Subgroups — #13066807

Order 1408.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇₆	C₂	1	1	yes	maximal, characteristic
2	C₁₁×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#10622144	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₈	—	2	1	no	—
5	C₂×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
6	#66883	—	2	1	no	—
7	D₁₇₆	—	2	1	no	—
8	D₁₇₆	C₄	1	2	yes	—
9	C₁₁×C₃₂	—	2	1	no	—
10	C₂×D₄₄	—	4	1	no	—
11	C₂×C₈×C₁₁	D₄	1	1	yes	characteristic
12	#4318	—	4	1	no	—
13	D₈₈	—	2	2	no	—
14	D₈₈	—	4	1	no	—
15	C₁₁×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₁₁×C₁₆	D₄	1	1	yes	characteristic
17	#1419	—	11	1	no	maximal, Sylow
18	C₂²×D₁₁	—	8	1	no	—
19	C₂×C₄×C₁₁	D₈	1	1	yes	characteristic
20	D₄₄	—	4	2	no	—
21	D₄₄	—	8	1	no	—
22	C₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₁	SD₁₆	1	1	yes	characteristic
24	C₁₁⋊C₈	—	8	1	no	—
25	C₂×D₁₆	—	11	1	no	—
26	M₆₄	D₁₁	1	1	yes	characteristic, complemented
27	#368	—	11	1	no	—
28	C₂²×C₁₁	SD₃₂	1	1	yes	characteristic
29	C₂×D₁₁	—	8	2	no	—
30	C₂×D₁₁	—	16	2	no	—
31	C₄×C₁₁	#102	1	1	yes	characteristic
32	C₄×C₁₁	D₁₆	1	1	yes	characteristic
33	C₂×D₈	—	22	1	no	—
34	D₁₆	—	11	2	no	—
35	D₁₆	—	22	1	no	—
36	C₂×C₁₆	C₂×D₁₁	1	1	yes	characteristic
37	#108	—	22	1	no	—
38	C₃₂	—	2	1	no	—
39	C₂×C₁₁	—	2	1	no	—
40	C₂×C₁₁	#357	1	1	yes	characteristic
41	D₁₁	—	16	2	no	—
42	C₂×D₄	—	44	1	no	—
43	D₈	—	22	2	no	—
44	D₈	—	44	1	no	—
45	M₁₆	—	44	1	no	—
46	C₂×C₈	D₄₄	1	1	yes	characteristic
47	C₁₆	C₄×D₁₁	1	1	yes	characteristic
48	C₁₆	C₂²⋊D₁₁	1	1	yes	characteristic
49	C₁₁	#1419	1	1	yes	characteristic, complemented, Sylow
50	C₂³	—	88	1	no	—
51	D₄	—	44	2	no	—
52	D₄	—	88	1	no	—
53	C₂×C₄	D₈₈	1	1	yes	characteristic
54	C₈	—	88	1	no	—
55	C₈	#4314	1	1	yes	characteristic
56	C₈	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
57	C₂²	—	88	2	no	—
58	C₂²	—	176	2	no	—
59	C₂²	#66864	1	1	yes	characteristic
60	C₄	D₁₇₆	1	1	yes	characteristic
61	C₄	#66885	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	176	2	no	—
64	C₂	#10622147	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10622147
commutator	C₁₁×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₁
Fitting	C₁₁×M₆₄	C₂
socle	C₂×C₁₁	#357
prime radical	C₂×D₁₇₆	C₂
cosocle	C₂×C₁₁×C₁₆	C₂²
hypercenter	M₆₄	D₁₁
hypocenter	C₁₁	#1419
LCS	G ▷ C₁₁×C₁₆ ▷ C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₁₁×M₆₄	C₂	semidirect
3. #10622144	C₂	semidirect
26. M₆₄	D₁₁	semidirect
49. C₁₁	#1419	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₇₆	2
C₁₁×M₆₄	2
#10622144	2
#1419	11