Subgroups — #13509351

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	#11737160	C₂	1	2	yes	maximal, complemented
2	C₄×C₇×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#11729258	C₂	1	1	yes	maximal, characteristic, complemented
4	#11729254	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#6402	C₂	1	2	yes	maximal, complemented
6	#11727081	C₂	1	1	yes	maximal, characteristic
7	#11727075	C₂	1	1	yes	maximal, characteristic, complemented
8	#11726982	C₂	1	2	yes	maximal, complemented
9	#11726970	C₂	1	2	yes	maximal, complemented
10	C₄×D₁₁₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₇×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#90271	—	2	2	no	—
13	C₂×C₇×Q₃₂	C₂²	1	1	yes	characteristic
14	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
15	#89963	C₂²	1	2	yes	—
16	#89960	C₂²	1	2	yes	—
17	C₂×#6402	C₂²	1	2	yes	—
18	C₄×#1143	—	2	2	no	—
19	C₂×D₁₁₂	C₂²	1	1	yes	characteristic
20	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
21	C₂×D₇×C₁₆	C₂²	1	1	yes	characteristic
22	#89655	C₂²	1	2	yes	—
23	#89650	C₂²	1	2	yes	—
24	#89579	—	2	2	no	—
25	#89572	—	2	2	no	—
26	C₄×D₅₆	C₂²	1	2	yes	—
27	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
28	C₇×#366	C₂²	1	1	yes	characteristic
29	C₇×#358	C₂²	1	2	yes	—
30	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
31	#89359	C₂²	1	2	yes	—
32	#89304	C₂²	1	2	yes	—
33	#89299	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₇	C₂²	1	1	yes	characteristic
36	#89286	C₂²	1	2	yes	—
37	#89285	C₂²	1	2	yes	—
38	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
39	#33735	—	7	1	no	maximal, Sylow
40	C₂×C₇×Q₁₆	C₂³	1	2	yes	—
41	C₄×C₇×Q₈	—	2	2	no	—
42	#6510	—	2	2	no	—
43	#6509	—	2	2	no	—
44	C₂×#1143	—	2	2	no	—
45	C₂×D₅₆	C₂³	1	2	yes	—
46	C₂×C₈×D₇	D₄	1	1	yes	characteristic
47	C₂×C₈×D₇	C₂³	1	1	yes	characteristic
48	#6461	—	4	2	no	—
49	#6458	—	2	2	no	—
50	#6457	—	4	2	no	—
51	#6453	—	2	2	no	—
52	C₄×D₂₈	—	2	2	no	—
53	C₄×D₂₈	—	4	2	no	—
54	#6435	—	4	2	no	—
55	C₄²×D₇	D₄	1	1	yes	characteristic
56	C₇×Q₃₂	—	2	2	no	—
57	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
59	C₇×#107	C₂³	1	2	yes	—
60	C₇×#103	—	2	2	no	—
61	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
62	#6409	—	2	2	no	—
63	#6402	—	2	4	no	—
64	#6395	—	2	2	no	—
65	#6392	C₂³	1	2	yes	—
66	C₈×Dic₇	D₄	1	1	yes	characteristic
67	C₈×Dic₇	C₂³	1	1	yes	characteristic
68	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₇⋊C₁₆	C₂³	1	1	yes	characteristic
70	#6383	—	2	2	no	—
71	#6382	—	2	2	no	—
72	C₄×C₇⋊C₈	D₄	1	1	yes	characteristic
73	D₁₁₂	—	2	2	no	—
74	D₇×C₁₆	—	2	2	no	—
75	D₇×C₁₆	—	2	2	no	—
76	#3168	—	7	2	no	—
77	#2253	—	7	1	no	—
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	D₇	1	1	yes	characteristic, complemented
84	C₄×SD₃₂	—	7	2	no	—
85	C₄×D₁₆	—	7	1	no	—
86	C₂×C₄×C₁₆	—	7	1	no	—
87	C₂×C₇×Q₈	—	2	2	no	—
88	C₂×D₂₈	—	2	2	no	—
89	C₂×D₂₈	—	4	2	no	—
90	C₂×C₄×D₇	—	4	2	no	—
91	C₂×C₄×D₇	D₈	1	1	yes	characteristic
92	C₂×C₄×D₇	D₈	1	1	yes	characteristic
93	C₂×C₄×D₇	C₂×D₄	1	1	yes	characteristic
94	C₇×Q₁₆	—	2	2	no	—
95	C₇×Q₁₆	—	2	2	no	—
96	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
98	C₇×C₄⋊C₄	—	2	2	no	—
99	C₇×C₄⋊C₄	—	4	2	no	—
100	C₄²×C₇	—	4	2	no	—
101	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
102	#1143	—	4	4	no	—
103	(C₂×D₇)⋊C₄	—	4	2	no	—
104	(C₂×D₇)⋊C₄	—	4	2	no	—
105	(C₂×D₇)⋊C₄	—	4	2	no	—
106	(C₂×D₇)⋊C₄	—	4	2	no	—
107	(C₂×D₇)⋊C₄	—	4	2	no	—
108	C₄⋊Dic₇	—	2	2	no	—
109	C₄⋊Dic₇	—	4	2	no	—
110	Dic₇⋊C₄	—	4	2	no	—
111	Dic₇⋊C₄	—	4	2	no	—
112	C₄×Dic₇	D₈	1	1	yes	characteristic
113	C₄×Dic₇	D₈	1	1	yes	characteristic
114	C₄×Dic₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
116	C₂×C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
117	D₅₆	—	2	2	no	—
118	D₅₆	—	2	2	no	—
119	C₈×D₇	—	2	1	no	—
120	C₈×D₇	—	2	1	no	—
121	C₈×D₇	—	2	2	no	—
122	C₇×C₁₆	—	2	1	no	—
123	C₇×C₁₆	#41	1	2	yes	—
124	C₇⋊C₁₆	—	2	1	no	—
125	C₇⋊C₁₆	#41	1	2	yes	—
126	#529	—	14	2	no	—
127	C₂×Q₃₂	C₂×D₇	1	1	yes	characteristic
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	—	7	2	no	—
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	C₂×D₇	1	2	yes	—
139	C₄×SD₁₆	—	14	2	no	—
140	C₄×D₈	—	7	2	no	—
141	C₂×C₄×C₈	—	7	1	no	—
142	#367	—	7	2	no	—
143	#366	—	7	1	no	—
144	#366	C₂×D₇	1	1	yes	characteristic
145	#358	—	7	2	no	—
146	#358	C₂×D₇	1	2	yes	—
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	C₇×Q₈	—	4	2	no	—
156	C₇×Q₈	—	4	2	no	—
157	C₂×C₄×C₇	—	4	2	no	—
158	C₂×C₄×C₇	—	4	2	no	—
159	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×Dic₇	—	4	2	no	—
163	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
164	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
165	C₂×Dic₇	C₂×D₈	1	1	yes	characteristic
166	D₂₈	—	4	2	no	—
167	D₂₈	—	4	2	no	—
168	D₂₈	—	8	4	no	—
169	C₄×D₇	—	2	2	no	—
170	C₄×D₇	—	2	1	no	—
171	C₄×D₇	—	2	1	no	—
172	C₄×D₇	—	2	2	no	—
173	C₄×D₇	—	8	4	no	—
174	C₇×C₈	—	2	1	no	—
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	#135	1	2	yes	—
179	C₂×Q₁₆	C₂²×D₇	1	2	yes	—
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	—	14	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	—	2	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	—	2	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	—	7	2	no	—
205	#107	C₂²×D₇	1	2	yes	—
206	#106	—	14	2	no	—
207	#103	—	2	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	—	8	2	no	—
224	C₄×C₇	—	8	2	no	—
225	C₄×C₇	#508	1	2	yes	—
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	—	2	1	no	—
231	Dic₇	—	8	2	no	—
232	C₂×Q₈	—	2	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	—	2	2	no	—
240	Q₁₆	—	2	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	—	4	2	no	—
253	C₄⋊C₄	—	14	2	no	—
254	C₄⋊C₄	—	28	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	—	4	2	no	—
263	C₄²	—	7	1	no	—
264	C₄²	—	7	1	no	—
265	C₄²	—	7	1	no	—
266	C₄²	D₄×D₇	1	1	yes	characteristic
267	C₁₆	—	2	1	no	—
268	C₁₆	—	7	2	no	—
269	C₁₆	—	14	1	no	—
270	C₁₆	#1157	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	—	4	2	no	—
279	Q₈	—	4	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	1	no	—
290	C₂×C₄	—	14	2	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	—	2	1	no	—
298	C₈	—	7	2	no	—
299	C₈	—	14	1	no	—
300	C₈	C₂×#1157	1	1	yes	characteristic
301	C₈	D₇×#41	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	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	14	1	no	—
314	C₄	—	14	1	no	—
315	C₄	—	14	1	no	—
316	C₄	—	56	2	no	—
317	C₄	#89691	1	2	yes	—
318	C₄	#90239	1	1	yes	characteristic
319	C₄	D₇×#135	1	1	yes	characteristic
320	C₂	—	14	2	no	—
321	C₂	—	112	2	no	—
322	C₂	#11737043	1	1	yes	characteristic
323	C₂	C₂×#89691	1	1	yes	characteristic
324	C₂	D₇×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₇×D₈
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₄×C₇×Q₃₂	C₂
socle	C₂²×C₇	C₂²×D₈
prime radical	#11727081	C₂
hypercenter	C₄×Q₃₂	D₇
hypocenter	C₇	#33735
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #11737160	C₂	semidirect
2. C₄×C₇×Q₃₂	C₂	semidirect
3. #11729258	C₂	semidirect
4. #11729254	C₂	semidirect
5. C₄×#6402	C₂	semidirect
7. #11727075	C₂	semidirect
8. #11726982	C₂	semidirect
9. #11726970	C₂	semidirect
10. C₄×D₁₁₂	C₂	semidirect
11. C₄×D₇×C₁₆	C₂	semidirect
83. C₄×Q₃₂	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
#11737160	2
C₄×C₇×Q₃₂	2
#11729258	2
#11729254	2
C₄×#6402	2
#11727081	2
#11727075	2
#11726982	2
#11726970	2
C₄×D₁₁₂	2
C₄×D₇×C₁₆	2
#33735	7