Subgroups — C₇×#22813 #13200029

Order 1792.

922 conjugacy classes of subgroups.

The subgroup lattice has 330 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×#2354	C₂	1	2	yes	maximal, complemented
3	C₇×#1972	C₂	1	2	yes	maximal
4	C₇×#1926	C₂	1	4	yes	maximal, complemented
5	C₇×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#1802	C₂	1	2	yes	maximal, complemented
7	C₇×#1776	C₂	1	2	yes	maximal, complemented
8	C₇×#1770	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	C₂×C₇×#117	—	2	2	no	—
12	C₄×C₇×D₈	C₄	1	8	yes	complemented
13	C₂×C₇×#107	C₂²	1	1	yes	characteristic
14	C₂×C₇×#106	C₂²	1	2	yes	—
15	C₂×C₇×#106	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₇×#102	C₂²	1	4	yes	—
18	C₂×C₇×#102	C₂²	1	2	yes	—
19	C₂×C₇×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
21	C₇×#390	C₂²	1	4	yes	—
22	C₇×#388	—	2	4	no	—
23	C₇×#384	C₂²	1	2	yes	—
24	C₇×#380	—	2	2	no	—
25	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
26	C₄×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₇×#340	C₂²	1	4	yes	—
28	C₇×#336	C₂²	1	2	yes	—
29	#22813	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	C₇×#117	—	4	8	no	—
41	C₂×C₇×C₄⋊C₄	—	2	2	no	—
42	C₂×C₇×C₄⋊C₄	—	2	2	no	—
43	C₂×C₇×C₄⋊C₄	D₄	1	2	yes	—
44	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
45	C₂×C₇×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₇×C₂²⋊C₄	—	2	2	no	—
47	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
48	C₂×C₇×C₂²⋊C₄	—	2	4	no	—
49	C₂×C₄²×C₇	—	2	4	no	—
50	C₂×C₄²×C₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
52	C₇×#107	C₂×C₄	1	4	yes	—
53	C₇×#106	—	2	4	no	—
54	C₇×#106	—	4	2	no	—
55	C₇×C₄⋊C₈	—	2	2	no	—
56	C₇×#102	—	4	4	no	—
57	C₇×#102	C₂×C₄	1	8	yes	—
58	C₇×#97	—	2	4	no	—
59	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
60	C₇×#95	—	2	4	no	—
61	C₇×#95	D₄	1	2	yes	—
62	C₂×C₄×D₈	C₂×C₇	1	1	yes	characteristic
63	#2354	C₂×C₇	1	2	yes	complemented
64	#1972	C₂×C₇	1	2	yes	—
65	#1926	C₂×C₇	1	4	yes	complemented
66	#1844	C₂×C₇	1	1	yes	characteristic, complemented
67	#1802	C₂×C₇	1	2	yes	complemented
68	#1776	C₂×C₇	1	2	yes	complemented
69	#1770	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	—	4	4	no	—
81	C₂²×C₄×C₇	—	4	4	no	—
82	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
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	—	4	4	no	—
101	C₇×C₄⋊C₄	—	8	2	no	—
102	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
103	C₇×C₂²⋊C₄	—	2	8	no	—
104	C₇×C₂²⋊C₄	—	4	8	no	—
105	C₇×C₂²⋊C₄	—	8	4	no	—
106	C₄²×C₇	—	2	2	no	—
107	C₄²×C₇	—	2	2	no	—
108	C₄²×C₇	—	4	8	no	—
109	C₄²×C₇	C₂²×C₄	1	2	yes	—
110	C₄²×C₇	#41	1	2	yes	—
111	C₂²×D₈	C₂²×C₇	1	1	yes	characteristic
112	C₂×C₄×D₄	C₂²×C₇	1	2	yes	—
113	C₂×#117	—	2	2	no	—
114	C₄×D₈	C₄×C₇	1	8	yes	complemented
115	C₂×#107	C₂²×C₇	1	1	yes	characteristic
116	C₂×#106	C₂²×C₇	1	2	yes	—
117	C₂×#106	C₂²×C₇	1	2	yes	—
118	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
119	C₂×#102	C₂²×C₇	1	4	yes	—
120	C₂×#102	C₂²×C₇	1	2	yes	—
121	C₂×#97	C₂²×C₇	1	2	yes	—
122	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
123	#390	C₂²×C₇	1	4	yes	—
124	#388	—	2	4	no	—
125	#384	C₂²×C₇	1	2	yes	—
126	#380	—	2	2	no	—
127	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
128	C₄×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
129	#340	C₂²×C₇	1	4	yes	—
130	#336	C₂²×C₇	1	2	yes	—
131	C₂³×C₇	—	2	4	no	—
132	C₂³×C₇	—	4	2	no	—
133	C₂³×C₇	—	4	2	no	—
134	C₂³×C₇	—	4	2	no	—
135	C₂³×C₇	—	4	2	no	—
136	C₂³×C₇	—	4	2	no	—
137	C₂³×C₇	—	4	2	no	—
138	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
139	C₇×D₄	—	2	4	no	—
140	C₇×D₄	—	2	4	no	—
141	C₇×D₄	—	4	2	no	—
142	C₇×D₄	—	4	2	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	—	2	4	no	—
145	C₂×C₄×C₇	—	2	4	no	—
146	C₂×C₄×C₇	—	2	4	no	—
147	C₂×C₄×C₇	—	2	4	no	—
148	C₂×C₄×C₇	—	4	2	no	—
149	C₂×C₄×C₇	—	4	2	no	—
150	C₂×C₄×C₇	—	4	16	no	—
151	C₂×C₄×C₇	—	4	4	no	—
152	C₂×C₄×C₇	—	4	4	no	—
153	C₂×C₄×C₇	—	4	4	no	—
154	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
155	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
156	C₂×C₄×C₇	AGL(1,ℤ/8)	1	4	yes	—
157	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
163	C₇×C₈	—	2	2	no	—
164	C₇×C₈	—	4	2	no	—
165	C₇×C₈	#117	1	4	yes	complemented
166	C₂²×D₄	C₂³×C₇	1	2	yes	—
167	C₂³×C₄	—	2	2	no	—
168	C₂×D₈	—	2	2	no	—
169	C₂×D₈	—	2	2	no	—
170	C₂×D₈	C₂×C₄×C₇	1	4	yes	—
171	C₂²×C₈	C₂³×C₇	1	2	yes	—
172	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
173	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
174	C₄×D₄	—	2	4	no	—
175	C₄×D₄	C₂×C₄×C₇	1	8	yes	—
176	#117	—	4	8	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₇×D₄	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
181	C₂×C₄⋊C₄	C₂³×C₇	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₂²⋊C₄	—	2	4	no	—
184	C₂×C₂²⋊C₄	—	2	4	no	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	C₂³×C₇	1	2	yes	—
187	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
188	#107	C₂×C₄×C₇	1	4	yes	—
189	#106	—	2	4	no	—
190	#106	—	4	2	no	—
191	C₄⋊C₈	—	2	2	no	—
192	#102	—	4	4	no	—
193	#102	C₂×C₄×C₇	1	8	yes	—
194	#97	—	2	4	no	—
195	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
196	#95	—	2	4	no	—
197	#95	C₇×D₄	1	2	yes	—
198	C₂²×C₇	—	4	4	no	—
199	C₂²×C₇	—	4	4	no	—
200	C₂²×C₇	—	4	4	no	—
201	C₂²×C₇	—	8	2	no	—
202	C₂²×C₇	—	8	2	no	—
203	C₂²×C₇	—	8	2	no	—
204	C₂²×C₇	—	8	2	no	—
205	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
206	C₂²×C₇	#547	1	1	yes	characteristic
207	C₂²×C₇	#550	1	1	yes	characteristic
208	C₂²×C₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
209	C₂²×C₇	#577	1	2	yes	—
210	C₄×C₇	—	2	2	no	—
211	C₄×C₇	—	4	4	no	—
212	C₄×C₇	—	4	4	no	—
213	C₄×C₇	—	8	4	no	—
214	C₄×C₇	#442	1	4	yes	—
215	C₄×C₇	C₂×#117	1	1	yes	characteristic
216	C₄×C₇	#520	1	1	yes	characteristic
217	C₄×C₇	#555	1	1	yes	characteristic
218	C₄×C₇	#556	1	1	yes	characteristic
219	C₂⁴	—	2	2	no	—
220	C₂×D₄	—	2	2	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	C₂²×C₄×C₇	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	4	no	—
230	C₂²×C₄	—	4	4	no	—
231	C₂²×C₄	C₂×C₇×D₄	1	4	yes	—
232	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
235	D₈	—	2	8	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
243	C₂×C₈	C₇×#41	1	2	yes	—
244	C₂×C₈	C₇×#41	1	2	yes	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	8	2	no	—
251	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
252	C₂²⋊C₄	—	2	8	no	—
253	C₂²⋊C₄	—	4	8	no	—
254	C₂²⋊C₄	—	8	4	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	4	8	no	—
258	C₄²	C₂²×C₄×C₇	1	2	yes	—
259	C₄²	C₇×#41	1	2	yes	—
260	C₂×C₇	—	8	4	no	—
261	C₂×C₇	#2354	1	1	yes	characteristic
262	C₂×C₇	C₂×#442	1	1	yes	characteristic
263	C₂×C₇	#2952	1	1	yes	characteristic
264	C₂×C₇	#3354	1	2	yes	—
265	C₂×C₇	#3397	1	2	yes	—
266	C₂³	—	2	4	no	—
267	C₂³	—	4	2	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
274	D₄	—	2	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	16	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
290	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
291	C₂×C₄	C₇×AGL(1,ℤ/8)	1	4	yes	—
292	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
293	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
294	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
295	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
296	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
297	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	C₇×#117	1	4	yes	complemented
301	C₇	#22813	1	1	yes	characteristic, direct product factor, Sylow
302	C₂²	—	4	4	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
310	C₂²	C₇×#547	1	1	yes	characteristic
311	C₂²	C₇×#550	1	1	yes	characteristic
312	C₂²	C₂×C₇×AGL(1,ℤ/8)	1	2	yes	—
313	C₂²	C₇×#577	1	2	yes	—
314	C₄	—	2	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	4	no	—
318	C₄	C₇×#442	1	4	yes	—
319	C₄	C₂×C₇×#117	1	1	yes	characteristic
320	C₄	C₇×#520	1	1	yes	characteristic
321	C₄	C₇×#555	1	1	yes	characteristic
322	C₄	C₇×#556	1	1	yes	characteristic
323	C₂	—	8	4	no	—
324	C₂	C₇×#2354	1	1	yes	characteristic
325	C₂	C₂×C₇×#442	1	1	yes	characteristic
326	C₂	C₇×#2952	1	1	yes	characteristic
327	C₂	C₇×#3354	1	2	yes	—
328	C₂	C₇×#3397	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×#2354	C₂	semidirect
4. C₇×#1926	C₂	semidirect
5. C₇×#1844	C₂	semidirect
6. C₇×#1802	C₂	semidirect
7. C₇×#1776	C₂	semidirect
8. C₇×#1770	C₂	semidirect
12. C₄×C₇×D₈	C₄	semidirect
29. #22813	C₇	direct (N × H)
63. #2354	C₂×C₇	semidirect
65. #1926	C₂×C₇	semidirect
66. #1844	C₂×C₇	semidirect
67. #1802	C₂×C₇	semidirect
68. #1776	C₂×C₇	semidirect
69. #1770	C₂×C₇	semidirect
114. C₄×D₈	C₄×C₇	semidirect
165. C₇×C₈	#117	semidirect
300. C₈	C₇×#117	semidirect
301. C₇	#22813	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×#2354	2
C₇×#1972	2
C₇×#1926	2
C₇×#1844	2
C₇×#1802	2
C₇×#1776	2
C₇×#1770	2
#22813	7

Subgroups — C7×#22813 #13200029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#22813 #13200029