Subgroups — #13509352

Order 1792.

526 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11739352	C₂	1	2	yes	maximal, complemented
2	C₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729105	C₂	1	2	yes	maximal, complemented
4	#11727101	C₂	1	2	yes	maximal, complemented
5	#11727091	C₂	1	2	yes	maximal, complemented
6	#11727082	C₂	1	1	yes	maximal, characteristic, complemented
7	#11727081	C₂	1	1	yes	maximal, characteristic
8	#11727072	C₂	1	1	yes	maximal, characteristic, complemented
9	#11727071	C₂	1	1	yes	maximal, characteristic, complemented
10	#11726816	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#90382	—	2	2	no	—
13	C₇×#499	C₂²	1	2	yes	—
14	#89840	—	2	2	no	—
15	C₂×Dic₅₆	C₂²	1	1	yes	characteristic
16	C₂×#6374	C₂²	1	2	yes	—
17	C₂×D₁₁₂	C₂²	1	1	yes	characteristic
18	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
19	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
20	#89659	C₂²	1	2	yes	—
21	#89655	C₂²	1	2	yes	—
22	#89645	C₂²	1	2	yes	—
23	#89644	C₂²	1	2	yes	—
24	#89636	—	2	2	no	—
25	#89624	—	2	2	no	—
26	#89468	C₂²	1	2	yes	—
27	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
28	C₇×#367	C₂²	1	2	yes	—
29	C₇×#366	C₂²	1	1	yes	characteristic
30	C₇×#366	C₂²	1	1	yes	characteristic
31	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
34	#89287	C₂²	1	2	yes	—
35	#89287	C₂²	1	2	yes	—
36	#89286	C₂²	1	2	yes	—
37	#89286	C₂²	1	2	yes	—
38	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
39	#33735	—	7	1	no	maximal, Sylow
40	C₇×#125	—	2	2	no	—
41	C₂×Dic₂₈	C₂³	1	2	yes	—
42	C₂×D₅₆	C₂³	1	2	yes	—
43	C₂×#1132	—	2	2	no	—
44	C₂×C₈×D₇	D₄	1	1	yes	characteristic
45	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
46	#6461	—	4	2	no	—
47	#6460	—	2	2	no	—
48	#6458	—	2	2	no	—
49	#6457	—	4	2	no	—
50	#6456	—	2	2	no	—
51	#6456	—	4	2	no	—
52	#6455	—	4	2	no	—
53	#6450	—	2	2	no	—
54	#6438	—	2	2	no	—
55	C₄²×D₇	D₄	1	1	yes	characteristic
56	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
58	C₇×#107	C₂³	1	2	yes	—
59	C₇×#107	C₂³	1	2	yes	—
60	C₇×#106	—	2	2	no	—
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₈×Dic₇	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	#6384	—	2	2	no	—
67	#6384	—	2	2	no	—
68	#6383	—	2	2	no	—
69	#6383	—	2	2	no	—
70	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
71	Dic₅₆	—	2	2	no	—
72	#6374	—	2	4	no	—
73	D₁₁₂	—	2	2	no	—
74	D₇×C₁₆	—	2	2	no	—
75	D₇×C₁₆	—	2	2	no	—
76	#3168	—	7	2	no	—
77	#2253	D₇	1	1	yes	characteristic, complemented
78	#2249	—	7	1	no	—
79	#2242	—	7	2	no	—
80	#2217	—	7	2	no	—
81	#2216	—	7	1	no	—
82	#2215	—	7	1	no	—
83	C₄×Q₃₂	—	7	1	no	—
84	C₄×SD₃₂	—	7	2	no	—
85	C₄×D₁₆	—	7	1	no	—
86	C₂×C₄×C₁₆	—	7	1	no	—
87	C₂×D₂₈	—	2	2	no	—
88	C₂×D₂₈	—	4	2	no	—
89	C₂×C₄×D₇	—	4	2	no	—
90	C₂×C₄×D₇	D₈	1	1	yes	characteristic
91	C₂×C₄×D₇	D₈	1	1	yes	characteristic
92	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₁₄	—	2	2	no	—
94	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
96	C₇×C₄⋊C₄	—	2	2	no	—
97	C₇×C₄⋊C₄	—	2	2	no	—
98	C₇×C₄⋊C₄	—	4	2	no	—
99	C₇×C₄⋊C₄	—	4	2	no	—
100	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
101	(C₂×D₇)⋊C₄	—	4	2	no	—
102	(C₂×D₇)⋊C₄	—	4	2	no	—
103	(C₂×D₇)⋊C₄	—	4	2	no	—
104	(C₂×D₇)⋊C₄	—	4	2	no	—
105	(C₂×D₇)⋊C₄	—	4	2	no	—
106	C₄⋊Dic₇	—	4	2	no	—
107	Dic₇⋊C₄	—	4	2	no	—
108	C₄×Dic₇	—	4	2	no	—
109	C₄×Dic₇	D₈	1	1	yes	characteristic
110	C₄×Dic₇	D₈	1	1	yes	characteristic
111	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
114	Dic₂₈	—	2	2	no	—
115	Dic₂₈	—	2	2	no	—
116	D₅₆	—	2	2	no	—
117	D₅₆	—	2	2	no	—
118	#1132	—	4	4	no	—
119	C₈×D₇	—	2	2	no	—
120	C₈×D₇	—	2	1	no	—
121	C₈×D₇	—	2	1	no	—
122	C₇×C₁₆	#41	1	2	yes	—
123	C₇×C₁₆	#41	1	2	yes	—
124	C₇⋊C₁₆	—	2	1	no	—
125	C₇⋊C₁₆	—	2	1	no	—
126	#529	—	14	2	no	—
127	C₂×Q₃₂	—	7	1	no	—
128	C₂×SD₃₂	—	7	2	no	—
129	C₂×D₁₆	—	7	1	no	—
130	C₂²×C₁₆	—	7	1	no	—
131	C₂²×C₁₆	—	7	1	no	—
132	#499	C₂×D₇	1	2	yes	—
133	#495	—	7	2	no	—
134	#488	—	14	2	no	—
135	#467	—	7	2	no	—
136	#466	—	7	2	no	—
137	#465	—	14	2	no	—
138	C₄×Q₁₆	—	7	2	no	—
139	C₄×SD₁₆	—	14	2	no	—
140	C₄×D₈	—	7	2	no	—
141	C₂×C₄×C₈	—	7	1	no	—
142	#367	C₂×D₇	1	2	yes	—
143	#366	C₂×D₇	1	1	yes	characteristic
144	#366	C₂×D₇	1	1	yes	characteristic
145	#358	—	7	2	no	—
146	#358	—	7	2	no	—
147	#357	—	7	2	no	—
148	#357	—	7	2	no	—
149	C₄×C₁₆	—	7	1	no	—
150	C₄×C₁₆	—	7	1	no	—
151	C₄×C₁₆	—	7	1	no	—
152	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
153	C₂²×D₇	—	4	2	no	—
154	C₂²×D₇	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₇	—	4	2	no	—
156	C₂×C₄×C₇	—	4	2	no	—
157	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
160	C₂×Dic₇	—	4	2	no	—
161	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
162	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
163	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
164	D₂₈	—	4	2	no	—
165	D₂₈	—	4	2	no	—
166	D₂₈	—	8	4	no	—
167	C₄×D₇	—	2	2	no	—
168	C₄×D₇	—	2	1	no	—
169	C₄×D₇	—	2	1	no	—
170	C₄×D₇	—	2	2	no	—
171	C₄×D₇	—	8	4	no	—
172	Dic₁₄	—	4	2	no	—
173	Dic₁₄	—	4	2	no	—
174	C₇×C₈	#135	1	2	yes	—
175	C₇×C₈	C₂×#41	1	1	yes	characteristic
176	C₇×C₈	C₂×#41	1	1	yes	characteristic
177	C₇⋊C₈	—	2	1	no	—
178	C₇⋊C₈	—	2	1	no	—
179	C₂×Q₁₆	—	7	2	no	—
180	C₂×SD₁₆	—	14	2	no	—
181	C₂×D₈	—	7	2	no	—
182	C₂²×C₈	—	7	1	no	—
183	C₂²×C₈	—	7	1	no	—
184	#126	—	28	2	no	—
185	#125	—	2	2	no	—
186	#124	—	14	2	no	—
187	#123	—	28	2	no	—
188	C₂²⋊Q₈	—	14	2	no	—
189	#121	—	14	2	no	—
190	C₄×Q₈	—	14	2	no	—
191	C₄×D₄	—	14	2	no	—
192	C₄×D₄	—	28	2	no	—
193	#117	—	28	2	no	—
194	C₂×C₄²	—	7	1	no	—
195	Q₃₂	—	14	2	no	—
196	SD₃₂	—	14	4	no	—
197	D₁₆	—	14	2	no	—
198	C₂×C₁₆	—	7	1	no	—
199	C₂×C₁₆	—	7	1	no	—
200	C₂×C₁₆	—	14	2	no	—
201	C₂×C₁₆	—	14	2	no	—
202	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
203	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
204	#107	C₂²×D₇	1	2	yes	—
205	#107	C₂²×D₇	1	2	yes	—
206	#106	—	2	2	no	—
207	#103	—	14	2	no	—
208	#103	—	14	2	no	—
209	#102	—	14	2	no	—
210	#102	—	14	2	no	—
211	C₄×C₈	—	7	1	no	—
212	C₄×C₈	—	7	1	no	—
213	C₄×C₈	—	7	1	no	—
214	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
215	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
216	C₂×D₇	—	2	1	no	—
217	C₂×D₇	—	2	1	no	—
218	C₂×D₇	—	8	2	no	—
219	C₂×D₇	—	8	2	no	—
220	C₂×D₇	—	8	2	no	—
221	C₂×D₇	#508	1	2	yes	—
222	C₄×C₇	—	2	1	no	—
223	C₄×C₇	—	2	1	no	—
224	C₄×C₇	—	8	2	no	—
225	C₄×C₇	—	8	2	no	—
226	C₄×C₇	#532	1	1	yes	characteristic
227	C₄×C₇	C₂×#135	1	1	yes	characteristic
228	Dic₇	—	2	1	no	—
229	Dic₇	—	2	1	no	—
230	Dic₇	—	8	2	no	—
231	Dic₇	#508	1	2	yes	—
232	C₂×Q₈	—	14	2	no	—
233	C₂×D₄	—	14	2	no	—
234	C₂×D₄	—	28	2	no	—
235	C₂²×C₄	—	7	1	no	—
236	C₂²×C₄	—	7	1	no	—
237	C₂²×C₄	—	7	1	no	—
238	C₂²×C₄	—	28	2	no	—
239	Q₁₆	—	14	2	no	—
240	Q₁₆	—	14	2	no	—
241	SD₁₆	—	28	4	no	—
242	D₈	—	14	2	no	—
243	D₈	—	14	2	no	—
244	C₂×C₈	—	7	1	no	—
245	C₂×C₈	—	7	1	no	—
246	C₂×C₈	—	14	2	no	—
247	C₂×C₈	—	14	1	no	—
248	C₂×C₈	—	14	1	no	—
249	C₂×C₈	D₄×D₇	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	28	2	no	—
256	C₄⋊C₄	—	28	2	no	—
257	C₂²⋊C₄	—	28	2	no	—
258	C₂²⋊C₄	—	28	2	no	—
259	C₂²⋊C₄	—	28	2	no	—
260	C₂²⋊C₄	—	28	2	no	—
261	C₂²⋊C₄	—	28	2	no	—
262	C₄²	—	7	1	no	—
263	C₄²	—	7	1	no	—
264	C₄²	—	7	1	no	—
265	C₄²	—	28	2	no	—
266	C₄²	D₄×D₇	1	1	yes	characteristic
267	C₁₆	—	14	1	no	—
268	C₁₆	—	14	1	no	—
269	C₁₆	#1161	1	2	yes	—
270	C₁₆	#1161	1	2	yes	—
271	C₂×C₇	#3169	1	1	yes	characteristic
272	C₂×C₇	C₂×#508	1	1	yes	characteristic
273	C₂×C₇	C₂×#508	1	1	yes	characteristic
274	D₇	—	2	2	no	—
275	D₇	—	16	2	no	—
276	C₂³	—	7	1	no	—
277	C₂³	—	28	2	no	—
278	Q₈	—	28	2	no	—
279	Q₈	—	28	2	no	—
280	D₄	—	28	2	no	—
281	D₄	—	28	2	no	—
282	D₄	—	56	4	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	7	1	no	—
286	C₂×C₄	—	7	1	no	—
287	C₂×C₄	—	7	1	no	—
288	C₂×C₄	—	14	2	no	—
289	C₂×C₄	—	14	2	no	—
290	C₂×C₄	—	14	1	no	—
291	C₂×C₄	—	14	1	no	—
292	C₂×C₄	—	28	2	no	—
293	C₂×C₄	—	56	4	no	—
294	C₂×C₄	D₇×D₈	1	1	yes	characteristic
295	C₂×C₄	D₇×D₈	1	1	yes	characteristic
296	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
297	C₈	—	14	1	no	—
298	C₈	—	14	1	no	—
299	C₈	#6482	1	2	yes	—
300	C₈	C₂×#1161	1	1	yes	characteristic
301	C₈	C₂×#1161	1	1	yes	characteristic
302	C₇	#33735	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	7	2	no	—
304	C₂²	—	14	1	no	—
305	C₂²	—	14	1	no	—
306	C₂²	—	56	2	no	—
307	C₂²	—	56	2	no	—
308	C₂²	—	56	2	no	—
309	C₂²	C₂×D₇×D₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	7	2	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	14	1	no	—
316	C₄	—	14	1	no	—
317	C₄	—	56	2	no	—
318	C₄	#90416	1	1	yes	characteristic
319	C₄	C₂×#6482	1	1	yes	characteristic
320	C₂	—	14	2	no	—
321	C₂	—	112	2	no	—
322	C₂	#11739378	1	1	yes	characteristic
323	C₂	D₇×#508	1	1	yes	characteristic
324	C₂	D₇×#508	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×#2253	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	#11727081	C₂
hypercenter	#2253	D₇
hypocenter	C₇	#33735
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
1. #11739352	C₂	semidirect
2. C₇×#2253	C₂	semidirect
3. #11729105	C₂	semidirect
4. #11727101	C₂	semidirect
5. #11727091	C₂	semidirect
6. #11727082	C₂	semidirect
8. #11727072	C₂	semidirect
9. #11727071	C₂	semidirect
10. #11726816	C₂	semidirect
11. C₄×D₇×C₁₆	C₂	semidirect
77. #2253	D₇	semidirect
302. C₇	#33735	semidirect

Maximal subgroups

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

subgroup	index
#11739352	2
C₇×#2253	2
#11729105	2
#11727101	2
#11727091	2
#11727082	2
#11727081	2
#11727072	2
#11727071	2
#11726816	2
C₄×D₇×C₁₆	2
#33735	7