	#468	—	2	1	no	—
260	#465	—	2	1	no	—
261	#465	—	2	1	no	—
262	#461	C₂²×C₇	1	1	yes	characteristic
263	#461	C₂²×C₇	1	1	yes	characteristic
264	#461	C₂²×C₇	1	1	yes	characteristic
265	#460	C₂²×C₇	1	1	yes	characteristic
266	#459	C₂²×C₇	1	1	yes	characteristic
267	#450	—	2	1	no	—
268	#450	—	2	1	no	—
269	#449	—	2	1	no	—
270	#448	—	2	1	no	—
271	#448	—	2	1	no	—
272	C₈×Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
273	C₄×SD₁₆	C₂²×C₇	1	1	yes	characteristic
274	C₄×D₈	—	2	1	no	—
275	C₄×D₈	C₂²×C₇	1	1	yes	characteristic
276	#436	C₂²×C₇	1	1	yes	characteristic
277	C₈×D₄	C₂²×C₇	1	1	yes	characteristic
278	C₈×D₄	C₂²×C₇	1	1	yes	characteristic, complemented
279	C₂×#103	—	2	1	no	—
280	#334	C₂²×C₇	1	1	yes	characteristic, complemented
281	#334	C₂²×C₇	1	1	yes	characteristic, complemented
282	#332	C₂²×C₇	1	1	yes	characteristic, complemented
283	#332	C₂²×C₇	1	1	yes	characteristic, complemented
284	#332	C₂²×C₇	1	1	yes	characteristic, complemented
285	#331	C₂²×C₇	1	1	yes	characteristic
286	Q₈⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
287	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
288	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
289	D₄⋊C₈	C₂²×C₇	1	1	yes	characteristic, complemented
290	#322	C₂²×C₇	1	1	yes	characteristic, complemented
291	C₂³×C₇	—	2	1	no	—
292	C₂³×C₇	—	2	1	no	—
293	C₂³×C₇	—	2	1	no	—
294	C₂³×C₇	—	4	1	no	—
295	C₂³×C₇	—	8	2	no	—
296	C₂³×C₇	—	8	2	no	—
297	C₂³×C₇	—	16	1	no	—
298	C₇×Q₈	—	2	1	no	—
299	C₇×Q₈	—	4	1	no	—
300	C₇×Q₈	—	4	1	no	—
301	C₇×Q₈	—	4	1	no	—
302	C₇×Q₈	—	4	1	no	—
303	C₇×Q₈	—	4	1	no	—
304	C₇×Q₈	—	4	1	no	—
305	C₇×Q₈	—	8	2	no	—
306	C₇×Q₈	—	8	2	no	—
307	C₇×Q₈	AGL(1,ℤ/8)	1	2	yes	complemented
308	C₇×D₄	—	2	1	no	—
309	C₇×D₄	—	2	1	no	—
310	C₇×D₄	—	2	1	no	—
311	C₇×D₄	—	2	1	no	—
312	C₇×D₄	—	4	1	no	—
313	C₇×D₄	—	4	1	no	—
314	C₇×D₄	—	4	1	no	—
315	C₇×D₄	—	4	1	no	—
316	C₇×D₄	—	8	1	no	—
317	C₇×D₄	—	8	2	no	—
318	C₇×D₄	—	8	2	no	—
319	C₇×D₄	—	8	2	no	—
320	C₇×D₄	C₂×SD₁₆	1	2	yes	—
321	C₇×D₄	C₂×SD₁₆	1	2	yes	complemented
322	C₂×C₄×C₇	—	2	1	no	—
323	C₂×C₄×C₇	—	2	1	no	—
324	C₂×C₄×C₇	—	2	1	no	—
325	C₂×C₄×C₇	—	2	1	no	—
326	C₂×C₄×C₇	—	2	1	no	—
327	C₂×C₄×C₇	—	4	1	no	—
328	C₂×C₄×C₇	—	4	1	no	—
329	C₂×C₄×C₇	—	4	1	no	—
330	C₂×C₄×C₇	—	4	1	no	—
331	C₂×C₄×C₇	—	4	1	no	—
332	C₂×C₄×C₇	—	4	1	no	—
333	C₂×C₄×C₇	—	4	1	no	—
334	C₂×C₄×C₇	—	4	1	no	—
335	C₂×C₄×C₇	—	4	1	no	—
336	C₂×C₄×C₇	—	8	2	no	—
337	C₂×C₄×C₇	—	8	2	no	—
338	C₂×C₄×C₇	—	8	2	no	—
339	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
340	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
341	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
342	C₇×C₈	—	2	1	no	—
343	C₇×C₈	—	2	1	no	—
344	C₇×C₈	—	2	1	no	—
345	C₇×C₈	—	4	1	no	—
346	C₇×C₈	—	4	1	no	—
347	C₇×C₈	—	4	1	no	—
348	C₇×C₈	—	4	1	no	—
349	C₇×C₈	—	4	1	no	—
350	C₇×C₈	C₂×SD₁₆	1	2	yes	complemented
351	C₂×#41	—	2	1	no	—
352	C₂²×Q₈	—	2	1	no	—
353	C₂²×Q₈	—	2	1	no	—
354	C₂²×D₄	—	2	1	no	—
355	C₂²×D₄	—	2	1	no	—
356	AGL(1,ℤ/8)	—	4	4	no	—
357	C₂×SD₁₆	—	2	1	no	—
358	C₂×SD₁₆	—	2	1	no	—
359	C₂×SD₁₆	—	2	1	no	—
360	C₂×SD₁₆	—	2	1	no	—
361	C₂×SD₁₆	—	2	1	no	—
362	C₂×SD₁₆	—	2	1	no	—
363	C₂×SD₁₆	—	2	1	no	—
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×SD₁₆	—	4	4	no	—
366	C₂×SD₁₆	C₇×D₄	1	1	yes	characteristic
367	C₂×D₈	—	2	1	no	—
368	C₂×D₈	C₇×D₄	1	1	yes	characteristic
369	C₂×M₁₆	—	2	1	no	—
370	C₂²×C₈	—	2	1	no	—
371	C₂²×C₈	—	2	1	no	—
372	C₄⋊Q₈	—	2	1	no	—
373	C₄⋊Q₈	—	2	1	no	—
374	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
375	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
376	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
377	#127	C₂³×C₇	1	1	yes	characteristic
378	#123	—	4	1	no	—
379	C₂²⋊Q₈	—	2	1	no	—
380	C₂²⋊Q₈	—	2	1	no	—
381	C₂²⋊Q₈	—	2	1	no	—
382	C₂²⋊Q₈	—	4	1	no	—
383	C₂²⋊Q₈	—	4	1	no	—
384	#121	—	2	1	no	—
385	#121	—	2	1	no	—
386	#121	—	2	1	no	—
387	#120	—	4	1	no	—
388	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	—	2	1	no	—
391	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
392	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
393	C₄×D₄	C₂³×C₇	1	1	yes	characteristic
394	C₂×C₄⋊C₄	—	2	1	no	—
395	#107	—	2	1	no	—
396	#107	—	2	1	no	—
397	#107	—	2	1	no	—
398	#107	—	2	1	no	—
399	#107	C₇×D₄	1	1	yes	characteristic
400	#106	—	2	1	no	—
401	#106	—	4	1	no	—
402	#106	C₇×D₄	1	1	yes	characteristic
403	C₄⋊C₈	—	2	1	no	—
404	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
407	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
408	#103	—	2	1	no	—
409	#103	—	2	1	no	—
410	#103	—	2	1	no	—
411	#103	—	2	1	no	—
412	#103	—	2	1	no	—
413	#103	—	2	1	no	—
414	#103	C₇×D₄	1	1	yes	characteristic
415	#102	—	2	1	no	—
416	#102	—	2	1	no	—
417	#102	—	2	1	no	—
418	#102	—	2	1	no	—
419	#102	—	4	1	no	—
420	#102	C₇×D₄	1	1	yes	characteristic
421	#102	C₇×D₄	1	1	yes	characteristic
422	#102	C₇×D₄	1	1	yes	characteristic
423	C₂²⋊C₈	—	2	1	no	—
424	C₂²⋊C₈	—	2	1	no	—
425	C₂²⋊C₈	—	2	1	no	—
426	C₄×C₈	—	2	1	no	—
427	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
428	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
429	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
430	C₂²×C₇	—	2	2	no	—
431	C₂²×C₇	—	2	2	no	—
432	C₂²×C₇	—	4	1	no	—
433	C₂²×C₇	—	4	1	no	—
434	C₂²×C₇	—	4	1	no	—
435	C₂²×C₇	—	4	1	no	—
436	C₂²×C₇	—	4	1	no	—
437	C₂²×C₇	—	4	1	no	—
438	C₂²×C₇	—	4	1	no	—
439	C₂²×C₇	—	8	1	no	—
440	C₂²×C₇	—	8	1	no	—
441	C₂²×C₇	—	8	1	no	—
442	C₂²×C₇	—	16	4	no	—
443	C₂²×C₇	D₄²	1	1	yes	characteristic
444	C₄×C₇	—	2	2	no	—
445	C₄×C₇	—	2	1	no	—
446	C₄×C₇	—	4	1	no	—
447	C₄×C₇	—	4	1	no	—
448	C₄×C₇	—	4	1	no	—
449	C₄×C₇	—	4	1	no	—
450	C₄×C₇	—	8	1	no	—
451	C₄×C₇	—	8	1	no	—
452	C₄×C₇	—	8	1	no	—
453	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
454	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
455	C₄×C₇	#576	1	1	yes	characteristic
456	C₄×C₇	#577	1	1	yes	characteristic
457	C₂⁴	—	4	1	no	—
458	#41	—	4	2	no	—
459	#41	—	8	1	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	4	1	no	—
464	C₂×Q₈	—	4	1	no	—
465	C₂×Q₈	—	4	2	no	—
466	C₂×Q₈	—	4	2	no	—
467	C₂×Q₈	—	8	1	no	—
468	C₂×Q₈	—	8	1	no	—
469	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	4	1	no	—
474	C₂×D₄	—	4	1	no	—
475	C₂×D₄	—	4	2	no	—
476	C₂×D₄	—	4	2	no	—
477	C₂×D₄	—	4	1	no	—
478	C₂×D₄	—	8	1	no	—
479	C₂×D₄	—	8	1	no	—
480	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
481	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
482	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	4	1	no	—
487	C₂²×C₄	—	4	1	no	—
488	C₂²×C₄	—	4	1	no	—
489	SD₁₆	—	2	2	no	—
490	SD₁₆	—	4	2	no	—
491	SD₁₆	—	4	2	no	—
492	SD₁₆	—	4	2	no	—
493	SD₁₆	—	4	2	no	—
494	SD₁₆	—	4	2	no	—
495	SD₁₆	—	8	1	no	—
496	SD₁₆	—	8	1	no	—
497	SD₁₆	—	8	1	no	—
498	D₈	—	4	2	no	—
499	D₈	C₇×SD₁₆	1	4	yes	complemented
500	M₁₆	—	4	2	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	1	no	—
503	C₂×C₈	—	2	1	no	—
504	C₂×C₈	—	2	1	no	—
505	C₂×C₈	—	2	1	no	—
506	C₂×C₈	—	4	2	no	—
507	C₂×C₈	—	8	1	no	—
508	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
509	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
510	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
511	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	—	2	1	no	—
519	C₄⋊C₄	—	4	1	no	—
520	C₄⋊C₄	—	4	1	no	—
521	C₄⋊C₄	—	4	1	no	—
522	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
523	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
524	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
525	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
526	C₂²⋊C₄	—	2	1	no	—
527	C₂²⋊C₄	—	2	1	no	—
528	C₂²⋊C₄	—	2	1	no	—
529	C₂²⋊C₄	—	4	1	no	—
530	C₂²⋊C₄	—	4	1	no	—
531	C₂²⋊C₄	—	4	1	no	—
532	C₂²⋊C₄	—	4	1	no	—
533	C₄²	—	2	1	no	—
534	C₄²	C₂⁴×C₇	1	1	yes	characteristic
535	C₂×C₇	—	4	2	no	—
536	C₂×C₇	—	4	2	no	—
537	C₂×C₇	—	8	1	no	—
538	C₂×C₇	—	16	1	no	—
539	C₂×C₇	#3264	1	1	yes	characteristic
540	C₂×C₇	D₄×SD₁₆	1	1	yes	characteristic
541	C₂×C₇	#3289	1	1	yes	characteristic
542	C₂³	—	2	1	no	—
543	C₂³	—	2	1	no	—
544	C₂³	—	2	1	no	—
545	C₂³	—	4	1	no	—
546	C₂³	—	8	2	no	—
547	C₂³	—	8	2	no	—
548	C₂³	—	16	1	no	—
549	Q₈	—	2	1	no	—
550	Q₈	—	4	1	no	—
551	Q₈	—	4	1	no	—
552	Q₈	—	4	1	no	—
553	Q₈	—	4	1	no	—
554	Q₈	—	4	1	no	—
555	Q₈	—	4	1	no	—
556	Q₈	—	8	2	no	—
557	Q₈	—	8	2	no	—
558	Q₈	C₇×AGL(1,ℤ/8)	1	2	yes	complemented
559	D₄	—	2	1	no	—
560	D₄	—	2	1	no	—
561	D₄	—	2	1	no	—
562	D₄	—	2	1	no	—
563	D₄	—	4	1	no	—
564	D₄	—	4	1	no	—
565	D₄	—	4	1	no	—
566	D₄	—	4	1	no	—
567	D₄	—	8	1	no	—
568	D₄	—	8	2	no	—
569	D₄	—	8	2	no	—
570	D₄	—	8	2	no	—
571	D₄	C₂×C₇×SD₁₆	1	2	yes	—
572	D₄	C₂×C₇×SD₁₆	1	2	yes	complemented
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	8	2	no	—
588	C₂×C₄	—	8	2	no	—
589	C₂×C₄	—	8	2	no	—
590	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
591	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
592	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
593	C₈	—	2	1	no	—
594	C₈	—	2	1	no	—
595	C₈	—	2	1	no	—
596	C₈	—	4	1	no	—
597	C₈	—	4	1	no	—
598	C₈	—	4	1	no	—
599	C₈	—	4	1	no	—
600	C₈	—	4	1	no	—
601	C₈	C₂×C₇×SD₁₆	1	2	yes	complemented
602	C₇	#19803	1	1	yes	characteristic, direct product factor, Sylow
603	C₂²	—	2	2	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	4	1	no	—
607	C₂²	—	4	1	no	—
608	C₂²	—	4	1	no	—
609	C₂²	—	4	1	no	—
610	C₂²	—	4	1	no	—
611	C₂²	—	4	1	no	—
612	C₂²	—	8	1	no	—
613	C₂²	—	8	1	no	—
614	C₂²	—	8	1	no	—
615	C₂²	—	16	4	no	—
616	C₂²	C₇×D₄²	1	1	yes	characteristic
617	C₄	—	2	2	no	—
618	C₄	—	2	1	no	—
619	C₄	—	4	1	no	—
620	C₄	—	4	1	no	—
621	C₄	—	4	1	no	—
622	C₄	—	4	1	no	—
623	C₄	—	8	1	no	—
624	C₄	—	8	1	no	—
625	C₄	—	8	1	no	—
626	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
627	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
628	C₄	C₇×#576	1	1	yes	characteristic
629	C₄	C₇×#577	1	1	yes	characteristic
630	C₂	—	4	2	no	—
631	C₂	—	4	2	no	—
632	C₂	—	8	1	no	—
633	C₂	—	16	1	no	—
634	C₂	C₇×#3264	1	1	yes	characteristic
635	C₂	C₇×D₄×SD₁₆	1	1	yes	characteristic
636	C₂	C₇×#3289	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3272	C₂
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×Q₈×D₈	C₂	semidirect
2. C₇×#3298	C₂	semidirect
3. C₇×D₄×SD₁₆	C₂	semidirect
5. C₇×#1714	C₂	semidirect
6. C₇×#1693	C₂	semidirect
7. C₇×#1679	C₂	semidirect
8. C₇×#1666	C₂	semidirect
9. C₇×#1665	C₂	semidirect
10. C₇×#1654	C₂	semidirect
11. C₇×#1651	C₂	semidirect
12. C₇×#1639	C₂	semidirect
13. C₇×#1627	C₂	semidirect
14. C₇×#1583	C₂	semidirect
15. C₇×#1580	C₂	semidirect
19. C₇×D₄×Q₈	C₂²	semidirect
23. C₇×#500	C₂²	semidirect
24. C₇×#498	C₂²	semidirect
27. C₇×#487	C₂²	semidirect
30. C₇×#475	C₂²	semidirect
34. C₇×#474	C₂²	semidirect
48. C₇×C₈×Q₈	C₂²	semidirect
54. C₇×C₈×D₄	C₂²	semidirect
56. C₇×#334	C₂²	semidirect
57. C₇×#334	C₂²	semidirect
58. C₇×#332	C₂²	semidirect
59. C₇×#332	C₂²	semidirect
60. C₇×#332	C₂²	semidirect
62. C₇×Q₈⋊C₈	C₂²	semidirect
65. C₇×D₄⋊C₈	C₂²	semidirect
66. C₇×#322	C₂²	semidirect
67. #19803	C₇	direct (N × H)
147. Q₈×D₈	C₂×C₇	semidirect
148. #3298	C₂×C₇	semidirect
149. D₄×SD₁₆	C₂×C₇	semidirect
151. #1714	C₂×C₇	semidirect
152. #1693	C₂×C₇	semidirect
153. #1679	C₂×C₇	semidirect
154. #1666	C₂×C₇	semidirect
155. #1665	C₂×C₇	semidirect
156. #1654	C₂×C₇	semidirect
157. #1651	C₂×C₇	semidirect
158. #1639	C₂×C₇	semidirect
159. #1627	C₂×C₇	semidirect
160. #1583	C₂×C₇	semidirect
161. #1580	C₂×C₇	semidirect
204. C₇×D₈	SD₁₆	semidirect
243. D₄×Q₈	C₂²×C₇	semidirect
247. #500	C₂²×C₇	semidirect
248. #498	C₂²×C₇	semidirect
251. #487	C₂²×C₇	semidirect
254. #475	C₂²×C₇	semidirect
258. #474	C₂²×C₇	semidirect
272. C₈×Q₈	C₂²×C₇	semidirect
278. C₈×D₄	C₂²×C₇	semidirect
280. #334	C₂²×C₇	semidirect
281. #334	C₂²×C₇	semidirect
282. #332	C₂²×C₇	semidirect
283. #332	C₂²×C₇	semidirect
284. #332	C₂²×C₇	semidirect
286. Q₈⋊C₈	C₂²×C₇	semidirect
289. D₄⋊C₈	C₂²×C₇	semidirect
290. #322	C₂²×C₇	semidirect
307. C₇×Q₈	AGL(1,ℤ/8)	semidirect
321. C₇×D₄	C₂×SD₁₆	semidirect
350. C₇×C₈	C₂×SD₁₆	semidirect
499. D₈	C₇×SD₁₆	semidirect
558. Q₈	C₇×AGL(1,ℤ/8)	semidirect
572. D₄	C₂×C₇×SD₁₆	semidirect
601. C₈	C₂×C₇×SD₁₆	semidirect
602. C₇	#19803	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×Q₈×D₈	2
C₇×#3298	2
C₇×D₄×SD₁₆	2
C₇×#3272	2
C₇×#1714	2
C₇×#1693	2
C₇×#1679	2
C₇×#1666	2
C₇×#1665	2
C₇×#1654	2
C₇×#1651	2
C₇×#1639	2
C₇×#1627	2
C₇×#1583	2
C₇×#1580	2
#19803	7

Subgroups — C7×#19803 #13197019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19803 #13197019