Subgroups — #14725533

Order 2816.

722 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13079195	C₂	1	1	yes	maximal, characteristic
2	#13077362	C₂	1	2	yes	maximal, complemented
3	#13067502	C₂	1	2	yes	maximal, complemented
4	#13067447	C₂	1	4	yes	maximal, complemented
5	C₈×D₈₈	C₂	1	2	yes	maximal, complemented
6	C₂×#10622096	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067016	C₂	1	2	yes	maximal, complemented
8	C₈×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#66960	—	2	2	no	—
10	#10623008	C₂²	1	2	yes	—
11	C₁₁×#431	C₂²	1	1	yes	characteristic
12	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	#10622742	—	2	2	no	—
15	#10622728	C₄	1	2	yes	—
16	#10622728	C₄	1	2	yes	—
17	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	2	yes	—
18	#10622319	—	2	4	no	—
19	#10622308	C₄	1	1	yes	characteristic
20	#10622307	C₄	1	2	yes	—
21	C₄×D₈₈	C₂²	1	2	yes	—
22	C₄×D₈₈	C₂²	1	2	yes	—
23	#10622301	C₂²	1	4	yes	—
24	C₈×D₄₄	C₂²	1	4	yes	—
25	#10622295	C₄	1	1	yes	characteristic
26	C₁₁×#322	C₂²	1	2	yes	—
27	C₈²×C₁₁	C₂²	1	2	yes	—
28	#10622097	C₂²	1	4	yes	—
29	#10622097	C₂²	1	4	yes	—
30	#10622096	C₂²	1	1	yes	characteristic
31	#10622096	C₂²	1	2	yes	—
32	#10622096	C₂²	1	1	yes	characteristic
33	C₂×#4339	—	2	2	no	—
34	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
35	C₂×C₁₁×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
39	#66983	—	2	2	no	—
40	#66983	C₂×C₄	1	4	yes	—
41	C₂²×C₁₁⋊C₈	—	2	2	no	—
42	#66960	—	4	8	no	—
43	C₂×D₈₈	C₂×C₄	1	2	yes	—
44	C₂×D₈₈	C₂×C₄	1	2	yes	—
45	C₂×#4313	—	2	4	no	—
46	C₂×C₈×D₁₁	—	2	4	no	—
47	#66928	—	2	2	no	—
48	#66927	C₂×C₄	1	2	yes	—
49	C₄×D₄₄	—	2	2	no	—
50	C₄×D₄₄	C₂³	1	4	yes	—
51	#66922	C₂×C₄	1	2	yes	—
52	C₁₁×C₄⋊C₈	—	2	1	no	—
53	C₁₁×C₂²⋊C₈	—	2	1	no	—
54	C₁₁×#97	D₄	1	2	yes	—
55	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	2	yes	—
58	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₁₁	C₂³	1	2	yes	—
61	#66885	C₂×C₄	1	4	yes	—
62	#66885	C₂×C₄	1	4	yes	—
63	(C₂×D₁₁)⋊C₈	—	2	4	no	—
64	(C₂×D₁₁)⋊C₈	—	2	4	no	—
65	#66882	C₂×C₄	1	2	yes	—
66	#66882	C₂×C₄	1	2	yes	—
67	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
68	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
69	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
70	#18066	—	11	1	no	maximal, Sylow
71	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₁	—	2	4	no	—
75	#4339	—	4	4	no	—
76	#4339	—	4	4	no	—
77	C₂×D₄₄	—	2	2	no	—
78	C₂×D₄₄	—	2	2	no	—
79	C₂×D₄₄	C₂²×C₄	1	4	yes	—
80	C₂×C₄×D₁₁	—	2	4	no	—
81	C₂×Dic₂₂	—	2	2	no	—
82	C₁₁×M₁₆	—	2	2	no	—
83	C₁₁×M₁₆	D₈	1	4	yes	complemented
84	C₂×C₈×C₁₁	—	2	2	no	—
85	C₂×C₈×C₁₁	—	2	2	no	—
86	C₂×C₈×C₁₁	—	4	1	no	—
87	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
93	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₁	#41	1	1	yes	characteristic
96	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
97	(C₂×D₁₁)⋊C₄	—	2	4	no	—
98	(C₂×D₁₁)⋊C₄	—	2	4	no	—
99	C₄⋊Dic₁₁	—	2	2	no	—
100	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
101	C₂×C₁₁⋊C₈	—	2	2	no	—
102	C₂×C₁₁⋊C₈	—	2	2	no	—
103	C₂×C₁₁⋊C₈	—	4	4	no	—
104	D₈₈	—	2	4	no	—
105	D₈₈	—	4	2	no	—
106	#4313	—	4	8	no	—
107	C₈×D₁₁	—	4	8	no	—
108	#3169	—	11	1	no	—
109	#2935	—	11	2	no	—
110	#1589	—	11	2	no	—
111	#1581	—	11	4	no	—
112	C₈×D₈	—	11	2	no	—
113	C₂×#335	—	11	1	no	—
114	#1480	—	11	2	no	—
115	C₈×M₁₆	D₁₁	1	1	yes	characteristic, complemented
116	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₁₁	—	2	4	no	—
118	C₂×C₄×C₁₁	—	2	1	no	—
119	C₂×C₄×C₁₁	—	2	1	no	—
120	C₂×C₄×C₁₁	—	4	1	no	—
121	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₁₁	#135	1	2	yes	—
126	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
129	C₂²⋊D₁₁	—	4	8	no	—
130	C₂×Dic₁₁	—	2	4	no	—
131	D₄₄	—	2	4	no	—
132	D₄₄	—	2	4	no	—
133	D₄₄	—	4	4	no	—
134	D₄₄	—	8	2	no	—
135	C₄×D₁₁	—	4	4	no	—
136	C₄×D₁₁	—	4	4	no	—
137	Dic₂₂	—	4	4	no	—
138	C₈×C₁₁	—	2	1	no	—
139	C₈×C₁₁	—	2	1	no	—
140	C₈×C₁₁	—	2	1	no	—
141	C₈×C₁₁	—	2	2	no	—
142	C₈×C₁₁	—	4	2	no	—
143	C₈×C₁₁	#131	1	2	yes	complemented
144	C₈×C₁₁	C₂×D₈	1	4	yes	—
145	C₁₁⋊C₈	—	4	2	no	—
146	C₁₁⋊C₈	—	4	2	no	—
147	C₂×#131	—	22	2	no	—
148	#532	—	11	2	no	—
149	#500	—	11	1	no	—
150	#493	—	11	1	no	—
151	#486	—	11	2	no	—
152	#466	—	11	2	no	—
153	#466	—	11	2	no	—
154	C₄×D₈	—	11	2	no	—
155	C₄×D₈	—	11	2	no	—
156	#436	—	11	4	no	—
157	#435	—	22	4	no	—
158	C₈×D₄	—	11	4	no	—
159	#431	C₂×D₁₁	1	1	yes	characteristic
160	C₂×C₄⋊C₈	—	11	2	no	—
161	#408	—	22	2	no	—
162	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
164	#335	—	11	1	no	—
165	#335	—	11	2	no	—
166	#335	—	11	1	no	—
167	D₄⋊C₈	—	11	4	no	—
168	D₄⋊C₈	—	11	4	no	—
169	#322	C₂×D₁₁	1	2	yes	—
170	C₈²	C₂×D₁₁	1	2	yes	—
171	C₂²×C₁₁	—	2	1	no	—
172	C₂²×C₁₁	—	2	1	no	—
173	C₂²×C₁₁	C₄×D₈	1	2	yes	—
174	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₁₁	—	4	4	no	—
176	C₂×D₁₁	—	4	4	no	—
177	C₂×D₁₁	—	4	4	no	—
178	C₄×C₁₁	—	2	1	no	—
179	C₄×C₁₁	—	2	1	no	—
180	C₄×C₁₁	—	2	1	no	—
181	C₄×C₁₁	C₄×D₈	1	2	yes	—
182	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
183	C₄×C₁₁	#568	1	1	yes	characteristic
184	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₁₁	C₂×#135	1	1	yes	characteristic
186	Dic₁₁	—	4	4	no	—
187	C₂×#41	—	22	2	no	—
188	C₂×D₈	—	11	2	no	—
189	C₂×D₈	—	11	2	no	—
190	#131	—	44	8	no	—
191	C₂×M₁₆	—	22	4	no	—
192	C₂×M₁₆	D₄₄	1	1	yes	characteristic
193	C₂×M₁₆	D₄₄	1	1	yes	characteristic
194	C₂²×C₈	—	22	2	no	—
195	C₂²×C₈	—	22	4	no	—
196	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
197	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
198	C₄⋊Q₈	—	11	2	no	—
199	#127	—	11	2	no	—
200	#124	—	22	2	no	—
201	#121	—	11	4	no	—
202	#121	—	22	2	no	—
203	C₄×D₄	—	11	4	no	—
204	C₄×D₄	—	22	2	no	—
205	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
206	#107	—	11	2	no	—
207	#107	—	11	2	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	11	2	no	—
210	C₄⋊C₈	—	11	2	no	—
211	C₄⋊C₈	—	22	2	no	—
212	#102	—	11	4	no	—
213	#102	—	11	4	no	—
214	C₂²⋊C₈	—	2	1	no	—
215	C₂²⋊C₈	—	22	4	no	—
216	C₂²⋊C₈	—	22	4	no	—
217	#97	D₄₄	1	2	yes	—
218	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
219	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
220	C₄×C₈	C₂²×D₁₁	1	2	yes	—
221	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₁₁	1	2	yes	—
224	C₂×C₁₁	—	2	2	no	—
225	C₂×C₁₁	#2935	1	1	yes	characteristic
226	C₂×C₁₁	C₂×C₄×D₈	1	1	yes	characteristic
227	C₂×C₁₁	#2957	1	1	yes	characteristic
228	D₁₁	—	8	4	no	—
229	#41	—	44	4	no	—
230	#41	—	44	4	no	—
231	C₂×Q₈	—	22	2	no	—
232	C₂×D₄	—	11	4	no	—
233	C₂×D₄	—	22	4	no	—
234	C₂×D₄	—	22	2	no	—
235	C₂×D₄	—	22	2	no	—
236	C₂²×C₄	—	22	4	no	—
237	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
239	C₂²×C₄	#4339	1	1	yes	characteristic
240	D₈	—	22	4	no	—
241	D₈	—	44	2	no	—
242	M₁₆	—	2	2	no	—
243	M₁₆	—	44	8	no	—
244	M₁₆	D₈₈	1	4	yes	complemented
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	4	1	no	—
248	C₂×C₈	—	22	2	no	—
249	C₂×C₈	—	22	2	no	—
250	C₂×C₈	—	44	4	no	—
251	C₂×C₈	—	44	8	no	—
252	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
253	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
254	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
255	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
256	C₂×C₈	C₂×D₄₄	1	2	yes	—
257	C₂×C₈	C₂×D₄₄	1	2	yes	—
258	C₄⋊C₄	—	11	4	no	—
259	C₄⋊C₄	—	22	2	no	—
260	C₂²⋊C₄	—	22	4	no	—
261	C₂²⋊C₄	—	22	4	no	—
262	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
263	C₄²	C₂×D₄₄	1	1	yes	characteristic
264	C₄²	#4339	1	1	yes	characteristic
265	C₄²	C₂³×D₁₁	1	1	yes	characteristic
266	C₁₁	#18066	1	1	yes	characteristic, complemented, Sylow
267	C₂³	—	22	4	no	—
268	C₂³	C₄×D₄₄	1	1	yes	characteristic
269	Q₈	—	44	4	no	—
270	D₄	—	22	4	no	—
271	D₄	—	22	4	no	—
272	D₄	—	44	8	no	—
273	D₄	—	44	4	no	—
274	D₄	—	88	2	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	22	4	no	—
279	C₂×C₄	—	44	4	no	—
280	C₂×C₄	—	44	4	no	—
281	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
282	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
284	C₂×C₄	C₂×D₈₈	1	2	yes	—
285	C₂×C₄	#66957	1	2	yes	—
286	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
288	C₂×C₄	C₂×#4339	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	4	2	no	—
294	C₈	—	44	2	no	—
295	C₈	—	44	2	no	—
296	C₈	C₂×D₈₈	1	4	yes	—
297	C₈	#66960	1	2	yes	complemented
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	44	4	no	—
301	C₂²	—	44	4	no	—
302	C₂²	—	44	4	no	—
303	C₂²	C₄×D₈₈	1	2	yes	—
304	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	44	4	no	—
309	C₄	C₄×D₈₈	1	2	yes	—
310	C₄	C₂²×D₈₈	1	1	yes	characteristic
311	C₄	C₂×#66957	1	1	yes	characteristic
312	C₄	C₂×#66960	1	1	yes	characteristic
313	C₄	#10623276	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	88	4	no	—
316	C₂	#13077362	1	1	yes	characteristic
317	C₂	C₂×C₄×D₈₈	1	1	yes	characteristic
318	C₂	#13077534	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₈×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	#13079195	C₂
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₈×M₁₆	D₁₁
hypocenter	C₁₁	#18066
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13077362	C₂	semidirect
3. #13067502	C₂	semidirect
4. #13067447	C₂	semidirect
5. C₈×D₈₈	C₂	semidirect
6. C₂×#10622096	C₂	semidirect
7. #13067016	C₂	semidirect
8. C₈×C₁₁×M₁₆	C₂	semidirect
83. C₁₁×M₁₆	D₈	semidirect
115. C₈×M₁₆	D₁₁	semidirect
143. C₈×C₁₁	#131	semidirect
244. M₁₆	D₈₈	semidirect
266. C₁₁	#18066	semidirect
297. C₈	#66960	semidirect

Maximal subgroups

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

subgroup	index
#13079195	2
#13077362	2
#13067502	2
#13067447	2
C₈×D₈₈	2
C₂×#10622096	2
#13067016	2
C₈×C₁₁×M₁₆	2
#18066	11