Subgroups — #14725747

Order 2816.

652 conjugacy classes of subgroups.

The subgroup lattice has 314 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	#13077682	C₂	1	2	yes	maximal, complemented
3	#13067502	C₂	1	2	yes	maximal, complemented
4	#13067482	C₂	1	4	yes	maximal, complemented
5	C₈×D₈₈	C₂	1	2	yes	maximal, complemented
6	#13067364	C₂	1	1	yes	maximal, characteristic, complemented
7	#13067034	C₂	1	2	yes	maximal, complemented
8	C₈×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#10623008	C₂²	1	2	yes	—
10	C₁₁×#431	C₂²	1	1	yes	characteristic
11	C₄×C₁₁×M₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	#10622728	C₄	1	2	yes	—
14	#10622728	C₄	1	2	yes	—
15	#10622538	C₂²	1	2	yes	—
16	#10622451	—	2	2	no	—
17	#10622447	—	2	2	no	—
18	#10622340	—	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	1	yes	characteristic
32	#10622095	C₂²	1	2	yes	—
33	C₂×#4339	—	2	2	no	—
34	C₂×C₁₁×M₁₆	—	2	1	no	—
35	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
38	#66983	—	2	2	no	—
39	#66983	C₂×C₄	1	4	yes	—
40	C₂×D₈₈	C₂×C₄	1	2	yes	—
41	C₂×D₈₈	C₂×C₄	1	2	yes	—
42	C₂×#4313	—	2	4	no	—
43	C₂×C₈×D₁₁	—	2	4	no	—
44	#66928	—	2	2	no	—
45	#66927	C₂×C₄	1	2	yes	—
46	C₄×D₄₄	—	2	2	no	—
47	C₄×D₄₄	C₂³	1	4	yes	—
48	#66922	C₂×C₄	1	2	yes	—
49	C₁₁×C₄⋊C₈	D₄	1	1	yes	characteristic
50	C₁₁×C₄⋊C₈	D₄	1	1	yes	characteristic
51	C₁₁×C₂²⋊C₈	D₄	1	2	yes	—
52	C₁₁×#97	—	2	1	no	—
53	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₁	C₂³	1	2	yes	—
56	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂³	1	2	yes	—
59	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
60	#66885	C₂×C₄	1	4	yes	—
61	#66885	C₂×C₄	1	4	yes	—
62	(C₂×D₁₁)⋊C₈	—	2	4	no	—
63	(C₂×D₁₁)⋊C₈	—	2	4	no	—
64	#66882	C₂×C₄	1	2	yes	—
65	#66882	C₂×C₄	1	2	yes	—
66	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
67	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
68	C₄×C₁₁⋊C₈	—	2	2	no	—
69	#18084	—	11	1	no	maximal, Sylow
70	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
73	C₂×C₂²⋊D₁₁	—	2	4	no	—
74	#4339	—	8	2	no	—
75	#4339	—	8	2	no	—
76	C₂×D₄₄	—	2	2	no	—
77	C₂×D₄₄	—	2	2	no	—
78	C₂×D₄₄	C₂²×C₄	1	4	yes	—
79	C₂×C₄×D₁₁	—	2	4	no	—
80	C₂×Dic₂₂	—	2	2	no	—
81	C₁₁×M₁₆	—	2	4	no	—
82	C₂×C₈×C₁₁	—	2	2	no	—
83	C₂×C₈×C₁₁	—	2	2	no	—
84	C₂×C₈×C₁₁	—	4	1	no	—
85	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₁	C₂×D₄	1	2	yes	—
91	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₁₁	#41	1	1	yes	characteristic
94	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
95	(C₂×D₁₁)⋊C₄	—	2	4	no	—
96	(C₂×D₁₁)⋊C₄	—	2	4	no	—
97	C₄⋊Dic₁₁	—	2	2	no	—
98	C₄⋊Dic₁₁	C₂²×C₄	1	4	yes	—
99	C₂×C₁₁⋊C₈	—	2	2	no	—
100	C₂×C₁₁⋊C₈	—	2	2	no	—
101	D₈₈	—	2	4	no	—
102	D₈₈	—	4	2	no	—
103	#4313	—	8	4	no	—
104	C₈×D₁₁	—	8	4	no	—
105	#3169	—	11	1	no	—
106	#2979	—	11	2	no	—
107	#1589	—	11	2	no	—
108	#1586	—	11	4	no	—
109	C₈×D₈	—	11	2	no	—
110	#1564	—	11	1	no	—
111	#1484	—	11	2	no	—
112	C₈×M₁₆	D₁₁	1	1	yes	characteristic, complemented
113	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
114	C₂²×D₁₁	—	2	4	no	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	1	no	—
117	C₂×C₄×C₁₁	—	2	1	no	—
118	C₂×C₄×C₁₁	—	2	1	no	—
119	C₂×C₄×C₁₁	—	2	2	no	—
120	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
126	C₂²⋊D₁₁	—	8	4	no	—
127	C₂×Dic₁₁	—	2	4	no	—
128	D₄₄	—	2	4	no	—
129	D₄₄	—	2	4	no	—
130	D₄₄	—	4	4	no	—
131	D₄₄	—	8	2	no	—
132	C₄×D₁₁	—	4	4	no	—
133	C₄×D₁₁	—	4	4	no	—
134	Dic₂₂	—	8	2	no	—
135	C₈×C₁₁	—	2	4	no	—
136	C₈×C₁₁	—	2	1	no	—
137	C₈×C₁₁	—	2	1	no	—
138	C₈×C₁₁	—	2	1	no	—
139	C₈×C₁₁	—	2	2	no	—
140	C₈×C₁₁	—	2	2	no	—
141	C₈×C₁₁	#131	1	2	yes	—
142	C₁₁⋊C₈	—	4	2	no	—
143	C₁₁⋊C₈	—	4	2	no	—
144	#532	—	11	2	no	—
145	#500	—	11	1	no	—
146	#493	—	11	1	no	—
147	#486	—	11	2	no	—
148	#466	—	11	2	no	—
149	#466	—	11	2	no	—
150	C₄×D₈	—	11	2	no	—
151	C₄×D₈	—	11	2	no	—
152	#436	—	11	4	no	—
153	#436	—	22	2	no	—
154	C₈×D₄	—	11	4	no	—
155	C₈×D₄	—	22	2	no	—
156	#431	—	11	2	no	—
157	#431	C₂×D₁₁	1	1	yes	characteristic
158	#408	—	22	4	no	—
159	C₄×M₁₆	C₂×D₁₁	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
161	#335	—	11	1	no	—
162	#335	—	11	1	no	—
163	#334	—	11	2	no	—
164	D₄⋊C₈	—	11	4	no	—
165	D₄⋊C₈	—	11	4	no	—
166	#322	C₂×D₁₁	1	2	yes	—
167	C₈²	C₂×D₁₁	1	2	yes	—
168	C₂²×C₁₁	—	2	1	no	—
169	C₂²×C₁₁	—	2	1	no	—
170	C₂²×C₁₁	—	2	1	no	—
171	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂×D₁₁	—	4	4	no	—
173	C₂×D₁₁	—	4	4	no	—
174	C₂×D₁₁	—	4	4	no	—
175	C₄×C₁₁	—	2	1	no	—
176	C₄×C₁₁	—	2	1	no	—
177	C₄×C₁₁	—	2	1	no	—
178	C₄×C₁₁	#443	1	2	yes	—
179	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
180	C₄×C₁₁	#568	1	1	yes	characteristic
181	C₄×C₁₁	#576	1	1	yes	characteristic
182	C₄×C₁₁	#576	1	1	yes	characteristic
183	Dic₁₁	—	4	4	no	—
184	C₂×#41	—	22	2	no	—
185	C₂×D₈	—	11	2	no	—
186	C₂×D₈	—	11	2	no	—
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	22	4	no	—
189	C₂²×C₈	—	22	4	no	—
190	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
191	C₂²×C₈	C₄×D₁₁	1	1	yes	characteristic
192	C₄⋊Q₈	—	11	2	no	—
193	#127	—	11	2	no	—
194	#124	—	22	2	no	—
195	#121	—	11	4	no	—
196	#121	—	22	2	no	—
197	C₄×D₄	—	11	4	no	—
198	C₄×D₄	—	22	2	no	—
199	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
200	#107	—	11	2	no	—
201	#107	—	11	2	no	—
202	C₄⋊C₈	—	11	2	no	—
203	C₄⋊C₈	—	11	2	no	—
204	C₄⋊C₈	D₄₄	1	1	yes	characteristic
205	C₄⋊C₈	D₄₄	1	1	yes	characteristic
206	#102	—	11	4	no	—
207	#102	—	11	4	no	—
208	C₂²⋊C₈	—	22	2	no	—
209	C₂²⋊C₈	—	22	4	no	—
210	C₂²⋊C₈	—	22	4	no	—
211	C₂²⋊C₈	D₄₄	1	2	yes	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	22	2	no	—
214	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
215	C₄×C₈	C₄×D₁₁	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₁₁	1	2	yes	—
217	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
218	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
219	C₄×C₈	C₂²×D₁₁	1	2	yes	—
220	C₂×C₁₁	—	4	1	no	—
221	C₂×C₁₁	#2935	1	1	yes	characteristic
222	C₂×C₁₁	#2948	1	1	yes	characteristic
223	C₂×C₁₁	C₂×#443	1	1	yes	characteristic
224	D₁₁	—	8	4	no	—
225	#41	—	88	2	no	—
226	#41	—	88	2	no	—
227	C₂×Q₈	—	22	2	no	—
228	C₂×D₄	—	11	4	no	—
229	C₂×D₄	—	22	4	no	—
230	C₂×D₄	—	22	2	no	—
231	C₂×D₄	—	22	2	no	—
232	C₂²×C₄	—	22	4	no	—
233	C₂²×C₄	C₂×C₄×D₁₁	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
235	C₂²×C₄	#4339	1	1	yes	characteristic
236	D₈	—	22	4	no	—
237	D₈	—	44	2	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	88	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	22	2	no	—
244	C₂×C₈	—	22	2	no	—
245	C₂×C₈	—	88	4	no	—
246	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₁₁	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄₄	1	2	yes	—
251	C₂×C₈	C₂×D₄₄	1	2	yes	—
252	C₄⋊C₄	—	11	4	no	—
253	C₄⋊C₄	—	22	2	no	—
254	C₂²⋊C₄	—	22	4	no	—
255	C₂²⋊C₄	—	22	4	no	—
256	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
257	C₄²	C₂×D₄₄	1	1	yes	characteristic
258	C₄²	#4339	1	1	yes	characteristic
259	C₄²	C₂³×D₁₁	1	1	yes	characteristic
260	C₁₁	#18084	1	1	yes	characteristic, complemented, Sylow
261	C₂³	—	22	4	no	—
262	C₂³	C₄×D₄₄	1	1	yes	characteristic
263	Q₈	—	88	2	no	—
264	D₄	—	22	4	no	—
265	D₄	—	22	4	no	—
266	D₄	—	44	4	no	—
267	D₄	—	88	2	no	—
268	D₄	—	88	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	22	4	no	—
275	C₂×C₄	—	44	4	no	—
276	C₂×C₄	—	44	4	no	—
277	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
278	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
279	C₂×C₄	C₄×D₄₄	1	1	yes	characteristic
280	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
282	C₂×C₄	C₂×#4339	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	44	2	no	—
290	C₈	—	44	2	no	—
291	C₈	#66960	1	2	yes	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	44	4	no	—
296	C₂²	—	44	4	no	—
297	C₂²	—	44	4	no	—
298	C₂²	C₂×C₄×D₄₄	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	44	4	no	—
303	C₄	#10622313	1	2	yes	—
304	C₄	C₂×#66960	1	1	yes	characteristic
305	C₄	#10623276	1	1	yes	characteristic
306	C₄	#10623283	1	1	yes	characteristic
307	C₄	#10623283	1	1	yes	characteristic
308	C₂	—	4	1	no	—
309	C₂	—	88	4	no	—
310	C₂	#13077362	1	1	yes	characteristic
311	C₂	#13077469	1	1	yes	characteristic
312	C₂	C₂×#10622313	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	#18084
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13077682	C₂	semidirect
3. #13067502	C₂	semidirect
4. #13067482	C₂	semidirect
5. C₈×D₈₈	C₂	semidirect
6. #13067364	C₂	semidirect
7. #13067034	C₂	semidirect
8. C₈×C₁₁×M₁₆	C₂	semidirect
112. C₈×M₁₆	D₁₁	semidirect
260. C₁₁	#18084	semidirect

Maximal subgroups

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

subgroup	index
#13079195	2
#13077682	2
#13067502	2
#13067482	2
C₈×D₈₈	2
#13067364	2
#13067034	2
C₈×C₁₁×M₁₆	2
#18084	11