Subgroups — C₇×#60564 #13237780

Order 1792.

2218 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#518	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×#3123	C₂	1	4	yes	maximal, complemented
4	C₇×#3122	C₂	1	4	yes	maximal, complemented
5	C₇×#3118	C₂	1	8	yes	maximal, complemented
6	C₇×#3093	C₂	1	2	yes	maximal, complemented
7	C₇×#3077	C₂	1	2	yes	maximal, complemented
8	C₇×#3036	C₂	1	2	yes	maximal, complemented
9	C₇×#3003	C₂	1	2	yes	maximal, complemented
10	C₄×C₇×AGL(1,ℤ/8)	C₂	1	4	yes	maximal, complemented
11	C₇×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₇×AGL(1,ℤ/8)	C₂²	1	4	yes	—
14	C₇×#542	C₂²	1	4	yes	complemented
15	C₇×#541	C₂²	1	2	yes	complemented
16	C₇×#540	C₂²	1	2	yes	complemented
17	C₇×#539	—	2	2	no	—
18	C₇×#538	—	2	2	no	—
19	C₇×#533	C₂²	1	1	yes	characteristic, complemented
20	C₇×#532	C₂²	1	1	yes	characteristic, complemented
21	C₇×#529	—	2	4	no	—
22	C₇×#525	C₂²	1	2	yes	—
23	C₇×#518	—	2	4	no	—
24	C₇×#518	C₂²	1	4	yes	—
25	C₇×#518	C₂²	1	4	yes	—
26	C₄×C₇×#41	C₂²	1	2	yes	complemented
27	C₂×C₄×C₇×D₄	—	2	2	no	—
28	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
29	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
30	C₂×C₇×#117	C₂²	1	1	yes	characteristic
31	C₂×C₇×#117	C₂²	1	1	yes	characteristic
32	C₂²×C₇×C₂²⋊C₄	—	2	2	no	—
33	C₇×#485	C₂²	1	8	yes	complemented
34	C₇×#482	C₂²	1	8	yes	complemented
35	C₇×#478	C₂²	1	8	yes	complemented
36	C₇×#476	C₂²	1	4	yes	complemented
37	C₇×#474	C₂²	1	4	yes	complemented
38	C₇×#463	C₂²	1	8	yes	complemented
39	C₇×#461	C₂²	1	4	yes	complemented
40	C₇×#459	C₂²	1	4	yes	complemented
41	C₇×#450	C₂²	1	8	yes	—
42	C₇×#447	C₂²	1	8	yes	—
43	C₇×#442	C₂²	1	8	yes	complemented
44	C₇×#440	C₂²	1	8	yes	complemented
45	C₄×C₇×SD₁₆	C₂²	1	8	yes	complemented
46	C₄×C₇×D₈	C₂²	1	8	yes	complemented
47	C₇×#433	C₂²	1	4	yes	complemented
48	C₇×#433	C₂²	1	4	yes	complemented
49	C₇×#428	C₂²	1	4	yes	complemented
50	C₇×#423	C₂²	1	2	yes	complemented
51	C₇×#418	C₂²	1	4	yes	—
52	C₇×#417	C₂²	1	4	yes	complemented
53	C₇×#407	C₂²	1	2	yes	—
54	C₄×C₇×M₁₆	C₂²	1	2	yes	complemented
55	#60564	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁵×C₇	—	2	1	no	—
57	C₂×C₇×#41	C₂³	1	2	yes	—
58	C₂²×C₇×D₄	—	2	8	no	—
59	C₂²×C₇×D₄	—	2	1	no	—
60	C₂²×C₇×D₄	—	2	2	no	—
61	C₂²×C₇×D₄	C₂³	1	4	yes	—
62	C₂²×C₇×D₄	C₂³	1	2	yes	—
63	C₂³×C₄×C₇	—	2	2	no	—
64	C₂³×C₄×C₇	—	2	2	no	—
65	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
66	C₇×AGL(1,ℤ/8)	—	2	16	no	—
67	C₂×C₇×SD₁₆	C₂³	1	8	yes	—
68	C₂×C₇×D₈	C₂³	1	8	yes	—
69	C₂×C₇×M₁₆	C₂³	1	4	yes	—
70	C₇×C₄⋊Q₈	C₂³	1	1	yes	characteristic, complemented
71	C₇×C₄⋊Q₈	C₂³	1	2	yes	complemented
72	C₇×#127	C₂³	1	1	yes	characteristic, complemented
73	C₇×#126	—	2	4	no	—
74	C₇×#126	—	2	4	no	—
75	C₇×#125	—	2	4	no	—
76	C₇×#125	C₂³	1	2	yes	complemented
77	C₇×#124	—	2	4	no	—
78	C₇×#124	C₂³	1	2	yes	complemented
79	C₇×#123	—	2	2	no	—
80	C₇×#123	—	2	4	no	—
81	C₇×#123	—	2	4	no	—
82	C₇×C₂²⋊Q₈	—	2	2	no	—
83	C₇×C₂²⋊Q₈	—	2	4	no	—
84	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
85	C₇×#121	—	2	2	no	—
86	C₇×#121	—	2	4	no	—
87	C₇×#121	C₂³	1	4	yes	—
88	C₇×#120	—	2	2	no	—
89	C₄×C₇×Q₈	—	2	2	no	—
90	C₄×C₇×Q₈	C₂³	1	4	yes	complemented
91	C₄×C₇×D₄	—	2	4	no	—
92	C₄×C₇×D₄	—	2	4	no	—
93	C₄×C₇×D₄	—	2	8	no	—
94	C₄×C₇×D₄	—	2	8	no	—
95	C₄×C₇×D₄	—	2	4	no	—
96	C₄×C₇×D₄	—	2	2	no	—
97	C₄×C₇×D₄	—	2	2	no	—
98	C₄×C₇×D₄	—	2	2	no	—
99	C₄×C₇×D₄	C₂³	1	4	yes	—
100	C₄×C₇×D₄	C₂³	1	4	yes	—
101	C₄×C₇×D₄	C₂³	1	8	yes	—
102	C₄×C₇×D₄	C₂³	1	4	yes	complemented
103	C₇×#117	—	2	2	no	—
104	C₇×#117	—	2	2	no	—
105	C₇×#117	D₄	1	4	yes	—
106	C₇×#117	C₂³	1	4	yes	—
107	C₇×#117	C₂³	1	2	yes	—
108	C₇×#117	C₂³	1	2	yes	—
109	C₇×#117	C₂³	1	2	yes	—
110	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
111	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
112	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
113	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
114	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
115	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
116	C₂×C₇×C₂²⋊C₄	—	2	8	no	—
117	C₂×C₇×C₂²⋊C₄	D₄	1	2	yes	—
118	C₂×C₄²×C₇	—	2	1	no	—
119	C₂×C₄²×C₇	—	2	2	no	—
120	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
122	C₇×#107	C₂³	1	8	yes	complemented
123	C₇×#106	C₂³	1	8	yes	complemented
124	C₇×C₄⋊C₈	C₂³	1	4	yes	complemented
125	C₇×C₄⋊C₈	C₂³	1	4	yes	complemented
126	C₇×#103	C₂³	1	8	yes	complemented
127	C₇×#102	C₂³	1	8	yes	—
128	C₇×#102	C₂³	1	8	yes	—
129	C₇×#102	C₂³	1	8	yes	complemented
130	C₇×C₂²⋊C₈	C₂³	1	8	yes	—
131	C₇×#97	C₂³	1	4	yes	complemented
132	C₄×C₇×C₈	C₂³	1	4	yes	complemented
133	#3455	C₂×C₇	1	1	yes	characteristic, complemented
134	C₂×#518	C₂×C₇	1	1	yes	characteristic, complemented
135	#3123	C₂×C₇	1	4	yes	complemented
136	#3122	C₂×C₇	1	4	yes	complemented
137	#3118	C₂×C₇	1	8	yes	complemented
138	#3093	C₂×C₇	1	2	yes	complemented
139	#3077	C₂×C₇	1	2	yes	complemented
140	#3036	C₂×C₇	1	2	yes	complemented
141	#3003	C₂×C₇	1	2	yes	complemented
142	C₄×AGL(1,ℤ/8)	C₂×C₇	1	4	yes	complemented
143	#2921	C₂×C₇	1	1	yes	characteristic, complemented
144	C₂⁴×C₇	—	2	8	no	—
145	C₂⁴×C₇	—	2	2	no	—
146	C₂⁴×C₇	—	2	4	no	—
147	C₂⁴×C₇	—	8	4	no	—
148	C₂⁴×C₇	C₂×D₄	1	1	yes	characteristic
149	C₇×#41	—	2	8	no	—
150	C₂×C₇×Q₈	—	2	2	no	—
151	C₂×C₇×Q₈	C₂⁴	1	2	yes	—
152	C₂×C₇×D₄	—	2	8	no	—
153	C₂×C₇×D₄	—	2	16	no	—
154	C₂×C₇×D₄	—	2	16	no	—
155	C₂×C₇×D₄	—	2	2	no	—
156	C₂×C₇×D₄	—	2	4	no	—
157	C₂×C₇×D₄	—	2	4	no	—
158	C₂×C₇×D₄	—	2	2	no	—
159	C₂×C₇×D₄	—	2	2	no	—
160	C₂×C₇×D₄	—	2	4	no	—
161	C₂×C₇×D₄	#41	1	8	yes	—
162	C₂×C₇×D₄	C₂⁴	1	2	yes	—
163	C₂×C₇×D₄	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₇	—	2	4	no	—
165	C₂²×C₄×C₇	—	2	2	no	—
166	C₂²×C₄×C₇	—	2	2	no	—
167	C₂²×C₄×C₇	—	2	4	no	—
168	C₂²×C₄×C₇	—	2	8	no	—
169	C₂²×C₄×C₇	—	2	4	no	—
170	C₂²×C₄×C₇	—	2	4	no	—
171	C₂²×C₄×C₇	—	8	4	no	—
172	C₂²×C₄×C₇	—	8	4	no	—
173	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
179	C₇×SD₁₆	—	2	16	no	—
180	C₇×D₈	—	2	16	no	—
181	C₇×M₁₆	—	2	8	no	—
182	C₂×C₇×C₈	C₂⁴	1	8	yes	—
183	C₇×C₄⋊C₄	—	2	2	no	—
184	C₇×C₄⋊C₄	—	2	4	no	—
185	C₇×C₄⋊C₄	—	2	2	no	—
186	C₇×C₄⋊C₄	—	2	4	no	—
187	C₇×C₄⋊C₄	—	2	4	no	—
188	C₇×C₄⋊C₄	—	2	2	no	—
189	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
190	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
191	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
192	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
193	C₇×C₂²⋊C₄	—	2	4	no	—
194	C₇×C₂²⋊C₄	—	2	2	no	—
195	C₇×C₂²⋊C₄	—	2	4	no	—
196	C₇×C₂²⋊C₄	—	2	8	no	—
197	C₇×C₂²⋊C₄	—	2	4	no	—
198	C₇×C₂²⋊C₄	—	2	4	no	—
199	C₇×C₂²⋊C₄	—	2	4	no	—
200	C₇×C₂²⋊C₄	—	2	4	no	—
201	C₇×C₂²⋊C₄	—	2	4	no	—
202	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
203	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
204	C₄²×C₇	—	2	4	no	—
205	C₄²×C₇	—	2	2	no	—
206	C₄²×C₇	—	2	4	no	—
207	C₄²×C₇	C₂×D₄	1	2	yes	—
208	C₄²×C₇	C₂×D₄	1	2	yes	—
209	C₄²×C₇	C₂⁴	1	2	yes	—
210	C₄²×C₇	C₂⁴	1	1	yes	characteristic
211	C₄²×C₇	C₂⁴	1	1	yes	characteristic
212	C₂³×D₄	C₂²×C₇	1	1	yes	characteristic
213	C₂×AGL(1,ℤ/8)	C₂²×C₇	1	4	yes	—
214	#542	C₂²×C₇	1	4	yes	complemented
215	#541	C₂²×C₇	1	2	yes	complemented
216	#540	C₂²×C₇	1	2	yes	complemented
217	#539	—	2	2	no	—
218	#538	—	2	2	no	—
219	#533	C₂²×C₇	1	1	yes	characteristic, complemented
220	#532	C₂²×C₇	1	1	yes	characteristic, complemented
221	#529	—	2	4	no	—
222	#525	C₂²×C₇	1	2	yes	—
223	#518	—	2	4	no	—
224	#518	C₂²×C₇	1	4	yes	—
225	#518	C₂²×C₇	1	4	yes	—
226	C₄×#41	C₂²×C₇	1	2	yes	complemented
227	C₂×C₄×D₄	—	2	2	no	—
228	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
229	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
230	C₂×#117	C₂²×C₇	1	1	yes	characteristic
231	C₂×#117	C₂²×C₇	1	1	yes	characteristic
232	C₂²×C₂²⋊C₄	—	2	2	no	—
233	#485	C₂²×C₇	1	8	yes	complemented
234	#482	C₂²×C₇	1	8	yes	complemented
235	#478	C₂²×C₇	1	8	yes	complemented
236	#476	C₂²×C₇	1	4	yes	complemented
237	#474	C₂²×C₇	1	4	yes	complemented
238	#463	C₂²×C₇	1	8	yes	complemented
239	#461	C₂²×C₇	1	4	yes	complemented
240	#459	C₂²×C₇	1	4	yes	complemented
241	#450	C₂²×C₇	1	8	yes	—
242	#447	C₂²×C₇	1	8	yes	—
243	#442	C₂²×C₇	1	8	yes	complemented
244	#440	C₂²×C₇	1	8	yes	complemented
245	C₄×SD₁₆	C₂²×C₇	1	8	yes	complemented
246	C₄×D₈	C₂²×C₇	1	8	yes	complemented
247	#433	C₂²×C₇	1	4	yes	complemented
248	#433	C₂²×C₇	1	4	yes	complemented
249	#428	C₂²×C₇	1	4	yes	complemented
250	#423	C₂²×C₇	1	2	yes	complemented
251	#418	C₂²×C₇	1	4	yes	—
252	#417	C₂²×C₇	1	4	yes	complemented
253	#407	C₂²×C₇	1	2	yes	—
254	C₄×M₁₆	C₂²×C₇	1	2	yes	complemented
255	C₂³×C₇	—	2	16	no	—
256	C₂³×C₇	—	2	2	no	—
257	C₂³×C₇	—	2	8	no	—
258	C₂³×C₇	—	2	4	no	—
259	C₂³×C₇	—	2	2	no	—
260	C₂³×C₇	—	4	2	no	—
261	C₂³×C₇	—	4	4	no	—
262	C₂³×C₇	—	4	4	no	—
263	C₂³×C₇	—	8	8	no	—
264	C₂³×C₇	—	8	16	no	—
265	C₂³×C₇	C₂²×D₄	1	2	yes	—
266	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
267	C₇×Q₈	—	2	4	no	—
268	C₇×Q₈	—	8	2	no	—
269	C₇×D₄	—	2	16	no	—
270	C₇×D₄	—	2	8	no	—
271	C₇×D₄	—	2	4	no	—
272	C₇×D₄	—	2	4	no	—
273	C₇×D₄	—	4	4	no	—
274	C₇×D₄	—	8	2	no	—
275	C₇×D₄	—	8	4	no	—
276	C₇×D₄	C₂×#41	1	8	yes	—
277	C₂×C₄×C₇	—	2	2	no	—
278	C₂×C₄×C₇	—	2	4	no	—
279	C₂×C₄×C₇	—	2	4	no	—
280	C₂×C₄×C₇	—	2	4	no	—
281	C₂×C₄×C₇	—	2	2	no	—
282	C₂×C₄×C₇	—	2	4	no	—
283	C₂×C₄×C₇	—	4	2	no	—
284	C₂×C₄×C₇	—	4	2	no	—
285	C₂×C₄×C₇	—	4	4	no	—
286	C₂×C₄×C₇	—	4	4	no	—
287	C₂×C₄×C₇	—	8	4	no	—
288	C₂×C₄×C₇	—	8	8	no	—
289	C₂×C₄×C₇	—	8	4	no	—
290	C₂×C₄×C₇	—	8	4	no	—
291	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
293	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₇	C₂×#41	1	4	yes	—
297	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
298	C₇×C₈	—	2	8	no	—
299	C₂⁵	—	2	1	no	—
300	C₂×#41	C₂³×C₇	1	2	yes	—
301	C₂²×D₄	—	2	8	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	—	2	2	no	—
304	C₂²×D₄	C₂³×C₇	1	4	yes	—
305	C₂²×D₄	C₂³×C₇	1	2	yes	—
306	C₂³×C₄	—	2	2	no	—
307	C₂³×C₄	—	2	2	no	—
308	C₂³×C₄	C₂³×C₇	1	1	yes	characteristic
309	AGL(1,ℤ/8)	—	2	16	no	—
310	C₂×SD₁₆	C₂³×C₇	1	8	yes	—
311	C₂×D₈	C₂³×C₇	1	8	yes	—
312	C₂×M₁₆	C₂³×C₇	1	4	yes	—
313	C₄⋊Q₈	C₂³×C₇	1	1	yes	characteristic, complemented
314	C₄⋊Q₈	C₂³×C₇	1	2	yes	complemented
315	#127	C₂³×C₇	1	1	yes	characteristic, complemented
316	#126	—	2	4	no	—
317	#126	—	2	4	no	—
318	#125	—	2	4	no	—
319	#125	C₂³×C₇	1	2	yes	complemented
320	#124	—	2	4	no	—
321	#124	C₂³×C₇	1	2	yes	complemented
322	#123	—	2	2	no	—
323	#123	—	2	4	no	—
324	#123	—	2	4	no	—
325	C₂²⋊Q₈	—	2	2	no	—
326	C₂²⋊Q₈	—	2	4	no	—
327	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
328	#121	—	2	2	no	—
329	#121	—	2	4	no	—
330	#121	C₂³×C₇	1	4	yes	—
331	#120	—	2	2	no	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×Q₈	C₂³×C₇	1	4	yes	complemented
334	C₄×D₄	—	2	4	no	—
335	C₄×D₄	—	2	4	no	—
336	C₄×D₄	—	2	8	no	—
337	C₄×D₄	—	2	8	no	—
338	C₄×D₄	—	2	4	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	C₂³×C₇	1	8	yes	—
343	C₄×D₄	C₂³×C₇	1	4	yes	—
344	C₄×D₄	C₂³×C₇	1	4	yes	—
345	C₄×D₄	C₂³×C₇	1	4	yes	complemented
346	#117	—	2	2	no	—
347	#117	—	2	2	no	—
348	#117	C₇×D₄	1	4	yes	—
349	#117	C₂³×C₇	1	4	yes	—
350	#117	C₂³×C₇	1	2	yes	—
351	#117	C₂³×C₇	1	2	yes	—
352	#117	C₂³×C₇	1	2	yes	—
353	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
354	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
355	C₂×C₂²⋊C₄	—	2	1	no	—
356	C₂×C₂²⋊C₄	—	2	2	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	—	2	8	no	—
359	C₂×C₂²⋊C₄	—	2	8	no	—
360	C₂×C₂²⋊C₄	C₇×D₄	1	2	yes	—
361	C₂×C₄²	—	2	1	no	—
362	C₂×C₄²	—	2	2	no	—
363	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
365	#107	C₂³×C₇	1	8	yes	complemented
366	#106	C₂³×C₇	1	8	yes	complemented
367	C₄⋊C₈	C₂³×C₇	1	4	yes	complemented
368	C₄⋊C₈	C₂³×C₇	1	4	yes	complemented
369	#103	C₂³×C₇	1	8	yes	complemented
370	#102	C₂³×C₇	1	8	yes	—
371	#102	C₂³×C₇	1	8	yes	—
372	#102	C₂³×C₇	1	8	yes	complemented
373	C₂²⋊C₈	C₂³×C₇	1	8	yes	—
374	#97	C₂³×C₇	1	4	yes	complemented
375	C₄×C₈	C₂³×C₇	1	4	yes	complemented
376	C₂²×C₇	—	2	2	no	—
377	C₂²×C₇	—	2	4	no	—
378	C₂²×C₇	—	2	2	no	—
379	C₂²×C₇	—	2	8	no	—
380	C₂²×C₇	—	4	16	no	—
381	C₂²×C₇	—	4	4	no	—
382	C₂²×C₇	—	4	4	no	—
383	C₂²×C₇	—	4	4	no	—
384	C₂²×C₇	—	4	4	no	—
385	C₂²×C₇	—	4	2	no	—
386	C₂²×C₇	—	8	16	no	—
387	C₂²×C₇	#525	1	2	yes	—
388	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
389	C₄×C₇	—	2	4	no	—
390	C₄×C₇	—	2	2	no	—
391	C₄×C₇	—	2	2	no	—
392	C₄×C₇	—	2	2	no	—
393	C₄×C₇	—	4	4	no	—
394	C₄×C₇	—	4	4	no	—
395	C₄×C₇	—	4	2	no	—
396	C₄×C₇	#525	1	2	yes	—
397	C₄×C₇	C₂²×#41	1	2	yes	—
398	C₂⁴	—	2	8	no	—
399	C₂⁴	—	2	2	no	—
400	C₂⁴	—	2	4	no	—
401	C₂⁴	—	8	4	no	—
402	C₂⁴	C₂×C₇×D₄	1	1	yes	characteristic
403	#41	—	2	8	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	C₂⁴×C₇	1	2	yes	—
406	C₂×D₄	—	2	8	no	—
407	C₂×D₄	—	2	16	no	—
408	C₂×D₄	—	2	16	no	—
409	C₂×D₄	—	2	2	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	2	no	—
413	C₂×D₄	—	2	2	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	C₇×#41	1	8	yes	—
416	C₂×D₄	C₂⁴×C₇	1	2	yes	—
417	C₂×D₄	C₂⁴×C₇	1	4	yes	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	8	4	no	—
426	C₂²×C₄	—	8	4	no	—
427	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
428	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
429	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
430	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
432	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
433	SD₁₆	—	2	16	no	—
434	D₈	—	2	16	no	—
435	M₁₆	—	2	8	no	—
436	C₂×C₈	C₂⁴×C₇	1	8	yes	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	4	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	4	no	—
441	C₄⋊C₄	—	2	4	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
444	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
445	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
446	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
447	C₂²⋊C₄	—	2	4	no	—
448	C₂²⋊C₄	—	2	2	no	—
449	C₂²⋊C₄	—	2	4	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	—	2	4	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	4	no	—
456	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
457	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
458	C₄²	—	2	4	no	—
459	C₄²	—	2	2	no	—
460	C₄²	—	2	4	no	—
461	C₄²	C₂×C₇×D₄	1	2	yes	—
462	C₄²	C₂×C₇×D₄	1	2	yes	—
463	C₄²	C₂⁴×C₇	1	2	yes	—
464	C₄²	C₂⁴×C₇	1	1	yes	characteristic
465	C₄²	C₂⁴×C₇	1	1	yes	characteristic
466	C₂×C₇	—	2	2	no	—
467	C₂×C₇	—	4	2	no	—
468	C₂×C₇	—	4	8	no	—
469	C₂×C₇	—	8	2	no	—
470	C₂×C₇	C₂×#525	1	1	yes	characteristic
471	C₂×C₇	#3585	1	2	yes	—
472	C₂³	—	2	16	no	—
473	C₂³	—	2	4	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	8	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	4	2	no	—
478	C₂³	—	4	4	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	8	8	no	—
481	C₂³	—	8	16	no	—
482	C₂³	C₂²×C₇×D₄	1	2	yes	—
483	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
484	Q₈	—	2	4	no	—
485	Q₈	—	8	2	no	—
486	D₄	—	2	16	no	—
487	D₄	—	2	4	no	—
488	D₄	—	2	4	no	—
489	D₄	—	2	8	no	—
490	D₄	—	4	4	no	—
491	D₄	—	8	2	no	—
492	D₄	—	8	4	no	—
493	D₄	C₂×C₇×#41	1	8	yes	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	4	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	4	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	8	4	no	—
505	C₂×C₄	—	8	8	no	—
506	C₂×C₄	—	8	4	no	—
507	C₂×C₄	—	8	4	no	—
508	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
509	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
510	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
511	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
512	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
513	C₂×C₄	C₂×C₇×#41	1	4	yes	—
514	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
515	C₈	—	2	8	no	—
516	C₇	#60564	1	1	yes	characteristic, direct product factor, Sylow
517	C₂²	—	2	2	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	8	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	16	no	—
523	C₂²	—	4	4	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	8	16	no	—
528	C₂²	C₇×#525	1	2	yes	—
529	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
530	C₄	—	2	4	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	2	no	—
534	C₄	—	4	4	no	—
535	C₄	—	4	4	no	—
536	C₄	—	4	2	no	—
537	C₄	C₇×#525	1	2	yes	—
538	C₄	C₂²×C₇×#41	1	2	yes	—
539	C₂	—	2	2	no	—
540	C₂	—	4	2	no	—
541	C₂	—	4	8	no	—
542	C₂	—	8	2	no	—
543	C₂	C₂×C₇×#525	1	1	yes	characteristic
544	C₂	C₇×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₇×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3455	C₂	semidirect
2. C₂×C₇×#518	C₂	semidirect
3. C₇×#3123	C₂	semidirect
4. C₇×#3122	C₂	semidirect
5. C₇×#3118	C₂	semidirect
6. C₇×#3093	C₂	semidirect
7. C₇×#3077	C₂	semidirect
8. C₇×#3036	C₂	semidirect
9. C₇×#3003	C₂	semidirect
10. C₄×C₇×AGL(1,ℤ/8)	C₂	semidirect
11. C₇×#2921	C₂	semidirect
14. C₇×#542	C₂²	semidirect
15. C₇×#541	C₂²	semidirect
16. C₇×#540	C₂²	semidirect
19. C₇×#533	C₂²	semidirect
20. C₇×#532	C₂²	semidirect
26. C₄×C₇×#41	C₂²	semidirect
33. C₇×#485	C₂²	semidirect
34. C₇×#482	C₂²	semidirect
35. C₇×#478	C₂²	semidirect
36. C₇×#476	C₂²	semidirect
37. C₇×#474	C₂²	semidirect
38. C₇×#463	C₂²	semidirect
39. C₇×#461	C₂²	semidirect
40. C₇×#459	C₂²	semidirect
43. C₇×#442	C₂²	semidirect
44. C₇×#440	C₂²	semidirect
45. C₄×C₇×SD₁₆	C₂²	semidirect
46. C₄×C₇×D₈	C₂²	semidirect
47. C₇×#433	C₂²	semidirect
48. C₇×#433	C₂²	semidirect
49. C₇×#428	C₂²	semidirect
50. C₇×#423	C₂²	semidirect
52. C₇×#417	C₂²	semidirect
54. C₄×C₇×M₁₆	C₂²	semidirect
55. #60564	C₇	direct (N × H)
70. C₇×C₄⋊Q₈	C₂³	semidirect
71. C₇×C₄⋊Q₈	C₂³	semidirect
72. C₇×#127	C₂³	semidirect
76. C₇×#125	C₂³	semidirect
78. C₇×#124	C₂³	semidirect
90. C₄×C₇×Q₈	C₂³	semidirect
102. C₄×C₇×D₄	C₂³	semidirect
122. C₇×#107	C₂³	semidirect
123. C₇×#106	C₂³	semidirect
124. C₇×C₄⋊C₈	C₂³	semidirect
125. C₇×C₄⋊C₈	C₂³	semidirect
126. C₇×#103	C₂³	semidirect
129. C₇×#102	C₂³	semidirect
131. C₇×#97	C₂³	semidirect
132. C₄×C₇×C₈	C₂³	semidirect
133. #3455	C₂×C₇	semidirect
134. C₂×#518	C₂×C₇	semidirect
135. #3123	C₂×C₇	semidirect
136. #3122	C₂×C₇	semidirect
137. #3118	C₂×C₇	semidirect
138. #3093	C₂×C₇	semidirect
139. #3077	C₂×C₇	semidirect
140. #3036	C₂×C₇	semidirect
141. #3003	C₂×C₇	semidirect
142. C₄×AGL(1,ℤ/8)	C₂×C₇	semidirect
143. #2921	C₂×C₇	semidirect
214. #542	C₂²×C₇	semidirect
215. #541	C₂²×C₇	semidirect
216. #540	C₂²×C₇	semidirect
219. #533	C₂²×C₇	semidirect
220. #532	C₂²×C₇	semidirect
226. C₄×#41	C₂²×C₇	semidirect
233. #485	C₂²×C₇	semidirect
234. #482	C₂²×C₇	semidirect
235. #478	C₂²×C₇	semidirect
236. #476	C₂²×C₇	semidirect
237. #474	C₂²×C₇	semidirect
238. #463	C₂²×C₇	semidirect
239. #461	C₂²×C₇	semidirect
240. #459	C₂²×C₇	semidirect
243. #442	C₂²×C₇	semidirect
244. #440	C₂²×C₇	semidirect
245. C₄×SD₁₆	C₂²×C₇	semidirect
246. C₄×D₈	C₂²×C₇	semidirect
247. #433	C₂²×C₇	semidirect
248. #433	C₂²×C₇	semidirect
249. #428	C₂²×C₇	semidirect
250. #423	C₂²×C₇	semidirect
252. #417	C₂²×C₇	semidirect
254. C₄×M₁₆	C₂²×C₇	semidirect
313. C₄⋊Q₈	C₂³×C₇	semidirect
314. C₄⋊Q₈	C₂³×C₇	semidirect
315. #127	C₂³×C₇	semidirect
319. #125	C₂³×C₇	semidirect
321. #124	C₂³×C₇	semidirect
333. C₄×Q₈	C₂³×C₇	semidirect
345. C₄×D₄	C₂³×C₇	semidirect
365. #107	C₂³×C₇	semidirect
366. #106	C₂³×C₇	semidirect
367. C₄⋊C₈	C₂³×C₇	semidirect
368. C₄⋊C₈	C₂³×C₇	semidirect
369. #103	C₂³×C₇	semidirect
372. #102	C₂³×C₇	semidirect
374. #97	C₂³×C₇	semidirect
375. C₄×C₈	C₂³×C₇	semidirect
516. C₇	#60564	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×#3455	2
C₂×C₇×#518	2
C₇×#3123	2
C₇×#3122	2
C₇×#3118	2
C₇×#3093	2
C₇×#3077	2
C₇×#3036	2
C₇×#3003	2
C₄×C₇×AGL(1,ℤ/8)	2
C₇×#2921	2
#60564	7

Subgroups — C7×#60564 #13237780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#60564 #13237780