Subgroups — #14767664

Order 2816.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13077334	C₂	1	2	yes	maximal, complemented
2	#13069449	C₂	1	2	yes	maximal, complemented
3	#13068945	C₂	1	8	yes	maximal, complemented
4	#13068754	C₂	1	1	yes	maximal, characteristic, complemented
5	#13068367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₁×#3147	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₁×#135	C₂²	1	4	yes	—
8	C₁₁×#532	C₂²	1	2	yes	—
9	C₂×#67001	C₂²	1	4	yes	—
10	C₁₁×#495	C₄	1	4	yes	complemented
11	C₁₁×#494	C₄	1	1	yes	characteristic, complemented
12	C₁₁×#493	C₄	1	1	yes	characteristic, complemented
13	C₁₁×#492	C₄	1	2	yes	complemented
14	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
15	C₂×#66901	C₂²	1	8	yes	—
16	#10622847	C₂²	1	4	yes	—
17	#10622624	—	2	8	no	—
18	#10622545	C₂²	1	1	yes	characteristic
19	#10622537	C₂²	1	2	yes	—
20	C₂×#66867	C₂²	1	1	yes	characteristic
21	#10622190	C₂²	1	4	yes	—
22	#10622188	C₂²	1	4	yes	—
23	C₂×C₁₁×#41	C₂³	1	4	yes	—
24	C₁₁×#135	—	4	8	no	—
25	C₂×C₁₁×Q₁₆	C₂×C₄	1	4	yes	—
26	C₂×C₁₁×SD₁₆	C₂×C₄	1	8	yes	—
27	C₂×C₁₁×D₈	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
29	C₁₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₁₁×#127	C₂×C₄	1	2	yes	—
31	C₁₁×#124	C₂×C₄	1	4	yes	—
32	C₁₁×#121	—	2	4	no	—
33	C₄×D₄×C₁₁	—	2	4	no	—
34	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
35	#67001	—	4	8	no	—
36	C₂×C₄×Dic₁₁	C₂³	1	2	yes	—
37	C₂×#4318	—	2	4	no	—
38	C₂×#4318	C₂³	1	4	yes	—
39	C₂²×C₁₁⋊C₈	—	2	4	no	—
40	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
41	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
44	#66901	—	4	8	no	—
45	C₁₁⋊#95	—	2	2	no	—
46	C₂²⋊(C₁₁⋊C₈)	—	2	8	no	—
47	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
48	#66867	D₄	1	4	yes	—
49	#22282	—	11	1	no	maximal, Sylow
50	C₁₁×#41	—	2	8	no	—
51	C₁₁×#41	—	2	8	no	—
52	C₂×Q₈×C₁₁	C₂²×C₄	1	4	yes	—
53	C₂×D₄×C₁₁	—	2	4	no	—
54	C₂×D₄×C₁₁	—	2	4	no	—
55	C₂×D₄×C₁₁	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₁₁	—	2	4	no	—
57	C₂²×C₄×C₁₁	#41	1	2	yes	—
58	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₁₁	—	2	2	no	—
60	C₁₁×Q₁₆	—	4	4	no	—
61	C₁₁×SD₁₆	—	4	8	no	—
62	C₁₁×D₈	—	4	4	no	—
63	C₂×C₈×C₁₁	—	2	4	no	—
64	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
66	C₁₁×C₄⋊C₄	—	2	4	no	—
67	C₁₁×C₂²⋊C₄	—	2	8	no	—
68	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
69	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₁	C₂×D₄	1	2	yes	—
71	C₄×Dic₁₁	—	2	4	no	—
72	#4318	—	2	8	no	—
73	#4318	—	4	8	no	—
74	C₂×C₁₁⋊C₈	—	2	4	no	—
75	C₂×C₁₁⋊C₈	—	2	4	no	—
76	C₂×C₁₁⋊C₈	—	4	8	no	—
77	C₂×C₁₁⋊C₈	C₂×D₄	1	4	yes	—
78	#3147	D₁₁	1	1	yes	characteristic
79	#2932	—	11	2	no	—
80	#1963	—	11	2	no	—
81	#1876	—	11	8	no	—
82	#1837	—	11	1	no	—
83	#1746	—	11	1	no	—
84	C₂³×C₁₁	—	2	4	no	—
85	C₂³×C₁₁	C₂×#41	1	1	yes	characteristic
86	Q₈×C₁₁	—	2	8	no	—
87	D₄×C₁₁	—	2	8	no	—
88	D₄×C₁₁	—	4	8	no	—
89	D₄×C₁₁	—	8	4	no	—
90	C₂×C₄×C₁₁	—	2	4	no	—
91	C₂×C₄×C₁₁	—	4	2	no	—
92	C₂×C₄×C₁₁	—	4	4	no	—
93	C₂×C₄×C₁₁	—	4	4	no	—
94	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₁₁	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₁	H(5,𝔽₂)	1	4	yes	—
99	C₂×Dic₁₁	—	4	4	no	—
100	C₂×Dic₁₁	—	4	2	no	—
101	C₈×C₁₁	—	2	2	no	—
102	C₈×C₁₁	—	2	2	no	—
103	C₁₁⋊C₈	—	2	4	no	—
104	C₁₁⋊C₈	—	4	4	no	—
105	C₂×#135	C₂×D₁₁	1	4	yes	—
106	C₂×#131	—	11	4	no	—
107	#532	C₂×D₁₁	1	2	yes	—
108	#495	Dic₁₁	1	4	yes	complemented
109	#494	Dic₁₁	1	1	yes	characteristic, complemented
110	#493	Dic₁₁	1	1	yes	characteristic, complemented
111	#492	Dic₁₁	1	2	yes	complemented
112	#435	—	22	8	no	—
113	#423	—	11	2	no	—
114	C₂×#104	—	11	8	no	—
115	#408	—	11	4	no	—
116	C₂×#97	—	11	1	no	—
117	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
118	#376	—	11	1	no	—
119	#341	—	11	4	no	—
120	#336	—	11	4	no	—
121	C₂²×C₁₁	—	2	2	no	—
122	C₂²×C₁₁	—	4	4	no	—
123	C₂²×C₁₁	—	4	8	no	—
124	C₂²×C₁₁	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₁₁	#540	1	2	yes	—
126	C₄×C₁₁	—	2	2	no	—
127	C₄×C₁₁	—	2	2	no	—
128	C₄×C₁₁	—	4	4	no	—
129	C₄×C₁₁	#518	1	2	yes	—
130	C₄×C₁₁	#535	1	2	yes	—
131	Dic₁₁	—	8	2	no	—
132	C₂×#41	C₂²×D₁₁	1	4	yes	—
133	#135	—	4	8	no	—
134	C₂×Q₁₆	C₂×Dic₁₁	1	4	yes	—
135	C₂×SD₁₆	C₂×Dic₁₁	1	8	yes	—
136	C₂×D₈	C₂×Dic₁₁	1	4	yes	—
137	#131	—	44	8	no	—
138	C₂×M₁₆	—	11	4	no	—
139	C₂×M₁₆	—	22	4	no	—
140	C₂²×C₈	—	11	2	no	—
141	C₂²×C₈	—	22	4	no	—
142	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
143	C₄⋊Q₈	C₂×Dic₁₁	1	2	yes	—
144	#127	C₂×Dic₁₁	1	2	yes	—
145	#124	C₂×Dic₁₁	1	4	yes	—
146	#121	—	2	4	no	—
147	C₄×D₄	—	2	4	no	—
148	C₂×C₄²	—	11	2	no	—
149	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
150	C₄⋊C₈	—	22	4	no	—
151	#104	—	44	8	no	—
152	C₂²⋊C₈	—	22	8	no	—
153	#97	—	11	4	no	—
154	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
155	C₄×C₈	C₂×Dic₁₁	1	1	yes	characteristic
156	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
157	#95	—	22	2	no	—
158	C₂×C₁₁	—	2	2	no	—
159	C₂×C₁₁	—	8	4	no	—
160	C₂×C₁₁	#2328	1	1	yes	characteristic
161	C₂×C₁₁	#2957	1	2	yes	—
162	#41	—	2	8	no	—
163	#41	—	2	8	no	—
164	C₂×Q₈	C₂²×Dic₁₁	1	4	yes	—
165	C₂×D₄	—	2	4	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	C₂²×Dic₁₁	1	4	yes	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	22	2	no	—
170	C₂²×C₄	#4341	1	2	yes	—
171	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
172	Q₁₆	—	4	4	no	—
173	SD₁₆	—	4	8	no	—
174	D₈	—	4	4	no	—
175	M₁₆	—	22	8	no	—
176	M₁₆	—	44	8	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	11	4	no	—
179	C₂×C₈	—	22	4	no	—
180	C₂×C₈	—	22	4	no	—
181	C₂×C₈	—	44	8	no	—
182	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
183	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	8	no	—
186	C₄²	—	22	4	no	—
187	C₄²	D₄×D₁₁	1	2	yes	—
188	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
189	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
190	C₁₁	#22282	1	1	yes	characteristic, complemented, Sylow
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#4341	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	44	4	no	—
202	C₂×C₄	—	44	2	no	—
203	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
204	C₂×C₄	D₄×Dic₁₁	1	2	yes	—
205	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
206	C₂×C₄	#67037	1	4	yes	—
207	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	22	4	no	—
211	C₈	—	44	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	8	no	—
214	C₂²	—	4	4	no	—
215	C₂²	#10623250	1	2	yes	—
216	C₂²	C₂×D₄×Dic₁₁	1	1	yes	characteristic
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	88	2	no	—
221	C₄	#10623246	1	2	yes	—
222	C₄	#10623329	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	#13071342	1	1	yes	characteristic
226	C₂	#13077533	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting, prime radical	C₁₁×#3147	C₂
socle	C₂²×C₁₁	C₂×C₄×D₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	#3147	D₁₁
hypocenter	C₁₁	#22282
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. #13077334	C₂	semidirect
2. #13069449	C₂	semidirect
3. #13068945	C₂	semidirect
4. #13068754	C₂	semidirect
5. #13068367	C₂	semidirect
10. C₁₁×#495	C₄	semidirect
11. C₁₁×#494	C₄	semidirect
12. C₁₁×#493	C₄	semidirect
13. C₁₁×#492	C₄	semidirect
108. #495	Dic₁₁	semidirect
109. #494	Dic₁₁	semidirect
110. #493	Dic₁₁	semidirect
111. #492	Dic₁₁	semidirect
190. C₁₁	#22282	semidirect

Maximal subgroups

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

subgroup	index
#13077334	2
#13069449	2
#13068945	2
#13068754	2
#13068367	2
C₁₁×#3147	2
#22282	11