Subgroups — C₇×#61007 #13238223

Order 1792.

1704 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3510	C₂	1	2	yes	maximal, complemented
2	C₇×#3263	C₂	1	2	yes	maximal, complemented
3	C₇×#3262	C₂	1	2	yes	maximal, complemented
4	C₇×#3233	C₂	1	4	yes	maximal, complemented
5	C₇×#3211	C₂	1	4	yes	maximal, complemented
6	C₇×#3202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇×#3200	C₂	1	2	yes	maximal, complemented
8	C₇×#3199	C₂	1	1	yes	maximal, characteristic
9	C₇×#3183	C₂	1	2	yes	maximal, complemented
10	C₇×#3168	C₂	1	2	yes	maximal, complemented
11	C₇×#3121	C₂	1	4	yes	maximal, complemented
12	C₇×#3118	C₂	1	4	yes	maximal, complemented
13	C₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇×#559	—	2	4	no	—
15	C₇×#556	—	2	4	no	—
16	C₇×#551	—	2	4	no	—
17	C₇×#551	—	2	4	no	—
18	C₇×#542	C₂²	1	4	yes	—
19	C₇×#542	C₂²	1	4	yes	complemented
20	C₇×#536	C₂²	1	2	yes	—
21	C₇×#535	C₂²	1	2	yes	—
22	C₇×#533	C₂²	1	2	yes	complemented
23	C₇×#529	C₂²	1	4	yes	—
24	C₇×#529	C₂²	1	4	yes	complemented
25	C₂×C₇×#124	C₂²	1	2	yes	complemented
26	C₇×#519	C₂²	1	2	yes	complemented
27	C₇×#518	C₂²	1	2	yes	complemented
28	C₄×C₇×#41	C₂²	1	1	yes	characteristic
29	C₇×#499	C₂²	1	2	yes	complemented
30	C₇×#499	C₂²	1	2	yes	complemented
31	C₇×#495	C₂²	1	2	yes	—
32	C₇×#495	C₂²	1	2	yes	—
33	C₇×#488	C₂²	1	4	yes	complemented
34	C₇×#488	C₂²	1	4	yes	complemented
35	C₇×#485	C₂²	1	4	yes	complemented
36	C₇×#485	C₂²	1	4	yes	complemented
37	C₇×#482	C₂²	1	4	yes	—
38	C₇×#482	C₂²	1	4	yes	—
39	C₇×#477	C₂²	1	4	yes	complemented
40	C₇×#476	C₂²	1	4	yes	complemented
41	C₇×#475	C₂²	1	4	yes	complemented
42	C₇×#474	C₂²	1	4	yes	complemented
43	C₇×#470	C₂²	1	2	yes	complemented
44	C₇×#469	C₂²	1	2	yes	—
45	C₇×#468	C₂²	1	4	yes	complemented
46	C₇×#467	C₂²	1	2	yes	complemented
47	C₇×#466	C₂²	1	2	yes	—
48	C₇×#465	C₂²	1	4	yes	complemented
49	C₇×#464	C₂²	1	4	yes	complemented
50	C₇×#464	C₂²	1	4	yes	complemented
51	C₇×#463	C₂²	1	4	yes	—
52	C₇×#463	C₂²	1	4	yes	—
53	C₇×#451	C₂²	1	4	yes	complemented
54	C₇×#450	C₂²	1	4	yes	complemented
55	C₇×#448	C₂²	1	4	yes	—
56	C₇×#447	C₂²	1	4	yes	—
57	C₇×#442	C₂²	1	2	yes	—
58	C₇×#441	C₂²	1	2	yes	complemented
59	C₇×#440	C₂²	1	4	yes	complemented
60	C₄×C₇×Q₁₆	C₂²	1	2	yes	complemented
61	C₄×C₇×SD₁₆	C₂²	1	4	yes	complemented
62	C₄×C₇×D₈	C₂²	1	2	yes	—
63	C₇×#433	C₂²	1	4	yes	complemented
64	C₇×#431	C₂²	1	4	yes	complemented
65	C₇×#424	C₂²	1	1	yes	characteristic, complemented
66	C₇×#424	C₂²	1	1	yes	characteristic, complemented
67	C₇×#408	C₂²	1	1	yes	characteristic
68	C₇×#408	C₂²	1	1	yes	characteristic
69	C₄×C₇×M₁₆	C₂²	1	1	yes	characteristic, complemented
70	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic, complemented
71	#61007	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₇×#41	C₂³	1	1	yes	characteristic
73	C₂²×C₇×Q₈	C₂³	1	2	yes	—
74	C₂²×C₇×D₄	C₂³	1	2	yes	—
75	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
76	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
77	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
78	C₂×C₇×SD₁₆	C₂³	1	4	yes	—
79	C₂×C₇×D₈	C₂³	1	2	yes	—
80	C₂×C₇×D₈	C₂³	1	2	yes	—
81	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
82	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
83	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
84	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
85	C₇×C₄⋊Q₈	—	2	2	no	—
86	C₇×C₄⋊Q₈	C₂³	1	4	yes	—
87	C₇×#127	—	2	2	no	—
88	C₇×#126	—	2	4	no	—
89	C₇×#126	—	2	4	no	—
90	C₇×#126	—	2	4	no	—
91	C₇×#126	—	2	4	no	—
92	C₇×#125	C₂³	1	4	yes	—
93	C₇×#124	—	2	2	no	—
94	C₇×#124	—	2	2	no	—
95	C₇×#124	—	2	4	no	—
96	C₇×#124	—	2	4	no	—
97	C₇×#124	—	2	4	no	—
98	C₇×#124	C₂³	1	4	yes	—
99	C₇×#124	C₂³	1	4	yes	—
100	C₇×#123	—	2	2	no	—
101	C₇×#123	—	2	2	no	—
102	C₇×#123	—	2	4	no	—
103	C₇×#123	—	2	4	no	—
104	C₇×#123	—	2	4	no	—
105	C₇×#123	—	2	4	no	—
106	C₇×C₂²⋊Q₈	—	2	4	no	—
107	C₇×C₂²⋊Q₈	—	2	4	no	—
108	C₇×C₂²⋊Q₈	—	2	4	no	—
109	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
110	C₇×C₂²⋊Q₈	C₂³	1	4	yes	—
111	C₇×#121	C₂³	1	4	yes	—
112	C₇×#121	C₂³	1	4	yes	—
113	C₇×#120	—	2	2	no	—
114	C₇×#120	—	2	2	no	—
115	C₄×C₇×Q₈	—	2	4	no	—
116	C₄×C₇×Q₈	—	2	4	no	—
117	C₄×C₇×Q₈	D₄	1	2	yes	—
118	C₄×C₇×Q₈	C₂³	1	4	yes	—
119	C₄×C₇×Q₈	C₂³	1	4	yes	—
120	C₄×C₇×D₄	—	2	4	no	—
121	C₄×C₇×D₄	—	2	4	no	—
122	C₄×C₇×D₄	—	2	4	no	—
123	C₄×C₇×D₄	D₄	1	2	yes	—
124	C₄×C₇×D₄	D₄	1	4	yes	—
125	C₄×C₇×D₄	C₂³	1	4	yes	—
126	C₄×C₇×D₄	C₂³	1	4	yes	—
127	C₇×#117	—	2	4	no	—
128	C₇×#117	—	2	4	no	—
129	C₇×#117	—	2	4	no	—
130	C₇×#117	C₂³	1	2	yes	—
131	C₇×#117	C₂³	1	2	yes	—
132	C₇×#117	C₂³	1	2	yes	—
133	C₇×#117	C₂³	1	1	yes	characteristic
134	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
135	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
136	C₂×C₄²×C₇	C₂³	1	2	yes	—
137	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
138	C₇×#107	C₂³	1	2	yes	—
139	C₇×#107	C₂³	1	2	yes	—
140	C₇×#107	C₂³	1	4	yes	—
141	C₇×#106	C₂³	1	2	yes	—
142	C₇×#106	C₂³	1	2	yes	—
143	C₇×#106	C₂³	1	4	yes	—
144	C₇×C₄⋊C₈	C₂³	1	4	yes	—
145	C₇×C₄⋊C₈	C₂³	1	4	yes	—
146	C₇×#103	C₂³	1	4	yes	—
147	C₇×#103	C₂³	1	4	yes	—
148	C₇×#103	C₂³	1	4	yes	—
149	C₇×#103	C₂³	1	4	yes	—
150	C₇×#102	C₂³	1	4	yes	—
151	C₇×#102	C₂³	1	4	yes	—
152	C₇×#102	C₂³	1	4	yes	—
153	C₇×#102	C₂³	1	4	yes	—
154	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
155	C₇×C₂²⋊C₈	C₂³	1	4	yes	—
156	C₇×#97	C₂³	1	2	yes	—
157	C₄×C₇×C₈	C₂³	1	2	yes	—
158	C₄×C₇×C₈	C₂³	1	2	yes	—
159	C₄×C₇×C₈	C₂³	1	2	yes	—
160	#3510	C₂×C₇	1	2	yes	complemented
161	#3263	C₂×C₇	1	2	yes	complemented
162	#3262	C₂×C₇	1	2	yes	complemented
163	#3233	C₂×C₇	1	4	yes	complemented
164	#3211	C₂×C₇	1	4	yes	complemented
165	#3202	C₂×C₇	1	1	yes	characteristic, complemented
166	#3200	C₂×C₇	1	2	yes	complemented
167	#3199	C₂×C₇	1	1	yes	characteristic
168	#3183	C₂×C₇	1	2	yes	complemented
169	#3168	C₂×C₇	1	2	yes	complemented
170	#3121	C₂×C₇	1	4	yes	complemented
171	#3118	C₂×C₇	1	4	yes	complemented
172	#2922	C₂×C₇	1	1	yes	characteristic, complemented
173	C₂⁴×C₇	—	2	2	no	—
174	C₇×#41	—	4	1	no	—
175	C₇×#41	—	4	1	no	—
176	C₂×C₇×Q₈	—	2	2	no	—
177	C₂×C₇×Q₈	—	2	4	no	—
178	C₂×C₇×Q₈	—	4	2	no	—
179	C₂×C₇×Q₈	—	4	2	no	—
180	C₂×C₇×Q₈	C₂×D₄	1	1	yes	characteristic
181	C₂×C₇×Q₈	C₂⁴	1	4	yes	—
182	C₂×C₇×D₄	—	2	4	no	—
183	C₂×C₇×D₄	—	2	4	no	—
184	C₂×C₇×D₄	—	2	2	no	—
185	C₂×C₇×D₄	—	4	2	no	—
186	C₂×C₇×D₄	—	4	2	no	—
187	C₂×C₇×D₄	C₂×D₄	1	1	yes	characteristic
188	C₂×C₇×D₄	C₂×D₄	1	2	yes	—
189	C₂×C₇×D₄	C₂⁴	1	4	yes	—
190	C₂²×C₄×C₇	—	2	2	no	—
191	C₂²×C₄×C₇	—	2	4	no	—
192	C₂²×C₄×C₇	—	2	4	no	—
193	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
194	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
197	C₇×Q₁₆	—	4	2	no	—
198	C₇×Q₁₆	—	4	2	no	—
199	C₇×SD₁₆	—	4	4	no	—
200	C₇×SD₁₆	—	4	4	no	—
201	C₇×D₈	—	4	2	no	—
202	C₇×D₈	—	4	2	no	—
203	C₇×M₁₆	—	4	1	no	—
204	C₇×M₁₆	—	4	1	no	—
205	C₂×C₇×C₈	—	4	1	no	—
206	C₂×C₇×C₈	—	4	1	no	—
207	C₂×C₇×C₈	C₂⁴	1	2	yes	—
208	C₂×C₇×C₈	C₂⁴	1	2	yes	—
209	C₂×C₇×C₈	C₂⁴	1	2	yes	—
210	C₂×C₇×C₈	C₂⁴	1	2	yes	—
211	C₇×C₄⋊C₄	—	2	4	no	—
212	C₇×C₄⋊C₄	—	2	4	no	—
213	C₇×C₄⋊C₄	—	2	4	no	—
214	C₇×C₄⋊C₄	—	2	4	no	—
215	C₇×C₄⋊C₄	—	2	4	no	—
216	C₇×C₄⋊C₄	—	2	4	no	—
217	C₇×C₄⋊C₄	—	2	4	no	—
218	C₇×C₄⋊C₄	C₂×D₄	1	4	yes	—
219	C₇×C₄⋊C₄	C₂×D₄	1	2	yes	—
220	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
221	C₇×C₄⋊C₄	C₂⁴	1	4	yes	—
222	C₇×C₂²⋊C₄	—	2	2	no	—
223	C₇×C₂²⋊C₄	—	2	2	no	—
224	C₇×C₂²⋊C₄	—	2	4	no	—
225	C₇×C₂²⋊C₄	—	2	4	no	—
226	C₇×C₂²⋊C₄	—	2	4	no	—
227	C₇×C₂²⋊C₄	—	2	4	no	—
228	C₇×C₂²⋊C₄	—	2	4	no	—
229	C₇×C₂²⋊C₄	—	2	4	no	—
230	C₇×C₂²⋊C₄	—	2	4	no	—
231	C₇×C₂²⋊C₄	—	2	4	no	—
232	C₇×C₂²⋊C₄	—	2	4	no	—
233	C₇×C₂²⋊C₄	—	2	4	no	—
234	C₇×C₂²⋊C₄	C₂×D₄	1	4	yes	—
235	C₇×C₂²⋊C₄	C₂×D₄	1	2	yes	—
236	C₄²×C₇	—	2	4	no	—
237	C₄²×C₇	—	2	4	no	—
238	C₄²×C₇	—	2	2	no	—
239	C₄²×C₇	—	2	2	no	—
240	C₄²×C₇	C₂×D₄	1	2	yes	—
241	C₄²×C₇	C₂×D₄	1	4	yes	—
242	C₄²×C₇	C₂⁴	1	1	yes	characteristic
243	C₄²×C₇	C₂⁴	1	1	yes	characteristic
244	C₄²×C₇	C₂⁴	1	2	yes	—
245	#559	—	2	4	no	—
246	#556	—	2	4	no	—
247	#551	—	2	4	no	—
248	#551	—	2	4	no	—
249	#542	C₂²×C₇	1	4	yes	—
250	#542	C₂²×C₇	1	4	yes	complemented
251	#536	C₂²×C₇	1	2	yes	—
252	#535	C₂²×C₇	1	2	yes	—
253	#533	C₂²×C₇	1	2	yes	complemented
254	#529	C₂²×C₇	1	4	yes	—
255	#529	C₂²×C₇	1	4	yes	complemented
256	C₂×#124	C₂²×C₇	1	2	yes	complemented
257	#519	C₂²×C₇	1	2	yes	complemented
258	#518	C₂²×C₇	1	2	yes	complemented
259	C₄×#41	C₂²×C₇	1	1	yes	characteristic
260	#499	C₂²×C₇	1	2	yes	complemented
261	#499	C₂²×C₇	1	2	yes	complemented
262	#495	C₂²×C₇	1	2	yes	—
263	#495	C₂²×C₇	1	2	yes	—
264	#488	C₂²×C₇	1	4	yes	complemented
265	#488	C₂²×C₇	1	4	yes	complemented
266	#485	C₂²×C₇	1	4	yes	complemented
267	#485	C₂²×C₇	1	4	yes	complemented
268	#482	C₂²×C₇	1	4	yes	—
269	#482	C₂²×C₇	1	4	yes	—
270	#477	C₂²×C₇	1	4	yes	complemented
271	#476	C₂²×C₇	1	4	yes	complemented
272	#475	C₂²×C₇	1	4	yes	complemented
273	#474	C₂²×C₇	1	4	yes	complemented
274	#470	C₂²×C₇	1	2	yes	complemented
275	#469	C₂²×C₇	1	2	yes	—
276	#468	C₂²×C₇	1	4	yes	complemented
277	#467	C₂²×C₇	1	2	yes	complemented
278	#466	C₂²×C₇	1	2	yes	—
279	#465	C₂²×C₇	1	4	yes	complemented
280	#464	C₂²×C₇	1	4	yes	complemented
281	#464	C₂²×C₇	1	4	yes	complemented
282	#463	C₂²×C₇	1	4	yes	—
283	#463	C₂²×C₇	1	4	yes	—
284	#451	C₂²×C₇	1	4	yes	complemented
285	#450	C₂²×C₇	1	4	yes	complemented
286	#448	C₂²×C₇	1	4	yes	—
287	#447	C₂²×C₇	1	4	yes	—
288	#442	C₂²×C₇	1	2	yes	—
289	#441	C₂²×C₇	1	2	yes	complemented
290	#440	C₂²×C₇	1	4	yes	complemented
291	C₄×Q₁₆	C₂²×C₇	1	2	yes	complemented
292	C₄×SD₁₆	C₂²×C₇	1	4	yes	complemented
293	C₄×D₈	C₂²×C₇	1	2	yes	—
294	#433	C₂²×C₇	1	4	yes	complemented
295	#431	C₂²×C₇	1	4	yes	complemented
296	#424	C₂²×C₇	1	1	yes	characteristic, complemented
297	#424	C₂²×C₇	1	1	yes	characteristic, complemented
298	#408	C₂²×C₇	1	1	yes	characteristic
299	#408	C₂²×C₇	1	1	yes	characteristic
300	C₄×M₁₆	C₂²×C₇	1	1	yes	characteristic, complemented
301	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic, complemented
302	C₂³×C₇	—	2	4	no	—
303	C₂³×C₇	—	4	4	no	—
304	C₂³×C₇	—	8	4	no	—
305	C₂³×C₇	C₂²×D₄	1	2	yes	—
306	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
307	C₇×Q₈	—	2	4	no	—
308	C₇×Q₈	—	2	4	no	—
309	C₇×Q₈	—	4	1	no	—
310	C₇×Q₈	—	4	4	no	—
311	C₇×Q₈	—	8	4	no	—
312	C₇×D₄	—	2	4	no	—
313	C₇×D₄	—	2	4	no	—
314	C₇×D₄	—	4	2	no	—
315	C₇×D₄	—	4	1	no	—
316	C₇×D₄	—	4	4	no	—
317	C₇×D₄	—	8	4	no	—
318	C₇×D₄	—	8	4	no	—
319	C₂×C₄×C₇	—	2	1	no	—
320	C₂×C₄×C₇	—	2	1	no	—
321	C₂×C₄×C₇	—	2	2	no	—
322	C₂×C₄×C₇	—	2	2	no	—
323	C₂×C₄×C₇	—	2	4	no	—
324	C₂×C₄×C₇	—	2	4	no	—
325	C₂×C₄×C₇	—	2	4	no	—
326	C₂×C₄×C₇	—	4	2	no	—
327	C₂×C₄×C₇	—	4	4	no	—
328	C₂×C₄×C₇	—	4	4	no	—
329	C₂×C₄×C₇	—	8	4	no	—
330	C₂×C₄×C₇	—	8	4	no	—
331	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
332	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
333	C₂×C₄×C₇	C₂²×D₄	1	4	yes	—
334	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
335	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
336	C₂×C₄×C₇	C₂⁵	1	1	yes	characteristic
337	C₇×C₈	—	2	2	no	—
338	C₇×C₈	—	2	2	no	—
339	C₇×C₈	—	2	2	no	—
340	C₇×C₈	—	2	2	no	—
341	C₂×#41	C₂³×C₇	1	1	yes	characteristic
342	C₂²×Q₈	C₂³×C₇	1	2	yes	—
343	C₂²×D₄	C₂³×C₇	1	2	yes	—
344	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
345	C₂×Q₁₆	C₂³×C₇	1	2	yes	—
346	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
347	C₂×SD₁₆	C₂³×C₇	1	4	yes	—
348	C₂×D₈	C₂³×C₇	1	2	yes	—
349	C₂×D₈	C₂³×C₇	1	2	yes	—
350	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
351	C₂×M₁₆	C₂³×C₇	1	1	yes	characteristic
352	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
353	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
354	C₄⋊Q₈	—	2	2	no	—
355	C₄⋊Q₈	C₂³×C₇	1	4	yes	—
356	#127	—	2	2	no	—
357	#126	—	2	4	no	—
358	#126	—	2	4	no	—
359	#126	—	2	4	no	—
360	#126	—	2	4	no	—
361	#125	C₂³×C₇	1	4	yes	—
362	#124	—	2	2	no	—
363	#124	—	2	2	no	—
364	#124	—	2	4	no	—
365	#124	—	2	4	no	—
366	#124	—	2	4	no	—
367	#124	C₂³×C₇	1	4	yes	—
368	#124	C₂³×C₇	1	4	yes	—
369	#123	—	2	2	no	—
370	#123	—	2	2	no	—
371	#123	—	2	4	no	—
372	#123	—	2	4	no	—
373	#123	—	2	4	no	—
374	#123	—	2	4	no	—
375	C₂²⋊Q₈	—	2	4	no	—
376	C₂²⋊Q₈	—	2	4	no	—
377	C₂²⋊Q₈	—	2	4	no	—
378	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
379	C₂²⋊Q₈	C₂³×C₇	1	4	yes	—
380	#121	C₂³×C₇	1	4	yes	—
381	#121	C₂³×C₇	1	4	yes	—
382	#120	—	2	2	no	—
383	#120	—	2	2	no	—
384	C₄×Q₈	—	2	4	no	—
385	C₄×Q₈	—	2	4	no	—
386	C₄×Q₈	C₇×D₄	1	2	yes	—
387	C₄×Q₈	C₂³×C₇	1	4	yes	—
388	C₄×Q₈	C₂³×C₇	1	4	yes	—
389	C₄×D₄	—	2	4	no	—
390	C₄×D₄	—	2	4	no	—
391	C₄×D₄	—	2	4	no	—
392	C₄×D₄	C₇×D₄	1	2	yes	—
393	C₄×D₄	C₇×D₄	1	4	yes	—
394	C₄×D₄	C₂³×C₇	1	4	yes	—
395	C₄×D₄	C₂³×C₇	1	4	yes	—
396	#117	—	2	4	no	—
397	#117	—	2	4	no	—
398	#117	—	2	4	no	—
399	#117	C₂³×C₇	1	2	yes	—
400	#117	C₂³×C₇	1	2	yes	—
401	#117	C₂³×C₇	1	2	yes	—
402	#117	C₂³×C₇	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	—	2	4	no	—
404	C₂×C₂²⋊C₄	—	2	4	no	—
405	C₂×C₄²	C₂³×C₇	1	2	yes	—
406	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
407	#107	C₂³×C₇	1	2	yes	—
408	#107	C₂³×C₇	1	2	yes	—
409	#107	C₂³×C₇	1	4	yes	—
410	#106	C₂³×C₇	1	4	yes	—
411	#106	C₂³×C₇	1	2	yes	—
412	#106	C₂³×C₇	1	2	yes	—
413	C₄⋊C₈	C₂³×C₇	1	4	yes	—
414	C₄⋊C₈	C₂³×C₇	1	4	yes	—
415	#103	C₂³×C₇	1	4	yes	—
416	#103	C₂³×C₇	1	4	yes	—
417	#103	C₂³×C₇	1	4	yes	—
418	#103	C₂³×C₇	1	4	yes	—
419	#102	C₂³×C₇	1	4	yes	—
420	#102	C₂³×C₇	1	4	yes	—
421	#102	C₂³×C₇	1	4	yes	—
422	#102	C₂³×C₇	1	4	yes	—
423	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
424	C₂²⋊C₈	C₂³×C₇	1	4	yes	—
425	#97	C₂³×C₇	1	2	yes	—
426	C₄×C₈	C₂³×C₇	1	2	yes	—
427	C₄×C₈	C₂³×C₇	1	2	yes	—
428	C₄×C₈	C₂³×C₇	1	2	yes	—
429	C₂²×C₇	—	2	2	no	—
430	C₂²×C₇	—	2	4	no	—
431	C₂²×C₇	—	2	2	no	—
432	C₂²×C₇	—	2	1	no	—
433	C₂²×C₇	—	4	4	no	—
434	C₂²×C₇	—	4	4	no	—
435	C₂²×C₇	—	4	4	no	—
436	C₂²×C₇	—	16	4	no	—
437	C₂²×C₇	C₂³×D₄	1	1	yes	characteristic
438	C₄×C₇	—	2	2	no	—
439	C₄×C₇	—	2	2	no	—
440	C₄×C₇	—	2	4	no	—
441	C₄×C₇	—	2	2	no	—
442	C₄×C₇	—	2	1	no	—
443	C₄×C₇	—	4	4	no	—
444	C₄×C₇	—	4	4	no	—
445	C₄×C₇	—	4	4	no	—
446	C₄×C₇	#585	1	1	yes	characteristic
447	C₄×C₇	#585	1	1	yes	characteristic
448	C₂⁴	—	2	2	no	—
449	#41	—	4	1	no	—
450	#41	—	4	1	no	—
451	C₂×Q₈	—	2	2	no	—
452	C₂×Q₈	—	2	4	no	—
453	C₂×Q₈	—	4	2	no	—
454	C₂×Q₈	—	4	2	no	—
455	C₂×Q₈	C₂×C₇×D₄	1	1	yes	characteristic
456	C₂×Q₈	C₂⁴×C₇	1	4	yes	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	—	4	2	no	—
462	C₂×D₄	C₂×C₇×D₄	1	1	yes	characteristic
463	C₂×D₄	C₂×C₇×D₄	1	2	yes	—
464	C₂×D₄	C₂⁴×C₇	1	4	yes	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
469	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
470	C₂²×C₄	C₂⁴×C₇	1	2	yes	—
471	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
472	Q₁₆	—	4	2	no	—
473	Q₁₆	—	4	2	no	—
474	SD₁₆	—	4	4	no	—
475	SD₁₆	—	4	4	no	—
476	D₈	—	4	2	no	—
477	D₈	—	4	2	no	—
478	M₁₆	—	4	1	no	—
479	M₁₆	—	4	1	no	—
480	C₂×C₈	—	4	1	no	—
481	C₂×C₈	—	4	1	no	—
482	C₂×C₈	C₂⁴×C₇	1	2	yes	—
483	C₂×C₈	C₂⁴×C₇	1	2	yes	—
484	C₂×C₈	C₂⁴×C₇	1	2	yes	—
485	C₂×C₈	C₂⁴×C₇	1	2	yes	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	4	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	C₂×C₇×D₄	1	4	yes	—
494	C₄⋊C₄	C₂×C₇×D₄	1	2	yes	—
495	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
496	C₄⋊C₄	C₂⁴×C₇	1	4	yes	—
497	C₂²⋊C₄	—	2	2	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	C₂×C₇×D₄	1	4	yes	—
510	C₂²⋊C₄	C₂×C₇×D₄	1	2	yes	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	—	2	2	no	—
514	C₄²	—	2	2	no	—
515	C₄²	C₂×C₇×D₄	1	2	yes	—
516	C₄²	C₂×C₇×D₄	1	4	yes	—
517	C₄²	C₂⁴×C₇	1	1	yes	characteristic
518	C₄²	C₂⁴×C₇	1	1	yes	characteristic
519	C₄²	C₂⁴×C₇	1	2	yes	—
520	C₂×C₇	—	4	1	no	—
521	C₂×C₇	—	4	2	no	—
522	C₂×C₇	—	8	4	no	—
523	C₂×C₇	#3449	1	1	yes	characteristic
524	C₂×C₇	#3584	1	2	yes	—
525	C₂³	—	2	4	no	—
526	C₂³	—	4	4	no	—
527	C₂³	—	8	4	no	—
528	C₂³	C₂²×C₇×D₄	1	2	yes	—
529	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
530	Q₈	—	2	4	no	—
531	Q₈	—	2	4	no	—
532	Q₈	—	4	1	no	—
533	Q₈	—	4	4	no	—
534	Q₈	—	8	4	no	—
535	D₄	—	2	4	no	—
536	D₄	—	2	4	no	—
537	D₄	—	4	2	no	—
538	D₄	—	4	1	no	—
539	D₄	—	4	4	no	—
540	D₄	—	8	4	no	—
541	D₄	—	8	4	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	4	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	8	4	no	—
553	C₂×C₄	—	8	4	no	—
554	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
555	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
556	C₂×C₄	C₂²×C₇×D₄	1	4	yes	—
557	C₂×C₄	C₂²×C₇×D₄	1	2	yes	—
558	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
559	C₂×C₄	C₂⁵×C₇	1	1	yes	characteristic
560	C₈	—	2	2	no	—
561	C₈	—	2	2	no	—
562	C₈	—	2	2	no	—
563	C₈	—	2	2	no	—
564	C₇	#61007	1	1	yes	characteristic, direct product factor, Sylow
565	C₂²	—	2	2	no	—
566	C₂²	—	2	4	no	—
567	C₂²	—	2	2	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	4	4	no	—
571	C₂²	—	4	4	no	—
572	C₂²	—	16	4	no	—
573	C₂²	C₂³×C₇×D₄	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	4	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	1	no	—
579	C₄	—	4	4	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	4	no	—
582	C₄	C₇×#585	1	1	yes	characteristic
583	C₄	C₇×#585	1	1	yes	characteristic
584	C₂	—	4	2	no	—
585	C₂	—	4	1	no	—
586	C₂	—	8	4	no	—
587	C₂	C₇×#3449	1	1	yes	characteristic
588	C₂	C₇×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇
prime radical	C₇×#3199	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₄×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#3510	C₂	semidirect
2. C₇×#3263	C₂	semidirect
3. C₇×#3262	C₂	semidirect
4. C₇×#3233	C₂	semidirect
5. C₇×#3211	C₂	semidirect
6. C₇×#3202	C₂	semidirect
7. C₇×#3200	C₂	semidirect
9. C₇×#3183	C₂	semidirect
10. C₇×#3168	C₂	semidirect
11. C₇×#3121	C₂	semidirect
12. C₇×#3118	C₂	semidirect
13. C₇×#2922	C₂	semidirect
19. C₇×#542	C₂²	semidirect
22. C₇×#533	C₂²	semidirect
24. C₇×#529	C₂²	semidirect
25. C₂×C₇×#124	C₂²	semidirect
26. C₇×#519	C₂²	semidirect
27. C₇×#518	C₂²	semidirect
29. C₇×#499	C₂²	semidirect
30. C₇×#499	C₂²	semidirect
33. C₇×#488	C₂²	semidirect
34. C₇×#488	C₂²	semidirect
35. C₇×#485	C₂²	semidirect
36. C₇×#485	C₂²	semidirect
39. C₇×#477	C₂²	semidirect
40. C₇×#476	C₂²	semidirect
41. C₇×#475	C₂²	semidirect
42. C₇×#474	C₂²	semidirect
43. C₇×#470	C₂²	semidirect
45. C₇×#468	C₂²	semidirect
46. C₇×#467	C₂²	semidirect
48. C₇×#465	C₂²	semidirect
49. C₇×#464	C₂²	semidirect
50. C₇×#464	C₂²	semidirect
53. C₇×#451	C₂²	semidirect
54. C₇×#450	C₂²	semidirect
58. C₇×#441	C₂²	semidirect
59. C₇×#440	C₂²	semidirect
60. C₄×C₇×Q₁₆	C₂²	semidirect
61. C₄×C₇×SD₁₆	C₂²	semidirect
63. C₇×#433	C₂²	semidirect
64. C₇×#431	C₂²	semidirect
65. C₇×#424	C₂²	semidirect
66. C₇×#424	C₂²	semidirect
69. C₄×C₇×M₁₆	C₂²	semidirect
70. C₂×C₄×C₇×C₈	C₂²	semidirect
71. #61007	C₇	direct (N × H)
160. #3510	C₂×C₇	semidirect
161. #3263	C₂×C₇	semidirect
162. #3262	C₂×C₇	semidirect
163. #3233	C₂×C₇	semidirect
164. #3211	C₂×C₇	semidirect
165. #3202	C₂×C₇	semidirect
166. #3200	C₂×C₇	semidirect
168. #3183	C₂×C₇	semidirect
169. #3168	C₂×C₇	semidirect
170. #3121	C₂×C₇	semidirect
171. #3118	C₂×C₇	semidirect
172. #2922	C₂×C₇	semidirect
250. #542	C₂²×C₇	semidirect
253. #533	C₂²×C₇	semidirect
255. #529	C₂²×C₇	semidirect
256. C₂×#124	C₂²×C₇	semidirect
257. #519	C₂²×C₇	semidirect
258. #518	C₂²×C₇	semidirect
260. #499	C₂²×C₇	semidirect
261. #499	C₂²×C₇	semidirect
264. #488	C₂²×C₇	semidirect
265. #488	C₂²×C₇	semidirect
266. #485	C₂²×C₇	semidirect
267. #485	C₂²×C₇	semidirect
270. #477	C₂²×C₇	semidirect
271. #476	C₂²×C₇	semidirect
272. #475	C₂²×C₇	semidirect
273. #474	C₂²×C₇	semidirect
274. #470	C₂²×C₇	semidirect
276. #468	C₂²×C₇	semidirect
277. #467	C₂²×C₇	semidirect
279. #465	C₂²×C₇	semidirect
280. #464	C₂²×C₇	semidirect
281. #464	C₂²×C₇	semidirect
284. #451	C₂²×C₇	semidirect
285. #450	C₂²×C₇	semidirect
289. #441	C₂²×C₇	semidirect
290. #440	C₂²×C₇	semidirect
291. C₄×Q₁₆	C₂²×C₇	semidirect
292. C₄×SD₁₆	C₂²×C₇	semidirect
294. #433	C₂²×C₇	semidirect
295. #431	C₂²×C₇	semidirect
296. #424	C₂²×C₇	semidirect
297. #424	C₂²×C₇	semidirect
300. C₄×M₁₆	C₂²×C₇	semidirect
301. C₂×C₄×C₈	C₂²×C₇	semidirect
564. C₇	#61007	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×#3510	2
C₇×#3263	2
C₇×#3262	2
C₇×#3233	2
C₇×#3211	2
C₇×#3202	2
C₇×#3200	2
C₇×#3199	2
C₇×#3183	2
C₇×#3168	2
C₇×#3121	2
C₇×#3118	2
C₇×#2922	2
#61007	7

Subgroups — C7×#61007 #13238223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#61007 #13238223