Subgroups — #13336433

Order 1792.

586 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#3160	C₂	1	1	yes	maximal, characteristic
2	#11740813	C₂	1	2	yes	maximal
3	C₂×#89451	C₂	1	1	yes	maximal, characteristic
4	#11729582	C₂	1	2	yes	maximal
5	#11729134	C₂	1	2	yes	maximal, complemented
6	#11729133	C₂	1	2	yes	maximal, complemented
7	#11729103	C₂	1	4	yes	maximal, complemented
8	#11727483	C₂	1	1	yes	maximal, characteristic
9	C₇×#533	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₄	C₂²	1	1	yes	characteristic
11	C₇×#500	C₂²	1	1	yes	characteristic
12	C₇×#500	C₂²	1	1	yes	characteristic
13	C₇×#499	—	2	1	no	—
14	C₇×#499	—	2	1	no	—
15	C₇×#498	C₂²	1	2	yes	—
16	C₇×#427	—	2	1	no	—
17	C₇×#427	—	2	1	no	—
18	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
19	C₂×#6389	—	2	1	no	—
20	C₂×Dic₇⋊C₈	—	2	1	no	—
21	#89863	C₂²	1	4	yes	—
22	#89862	C₂²	1	4	yes	—
23	#89822	—	2	2	no	—
24	#89816	—	2	2	no	—
25	C₄×#1144	—	2	4	no	—
26	#89697	C₂²	1	2	yes	—
27	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
28	#89451	—	2	1	no	—
29	#89451	—	2	1	no	—
30	#89451	C₂²	1	2	yes	—
31	#89451	C₂²	1	2	yes	—
32	#89342	C₂²	1	4	yes	—
33	#89341	C₂²	1	4	yes	—
34	#89253	C₂²	1	2	yes	—
35	#89252	C₂²	1	2	yes	—
36	#19150	—	7	1	no	maximal, Sylow
37	C₂²×Dic₁₄	D₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
40	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
41	C₇×C₄⋊Q₈	C₂³	1	2	yes	—
42	C₇×#125	—	2	2	no	—
43	C₇×C₂²⋊Q₈	—	2	2	no	—
44	C₇×C₂²⋊Q₈	—	2	2	no	—
45	C₄×C₇×Q₈	—	2	2	no	—
46	C₄×C₇×Q₈	—	2	2	no	—
47	C₇×#117	—	2	2	no	—
48	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
49	C₂×#1144	—	2	4	no	—
50	C₂×C₄⋊Dic₇	D₄	1	1	yes	characteristic
51	C₂×Dic₇⋊C₄	—	2	1	no	—
52	C₂×C₄×Dic₇	—	2	1	no	—
53	C₂²×C₇⋊C₈	—	2	1	no	—
54	C₄×Dic₁₄	—	2	1	no	—
55	C₄×Dic₁₄	—	2	1	no	—
56	C₄×Dic₁₄	D₄	1	2	yes	—
57	C₄×Dic₁₄	C₂³	1	2	yes	—
58	C₇×#107	—	2	1	no	—
59	C₇×#107	—	2	1	no	—
60	C₇×#107	—	2	1	no	—
61	C₇×#107	—	2	1	no	—
62	C₇×#106	—	2	2	no	—
63	C₇×#106	—	2	2	no	—
64	C₄×C₇×C₈	D₄	1	1	yes	characteristic
65	C₄×C₇×C₈	D₄	1	1	yes	characteristic
66	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
68	#6409	—	2	4	no	—
69	C₂²⋊(C₇⋊C₈)	—	2	2	no	—
70	#6389	—	2	2	no	—
71	#6389	—	2	2	no	—
72	Dic₇⋊C₈	—	2	2	no	—
73	Dic₇⋊C₈	—	2	2	no	—
74	#6384	—	2	4	no	—
75	#6381	—	2	4	no	—
76	C₄⋊(C₇⋊C₈)	—	2	1	no	—
77	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
78	C₄⋊(C₇⋊C₈)	C₂³	1	2	yes	—
79	C₄×C₇⋊C₈	—	2	2	no	—
80	#3160	D₇	1	1	yes	characteristic
81	#3104	—	7	2	no	—
82	C₂×#446	—	7	1	no	—
83	#1696	—	7	2	no	—
84	#1694	—	7	2	no	—
85	#1692	—	7	4	no	—
86	#1564	—	7	1	no	—
87	#1542	—	7	2	no	—
88	C₂×C₇×Q₈	—	2	2	no	—
89	C₂×C₇×Q₈	—	2	2	no	—
90	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇	#41	1	1	yes	characteristic
93	C₂²×Dic₇	—	2	1	no	—
94	C₂²×Dic₇	—	2	1	no	—
95	C₂×Dic₁₄	—	2	1	no	—
96	C₂×Dic₁₄	—	4	2	no	—
97	C₂×Dic₁₄	C₂×D₄	1	2	yes	—
98	C₂×C₇×C₈	—	2	2	no	—
99	C₂×C₇×C₈	Q₁₆	1	4	yes	complemented
100	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
101	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
102	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
103	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
104	C₇×C₄⋊C₄	—	2	2	no	—
105	C₇×C₄⋊C₄	—	2	2	no	—
106	C₇×C₄⋊C₄	—	2	2	no	—
107	C₇×C₄⋊C₄	—	2	2	no	—
108	C₇×C₄⋊C₄	—	4	2	no	—
109	C₇×C₄⋊C₄	—	4	2	no	—
110	C₇×C₂²⋊C₄	—	4	2	no	—
111	C₄²×C₇	—	4	2	no	—
112	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇	#41	1	1	yes	characteristic
115	C₄²×C₇	C₂⁴	1	1	yes	characteristic
116	#1144	—	4	8	no	—
117	C₄⋊Dic₇	—	2	1	no	—
118	C₄⋊Dic₇	C₂×D₄	1	2	yes	—
119	Dic₇⋊C₄	—	2	2	no	—
120	Dic₇⋊C₄	—	2	2	no	—
121	C₄×Dic₇	—	2	2	no	—
122	C₄×Dic₇	—	2	2	no	—
123	C₂×C₇⋊C₈	—	2	4	no	—
124	C₂×C₇⋊C₈	—	2	2	no	—
125	C₂×C₇⋊C₈	—	4	2	no	—
126	#533	C₂×D₇	1	2	yes	—
127	C₂×C₄×Q₈	—	7	1	no	—
128	#500	C₂×D₇	1	1	yes	characteristic
129	#500	C₂×D₇	1	1	yes	characteristic
130	#499	—	2	1	no	—
131	#499	—	2	1	no	—
132	#498	C₂×D₇	1	2	yes	—
133	#484	—	14	2	no	—
134	#477	—	7	4	no	—
135	#462	—	7	4	no	—
136	#451	—	14	2	no	—
137	#446	—	7	2	no	—
138	#446	—	7	2	no	—
139	#446	—	14	1	no	—
140	#446	—	14	1	no	—
141	C₄×Q₁₆	—	14	4	no	—
142	#431	—	7	2	no	—
143	#427	—	2	1	no	—
144	#427	—	2	1	no	—
145	C₂×C₄⋊C₈	—	7	1	no	—
146	C₂×C₄⋊C₈	—	14	1	no	—
147	C₂×#97	—	14	1	no	—
148	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
149	#335	—	7	2	no	—
150	#334	—	7	2	no	—
151	#333	—	7	4	no	—
152	#332	—	7	4	no	—
153	C₂³×C₇	#121	1	1	yes	characteristic
154	C₇×Q₈	—	4	4	no	—
155	C₇×Q₈	—	8	2	no	—
156	C₂×C₄×C₇	—	2	2	no	—
157	C₂×C₄×C₇	—	4	1	no	—
158	C₂×C₄×C₇	—	4	2	no	—
159	C₂×C₄×C₇	—	4	2	no	—
160	C₂×C₄×C₇	#121	1	1	yes	characteristic
161	C₂×C₄×C₇	#121	1	1	yes	characteristic
162	C₂×C₄×C₇	#121	1	1	yes	characteristic
163	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
164	C₂×C₄×C₇	#137	1	2	yes	—
165	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
168	C₂×Dic₇	—	2	2	no	—
169	C₂×Dic₇	—	2	2	no	—
170	C₂×Dic₇	—	4	2	no	—
171	C₂×Dic₇	—	4	2	no	—
172	Dic₁₄	—	2	4	no	—
173	Dic₁₄	—	4	2	no	—
174	C₇×C₈	—	2	1	no	—
175	C₇×C₈	—	2	1	no	—
176	C₇×C₈	C₂×Q₁₆	1	2	yes	—
177	C₇×C₈	C₂×Q₁₆	1	2	yes	—
178	C₇⋊C₈	—	4	2	no	—
179	C₇⋊C₈	—	4	4	no	—
180	C₂²×Q₈	—	7	1	no	—
181	C₂×Q₁₆	—	14	4	no	—
182	C₂²×C₈	—	14	1	no	—
183	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
184	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
185	C₄⋊Q₈	C₂²×D₇	1	2	yes	—
186	C₄⋊Q₈	C₂²×D₇	1	2	yes	—
187	#125	—	2	2	no	—
188	C₂²⋊Q₈	—	2	2	no	—
189	C₂²⋊Q₈	—	2	2	no	—
190	C₄×Q₈	—	2	2	no	—
191	C₄×Q₈	—	2	2	no	—
192	C₄×Q₈	—	7	2	no	—
193	C₄×Q₈	—	7	2	no	—
194	C₄×Q₈	—	14	1	no	—
195	C₄×Q₈	—	14	1	no	—
196	#117	—	2	2	no	—
197	C₂×C₄⋊C₄	—	7	1	no	—
198	C₂×C₄⋊C₄	—	14	1	no	—
199	C₂×C₄²	—	14	1	no	—
200	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
201	#107	—	2	1	no	—
202	#107	—	2	1	no	—
203	#107	—	2	1	no	—
204	#107	—	2	1	no	—
205	#107	—	14	4	no	—
206	#106	—	2	2	no	—
207	#106	—	2	2	no	—
208	C₄⋊C₈	—	7	2	no	—
209	C₄⋊C₈	—	7	4	no	—
210	C₄⋊C₈	—	14	2	no	—
211	C₄⋊C₈	—	14	2	no	—
212	C₄⋊C₈	—	14	1	no	—
213	#103	—	14	4	no	—
214	#103	—	14	4	no	—
215	C₂²⋊C₈	—	14	2	no	—
216	#97	—	14	2	no	—
217	#97	—	14	2	no	—
218	C₄×C₈	—	14	2	no	—
219	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
220	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
221	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
223	C₂²×C₇	—	2	1	no	—
224	C₂²×C₇	—	2	1	no	—
225	C₂²×C₇	#462	1	2	yes	—
226	C₂²×C₇	C₂×#121	1	1	yes	characteristic
227	C₄×C₇	—	2	1	no	—
228	C₄×C₇	—	2	1	no	—
229	C₄×C₇	—	2	1	no	—
230	C₄×C₇	—	8	2	no	—
231	C₄×C₇	—	8	2	no	—
232	C₄×C₇	#462	1	2	yes	—
233	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
234	C₄×C₇	C₂×#137	1	1	yes	characteristic
235	C₄×C₇	#575	1	2	yes	—
236	Dic₇	—	4	2	no	—
237	Dic₇	—	4	2	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	7	2	no	—
241	C₂×Q₈	—	14	1	no	—
242	C₂×Q₈	—	28	2	no	—
243	C₂²×C₄	—	14	1	no	—
244	C₂²×C₄	—	14	1	no	—
245	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
246	C₂²×C₄	#1159	1	1	yes	characteristic
247	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
248	Q₁₆	—	28	8	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	14	4	no	—
251	C₂×C₈	—	14	2	no	—
252	C₂×C₈	—	28	2	no	—
253	C₂×C₈	#1144	1	4	yes	complemented
254	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
255	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
256	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
257	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	7	2	no	—
265	C₄⋊C₄	—	14	2	no	—
266	C₄⋊C₄	—	14	1	no	—
267	C₄⋊C₄	—	14	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	14	2	no	—
271	C₄²	—	14	2	no	—
272	C₄²	D₄×D₇	1	1	yes	characteristic
273	C₄²	#1159	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
275	C₄²	C₂³×D₇	1	1	yes	characteristic
276	C₂×C₇	—	2	2	no	—
277	C₂×C₇	C₂×#462	1	1	yes	characteristic
278	C₂×C₇	#3054	1	1	yes	characteristic
279	C₂×C₇	#3067	1	1	yes	characteristic
280	C₂³	#6501	1	1	yes	characteristic
281	Q₈	—	4	4	no	—
282	Q₈	—	8	2	no	—
283	Q₈	—	14	4	no	—
284	Q₈	—	28	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	14	2	no	—
290	C₂×C₄	—	14	2	no	—
291	C₂×C₄	—	28	2	no	—
292	C₂×C₄	—	28	2	no	—
293	C₂×C₄	#6501	1	1	yes	characteristic
294	C₂×C₄	#6501	1	1	yes	characteristic
295	C₂×C₄	#6501	1	1	yes	characteristic
296	C₂×C₄	#6505	1	2	yes	—
297	C₂×C₄	C₂×#1144	1	2	yes	—
298	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
299	C₂×C₄	C₂×#1159	1	1	yes	characteristic
300	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	28	4	no	—
304	C₈	—	28	2	no	—
305	C₈	C₂×#1144	1	2	yes	—
306	C₈	C₂×#1144	1	2	yes	—
307	C₇	#19150	1	1	yes	characteristic, complemented, Sylow
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	#89862	1	2	yes	—
311	C₂²	C₂×#6501	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	28	2	no	—
318	C₄	—	28	2	no	—
319	C₄	#89862	1	2	yes	—
320	C₄	#90459	1	2	yes	—
321	C₄	C₂×#6505	1	1	yes	characteristic
322	C₄	C₂²×#1144	1	1	yes	characteristic
323	C₂	—	2	2	no	—
324	C₂	C₂×#89862	1	1	yes	characteristic
325	C₂	#11741360	1	1	yes	characteristic
326	C₂	#11741373	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6501
commutator, cosocle	C₄²×C₇	C₂⁴
Frattini	C₄²	C₂³×D₇
Fitting	C₇×#3160	C₂
socle	C₂²×C₇	C₂×#121
prime radical	C₂³×C₇	#121
hypercenter	#3160	D₇
hypocenter	C₇	#19150
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

Complements

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

N	H ≅ G/N	type
5. #11729134	C₂	semidirect
6. #11729133	C₂	semidirect
7. #11729103	C₂	semidirect
99. C₂×C₇×C₈	Q₁₆	semidirect
253. C₂×C₈	#1144	semidirect
307. C₇	#19150	semidirect

Maximal subgroups

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

subgroup	index
C₇×#3160	2
#11740813	2
C₂×#89451	2
#11729582	2
#11729134	2
#11729133	2
#11729103	2
#11727483	2
#19150	7