Subgroups — #13509299

Order 1792.

474 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11739384	C₂	1	2	yes	maximal
2	C₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729140	C₂	1	2	yes	maximal, complemented
4	#11727080	C₂	1	2	yes	maximal
5	#11727079	C₂	1	1	yes	maximal, characteristic
6	D₇×#366	C₂	1	1	yes	maximal, characteristic
7	#11727076	C₂	1	2	yes	maximal, complemented
8	#11727075	C₂	1	1	yes	maximal, characteristic, complemented
9	#11727074	C₂	1	1	yes	maximal, characteristic, complemented
10	#11726805	C₂	1	1	yes	maximal, characteristic
11	#11726799	C₂	1	1	yes	maximal, characteristic, complemented
12	#90420	—	2	2	no	—
13	C₇×#500	C₂²	1	2	yes	—
14	#89852	—	2	2	no	—
15	C₂×#6372	C₂²	1	1	yes	characteristic
16	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
17	#89653	—	2	2	no	—
18	#89652	C₂²	1	2	yes	—
19	D₇×#107	C₂²	1	2	yes	—
20	#89650	C₂²	1	2	yes	—
21	#89648	—	2	2	no	—
22	#89646	C₂²	1	2	yes	—
23	#89461	C₂²	1	1	yes	characteristic
24	#89455	C₂²	1	2	yes	—
25	C₇×#366	C₂²	1	1	yes	characteristic
26	C₇×#366	C₂²	1	1	yes	characteristic
27	C₇×#366	C₂²	1	2	yes	—
28	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
29	#89300	C₂²	1	2	yes	—
30	#89299	C₂²	1	1	yes	characteristic
31	#89299	C₂²	1	1	yes	characteristic
32	#89297	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
34	#89285	C₂²	1	2	yes	—
35	#89285	C₂²	1	2	yes	—
36	#89284	C₂²	1	2	yes	—
37	#89284	C₂²	1	2	yes	—
38	#89256	C₂²	1	1	yes	characteristic
39	#33729	—	7	1	no	maximal, Sylow
40	C₇×C₄⋊Q₈	—	2	2	no	—
41	#6509	—	4	2	no	—
42	#6507	—	4	2	no	—
43	C₂×#1131	D₄	1	1	yes	characteristic
44	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
45	#6459	—	4	2	no	—
46	#6455	—	2	2	no	—
47	D₇×C₄⋊C₄	—	2	2	no	—
48	#6453	—	2	2	no	—
49	#6452	—	4	2	no	—
50	#6451	—	2	2	no	—
51	#6435	D₄	1	1	yes	characteristic
52	#6433	—	2	2	no	—
53	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
54	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
55	C₇×#107	—	2	2	no	—
56	C₇×#107	C₂³	1	2	yes	—
57	C₇×#107	C₂³	1	2	yes	—
58	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
59	#6392	C₂³	1	2	yes	—
60	#6392	C₂³	1	2	yes	—
61	#6391	—	2	2	no	—
62	#6389	D₄	1	1	yes	characteristic
63	C₈×Dic₇	C₂³	1	1	yes	characteristic
64	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
66	#6382	—	2	2	no	—
67	#6382	—	2	2	no	—
68	#6381	—	2	2	no	—
69	#6381	—	2	2	no	—
70	#6377	D₄	1	1	yes	characteristic
71	#6372	Q₈	1	4	yes	complemented
72	D₇×C₁₆	Q₈	1	4	yes	complemented
73	#3164	—	7	2	no	—
74	#2255	—	7	2	no	—
75	#2255	—	7	2	no	—
76	#2253	—	7	1	no	—
77	#2253	—	7	1	no	—
78	#2252	—	7	1	no	—
79	#2252	D₇	1	1	yes	characteristic, complemented
80	#2159	—	7	2	no	—
81	#2158	—	7	1	no	—
82	C₂×#366	—	7	1	no	—
83	#2108	—	7	1	no	—
84	C₂×C₇×Q₈	—	4	2	no	—
85	C₂×C₄×D₇	—	4	2	no	—
86	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
87	C₂×Dic₁₄	—	4	2	no	—
88	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
89	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	2	2	no	—
92	C₇×C₄⋊C₄	—	4	2	no	—
93	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
94	(C₂×D₇)⋊C₄	—	4	2	no	—
95	(C₂×D₇)⋊C₄	D₈	1	1	yes	characteristic
96	(C₂×D₇)⋊C₄	D₈	1	1	yes	characteristic
97	C₄⋊Dic₇	—	2	2	no	—
98	C₄⋊Dic₇	—	2	2	no	—
99	C₄⋊Dic₇	—	4	2	no	—
100	Dic₇⋊C₄	—	4	2	no	—
101	Dic₇⋊C₄	—	4	2	no	—
102	Dic₇⋊C₄	—	4	2	no	—
103	Dic₇⋊C₄	—	4	2	no	—
104	Dic₇⋊C₄	—	4	2	no	—
105	Dic₇⋊C₄	D₈	1	1	yes	characteristic
106	Dic₇⋊C₄	D₈	1	1	yes	characteristic
107	C₄×Dic₇	—	4	2	no	—
108	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
109	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
111	#1131	Q₁₆	1	4	yes	—
112	C₈×D₇	C₂×Q₈	1	2	yes	—
113	C₈×D₇	C₂×Q₈	1	2	yes	—
114	C₇×C₁₆	C₂×Q₈	1	2	yes	—
115	C₇×C₁₆	C₂×Q₈	1	2	yes	—
116	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
117	C₇⋊C₁₆	C₂×Q₈	1	2	yes	—
118	#544	—	14	2	no	—
119	C₂×M₃₂	—	7	1	no	—
120	C₂²×C₁₆	—	7	1	no	—
121	#501	—	14	2	no	—
122	#501	—	14	2	no	—
123	#500	—	7	2	no	—
124	#500	C₂×D₇	1	2	yes	—
125	#499	—	7	2	no	—
126	#499	—	7	2	no	—
127	#428	—	14	2	no	—
128	#427	—	7	2	no	—
129	C₂×#107	—	7	2	no	—
130	#405	—	7	1	no	—
131	#367	—	7	2	no	—
132	#367	—	7	2	no	—
133	#367	—	7	2	no	—
134	#366	—	7	1	no	—
135	#366	—	7	1	no	—
136	#366	—	7	2	no	—
137	#366	—	7	2	no	—
138	#366	C₂×D₇	1	1	yes	characteristic
139	#366	C₂×D₇	1	1	yes	characteristic
140	#366	C₂×D₇	1	2	yes	—
141	#346	—	7	1	no	—
142	#346	—	7	1	no	—
143	C₄×C₁₆	—	7	1	no	—
144	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
145	C₂²×D₇	C₂×D₈	1	1	yes	characteristic
146	C₇×Q₈	—	16	2	no	—
147	C₂×C₄×C₇	—	4	2	no	—
148	C₂×C₄×C₇	—	4	2	no	—
149	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₇	—	4	2	no	—
153	C₂×Dic₇	—	4	2	no	—
154	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
155	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
156	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
157	C₄×D₇	—	16	2	no	—
158	C₄×D₇	C₄⋊Q₈	1	2	yes	—
159	C₄×D₇	C₂×Q₁₆	1	2	yes	—
160	Dic₁₄	—	16	2	no	—
161	C₇×C₈	C₂×Q₁₆	1	2	yes	—
162	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
163	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
164	C₇⋊C₈	C₄⋊Q₈	1	2	yes	—
165	C₇⋊C₈	C₂×Q₁₆	1	2	yes	—
166	C₂×M₁₆	—	7	1	no	—
167	C₂²×C₈	—	7	1	no	—
168	C₄⋊Q₈	—	2	2	no	—
169	C₄⋊Q₈	—	14	2	no	—
170	C₄⋊Q₈	—	28	2	no	—
171	#125	—	14	2	no	—
172	#125	—	14	2	no	—
173	#125	—	28	2	no	—
174	C₂²⋊Q₈	—	28	2	no	—
175	C₂²⋊Q₈	—	28	2	no	—
176	#117	—	7	1	no	—
177	#117	—	14	2	no	—
178	C₂×C₄⋊C₄	—	14	2	no	—
179	M₃₂	—	7	4	no	—
180	C₂×C₁₆	—	7	1	no	—
181	C₂×C₁₆	—	7	1	no	—
182	C₂×C₁₆	—	7	4	no	—
183	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
184	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
185	#107	—	2	2	no	—
186	#107	—	7	2	no	—
187	#107	—	7	2	no	—
188	#107	—	14	2	no	—
189	#107	—	14	2	no	—
190	#107	C₂²×D₇	1	2	yes	—
191	#107	C₂²×D₇	1	2	yes	—
192	#106	—	14	2	no	—
193	#106	—	14	2	no	—
194	#106	—	14	2	no	—
195	#97	—	7	1	no	—
196	#97	—	7	1	no	—
197	C₄×C₈	—	7	1	no	—
198	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
199	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
200	C₂×D₇	—	2	1	no	—
201	C₂×D₇	—	2	1	no	—
202	C₂×D₇	#500	1	2	yes	—
203	C₄×C₇	—	2	1	no	—
204	C₄×C₇	—	2	1	no	—
205	C₄×C₇	—	8	2	no	—
206	C₄×C₇	—	8	2	no	—
207	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
208	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
209	Dic₇	—	2	1	no	—
210	Dic₇	—	2	1	no	—
211	Dic₇	—	8	2	no	—
212	Dic₇	—	8	2	no	—
213	Dic₇	#500	1	2	yes	—
214	C₂×Q₈	—	4	2	no	—
215	C₂×Q₈	—	28	2	no	—
216	C₂²×C₄	—	7	1	no	—
217	C₂²×C₄	—	28	2	no	—
218	M₁₆	—	7	4	no	—
219	C₂×C₈	—	7	1	no	—
220	C₂×C₈	—	7	1	no	—
221	C₂×C₈	—	7	2	no	—
222	C₂×C₈	—	7	2	no	—
223	C₂×C₈	D₄×D₇	1	1	yes	characteristic
224	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	7	1	no	—
229	C₄⋊C₄	—	7	1	no	—
230	C₄⋊C₄	—	14	2	no	—
231	C₄⋊C₄	—	14	2	no	—
232	C₄⋊C₄	—	28	2	no	—
233	C₄⋊C₄	—	28	2	no	—
234	C₄⋊C₄	—	28	2	no	—
235	C₄⋊C₄	—	28	2	no	—
236	C₄⋊C₄	—	28	2	no	—
237	C₄⋊C₄	—	28	2	no	—
238	C₂²⋊C₄	—	7	1	no	—
239	C₂²⋊C₄	—	7	1	no	—
240	C₂²⋊C₄	—	28	2	no	—
241	C₄²	—	7	1	no	—
242	C₄²	—	28	2	no	—
243	C₄²	D₄×D₇	1	1	yes	characteristic
244	C₁₆	—	7	2	no	—
245	C₁₆	—	7	2	no	—
246	C₁₆	Q₈×D₇	1	2	yes	—
247	C₁₆	Q₈×D₇	1	2	yes	—
248	C₂×C₇	C₂×#500	1	1	yes	characteristic
249	C₂×C₇	#3415	1	1	yes	characteristic
250	C₂×C₇	#3417	1	1	yes	characteristic
251	D₇	—	2	2	no	—
252	C₂³	—	7	1	no	—
253	Q₈	—	16	2	no	—
254	Q₈	—	112	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	7	1	no	—
258	C₂×C₄	—	7	2	no	—
259	C₂×C₄	—	7	2	no	—
260	C₂×C₄	—	7	1	no	—
261	C₂×C₄	—	7	1	no	—
262	C₂×C₄	—	28	2	no	—
263	C₂×C₄	—	28	2	no	—
264	C₂×C₄	—	112	2	no	—
265	C₂×C₄	D₇×D₈	1	1	yes	characteristic
266	C₂×C₄	D₇×D₈	1	1	yes	characteristic
267	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
268	C₈	—	7	2	no	—
269	C₈	—	7	2	no	—
270	C₈	D₇×Q₁₆	1	2	yes	—
271	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
272	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
273	C₇	#33729	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	7	2	no	—
275	C₂²	—	14	1	no	—
276	C₂²	—	14	1	no	—
277	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	7	2	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	14	1	no	—
284	C₄	—	14	1	no	—
285	C₄	—	56	2	no	—
286	C₄	—	56	2	no	—
287	C₄	D₇×C₄⋊Q₈	1	1	yes	characteristic
288	C₄	C₂×D₇×Q₁₆	1	1	yes	characteristic
289	C₂	—	14	2	no	—
290	C₂	D₇×#500	1	1	yes	characteristic
291	C₂	#11739464	1	1	yes	characteristic
292	C₂	#11739473	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₇×#2252	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	C₂²×D₇	C₂×D₈
hypercenter	#2252	D₇
hypocenter	C₇	#33729
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
2. C₇×#2252	C₂	semidirect
3. #11729140	C₂	semidirect
7. #11727076	C₂	semidirect
8. #11727075	C₂	semidirect
9. #11727074	C₂	semidirect
11. #11726799	C₂	semidirect
71. #6372	Q₈	semidirect
72. D₇×C₁₆	Q₈	semidirect
79. #2252	D₇	semidirect
273. C₇	#33729	semidirect

Maximal subgroups

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

subgroup	index
#11739384	2
C₇×#2252	2
#11729140	2
#11727080	2
#11727079	2
D₇×#366	2
#11727076	2
#11727075	2
#11727074	2
#11726805	2
#11726799	2
#33729	7