Subgroups — #14907435

Order 2816.

460 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070525	C₂	1	4	yes	maximal, complemented
2	Q₈×C₁₁⋊C₁₆	C₂	1	4	yes	maximal, complemented
3	#13070423	C₂	1	4	yes	maximal
4	#13070415	C₂	1	2	yes	maximal
5	C₁₁×#2960	C₂	1	1	yes	maximal, characteristic
6	C₁₁×#533	C₄	1	1	yes	characteristic
7	C₁₁×#446	C₄	1	4	yes	—
8	C₈×Q₈×C₁₁	C₂²	1	4	yes	—
9	C₁₁×#424	C₄	1	2	yes	—
10	C₁₁×#423	C₄	1	1	yes	characteristic
11	C₁₁×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₂²⋊(C₁₁⋊C₁₆)	C₂²	1	4	yes	—
14	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
15	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	2	yes	—
16	C₄⋊(C₁₁⋊C₁₆)	C₂²	1	8	yes	—
17	#10622099	C₂²	1	4	yes	—
18	C₄×C₁₁⋊C₁₆	C₂²	1	4	yes	—
19	C₄×C₁₁⋊C₁₆	C₂²	1	4	yes	—
20	C₂×C₁₁×M₁₆	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
23	C₁₁×C₄⋊Q₈	C₈	1	2	yes	—
24	C₁₁×#125	C₈	1	2	yes	—
25	C₁₁×C₂²⋊Q₈	C₈	1	4	yes	—
26	C₄×Q₈×C₁₁	C₂×C₄	1	4	yes	—
27	C₁₁×#117	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₁	C₂×C₄	1	1	yes	characteristic
29	C₁₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
30	C₁₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
31	C₁₁×#97	C₂×C₄	1	4	yes	—
32	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₁₁	C₂³	1	2	yes	—
34	C₄×C₈×C₁₁	C₂³	1	4	yes	—
35	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
36	C₂×C₁₁⋊C₁₆	C₂³	1	4	yes	—
37	#33500	—	11	1	no	maximal, Sylow
38	C₂×Q₈×C₁₁	C₂×C₈	1	4	yes	—
39	C₂²×C₄×C₁₁	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₁₁×M₁₆	—	4	2	no	—
42	C₂×C₈×C₁₁	—	2	2	no	—
43	C₂×C₈×C₁₁	—	4	1	no	—
44	C₂×C₈×C₁₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
46	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
47	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
48	C₁₁×C₄⋊C₄	C₂×C₈	1	4	yes	—
49	C₁₁×C₄⋊C₄	C₂×C₈	1	8	yes	—
50	C₁₁×C₄⋊C₄	C₂×C₈	1	4	yes	—
51	C₁₁×C₂²⋊C₄	C₂×C₈	1	4	yes	—
52	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₁₁	C₂×C₈	1	1	yes	characteristic
54	C₄²×C₁₁	C₂²×C₄	1	2	yes	—
55	C₄²×C₁₁	C₂²×C₄	1	4	yes	—
56	C₁₁⋊C₁₆	—	2	4	no	—
57	C₁₁⋊C₁₆	—	2	4	no	—
58	#2960	D₁₁	1	1	yes	characteristic
59	#2183	—	11	4	no	—
60	Q₈×C₁₆	—	11	4	no	—
61	#2164	—	11	4	no	—
62	#2162	—	11	2	no	—
63	C₂³×C₁₁	C₂²×C₈	1	1	yes	characteristic
64	Q₈×C₁₁	—	2	8	no	—
65	C₂×C₄×C₁₁	—	2	1	no	—
66	C₂×C₄×C₁₁	—	2	1	no	—
67	C₂×C₄×C₁₁	—	2	4	no	—
68	C₂×C₄×C₁₁	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
70	C₂×C₄×C₁₁	C₂²×C₈	1	8	yes	—
71	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
72	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
73	C₈×C₁₁	—	2	1	no	—
74	C₈×C₁₁	—	2	2	no	—
75	C₈×C₁₁	—	2	4	no	—
76	C₈×C₁₁	2₋¹⁺⁴	1	2	yes	—
77	#533	Dic₁₁	1	1	yes	characteristic
78	#446	Dic₁₁	1	4	yes	—
79	C₈×Q₈	C₂×D₁₁	1	4	yes	—
80	#424	Dic₁₁	1	2	yes	—
81	#423	Dic₁₁	1	1	yes	characteristic
82	#405	C₂×D₁₁	1	2	yes	—
83	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
84	C₄⋊C₁₆	—	11	2	no	—
85	C₄⋊C₁₆	—	11	2	no	—
86	C₄⋊C₁₆	—	11	8	no	—
87	C₂²⋊C₁₆	—	11	4	no	—
88	#346	—	11	4	no	—
89	C₄×C₁₆	—	11	4	no	—
90	C₄×C₁₆	—	11	4	no	—
91	C₂²×C₁₁	—	2	2	no	—
92	C₂²×C₁₁	—	2	1	no	—
93	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
94	C₄×C₁₁	—	2	2	no	—
95	C₄×C₁₁	—	2	8	no	—
96	C₄×C₁₁	—	2	1	no	—
97	C₄×C₁₁	#504	1	4	yes	—
98	C₄×C₁₁	#519	1	1	yes	characteristic
99	C₄×C₁₁	#568	1	1	yes	characteristic
100	C₂×M₁₆	C₂×Dic₁₁	1	2	yes	—
101	C₂²×C₈	C₂×Dic₁₁	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
103	C₄⋊Q₈	C₁₁⋊C₈	1	2	yes	—
104	#125	C₁₁⋊C₈	1	2	yes	—
105	C₂²⋊Q₈	C₁₁⋊C₈	1	4	yes	—
106	C₄×Q₈	C₂×Dic₁₁	1	4	yes	—
107	#117	C₂×Dic₁₁	1	2	yes	—
108	C₂×C₄²	C₂×Dic₁₁	1	1	yes	characteristic
109	C₂×C₁₆	—	11	4	no	—
110	C₂×C₁₆	—	11	4	no	—
111	C₄⋊C₈	C₂×Dic₁₁	1	4	yes	—
112	C₄⋊C₈	C₂×Dic₁₁	1	8	yes	—
113	#97	C₂×Dic₁₁	1	4	yes	—
114	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
115	C₄×C₈	C₂²×D₁₁	1	2	yes	—
116	C₄×C₈	C₂²×D₁₁	1	4	yes	—
117	C₂×C₁₁	—	4	1	no	—
118	C₂×C₁₁	#2982	1	1	yes	characteristic
119	C₂×C₁₁	C₂×#504	1	2	yes	—
120	C₂×Q₈	C₂×C₁₁⋊C₈	1	4	yes	—
121	C₂²×C₄	C₂×C₁₁⋊C₈	1	2	yes	—
122	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
123	M₁₆	—	4	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	4	1	no	—
126	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
127	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
128	C₂×C₈	C₂²×Dic₁₁	1	1	yes	characteristic
129	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
130	C₄⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
131	C₄⋊C₄	C₂×C₁₁⋊C₈	1	8	yes	—
132	C₄⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
133	C₂²⋊C₄	C₂×C₁₁⋊C₈	1	4	yes	—
134	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
135	C₄²	C₂×C₁₁⋊C₈	1	1	yes	characteristic
136	C₄²	C₂²×Dic₁₁	1	2	yes	—
137	C₄²	C₂²×Dic₁₁	1	4	yes	—
138	C₁₆	—	22	4	no	—
139	C₁₆	—	22	4	no	—
140	C₁₁	#33500	1	1	yes	characteristic, complemented, Sylow
141	C₂³	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
142	Q₈	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	C₂²×C₁₁⋊C₈	1	1	yes	characteristic
147	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
148	C₂×C₄	C₂²×C₁₁⋊C₈	1	8	yes	—
149	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
150	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	2	4	no	—
154	C₈	#67040	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	8	no	—
160	C₄	—	2	1	no	—
161	C₄	#10622753	1	4	yes	—
162	C₄	#10623343	1	1	yes	characteristic
163	C₄	#10623350	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	#13077697	1	1	yes	characteristic
166	C₂	C₂×#10622753	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₁₁
commutator, socle	C₂²×C₁₁	C₂³×C₈
Fitting	C₁₁×#2960	C₂
prime radical	C₂³×C₁₁	C₂²×C₈
cosocle	C₂×C₈×C₁₁	C₂⁴
hypercenter	#2960	D₁₁
hypocenter	C₁₁	#33500
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
1. #13070525	C₂	semidirect
2. Q₈×C₁₁⋊C₁₆	C₂	semidirect
140. C₁₁	#33500	semidirect

Maximal subgroups

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

subgroup	index
#13070525	2
Q₈×C₁₁⋊C₁₆	2
#13070423	2
#13070415	2
C₁₁×#2960	2
#33500	11