Subgroups — #13268637

Order 1792.

614 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	C₂²×#6372	C₂	1	1	yes	maximal, characteristic
2	C₂²×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#89461	C₂	1	1	yes	maximal, characteristic
4	C₇×#1374	C₂	1	1	yes	maximal, characteristic, complemented
5	#11725610	C₂	1	1	yes	maximal, characteristic, complemented
6	#11725610	C₂	1	1	yes	maximal, characteristic, complemented
7	#11725463	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#1131	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₇	C₂²	1	1	yes	characteristic
10	C₂×#6435	C₄	1	1	yes	characteristic
11	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6389	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
16	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×#6377	C₄	1	1	yes	characteristic
19	C₂×#6372	—	2	2	no	—
20	C₂×#6372	—	2	2	no	—
21	C₂×#6372	C₄	1	4	yes	complemented
22	C₂×D₇×C₁₆	—	2	2	no	—
23	C₂×D₇×C₁₆	—	2	2	no	—
24	C₂×D₇×C₁₆	C₄	1	4	yes	complemented
25	#89461	—	2	2	no	—
26	#89461	—	2	2	no	—
27	#12545	—	7	1	no	maximal, Sylow
28	C₂²×C₄×D₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×(C₂×D₇)⋊C₄	C₈	1	2	yes	—
33	C₂×Dic₇⋊C₄	C₈	1	2	yes	—
34	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×#1131	—	2	2	no	—
38	C₂×#1131	—	2	2	no	—
39	C₂×#1131	C₈	1	4	yes	—
40	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₇	D₄	1	2	yes	—
43	C₂×C₈×D₇	D₄	1	2	yes	—
44	C₂×C₈×D₇	D₄	1	2	yes	—
45	C₂×C₈×D₇	Q₈	1	2	yes	—
46	#6435	—	2	2	no	—
47	#6435	—	2	2	no	—
48	C₂×C₇×C₁₆	—	2	1	no	—
49	C₂×C₇×C₁₆	—	2	1	no	—
50	C₂×C₇×C₁₆	—	2	1	no	—
51	C₂×C₇×C₁₆	—	2	1	no	—
52	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
53	C₂×C₇×C₁₆	C₂×C₄	1	2	yes	—
54	C₄×C₇×C₈	—	2	1	no	—
55	C₄×C₇×C₈	—	2	1	no	—
56	#6389	—	2	2	no	—
57	C₈×Dic₇	—	2	1	no	—
58	C₈×Dic₇	—	2	1	no	—
59	C₂×C₇⋊C₁₆	—	2	1	no	—
60	C₂×C₇⋊C₁₆	—	2	1	no	—
61	C₂×C₇⋊C₁₆	—	2	1	no	—
62	C₂×C₇⋊C₁₆	—	2	1	no	—
63	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
64	C₂×C₇⋊C₁₆	C₂×C₄	1	2	yes	—
65	#6377	—	2	2	no	—
66	#6372	—	2	8	no	—
67	D₇×C₁₆	—	2	8	no	—
68	C₂²×M₃₂	—	7	1	no	—
69	C₂³×C₁₆	—	7	1	no	—
70	C₂×#405	—	7	1	no	—
71	#1374	—	7	1	no	—
72	#1374	—	7	1	no	—
73	#1374	—	7	1	no	—
74	#1374	D₇	1	1	yes	characteristic, complemented
75	C₂³×D₇	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
77	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×Dic₇	C₂×C₈	1	2	yes	—
79	C₂²×Dic₇	C₂×C₈	1	1	yes	characteristic
80	C₂×C₄×D₇	C₄²	1	2	yes	—
81	C₂×C₄×D₇	C₂²⋊C₄	1	2	yes	—
82	C₂×C₄×D₇	C₄⋊C₄	1	2	yes	—
83	C₂×C₄×D₇	C₂×C₈	1	2	yes	—
84	C₂×C₄×D₇	M₁₆	1	2	yes	—
85	C₂×C₄×D₇	M₁₆	1	2	yes	—
86	C₂×C₇×C₈	—	2	1	no	—
87	C₂×C₇×C₈	—	2	1	no	—
88	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
89	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
95	C₄²×C₇	—	2	1	no	—
96	C₄²×C₇	—	2	1	no	—
97	(C₂×D₇)⋊C₄	—	2	4	no	—
98	Dic₇⋊C₄	—	2	4	no	—
99	C₄×Dic₇	—	2	1	no	—
100	C₄×Dic₇	—	2	1	no	—
101	C₂×C₇⋊C₈	—	2	1	no	—
102	C₂×C₇⋊C₈	—	2	1	no	—
103	C₂×C₇⋊C₈	C₄²	1	2	yes	—
104	C₂×C₇⋊C₈	C₂²⋊C₄	1	2	yes	—
105	C₂×C₇⋊C₈	C₄⋊C₄	1	2	yes	—
106	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
107	#1131	—	2	8	no	—
108	C₈×D₇	C₂²⋊C₄	1	4	yes	—
109	C₈×D₇	C₂²⋊C₄	1	4	yes	—
110	C₈×D₇	C₄⋊C₄	1	4	yes	—
111	C₈×D₇	C₄⋊C₄	1	4	yes	—
112	C₇×C₁₆	—	2	2	no	—
113	C₇×C₁₆	—	2	2	no	—
114	C₇⋊C₁₆	—	2	2	no	—
115	C₇⋊C₁₆	—	2	2	no	—
116	C₂²×M₁₆	—	7	1	no	—
117	C₂³×C₈	—	7	1	no	—
118	C₂×#117	—	7	1	no	—
119	C₂×M₃₂	—	7	4	no	—
120	C₂×M₃₂	—	14	2	no	—
121	C₂×M₃₂	—	14	2	no	—
122	C₂²×C₁₆	—	7	1	no	—
123	C₂²×C₁₆	—	7	1	no	—
124	C₂²×C₁₆	—	7	4	no	—
125	C₂²×C₁₆	—	14	2	no	—
126	C₂²×C₁₆	—	14	2	no	—
127	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
128	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
129	#405	—	14	2	no	—
130	#405	—	14	2	no	—
131	C₂×#97	—	7	1	no	—
132	C₂×#97	—	7	1	no	—
133	C₂×C₄×C₈	—	7	1	no	—
134	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
135	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
136	C₂²×D₇	—	2	2	no	—
137	C₂²×D₇	—	2	2	no	—
138	C₂²×D₇	C₄×C₈	1	2	yes	—
139	C₂²×D₇	C₂²⋊C₈	1	2	yes	—
140	C₂²×D₇	C₄⋊C₈	1	2	yes	—
141	C₂×C₄×C₇	—	2	2	no	—
142	C₂×C₄×C₇	—	2	2	no	—
143	C₂×C₄×C₇	—	2	2	no	—
144	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
145	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
148	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
149	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
150	C₂×Dic₇	—	2	2	no	—
151	C₂×Dic₇	—	2	2	no	—
152	C₂×Dic₇	—	2	2	no	—
153	C₂×Dic₇	C₄×C₈	1	2	yes	—
154	C₂×Dic₇	C₂²⋊C₈	1	2	yes	—
155	C₂×Dic₇	C₄⋊C₈	1	2	yes	—
156	C₄×D₇	#95	1	4	yes	—
157	C₄×D₇	#97	1	4	yes	—
158	C₄×D₇	C₂²⋊C₈	1	4	yes	—
159	C₄×D₇	C₄⋊C₈	1	4	yes	—
160	C₇×C₈	—	2	2	no	—
161	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₇×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₇⋊C₈	—	2	2	no	—
166	C₇⋊C₈	#95	1	4	yes	—
167	C₂³×C₄	—	7	1	no	—
168	C₂×M₁₆	—	7	4	no	—
169	C₂×M₁₆	—	14	2	no	—
170	C₂×M₁₆	—	14	2	no	—
171	C₂²×C₈	—	7	1	no	—
172	C₂²×C₈	—	7	2	no	—
173	C₂²×C₈	—	7	2	no	—
174	C₂²×C₈	—	7	2	no	—
175	C₂²×C₈	—	7	2	no	—
176	C₂²×C₈	—	7	2	no	—
177	C₂²×C₈	—	7	2	no	—
178	C₂²×C₈	—	7	1	no	—
179	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
181	#117	—	14	2	no	—
182	#117	—	14	2	no	—
183	C₂×C₄⋊C₄	—	7	2	no	—
184	C₂×C₂²⋊C₄	—	7	2	no	—
185	C₂×C₄²	—	7	1	no	—
186	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
187	M₃₂	—	14	8	no	—
188	C₂×C₁₆	—	2	1	no	—
189	C₂×C₁₆	—	2	1	no	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	7	2	no	—
193	C₂×C₁₆	—	7	2	no	—
194	C₂×C₁₆	—	14	8	no	—
195	C₂×C₁₆	—	14	1	no	—
196	C₂×C₁₆	—	14	1	no	—
197	C₂×C₁₆	—	14	1	no	—
198	C₂×C₁₆	—	14	1	no	—
199	C₂×C₁₆	C₄×D₇	1	2	yes	—
200	C₂×C₁₆	C₄×D₇	1	2	yes	—
201	#97	—	14	2	no	—
202	#97	—	14	2	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	14	1	no	—
206	C₄×C₈	—	14	1	no	—
207	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
208	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
209	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
210	C₂²×C₇	#504	1	2	yes	—
211	C₂²×C₇	#504	1	2	yes	—
212	C₂×D₇	—	2	2	no	—
213	C₂×D₇	—	2	2	no	—
214	C₂×D₇	—	2	4	no	—
215	C₂×D₇	—	2	4	no	—
216	C₂×D₇	#336	1	4	yes	—
217	C₄×C₇	—	4	2	no	—
218	C₄×C₇	C₂×#95	1	1	yes	characteristic
219	C₄×C₇	C₂×#97	1	1	yes	characteristic
220	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
221	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
222	Dic₇	—	4	2	no	—
223	Dic₇	#336	1	4	yes	—
224	C₂⁴	—	7	1	no	—
225	C₂²×C₄	—	7	1	no	—
226	C₂²×C₄	—	7	2	no	—
227	C₂²×C₄	—	7	2	no	—
228	C₂²×C₄	—	7	2	no	—
229	C₂²×C₄	—	7	2	no	—
230	C₂²×C₄	—	7	2	no	—
231	C₂²×C₄	—	7	2	no	—
232	C₂²×C₄	—	7	2	no	—
233	C₂²×C₄	C₈×D₇	1	2	yes	—
234	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
235	M₁₆	—	14	8	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	7	2	no	—
239	C₂×C₈	—	7	2	no	—
240	C₂×C₈	—	7	2	no	—
241	C₂×C₈	—	7	4	no	—
242	C₂×C₈	—	7	4	no	—
243	C₂×C₈	—	7	4	no	—
244	C₂×C₈	—	7	4	no	—
245	C₂×C₈	—	7	2	no	—
246	C₂×C₈	—	14	1	no	—
247	C₂×C₈	—	14	1	no	—
248	C₂×C₈	C₈×D₇	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
250	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
251	C₂×C₈	D₄×D₇	1	1	yes	characteristic
252	C₂×C₈	D₄×D₇	1	1	yes	characteristic
253	C₂×C₈	D₄×D₇	1	1	yes	characteristic
254	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
255	C₄⋊C₄	—	14	4	no	—
256	C₂²⋊C₄	—	14	4	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	14	1	no	—
260	C₄²	—	14	1	no	—
261	C₁₆	—	2	2	no	—
262	C₁₆	—	2	2	no	—
263	C₁₆	—	14	2	no	—
264	C₁₆	—	14	2	no	—
265	C₂×C₇	C₂×#336	1	1	yes	characteristic
266	C₂×C₇	#2108	1	1	yes	characteristic
267	C₂×C₇	#2108	1	1	yes	characteristic
268	C₂×C₇	#2113	1	2	yes	—
269	C₂×C₇	#2151	1	2	yes	—
270	D₇	—	2	4	no	—
271	C₂³	—	7	2	no	—
272	C₂³	—	7	2	no	—
273	C₂³	—	7	2	no	—
274	C₂³	—	14	2	no	—
275	C₂³	—	14	2	no	—
276	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	7	2	no	—
281	C₂×C₄	—	7	2	no	—
282	C₂×C₄	—	7	2	no	—
283	C₂×C₄	—	7	4	no	—
284	C₂×C₄	—	7	4	no	—
285	C₂×C₄	—	7	4	no	—
286	C₂×C₄	—	7	4	no	—
287	C₂×C₄	—	14	2	no	—
288	C₂×C₄	—	14	2	no	—
289	C₂×C₄	—	14	2	no	—
290	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
291	C₂×C₄	D₇×C₂²⋊C₄	1	1	yes	characteristic
292	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
293	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
294	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
295	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	7	4	no	—
298	C₈	—	14	2	no	—
299	C₈	D₇×C₂²⋊C₄	1	1	yes	characteristic
300	C₈	D₇×C₂²⋊C₄	1	1	yes	characteristic
301	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
302	C₈	D₇×C₄⋊C₄	1	1	yes	characteristic
303	C₇	#12545	1	1	yes	characteristic, complemented, Sylow
304	C₂²	—	7	4	no	—
305	C₂²	—	14	4	no	—
306	C₂²	—	14	4	no	—
307	C₂²	—	14	2	no	—
308	C₂²	—	14	2	no	—
309	C₂²	C₄×C₈×D₇	1	1	yes	characteristic
310	C₂²	D₇×C₂²⋊C₈	1	1	yes	characteristic
311	C₂²	D₇×C₄⋊C₈	1	1	yes	characteristic
312	C₂²	#89673	1	2	yes	—
313	C₂²	#89679	1	2	yes	—
314	C₄	—	4	2	no	—
315	C₄	—	7	4	no	—
316	C₄	—	28	2	no	—
317	C₄	D₇×#95	1	1	yes	characteristic
318	C₄	D₇×#97	1	1	yes	characteristic
319	C₄	D₇×C₂²⋊C₈	1	1	yes	characteristic
320	C₄	D₇×C₄⋊C₈	1	1	yes	characteristic
321	C₂	—	14	4	no	—
322	C₂	D₇×#336	1	1	yes	characteristic
323	C₂	#11726805	1	1	yes	characteristic
324	C₂	#11726826	1	1	yes	characteristic
325	C₂	#11726861	1	2	yes	—
326	C₂	#11727039	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇
commutator	C₂²×C₇	C₂×C₄×C₈
Fitting	C₇×#1374	C₂
socle	C₂³×C₇	C₂²×C₈
prime radical	C₂³×D₇	C₂×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1374	D₇
hypocenter	C₇	#12545
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. C₇×#1374	C₂	semidirect
5. #11725610	C₂	semidirect
6. #11725610	C₂	semidirect
7. #11725463	C₂	semidirect
21. C₂×#6372	C₄	semidirect
24. C₂×D₇×C₁₆	C₄	semidirect
74. #1374	D₇	semidirect
303. C₇	#12545	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#6372	2
C₂²×D₇×C₁₆	2
C₂×#89461	2
C₇×#1374	2
#11725610	2
#11725463	2
#12545	7