Subgroups — #14674813

Order 2816.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13070909	C₂	1	1	yes	maximal, characteristic
2	#13066826	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	D₁₁×C₆₄	C₂	1	2	yes	maximal, complemented
5	C₁₁×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#10622514	C₄	1	1	yes	characteristic
7	C₂×C₁₁×C₃₂	C₂²	1	1	yes	characteristic
8	#10622137	C₄	1	1	yes	characteristic
9	#10622085	C₄	1	2	yes	—
10	D₁₁×C₃₂	C₂²	1	2	yes	—
11	C₁₁×C₆₄	C₂²	1	2	yes	—
12	C₁₁⋊C₆₄	C₂²	1	1	yes	characteristic
13	C₁₁⋊C₆₄	C₂²	1	1	yes	characteristic
14	#66954	C₈	1	1	yes	characteristic
15	C₂×C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
16	#66876	C₈	1	1	yes	characteristic
17	#66862	C₈	1	2	yes	—
18	C₁₆×D₁₁	C₂×C₄	1	2	yes	—
19	C₁₁×C₃₂	C₂×C₄	1	1	yes	characteristic
20	C₁₁×C₃₂	C₂³	1	1	yes	characteristic
21	C₁₁⋊C₃₂	C₂×C₄	1	2	yes	—
22	#13737	—	11	1	no	maximal, Sylow
23	#4339	C₁₆	1	1	yes	characteristic
24	C₂×C₈×C₁₁	C₂×C₈	1	1	yes	characteristic
25	#4318	C₁₆	1	1	yes	characteristic
26	#4313	C₁₆	1	2	yes	—
27	C₈×D₁₁	C₂×C₈	1	2	yes	—
28	C₁₁×C₁₆	C₂×C₈	1	1	yes	characteristic
29	C₁₁×C₁₆	C₂²×C₄	1	1	yes	characteristic
30	C₁₁⋊C₁₆	C₂×C₈	1	2	yes	—
31	#2258	—	11	1	no	—
32	M₁₂₈	—	11	2	no	—
33	M₁₂₈	D₁₁	1	1	yes	characteristic, complemented
34	C₂×C₆₄	—	11	1	no	—
35	C₂×C₆₄	—	11	2	no	—
36	C₂×C₄×C₁₁	C₂×C₁₆	1	1	yes	characteristic
37	C₂²⋊D₁₁	C₃₂	1	2	yes	—
38	D₄₄	C₃₂	1	1	yes	characteristic
39	C₄×D₁₁	C₂×C₁₆	1	2	yes	—
40	Dic₂₂	C₃₂	1	1	yes	characteristic
41	C₈×C₁₁	C₂×C₁₆	1	1	yes	characteristic
42	C₈×C₁₁	C₂²×C₈	1	1	yes	characteristic
43	C₁₁⋊C₈	C₂×C₁₆	1	2	yes	—
44	#504	—	11	1	no	—
45	M₆₄	—	11	1	no	—
46	M₆₄	—	11	2	no	—
47	C₂×C₃₂	—	11	2	no	—
48	C₂×C₃₂	C₂×D₁₁	1	1	yes	characteristic
49	C₆₄	—	11	1	no	—
50	C₆₄	—	11	1	no	—
51	C₆₄	C₂×D₁₁	1	2	yes	—
52	C₂²×C₁₁	C₂×C₃₂	1	1	yes	characteristic
53	C₂×D₁₁	C₂×C₃₂	1	2	yes	—
54	C₄×C₁₁	C₂×C₃₂	1	1	yes	characteristic
55	C₄×C₁₁	C₂²×C₁₆	1	1	yes	characteristic
56	Dic₁₁	C₂×C₃₂	1	2	yes	—
57	#131	—	11	1	no	—
58	M₃₂	—	11	1	no	—
59	M₃₂	—	11	2	no	—
60	C₂×C₁₆	—	11	2	no	—
61	C₂×C₁₆	C₄×D₁₁	1	1	yes	characteristic
62	C₃₂	—	11	2	no	—
63	C₃₂	C₄×D₁₁	1	1	yes	characteristic
64	C₃₂	C₂²×D₁₁	1	1	yes	characteristic
65	C₂×C₁₁	—	2	1	no	—
66	C₂×C₁₁	C₂²×C₃₂	1	1	yes	characteristic
67	D₁₁	—	2	2	no	—
68	#41	—	11	1	no	—
69	M₁₆	—	11	1	no	—
70	M₁₆	—	11	2	no	—
71	C₂×C₈	—	11	2	no	—
72	C₂×C₈	C₈×D₁₁	1	1	yes	characteristic
73	C₁₆	—	11	2	no	—
74	C₁₆	C₈×D₁₁	1	1	yes	characteristic
75	C₁₆	C₂×C₄×D₁₁	1	1	yes	characteristic
76	C₁₁	#13737	1	1	yes	characteristic, complemented, Sylow
77	Q₈	—	11	1	no	—
78	D₄	—	11	2	no	—
79	D₄	—	11	1	no	—
80	C₂×C₄	—	11	2	no	—
81	C₂×C₄	C₁₆×D₁₁	1	1	yes	characteristic
82	C₈	—	11	2	no	—
83	C₈	C₁₆×D₁₁	1	1	yes	characteristic
84	C₈	C₂×C₈×D₁₁	1	1	yes	characteristic
85	C₂²	—	11	2	no	—
86	C₂²	D₁₁×C₃₂	1	1	yes	characteristic
87	C₄	—	11	2	no	—
88	C₄	D₁₁×C₃₂	1	1	yes	characteristic
89	C₄	C₂×C₁₆×D₁₁	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	22	2	no	—
92	C₂	C₂×D₁₁×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₁₁
commutator, socle	C₂×C₁₁	C₂²×C₃₂
Fitting	C₁₁×M₁₂₈	C₂
prime radical	#4339	C₁₆
cosocle	C₁₁×C₃₂	C₂³
hypercenter	M₁₂₈	D₁₁
hypocenter	C₁₁	#13737
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #13066826	C₂	semidirect
3. C₂×C₁₁⋊C₆₄	C₂	semidirect
4. D₁₁×C₆₄	C₂	semidirect
5. C₁₁×M₁₂₈	C₂	semidirect
33. M₁₂₈	D₁₁	semidirect
76. C₁₁	#13737	semidirect

Maximal subgroups

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

subgroup	index
#13070909	2
#13066826	2
C₂×C₁₁⋊C₆₄	2
D₁₁×C₆₄	2
C₁₁×M₁₂₈	2
#13737	11