Subgroups — #14907404

Order 2816.

556 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070442	C₂	1	8	yes	maximal, complemented
2	#13070419	C₂	1	2	yes	maximal, complemented
3	#13070230	C₂	1	4	yes	maximal, complemented
4	C₁₁×#2931	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₁×#131	C₂²	1	2	yes	—
6	C₁₁×#532	C₄	1	1	yes	characteristic
7	C₁₁×#436	C₄	1	4	yes	—
8	C₈×D₄×C₁₁	C₂²	1	4	yes	—
9	C₁₁×#423	C₄	1	1	yes	characteristic
10	C₁₁×#408	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
12	C₂×#66876	C₂²	1	4	yes	—
13	C₂²×C₁₁⋊C₁₆	C₂²	1	4	yes	—
14	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
15	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	8	yes	—
16	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
17	#10622099	C₂²	1	4	yes	—
18	C₂×C₁₁×#41	C₂×C₄	1	2	yes	—
19	C₁₁×#131	—	4	2	no	—
20	C₁₁×#131	—	4	2	no	—
21	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₁×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₁	C₂³	1	4	yes	—
26	C₁₁×C₄⋊Q₈	C₈	1	1	yes	characteristic
27	C₁₁×#127	C₈	1	1	yes	characteristic
28	C₁₁×#124	C₈	1	2	yes	—
29	C₁₁×#121	C₈	1	4	yes	—
30	C₄×D₄×C₁₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
32	C₁₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
33	C₁₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₁₁	C₂³	1	2	yes	—
36	#66876	—	4	4	no	—
37	C₂×C₁₁⋊C₁₆	—	4	4	no	—
38	C₂×C₁₁⋊C₁₆	C₂³	1	8	yes	—
39	#33494	—	11	1	no	maximal, Sylow
40	C₁₁×#41	—	4	2	no	—
41	C₁₁×#41	—	4	2	no	—
42	C₂×Q₈×C₁₁	C₂×C₈	1	2	yes	—
43	C₂×D₄×C₁₁	C₂×C₈	1	4	yes	—
44	C₂×D₄×C₁₁	C₂×C₈	1	4	yes	—
45	C₂×D₄×C₁₁	C₂×C₈	1	2	yes	—
46	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₁₁	C₂²×C₄	1	4	yes	—
48	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
49	C₁₁×M₁₆	—	4	2	no	—
50	C₁₁×M₁₆	—	4	4	no	—
51	C₂×C₈×C₁₁	—	2	4	no	—
52	C₂×C₈×C₁₁	—	2	4	no	—
53	C₂×C₈×C₁₁	—	2	2	no	—
54	C₂×C₈×C₁₁	—	4	1	no	—
55	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
57	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
59	C₁₁×C₄⋊C₄	C₂×C₈	1	4	yes	—
60	C₁₁×C₂²⋊C₄	C₂×C₈	1	8	yes	—
61	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
62	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
63	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
64	C₁₁⋊C₁₆	—	2	8	no	—
65	#2931	D₁₁	1	1	yes	characteristic
66	#2168	—	11	8	no	—
67	#2163	—	11	2	no	—
68	#2113	—	11	4	no	—
69	C₂³×C₁₁	C₂²×C₈	1	4	yes	—
70	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
71	Q₈×C₁₁	—	4	2	no	—
72	D₄×C₁₁	—	4	4	no	—
73	D₄×C₁₁	—	4	2	no	—
74	D₄×C₁₁	—	4	4	no	—
75	C₂×C₄×C₁₁	—	2	1	no	—
76	C₂×C₄×C₁₁	—	2	4	no	—
77	C₂×C₄×C₁₁	—	2	1	no	—
78	C₂×C₄×C₁₁	—	2	4	no	—
79	C₂×C₄×C₁₁	—	4	2	no	—
80	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
81	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
82	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₁₁	C₂²×C₈	1	4	yes	—
84	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
85	C₈×C₁₁	—	2	4	no	—
86	C₈×C₁₁	—	2	1	no	—
87	C₈×C₁₁	—	2	2	no	—
88	C₈×C₁₁	H(5,𝔽₂)	1	2	yes	—
89	C₂×#131	C₂×D₁₁	1	2	yes	—
90	#532	Dic₁₁	1	1	yes	characteristic
91	C₂×M₃₂	—	11	4	no	—
92	C₂²×C₁₆	—	11	4	no	—
93	#436	Dic₁₁	1	4	yes	—
94	C₈×D₄	C₂×D₁₁	1	4	yes	—
95	#423	Dic₁₁	1	1	yes	characteristic
96	#408	Dic₁₁	1	2	yes	—
97	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
98	C₄⋊C₁₆	—	11	4	no	—
99	C₂²⋊C₁₆	—	11	4	no	—
100	C₂²⋊C₁₆	—	11	8	no	—
101	#346	—	11	4	no	—
102	C₂²×C₁₁	—	2	2	no	—
103	C₂²×C₁₁	—	2	8	no	—
104	C₂²×C₁₁	—	2	1	no	—
105	C₂²×C₁₁	—	2	4	no	—
106	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
107	C₄×C₁₁	—	2	2	no	—
108	C₄×C₁₁	—	2	2	no	—
109	C₄×C₁₁	—	2	1	no	—
110	C₄×C₁₁	—	2	4	no	—
111	C₄×C₁₁	#518	1	1	yes	characteristic
112	C₄×C₁₁	#568	1	1	yes	characteristic
113	C₂×#41	C₂×Dic₁₁	1	2	yes	—
114	#131	—	4	2	no	—
115	#131	—	4	2	no	—
116	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
117	C₂×M₁₆	C₂×Dic₁₁	1	4	yes	—
118	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
120	C₂²×C₈	C₂²×D₁₁	1	4	yes	—
121	C₄⋊Q₈	C₁₁⋊C₈	1	1	yes	characteristic
122	#127	C₁₁⋊C₈	1	1	yes	characteristic
123	#124	C₁₁⋊C₈	1	2	yes	—
124	#121	C₁₁⋊C₈	1	4	yes	—
125	C₄×D₄	C₂×Dic₁₁	1	4	yes	—
126	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
127	M₃₂	—	44	4	no	—
128	C₂×C₁₆	—	11	8	no	—
129	C₂×C₁₆	—	44	4	no	—
130	C₄⋊C₈	C₂×Dic₁₁	1	4	yes	—
131	C₂²⋊C₈	C₂×Dic₁₁	1	8	yes	—
132	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
133	C₄×C₈	C₂²×D₁₁	1	2	yes	—
134	C₂×C₁₁	—	4	1	no	—
135	C₂×C₁₁	—	4	4	no	—
136	C₂×C₁₁	#2972	1	1	yes	characteristic
137	C₂×C₁₁	#3407	1	2	yes	—
138	#41	—	4	2	no	—
139	#41	—	4	2	no	—
140	C₂×Q₈	C₂×C₁₁⋊C₈	1	2	yes	—
141	C₂×D₄	C₂×C₁₁⋊C₈	1	4	yes	—
142	C₂×D₄	C₂×C₁₁⋊C₈	1	4	yes	—
143	C₂×D₄	C₂×C₁₁⋊C₈	1	2	yes	—
144	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
145	C₂²×C₄	C₂²×Dic₁₁	1	4	yes	—
146	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
147	M₁₆	—	4	4	no	—
148	M₁₆	—	4	2	no	—
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	1	no	—
153	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
154	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
155	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
156	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
157	C₄⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
158	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	8	yes	—
159	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
160	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
161	C₄²	C₂²×Dic₁₁	1	2	yes	—
162	C₁₆	—	22	8	no	—
163	C₁₁	#33494	1	1	yes	characteristic, complemented, Sylow
164	C₂³	C₂²×C₁₁⋊C₈	1	4	yes	—
165	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
166	Q₈	—	4	2	no	—
167	D₄	—	4	4	no	—
168	D₄	—	4	2	no	—
169	D₄	—	4	4	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	2	no	—
175	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
176	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
177	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
178	C₂×C₄	C₂²×C₁₁⋊C₈	1	4	yes	—
179	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
180	C₈	—	2	1	no	—
181	C₈	—	2	4	no	—
182	C₈	—	2	2	no	—
183	C₈	#67037	1	2	yes	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	8	no	—
186	C₂²	—	2	1	no	—
187	C₂²	—	2	4	no	—
188	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	1	no	—
192	C₄	—	2	4	no	—
193	C₄	#10623329	1	1	yes	characteristic
194	C₄	#10623350	1	1	yes	characteristic
195	C₂	—	4	1	no	—
196	C₂	—	4	4	no	—
197	C₂	#13077640	1	1	yes	characteristic
198	C₂	#13081719	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2931	C₂
prime radical	C₁₁×#532	C₄
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2931	D₁₁
hypocenter	C₁₁	#33494
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #13070442	C₂	semidirect
2. #13070419	C₂	semidirect
3. #13070230	C₂	semidirect
163. C₁₁	#33494	semidirect

Maximal subgroups

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

subgroup	index
#13070442	2
#13070419	2
#13070230	2
C₁₁×#2931	2
#33494	11