Subgroups — C₇×#19170 #13196386

Order 1792.

766 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3151	C₂	1	1	yes	maximal, characteristic
2	C₇×#3103	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#460	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×Q₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#1688	C₂	1	2	yes	maximal, complemented
6	C₇×#1686	C₂	1	2	yes	maximal, complemented
7	C₇×#1668	C₂	1	2	yes	maximal, complemented
8	C₇×#1666	C₂	1	2	yes	maximal, complemented
9	C₇×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇×#1547	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇×#1545	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₇×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂×C₇×AGL(1,ℤ/8)	—	2	1	no	—
14	C₂²×C₇×SD₁₆	—	2	1	no	—
15	C₇×#532	C₂²	1	1	yes	characteristic
16	C₂×C₇×#127	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
18	C₇×#496	—	2	2	no	—
19	C₇×#493	C₂²	1	2	yes	—
20	C₇×#492	C₂²	1	2	yes	—
21	C₇×#481	—	2	1	no	—
22	C₇×#475	C₂²	1	1	yes	characteristic, complemented
23	C₇×#475	C₂²	1	1	yes	characteristic, complemented
24	C₇×#460	—	2	1	no	—
25	C₇×#460	—	2	1	no	—
26	C₇×#460	C₂²	1	2	yes	—
27	C₇×#460	C₂²	1	2	yes	—
28	C₇×#460	C₂²	1	1	yes	characteristic, complemented
29	C₇×#460	C₂²	1	1	yes	characteristic, complemented
30	C₇×#448	—	2	1	no	—
31	C₇×#446	—	2	2	no	—
32	C₇×#446	C₂²	1	2	yes	complemented
33	C₇×C₈×Q₈	C₂²	1	2	yes	complemented
34	C₄×C₇×SD₁₆	—	2	1	no	—
35	C₄×C₇×SD₁₆	—	2	1	no	—
36	C₇×#431	C₂²	1	1	yes	characteristic
37	C₇×#423	—	2	1	no	—
38	C₇×#423	C₂²	1	1	yes	characteristic
39	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₇×#102	—	2	1	no	—
41	C₄×C₇×M₁₆	—	2	1	no	—
42	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic
43	C₇×#335	C₂²	1	2	yes	complemented
44	C₇×#334	C₂²	1	2	yes	complemented
45	C₇×#332	C₂²	1	2	yes	complemented
46	C₇×#331	C₂²	1	2	yes	—
47	C₇×#331	C₂²	1	2	yes	—
48	C₇×#331	C₂²	1	2	yes	complemented
49	#19170	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×C₇×#41	—	2	1	no	—
51	C₂²×C₇×Q₈	D₄	1	1	yes	characteristic
52	C₂²×C₇×D₄	—	2	1	no	—
53	C₂²×C₇×D₄	—	2	1	no	—
54	C₂²×C₇×D₄	—	2	1	no	—
55	C₇×AGL(1,ℤ/8)	—	4	4	no	—
56	C₇×AGL(1,ℤ/8)	—	4	4	no	—
57	C₂×C₇×SD₁₆	—	2	2	no	—
58	C₂×C₇×SD₁₆	—	2	2	no	—
59	C₂×C₇×SD₁₆	—	2	1	no	—
60	C₂×C₇×SD₁₆	—	2	1	no	—
61	C₂×C₇×SD₁₆	—	2	2	no	—
62	C₂×C₇×SD₁₆	—	2	2	no	—
63	C₂×C₇×SD₁₆	—	4	4	no	—
64	C₂×C₇×D₈	—	2	2	no	—
65	C₂×C₇×D₈	—	2	2	no	—
66	C₂×C₇×M₁₆	—	2	1	no	—
67	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
68	C₂×C₇×M₁₆	D₄	1	1	yes	characteristic
69	C₂²×C₇×C₈	—	2	1	no	—
70	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic
71	C₇×#127	—	2	1	no	—
72	C₇×#127	—	2	1	no	—
73	C₇×#127	—	2	1	no	—
74	C₇×#127	C₂³	1	2	yes	—
75	C₇×#127	C₂³	1	1	yes	characteristic
76	C₇×#124	—	2	1	no	—
77	C₇×#121	—	2	1	no	—
78	C₇×#121	—	2	1	no	—
79	C₄×C₇×Q₈	—	2	1	no	—
80	C₄×C₇×Q₈	—	2	1	no	—
81	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
82	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
83	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
84	C₄×C₇×Q₈	C₂³	1	1	yes	characteristic
85	C₄×C₇×D₄	—	2	1	no	—
86	C₄×C₇×D₄	—	2	1	no	—
87	C₂×C₇×C₄⋊C₄	—	2	1	no	—
88	C₂×C₇×C₄⋊C₄	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₇	—	2	1	no	—
90	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
91	C₇×#106	—	2	1	no	—
92	C₇×#106	—	2	1	no	—
93	C₇×C₄⋊C₈	—	2	2	no	—
94	C₇×C₄⋊C₈	—	2	2	no	—
95	C₇×C₄⋊C₈	—	2	1	no	—
96	C₇×C₄⋊C₈	—	2	1	no	—
97	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
98	C₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
99	C₇×C₄⋊C₈	C₂³	1	2	yes	—
100	C₇×C₄⋊C₈	C₂³	1	2	yes	—
101	C₇×#103	—	2	1	no	—
102	C₇×#103	—	2	1	no	—
103	C₇×#102	—	2	1	no	—
104	C₇×#102	—	2	1	no	—
105	C₇×#102	—	2	2	no	—
106	C₇×#102	—	2	2	no	—
107	C₇×C₂²⋊C₈	—	2	1	no	—
108	C₇×#97	—	2	2	no	—
109	C₇×#97	D₄	1	2	yes	—
110	C₄×C₇×C₈	—	2	2	no	—
111	C₄×C₇×C₈	—	2	1	no	—
112	C₄×C₇×C₈	C₂³	1	2	yes	—
113	#3151	C₂×C₇	1	1	yes	characteristic
114	#3103	C₂×C₇	1	1	yes	characteristic, complemented
115	C₂×#460	C₂×C₇	1	1	yes	characteristic, complemented
116	Q₈×M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
117	#1688	C₂×C₇	1	2	yes	complemented
118	#1686	C₂×C₇	1	2	yes	complemented
119	#1668	C₂×C₇	1	2	yes	complemented
120	#1666	C₂×C₇	1	2	yes	complemented
121	#1565	C₂×C₇	1	1	yes	characteristic, complemented
122	#1547	C₂×C₇	1	1	yes	characteristic, complemented
123	#1545	C₂×C₇	1	1	yes	characteristic, complemented
124	C₂⁴×C₇	—	4	1	no	—
125	C₇×#41	—	4	2	no	—
126	C₇×#41	—	4	2	no	—
127	C₂×C₇×Q₈	—	2	1	no	—
128	C₂×C₇×Q₈	—	2	2	no	—
129	C₂×C₇×Q₈	—	2	1	no	—
130	C₂×C₇×Q₈	—	4	1	no	—
131	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
132	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
133	C₂×C₇×D₄	—	2	1	no	—
134	C₂×C₇×D₄	—	2	2	no	—
135	C₂×C₇×D₄	—	2	1	no	—
136	C₂×C₇×D₄	—	2	2	no	—
137	C₂×C₇×D₄	—	2	1	no	—
138	C₂×C₇×D₄	—	2	2	no	—
139	C₂×C₇×D₄	—	4	2	no	—
140	C₂×C₇×D₄	—	4	2	no	—
141	C₂×C₇×D₄	—	4	1	no	—
142	C₂×C₇×D₄	—	4	1	no	—
143	C₂×C₇×D₄	—	4	1	no	—
144	C₂×C₇×D₄	—	4	1	no	—
145	C₂×C₇×D₄	—	4	2	no	—
146	C₂×C₇×D₄	—	4	2	no	—
147	C₂×C₇×D₄	—	8	2	no	—
148	C₂²×C₄×C₇	—	2	1	no	—
149	C₂²×C₄×C₇	—	2	1	no	—
150	C₂²×C₄×C₇	—	4	1	no	—
151	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₇	#41	1	1	yes	characteristic
154	C₇×SD₁₆	—	4	4	no	—
155	C₇×SD₁₆	—	4	2	no	—
156	C₇×SD₁₆	—	4	4	no	—
157	C₇×SD₁₆	—	4	2	no	—
158	C₇×SD₁₆	—	4	4	no	—
159	C₇×SD₁₆	—	4	4	no	—
160	C₇×D₈	—	4	4	no	—
161	C₇×D₈	—	4	4	no	—
162	C₇×M₁₆	—	2	2	no	—
163	C₇×M₁₆	—	4	2	no	—
164	C₇×M₁₆	SD₁₆	1	4	yes	complemented
165	C₂×C₇×C₈	—	2	2	no	—
166	C₂×C₇×C₈	—	2	1	no	—
167	C₂×C₇×C₈	—	2	1	no	—
168	C₂×C₇×C₈	—	2	2	no	—
169	C₂×C₇×C₈	—	4	2	no	—
170	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
171	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
172	C₇×C₄⋊C₄	—	2	1	no	—
173	C₇×C₄⋊C₄	—	2	1	no	—
174	C₇×C₄⋊C₄	—	2	1	no	—
175	C₇×C₄⋊C₄	—	2	1	no	—
176	C₇×C₄⋊C₄	—	2	1	no	—
177	C₇×C₄⋊C₄	—	2	1	no	—
178	C₇×C₄⋊C₄	—	2	1	no	—
179	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
180	C₇×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
181	C₇×C₂²⋊C₄	—	4	1	no	—
182	C₇×C₂²⋊C₄	—	4	1	no	—
183	C₄²×C₇	—	2	1	no	—
184	C₄²×C₇	—	2	1	no	—
185	C₄²×C₇	—	2	1	no	—
186	C₄²×C₇	—	2	1	no	—
187	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₇	#41	1	1	yes	characteristic
190	C₄²×C₇	C₂⁴	1	1	yes	characteristic
191	C₂×AGL(1,ℤ/8)	—	2	1	no	—
192	C₂×AGL(1,ℤ/8)	—	2	1	no	—
193	C₂²×SD₁₆	—	2	1	no	—
194	#532	C₂²×C₇	1	1	yes	characteristic
195	C₂×#127	C₂²×C₇	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
197	#496	—	2	2	no	—
198	#493	C₂²×C₇	1	2	yes	—
199	#492	C₂²×C₇	1	2	yes	—
200	#481	—	2	1	no	—
201	#475	C₂²×C₇	1	1	yes	characteristic, complemented
202	#475	C₂²×C₇	1	1	yes	characteristic, complemented
203	#460	—	2	1	no	—
204	#460	—	2	1	no	—
205	#460	C₂²×C₇	1	2	yes	—
206	#460	C₂²×C₇	1	2	yes	—
207	#460	C₂²×C₇	1	1	yes	characteristic, complemented
208	#460	C₂²×C₇	1	1	yes	characteristic, complemented
209	#448	—	2	1	no	—
210	#446	—	2	2	no	—
211	#446	C₂²×C₇	1	2	yes	complemented
212	C₈×Q₈	C₂²×C₇	1	2	yes	complemented
213	C₄×SD₁₆	—	2	1	no	—
214	C₄×SD₁₆	—	2	1	no	—
215	#431	C₂²×C₇	1	1	yes	characteristic
216	#423	—	2	1	no	—
217	#423	C₂²×C₇	1	1	yes	characteristic
218	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
219	C₂×#102	—	2	1	no	—
220	C₄×M₁₆	—	2	1	no	—
221	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic
222	#335	C₂²×C₇	1	2	yes	complemented
223	#334	C₂²×C₇	1	2	yes	complemented
224	#332	C₂²×C₇	1	2	yes	complemented
225	#331	C₂²×C₇	1	2	yes	—
226	#331	C₂²×C₇	1	2	yes	—
227	#331	C₂²×C₇	1	2	yes	complemented
228	C₂³×C₇	—	4	1	no	—
229	C₂³×C₇	—	4	2	no	—
230	C₂³×C₇	—	8	2	no	—
231	C₂³×C₇	—	8	2	no	—
232	C₂³×C₇	—	8	2	no	—
233	C₂³×C₇	#121	1	1	yes	characteristic
234	C₇×Q₈	—	2	1	no	—
235	C₇×Q₈	—	2	1	no	—
236	C₇×Q₈	—	2	1	no	—
237	C₇×Q₈	—	4	2	no	—
238	C₇×Q₈	—	4	1	no	—
239	C₇×Q₈	—	4	1	no	—
240	C₇×Q₈	AGL(1,ℤ/8)	1	2	yes	complemented
241	C₇×D₄	—	4	1	no	—
242	C₇×D₄	—	4	2	no	—
243	C₇×D₄	—	4	1	no	—
244	C₇×D₄	—	4	2	no	—
245	C₇×D₄	—	4	2	no	—
246	C₇×D₄	—	4	1	no	—
247	C₇×D₄	—	4	1	no	—
248	C₇×D₄	—	4	2	no	—
249	C₇×D₄	—	8	1	no	—
250	C₇×D₄	—	8	2	no	—
251	C₇×D₄	—	8	2	no	—
252	C₇×D₄	—	8	2	no	—
253	C₇×D₄	—	8	2	no	—
254	C₇×D₄	—	8	2	no	—
255	C₂×C₄×C₇	—	2	1	no	—
256	C₂×C₄×C₇	—	2	1	no	—
257	C₂×C₄×C₇	—	2	1	no	—
258	C₂×C₄×C₇	—	2	1	no	—
259	C₂×C₄×C₇	—	2	1	no	—
260	C₂×C₄×C₇	—	2	1	no	—
261	C₂×C₄×C₇	—	4	1	no	—
262	C₂×C₄×C₇	—	4	2	no	—
263	C₂×C₄×C₇	—	4	1	no	—
264	C₂×C₄×C₇	—	8	1	no	—
265	C₂×C₄×C₇	—	8	2	no	—
266	C₂×C₄×C₇	#121	1	1	yes	characteristic
267	C₂×C₄×C₇	#121	1	1	yes	characteristic
268	C₂×C₄×C₇	#121	1	1	yes	characteristic
269	C₂×C₄×C₇	C₂×SD₁₆	1	2	yes	—
270	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
271	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
274	C₇×C₈	—	2	2	no	—
275	C₇×C₈	—	4	2	no	—
276	C₇×C₈	—	4	2	no	—
277	C₇×C₈	—	4	1	no	—
278	C₇×C₈	—	4	1	no	—
279	C₇×C₈	C₂×SD₁₆	1	4	yes	complemented
280	C₂×#41	—	2	1	no	—
281	C₂²×Q₈	C₇×D₄	1	1	yes	characteristic
282	C₂²×D₄	—	2	1	no	—
283	C₂²×D₄	—	2	1	no	—
284	C₂²×D₄	—	2	1	no	—
285	AGL(1,ℤ/8)	—	4	4	no	—
286	AGL(1,ℤ/8)	—	4	4	no	—
287	C₂×SD₁₆	—	2	2	no	—
288	C₂×SD₁₆	—	2	2	no	—
289	C₂×SD₁₆	—	2	1	no	—
290	C₂×SD₁₆	—	2	1	no	—
291	C₂×SD₁₆	—	2	2	no	—
292	C₂×SD₁₆	—	2	2	no	—
293	C₂×SD₁₆	—	4	4	no	—
294	C₂×D₈	—	2	2	no	—
295	C₂×D₈	—	2	2	no	—
296	C₂×M₁₆	—	2	1	no	—
297	C₂×M₁₆	C₇×D₄	1	1	yes	characteristic
298	C₂×M₁₆	C₇×D₄	1	1	yes	characteristic
299	C₂²×C₈	—	2	1	no	—
300	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic
301	#127	—	2	1	no	—
302	#127	—	2	1	no	—
303	#127	—	2	1	no	—
304	#127	C₂³×C₇	1	2	yes	—
305	#127	C₂³×C₇	1	1	yes	characteristic
306	#124	—	2	1	no	—
307	#121	—	2	1	no	—
308	#121	—	2	1	no	—
309	C₄×Q₈	—	2	1	no	—
310	C₄×Q₈	—	2	1	no	—
311	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
312	C₄×Q₈	C₇×D₄	1	1	yes	characteristic
313	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
314	C₄×Q₈	C₂³×C₇	1	1	yes	characteristic
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	C₇×D₄	1	1	yes	characteristic
319	C₂×C₄²	—	2	1	no	—
320	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
321	#106	—	2	1	no	—
322	#106	—	2	1	no	—
323	C₄⋊C₈	—	2	2	no	—
324	C₄⋊C₈	—	2	2	no	—
325	C₄⋊C₈	—	2	1	no	—
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	C₂³×C₇	1	2	yes	—
328	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
329	C₄⋊C₈	C₂³×C₇	1	1	yes	characteristic
330	C₄⋊C₈	C₂³×C₇	1	2	yes	—
331	#103	—	2	1	no	—
332	#103	—	2	1	no	—
333	#102	—	2	2	no	—
334	#102	—	2	2	no	—
335	#102	—	2	1	no	—
336	#102	—	2	1	no	—
337	C₂²⋊C₈	—	2	1	no	—
338	#97	—	2	2	no	—
339	#97	C₇×D₄	1	2	yes	—
340	C₄×C₈	—	2	2	no	—
341	C₄×C₈	—	2	1	no	—
342	C₄×C₈	C₂³×C₇	1	2	yes	—
343	C₂²×C₇	—	2	1	no	—
344	C₂²×C₇	—	2	1	no	—
345	C₂²×C₇	—	8	1	no	—
346	C₂²×C₇	—	8	2	no	—
347	C₂²×C₇	—	8	2	no	—
348	C₂²×C₇	—	8	1	no	—
349	C₂²×C₇	—	8	2	no	—
350	C₂²×C₇	—	8	1	no	—
351	C₂²×C₇	—	16	4	no	—
352	C₂²×C₇	#460	1	2	yes	—
353	C₂²×C₇	C₂×#121	1	1	yes	characteristic
354	C₄×C₇	—	2	1	no	—
355	C₄×C₇	—	2	2	no	—
356	C₄×C₇	—	2	1	no	—
357	C₄×C₇	—	2	1	no	—
358	C₄×C₇	—	4	1	no	—
359	C₄×C₇	—	4	1	no	—
360	C₄×C₇	—	4	1	no	—
361	C₄×C₇	—	8	1	no	—
362	C₄×C₇	#460	1	2	yes	—
363	C₄×C₇	C₂²×SD₁₆	1	1	yes	characteristic
364	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
365	C₄×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
366	C₄×C₇	#575	1	1	yes	characteristic
367	C₂⁴	—	4	1	no	—
368	#41	—	4	2	no	—
369	#41	—	4	2	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	4	1	no	—
374	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
375	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	4	2	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	1	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	—	4	1	no	—
387	C₂×D₄	—	4	1	no	—
388	C₂×D₄	—	4	2	no	—
389	C₂×D₄	—	4	2	no	—
390	C₂×D₄	—	8	2	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	4	1	no	—
394	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
395	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
396	C₂²×C₄	C₇×#41	1	1	yes	characteristic
397	SD₁₆	—	4	4	no	—
398	SD₁₆	—	4	2	no	—
399	SD₁₆	—	4	4	no	—
400	SD₁₆	—	4	2	no	—
401	SD₁₆	—	4	4	no	—
402	SD₁₆	—	4	4	no	—
403	D₈	—	4	4	no	—
404	D₈	—	4	4	no	—
405	M₁₆	—	2	2	no	—
406	M₁₆	—	4	2	no	—
407	M₁₆	C₇×SD₁₆	1	4	yes	complemented
408	C₂×C₈	—	2	2	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	4	2	no	—
413	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
414	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
423	C₄⋊C₄	C₂×C₇×D₄	1	1	yes	characteristic
424	C₂²⋊C₄	—	4	1	no	—
425	C₂²⋊C₄	—	4	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	1	no	—
428	C₄²	—	2	1	no	—
429	C₄²	—	2	1	no	—
430	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
431	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
432	C₄²	C₇×#41	1	1	yes	characteristic
433	C₄²	C₂⁴×C₇	1	1	yes	characteristic
434	C₂×C₇	—	2	2	no	—
435	C₂×C₇	—	16	1	no	—
436	C₂×C₇	—	16	2	no	—
437	C₂×C₇	C₂×#460	1	1	yes	characteristic
438	C₂×C₇	#3055	1	1	yes	characteristic
439	C₂×C₇	#3068	1	1	yes	characteristic
440	C₂³	—	4	1	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	8	2	no	—
443	C₂³	—	8	2	no	—
444	C₂³	—	8	2	no	—
445	C₂³	C₇×#121	1	1	yes	characteristic
446	Q₈	—	2	1	no	—
447	Q₈	—	2	1	no	—
448	Q₈	—	2	1	no	—
449	Q₈	—	4	2	no	—
450	Q₈	—	4	1	no	—
451	Q₈	—	4	1	no	—
452	Q₈	C₇×AGL(1,ℤ/8)	1	2	yes	complemented
453	D₄	—	4	1	no	—
454	D₄	—	4	2	no	—
455	D₄	—	4	1	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	2	no	—
461	D₄	—	8	1	no	—
462	D₄	—	8	2	no	—
463	D₄	—	8	2	no	—
464	D₄	—	8	2	no	—
465	D₄	—	8	2	no	—
466	D₄	—	8	2	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	8	1	no	—
477	C₂×C₄	—	8	2	no	—
478	C₂×C₄	C₇×#121	1	1	yes	characteristic
479	C₂×C₄	C₇×#121	1	1	yes	characteristic
480	C₂×C₄	C₇×#121	1	1	yes	characteristic
481	C₂×C₄	C₂×C₇×SD₁₆	1	2	yes	—
482	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
483	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
484	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
485	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
486	C₈	—	2	2	no	—
487	C₈	—	4	2	no	—
488	C₈	—	4	2	no	—
489	C₈	—	4	1	no	—
490	C₈	—	4	1	no	—
491	C₈	C₂×C₇×SD₁₆	1	4	yes	complemented
492	C₇	#19170	1	1	yes	characteristic, direct product factor, Sylow
493	C₂²	—	2	1	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	8	1	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	2	no	—
498	C₂²	—	8	1	no	—
499	C₂²	—	8	2	no	—
500	C₂²	—	8	1	no	—
501	C₂²	—	16	4	no	—
502	C₂²	C₇×#460	1	2	yes	—
503	C₂²	C₂×C₇×#121	1	1	yes	characteristic
504	C₄	—	2	1	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	1	no	—
507	C₄	—	2	1	no	—
508	C₄	—	4	1	no	—
509	C₄	—	4	1	no	—
510	C₄	—	4	1	no	—
511	C₄	—	8	1	no	—
512	C₄	C₇×#460	1	2	yes	—
513	C₄	C₂²×C₇×SD₁₆	1	1	yes	characteristic
514	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
515	C₄	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
516	C₄	C₇×#575	1	1	yes	characteristic
517	C₂	—	2	2	no	—
518	C₂	—	16	2	no	—
519	C₂	—	16	1	no	—
520	C₂	C₂×C₇×#460	1	1	yes	characteristic
521	C₂	C₇×#3055	1	1	yes	characteristic
522	C₂	C₇×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇
prime radical	C₇×#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#3103	C₂	semidirect
3. C₂×C₇×#460	C₂	semidirect
4. C₇×Q₈×M₁₆	C₂	semidirect
5. C₇×#1688	C₂	semidirect
6. C₇×#1686	C₂	semidirect
7. C₇×#1668	C₂	semidirect
8. C₇×#1666	C₂	semidirect
9. C₇×#1565	C₂	semidirect
10. C₇×#1547	C₂	semidirect
11. C₇×#1545	C₂	semidirect
22. C₇×#475	C₂²	semidirect
23. C₇×#475	C₂²	semidirect
28. C₇×#460	C₂²	semidirect
29. C₇×#460	C₂²	semidirect
32. C₇×#446	C₂²	semidirect
33. C₇×C₈×Q₈	C₂²	semidirect
43. C₇×#335	C₂²	semidirect
44. C₇×#334	C₂²	semidirect
45. C₇×#332	C₂²	semidirect
48. C₇×#331	C₂²	semidirect
49. #19170	C₇	direct (N × H)
114. #3103	C₂×C₇	semidirect
115. C₂×#460	C₂×C₇	semidirect
116. Q₈×M₁₆	C₂×C₇	semidirect
117. #1688	C₂×C₇	semidirect
118. #1686	C₂×C₇	semidirect
119. #1668	C₂×C₇	semidirect
120. #1666	C₂×C₇	semidirect
121. #1565	C₂×C₇	semidirect
122. #1547	C₂×C₇	semidirect
123. #1545	C₂×C₇	semidirect
164. C₇×M₁₆	SD₁₆	semidirect
201. #475	C₂²×C₇	semidirect
202. #475	C₂²×C₇	semidirect
207. #460	C₂²×C₇	semidirect
208. #460	C₂²×C₇	semidirect
211. #446	C₂²×C₇	semidirect
212. C₈×Q₈	C₂²×C₇	semidirect
222. #335	C₂²×C₇	semidirect
223. #334	C₂²×C₇	semidirect
224. #332	C₂²×C₇	semidirect
227. #331	C₂²×C₇	semidirect
240. C₇×Q₈	AGL(1,ℤ/8)	semidirect
279. C₇×C₈	C₂×SD₁₆	semidirect
407. M₁₆	C₇×SD₁₆	semidirect
452. Q₈	C₇×AGL(1,ℤ/8)	semidirect
491. C₈	C₂×C₇×SD₁₆	semidirect
492. C₇	#19170	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×#3151	2
C₇×#3103	2
C₂×C₇×#460	2
C₇×Q₈×M₁₆	2
C₇×#1688	2
C₇×#1686	2
C₇×#1668	2
C₇×#1666	2
C₇×#1565	2
C₇×#1547	2
C₇×#1545	2
#19170	7

Subgroups — C7×#19170 #13196386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#19170 #13196386