Subgroups — #14725764

Order 2816.

572 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077718	C₂	1	2	yes	maximal
2	#13067513	C₂	1	2	yes	maximal, complemented
3	#13067488	C₂	1	4	yes	maximal, complemented
4	Q₁₆×C₁₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	#13067364	C₂	1	1	yes	maximal, characteristic
6	#13067037	C₂	1	2	yes	maximal
7	C₈×#4318	C₂	1	1	yes	maximal, characteristic
8	C₁₁×#3170	C₂	1	1	yes	maximal, characteristic
9	C₁₁×#533	C₂²	1	2	yes	—
10	C₁₁×#500	C₄	1	1	yes	characteristic
11	C₁₁×#494	C₄	1	1	yes	characteristic
12	C₁₁×#487	C₄	1	2	yes	—
13	C₁₁×#467	C₄	1	2	yes	—
14	C₁₁×#467	C₄	1	2	yes	—
15	C₄×C₁₁×Q₁₆	C₂²	1	2	yes	—
16	C₄×C₁₁×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
18	#10622637	—	2	2	no	—
19	#10622637	C₂²	1	4	yes	—
20	Q₈×C₁₁⋊C₈	—	2	2	no	—
21	Q₈×C₁₁⋊C₈	C₂²	1	4	yes	—
22	#10622613	—	2	4	no	—
23	#10622538	C₂²	1	2	yes	—
24	#10622538	C₂²	1	1	yes	characteristic
25	C₄×#4318	C₂²	1	1	yes	characteristic
26	Q₈⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
27	Q₈⋊(C₁₁⋊C₈)	C₂²	1	4	yes	—
28	#10622096	C₂²	1	1	yes	characteristic
29	#10622096	C₂²	1	1	yes	characteristic
30	#10622095	C₂²	1	2	yes	—
31	#10622092	C₂²	1	2	yes	—
32	C₈×C₁₁⋊C₈	C₂²	1	2	yes	—
33	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
34	C₂×C₁₁×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
37	C₁₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
38	C₁₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₁₁×#125	—	2	2	no	—
40	C₁₁×C₂²⋊Q₈	—	2	2	no	—
41	C₁₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
42	C₄×Q₈×C₁₁	—	2	2	no	—
43	C₄×Q₈×C₁₁	C₂³	1	4	yes	—
44	C₁₁×#117	—	2	2	no	—
45	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
46	C₂×#4318	—	2	1	no	—
47	C₁₁×#107	C₂×C₄	1	2	yes	—
48	C₁₁×#107	C₂×C₄	1	2	yes	—
49	C₁₁×#103	C₂×C₄	1	4	yes	—
50	C₁₁×#103	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
55	C₂²⋊(C₁₁⋊C₈)	—	2	2	no	—
56	C₂²⋊(C₁₁⋊C₈)	D₄	1	2	yes	—
57	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
58	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
59	C₄⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
60	C₄⋊(C₁₁⋊C₈)	D₄	1	1	yes	characteristic
61	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
62	C₄⋊(C₁₁⋊C₈)	C₂³	1	2	yes	—
63	#66867	—	2	1	no	—
64	#66867	—	2	4	no	—
65	C₄×C₁₁⋊C₈	—	2	2	no	—
66	C₄×C₁₁⋊C₈	—	2	4	no	—
67	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
68	C₄×C₁₁⋊C₈	C₂³	1	2	yes	—
69	#18086	—	11	1	no	maximal, Sylow
70	C₂×Q₈×C₁₁	—	2	2	no	—
71	C₂×Q₈×C₁₁	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₁	#41	1	1	yes	characteristic
75	C₁₁×Q₁₆	—	2	4	no	—
76	C₁₁×Q₁₆	—	4	2	no	—
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	—	4	1	no	—
79	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
83	C₁₁×C₄⋊C₄	—	2	2	no	—
84	C₁₁×C₄⋊C₄	—	2	2	no	—
85	C₁₁×C₄⋊C₄	—	2	2	no	—
86	C₁₁×C₄⋊C₄	—	2	4	no	—
87	C₁₁×C₄⋊C₄	—	2	4	no	—
88	C₁₁×C₄⋊C₄	C₂²×C₄	1	4	yes	—
89	C₁₁×C₂²⋊C₄	—	2	4	no	—
90	C₄²×C₁₁	—	2	4	no	—
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	#4318	—	2	4	no	—
96	C₂×C₁₁⋊C₈	—	2	2	no	—
97	C₂×C₁₁⋊C₈	—	2	2	no	—
98	C₂×C₁₁⋊C₈	—	2	2	no	—
99	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
100	C₂×C₁₁⋊C₈	C₂×D₄	1	2	yes	—
101	#3170	D₁₁	1	1	yes	characteristic
102	#2986	—	11	2	no	—
103	#1591	—	11	2	no	—
104	#1587	—	11	4	no	—
105	C₈×Q₁₆	—	11	2	no	—
106	#1564	—	11	1	no	—
107	#1485	—	11	2	no	—
108	C₈×M₁₆	—	11	1	no	—
109	C₂³×C₁₁	C₄×D₄	1	1	yes	characteristic
110	Q₈×C₁₁	—	2	4	no	—
111	Q₈×C₁₁	—	2	4	no	—
112	Q₈×C₁₁	—	4	4	no	—
113	C₂×C₄×C₁₁	—	2	1	no	—
114	C₂×C₄×C₁₁	—	2	1	no	—
115	C₂×C₄×C₁₁	—	2	1	no	—
116	C₂×C₄×C₁₁	—	2	1	no	—
117	C₂×C₄×C₁₁	—	2	2	no	—
118	C₂×C₄×C₁₁	—	2	4	no	—
119	C₂×C₄×C₁₁	—	2	4	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	C₈×C₁₁	—	2	1	no	—
127	C₈×C₁₁	—	2	1	no	—
128	C₈×C₁₁	—	2	1	no	—
129	C₈×C₁₁	#131	1	2	yes	—
130	C₁₁⋊C₈	—	2	4	no	—
131	C₁₁⋊C₈	—	2	2	no	—
132	C₁₁⋊C₈	—	2	2	no	—
133	C₁₁⋊C₈	—	4	2	no	—
134	C₁₁⋊C₈	—	4	2	no	—
135	#533	C₂×D₁₁	1	2	yes	—
136	#500	Dic₁₁	1	1	yes	characteristic
137	#494	Dic₁₁	1	1	yes	characteristic
138	#487	Dic₁₁	1	2	yes	—
139	#467	Dic₁₁	1	2	yes	—
140	#467	Dic₁₁	1	2	yes	—
141	#446	—	11	4	no	—
142	#446	—	22	2	no	—
143	C₈×Q₈	—	11	4	no	—
144	C₈×Q₈	—	22	2	no	—
145	C₄×Q₁₆	C₂×D₁₁	1	2	yes	—
146	C₄×Q₁₆	C₂×D₁₁	1	2	yes	—
147	#433	—	22	4	no	—
148	#431	—	11	1	no	—
149	#431	—	11	2	no	—
150	C₄×M₁₆	—	11	1	no	—
151	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
152	#335	—	11	1	no	—
153	#335	—	11	1	no	—
154	#334	—	11	2	no	—
155	Q₈⋊C₈	—	11	4	no	—
156	Q₈⋊C₈	—	11	4	no	—
157	#322	—	11	2	no	—
158	C₈²	—	11	2	no	—
159	C₂²×C₁₁	—	2	1	no	—
160	C₂²×C₁₁	—	2	1	no	—
161	C₂²×C₁₁	—	2	1	no	—
162	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₁	—	2	1	no	—
164	C₄×C₁₁	—	2	1	no	—
165	C₄×C₁₁	—	2	1	no	—
166	C₄×C₁₁	—	4	4	no	—
167	C₄×C₁₁	—	4	4	no	—
168	C₄×C₁₁	#443	1	2	yes	—
169	C₄×C₁₁	C₂×#131	1	1	yes	characteristic
170	C₄×C₁₁	#568	1	1	yes	characteristic
171	C₄×C₁₁	#578	1	1	yes	characteristic
172	C₄×C₁₁	#578	1	1	yes	characteristic
173	C₂×Q₁₆	C₂×Dic₁₁	1	2	yes	—
174	C₂×Q₁₆	C₂×Dic₁₁	1	2	yes	—
175	C₂×M₁₆	—	22	1	no	—
176	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
177	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
178	C₄⋊Q₈	C₂×Dic₁₁	1	2	yes	—
179	C₄⋊Q₈	C₂×Dic₁₁	1	2	yes	—
180	#125	—	2	2	no	—
181	C₂²⋊Q₈	—	2	2	no	—
182	C₂²⋊Q₈	C₂×Dic₁₁	1	4	yes	—
183	C₄×Q₈	—	2	2	no	—
184	C₄×Q₈	C₂²×D₁₁	1	4	yes	—
185	#117	—	2	2	no	—
186	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
187	#107	C₂×Dic₁₁	1	2	yes	—
188	#107	C₂×Dic₁₁	1	2	yes	—
189	C₄⋊C₈	—	11	1	no	—
190	C₄⋊C₈	—	11	1	no	—
191	C₄⋊C₈	—	11	2	no	—
192	C₄⋊C₈	—	11	2	no	—
193	C₄⋊C₈	—	22	4	no	—
194	C₄⋊C₈	—	22	4	no	—
195	#103	C₂×Dic₁₁	1	4	yes	—
196	#103	C₂×Dic₁₁	1	4	yes	—
197	C₂²⋊C₈	—	11	2	no	—
198	C₂²⋊C₈	—	22	2	no	—
199	#97	—	22	4	no	—
200	#97	—	22	1	no	—
201	C₄×C₈	—	11	2	no	—
202	C₄×C₈	—	11	2	no	—
203	C₄×C₈	—	22	4	no	—
204	C₄×C₈	—	22	2	no	—
205	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
206	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
209	C₂×C₁₁	—	4	1	no	—
210	C₂×C₁₁	#2935	1	1	yes	characteristic
211	C₂×C₁₁	#2950	1	1	yes	characteristic
212	C₂×C₁₁	C₂×#443	1	1	yes	characteristic
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	C₂²×Dic₁₁	1	4	yes	—
215	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
216	C₂²×C₄	#4341	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
218	Q₁₆	—	2	4	no	—
219	Q₁₆	—	4	2	no	—
220	M₁₆	—	22	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	11	2	no	—
224	C₂×C₈	—	11	2	no	—
225	C₂×C₈	—	22	2	no	—
226	C₂×C₈	—	22	2	no	—
227	C₂×C₈	—	22	2	no	—
228	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
229	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
230	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
231	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	C₂²×Dic₁₁	1	4	yes	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₄²	—	2	4	no	—
240	C₄²	D₄×D₁₁	1	1	yes	characteristic
241	C₄²	#4341	1	1	yes	characteristic
242	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
243	C₄²	C₂³×D₁₁	1	1	yes	characteristic
244	C₁₁	#18086	1	1	yes	characteristic, complemented, Sylow
245	C₂³	D₄×Dic₁₁	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	2	4	no	—
248	Q₈	—	4	4	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
257	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
258	C₂×C₄	D₄×Dic₁₁	1	1	yes	characteristic
259	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
260	C₂×C₄	C₂×#4341	1	1	yes	characteristic
261	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
262	C₈	—	2	1	no	—
263	C₈	—	2	1	no	—
264	C₈	—	2	1	no	—
265	C₈	—	22	4	no	—
266	C₈	—	22	2	no	—
267	C₈	—	22	2	no	—
268	C₈	—	44	2	no	—
269	C₈	—	44	2	no	—
270	C₈	#67001	1	2	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	4	4	no	—
279	C₄	—	4	4	no	—
280	C₄	#10622809	1	2	yes	—
281	C₄	#10623297	1	1	yes	characteristic
282	C₄	#10623297	1	1	yes	characteristic
283	C₄	C₂×#67001	1	1	yes	characteristic
284	C₄	#10623350	1	1	yes	characteristic
285	C₂	—	4	1	no	—
286	C₂	#13077360	1	1	yes	characteristic
287	C₂	#13077484	1	1	yes	characteristic
288	C₂	C₂×#10622809	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₄²	C₂³×D₁₁
Fitting	C₁₁×#3170	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
prime radical	C₂³×C₁₁	C₄×D₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	#3170	D₁₁
hypocenter	C₁₁	#18086
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
2. #13067513	C₂	semidirect
3. #13067488	C₂	semidirect
4. Q₁₆×C₁₁⋊C₈	C₂	semidirect
244. C₁₁	#18086	semidirect

Maximal subgroups

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

subgroup	index
#13077718	2
#13067513	2
#13067488	2
Q₁₆×C₁₁⋊C₈	2
#13067364	2
#13067037	2
C₈×#4318	2
C₁₁×#3170	2
#18086	11