Subgroups — C₇×#22810 #13200026

Order 1792.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×D₈	C₂	1	1	yes	maximal, characteristic
2	C₇×#2357	C₂	1	2	yes	maximal, complemented
3	C₇×#1970	C₂	1	2	yes	maximal
4	C₇×#1926	C₂	1	4	yes	maximal, complemented
5	C₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1804	C₂	1	2	yes	maximal, complemented
7	C₄×C₇×#107	C₂	1	2	yes	maximal, complemented
8	C₇×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₇×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×D₄	C₂²	1	2	yes	—
11	C₄×C₇×D₈	C₄	1	8	yes	complemented
12	C₂×C₇×#107	C₂²	1	2	yes	—
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#106	C₂²	1	2	yes	—
15	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₇×#102	C₂²	1	4	yes	—
17	C₂×C₇×#102	C₂²	1	2	yes	—
18	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
20	C₇×#390	C₂²	1	4	yes	—
21	C₇×#388	—	2	4	no	—
22	C₇×#382	C₂²	1	2	yes	—
23	C₇×#381	—	2	2	no	—
24	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₄×C₇×C₂²⋊C₄	—	2	2	no	—
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	#22810	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₇×D₄	C₂³	1	2	yes	—
31	C₂³×C₄×C₇	—	2	2	no	—
32	C₂×C₇×D₈	—	2	2	no	—
33	C₂×C₇×D₈	—	2	2	no	—
34	C₂×C₇×D₈	C₂×C₄	1	4	yes	—
35	C₂²×C₇×C₈	C₂³	1	2	yes	—
36	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₇×D₄	—	2	4	no	—
39	C₄×C₇×D₄	C₂×C₄	1	8	yes	—
40	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
41	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
42	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
46	C₂×C₄²×C₇	—	2	4	no	—
47	C₂×C₄²×C₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₇×#107	—	2	4	no	—
50	C₇×#107	C₂×C₄	1	4	yes	—
51	C₇×#106	—	4	2	no	—
52	C₇×C₄⋊C₈	—	2	2	no	—
53	C₇×#102	—	4	4	no	—
54	C₇×#102	C₂×C₄	1	8	yes	—
55	C₄×C₇×C₈	—	2	4	no	—
56	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
57	C₇×#95	—	2	2	no	—
58	C₇×#95	—	2	4	no	—
59	C₇×#95	—	2	2	no	—
60	C₇×#95	D₄	1	1	yes	characteristic
61	C₇×#95	D₄	1	1	yes	characteristic
62	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
63	#2357	C₂×C₇	1	2	yes	complemented
64	#1970	C₂×C₇	1	2	yes	—
65	#1926	C₂×C₇	1	4	yes	complemented
66	#1846	C₂×C₇	1	1	yes	characteristic, complemented
67	#1804	C₂×C₇	1	2	yes	complemented
68	C₄×#107	C₂×C₇	1	2	yes	complemented
69	C₈×C₄⋊C₄	C₂×C₇	1	1	yes	characteristic, complemented
70	C₂⁴×C₇	—	2	2	no	—
71	C₂×C₇×D₄	—	2	2	no	—
72	C₂×C₇×D₄	—	2	2	no	—
73	C₂×C₇×D₄	—	2	2	no	—
74	C₂×C₇×D₄	—	2	2	no	—
75	C₂×C₇×D₄	—	2	2	no	—
76	C₂×C₇×D₄	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₇	—	2	2	no	—
78	C₂²×C₄×C₇	—	2	8	no	—
79	C₂²×C₄×C₇	—	2	4	no	—
80	C₂²×C₄×C₇	—	8	4	no	—
81	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
86	C₇×D₈	—	2	8	no	—
87	C₂×C₇×C₈	—	2	1	no	—
88	C₂×C₇×C₈	—	2	1	no	—
89	C₂×C₇×C₈	—	2	2	no	—
90	C₂×C₇×C₈	—	2	2	no	—
91	C₂×C₇×C₈	—	2	2	no	—
92	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
93	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₇×C₈	#41	1	2	yes	—
95	C₂×C₇×C₈	#41	1	2	yes	—
96	C₇×C₄⋊C₄	—	2	4	no	—
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	2	4	no	—
99	C₇×C₄⋊C₄	—	2	4	no	—
100	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
101	C₇×C₂²⋊C₄	—	2	8	no	—
102	C₇×C₂²⋊C₄	—	4	8	no	—
103	C₇×C₂²⋊C₄	—	8	4	no	—
104	C₄²×C₇	—	2	2	no	—
105	C₄²×C₇	—	2	2	no	—
106	C₄²×C₇	—	8	4	no	—
107	C₄²×C₇	C₂²×C₄	1	2	yes	—
108	C₄²×C₇	#41	1	2	yes	—
109	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
110	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
111	C₄×D₈	C₄×C₇	1	8	yes	complemented
112	C₂×#107	C₂²×C₇	1	2	yes	—
113	C₂×#107	C₂²×C₇	1	1	yes	characteristic
114	C₂×#106	C₂²×C₇	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
116	C₂×#102	C₂²×C₇	1	4	yes	—
117	C₂×#102	C₂²×C₇	1	2	yes	—
118	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
119	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
120	#390	C₂²×C₇	1	4	yes	—
121	#388	—	2	4	no	—
122	#382	C₂²×C₇	1	2	yes	—
123	#381	—	2	2	no	—
124	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
125	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
126	C₄×C₂²⋊C₄	—	2	2	no	—
127	#340	C₂²×C₇	1	4	yes	—
128	#336	C₂²×C₇	1	2	yes	—
129	C₂³×C₇	—	2	4	no	—
130	C₂³×C₇	—	4	2	no	—
131	C₂³×C₇	—	4	2	no	—
132	C₂³×C₇	—	4	4	no	—
133	C₂³×C₇	—	4	4	no	—
134	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
135	C₇×D₄	—	2	4	no	—
136	C₇×D₄	—	2	4	no	—
137	C₇×D₄	—	4	2	no	—
138	C₇×D₄	—	4	2	no	—
139	C₂×C₄×C₇	—	2	2	no	—
140	C₂×C₄×C₇	—	2	2	no	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	4	no	—
144	C₂×C₄×C₇	—	2	2	no	—
145	C₂×C₄×C₇	—	2	2	no	—
146	C₂×C₄×C₇	—	2	2	no	—
147	C₂×C₄×C₇	—	4	4	no	—
148	C₂×C₄×C₇	—	4	4	no	—
149	C₂×C₄×C₇	—	4	4	no	—
150	C₂×C₄×C₇	—	4	4	no	—
151	C₂×C₄×C₇	—	8	8	no	—
152	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
153	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
154	C₂×C₄×C₇	#135	1	4	yes	—
155	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
161	C₇×C₈	—	2	2	no	—
162	C₇×C₈	—	4	2	no	—
163	C₇×C₈	#117	1	4	yes	—
164	C₂²×D₄	C₂³×C₇	1	2	yes	—
165	C₂³×C₄	—	2	2	no	—
166	C₂×D₈	—	2	2	no	—
167	C₂×D₈	—	2	2	no	—
168	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
169	C₂²×C₈	C₂³×C₇	1	2	yes	—
170	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
171	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
172	C₄×D₄	—	2	4	no	—
173	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
174	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
175	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
176	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
177	C₂×C₂²⋊C₄	—	2	2	no	—
178	C₂×C₂²⋊C₄	—	2	4	no	—
179	C₂×C₂²⋊C₄	—	2	4	no	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	C₂³×C₇	1	2	yes	—
182	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
183	#107	—	2	4	no	—
184	#107	C₂×C₄×C₇	1	4	yes	—
185	#106	—	4	2	no	—
186	C₄⋊C₈	—	2	2	no	—
187	#102	—	4	4	no	—
188	#102	C₂×C₄×C₇	1	8	yes	—
189	C₄×C₈	—	2	4	no	—
190	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
191	#95	—	2	2	no	—
192	#95	—	2	4	no	—
193	#95	—	2	2	no	—
194	#95	C₇×D₄	1	1	yes	characteristic
195	#95	C₇×D₄	1	1	yes	characteristic
196	C₂²×C₇	—	4	4	no	—
197	C₂²×C₇	—	4	4	no	—
198	C₂²×C₇	—	4	4	no	—
199	C₂²×C₇	—	8	4	no	—
200	C₂²×C₇	—	8	4	no	—
201	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
202	C₂²×C₇	#547	1	1	yes	characteristic
203	C₂²×C₇	#550	1	1	yes	characteristic
204	C₂²×C₇	C₂×#135	1	2	yes	—
205	C₂²×C₇	#576	1	2	yes	—
206	C₄×C₇	—	2	4	no	—
207	C₄×C₇	—	2	2	no	—
208	C₄×C₇	—	2	2	no	—
209	C₄×C₇	—	4	2	no	—
210	C₄×C₇	—	4	4	no	—
211	C₄×C₇	—	8	4	no	—
212	C₄×C₇	C₂×#117	1	1	yes	characteristic
213	C₄×C₇	#520	1	1	yes	characteristic
214	C₄×C₇	#555	1	1	yes	characteristic
215	C₄×C₇	#556	1	1	yes	characteristic
216	C₂⁴	—	2	2	no	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	8	4	no	—
227	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
228	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
231	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
232	D₈	—	2	8	no	—
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
240	C₂×C₈	C₇×#41	1	2	yes	—
241	C₂×C₈	C₇×#41	1	2	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
247	C₂²⋊C₄	—	2	8	no	—
248	C₂²⋊C₄	—	4	8	no	—
249	C₂²⋊C₄	—	8	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	8	4	no	—
253	C₄²	C₂²×C₄×C₇	1	2	yes	—
254	C₄²	C₇×#41	1	2	yes	—
255	C₂×C₇	—	8	4	no	—
256	C₂×C₇	#2354	1	1	yes	characteristic
257	C₂×C₇	C₄×#135	1	1	yes	characteristic
258	C₂×C₇	#2948	1	1	yes	characteristic
259	C₂×C₇	#3339	1	2	yes	—
260	C₂×C₇	#3396	1	2	yes	—
261	C₂³	—	2	4	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
267	D₄	—	2	4	no	—
268	D₄	—	2	4	no	—
269	D₄	—	4	2	no	—
270	D₄	—	4	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	8	8	no	—
284	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
285	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
286	C₂×C₄	C₇×#135	1	4	yes	—
287	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
288	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
289	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
290	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
291	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
292	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	4	2	no	—
295	C₈	C₇×#117	1	4	yes	—
296	C₇	#22810	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²	—	4	4	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
303	C₂²	C₇×#547	1	1	yes	characteristic
304	C₂²	C₇×#550	1	1	yes	characteristic
305	C₂²	C₂×C₇×#135	1	2	yes	—
306	C₂²	C₇×#576	1	2	yes	—
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	4	no	—
310	C₄	—	4	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	8	4	no	—
313	C₄	C₂×C₇×#117	1	1	yes	characteristic
314	C₄	C₇×#520	1	1	yes	characteristic
315	C₄	C₇×#555	1	1	yes	characteristic
316	C₄	C₇×#556	1	1	yes	characteristic
317	C₂	—	8	4	no	—
318	C₂	C₇×#2354	1	1	yes	characteristic
319	C₂	C₄×C₇×#135	1	1	yes	characteristic
320	C₂	C₇×#2948	1	1	yes	characteristic
321	C₂	C₇×#3339	1	2	yes	—
322	C₂	C₇×#3396	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
prime radical	C₂²×C₇×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄×C₇×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×#2357	C₂	semidirect
4. C₇×#1926	C₂	semidirect
5. C₇×#1846	C₂	semidirect
6. C₇×#1804	C₂	semidirect
7. C₄×C₇×#107	C₂	semidirect
8. C₇×C₈×C₄⋊C₄	C₂	semidirect
11. C₄×C₇×D₈	C₄	semidirect
29. #22810	C₇	direct (N × H)
63. #2357	C₂×C₇	semidirect
65. #1926	C₂×C₇	semidirect
66. #1846	C₂×C₇	semidirect
67. #1804	C₂×C₇	semidirect
68. C₄×#107	C₂×C₇	semidirect
69. C₈×C₄⋊C₄	C₂×C₇	semidirect
111. C₄×D₈	C₄×C₇	semidirect
296. C₇	#22810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×D₈	2
C₇×#2357	2
C₇×#1970	2
C₇×#1926	2
C₇×#1846	2
C₇×#1804	2
C₄×C₇×#107	2
C₇×C₈×C₄⋊C₄	2
#22810	7

Subgroups — C7×#22810 #13200026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22810 #13200026