Subgroups — #13070936

Order 1408.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#10622517	C₂	1	1	yes	maximal, characteristic
3	C₁₁×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₁₇₆	C₂	1	2	yes	maximal, complemented
5	#10622087	C₂	1	2	yes	maximal, complemented
6	C₂×Dic₄₄	—	2	1	no	—
7	#66957	—	2	1	no	—
8	C₂×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
9	Dic₈₈	—	2	1	no	—
10	Dic₈₈	C₂²	1	1	yes	characteristic
11	Dic₈₈	C₂²	1	2	yes	—
12	#66864	—	2	1	no	—
13	D₁₇₆	C₂²	1	1	yes	characteristic
14	C₁₁×C₃₂	C₂²	1	2	yes	—
15	#4339	—	4	1	no	—
16	C₂×Dic₂₂	—	4	1	no	—
17	C₂×C₈×C₁₁	D₄	1	1	yes	characteristic
18	Dic₄₄	—	2	1	no	—
19	Dic₄₄	—	2	2	no	—
20	Dic₄₄	—	4	1	no	—
21	D₈₈	—	2	1	no	—
22	#4314	—	4	1	no	—
23	C₁₁×C₁₆	D₄	1	1	yes	characteristic
24	C₁₁×C₁₆	C₂³	1	1	yes	characteristic
25	#2264	—	11	1	no	maximal, Sylow
26	C₂×C₄×C₁₁	D₈	1	1	yes	characteristic
27	C₂²⋊D₁₁	—	8	1	no	—
28	C₂×Dic₁₁	—	8	1	no	—
29	D₄₄	—	4	1	no	—
30	C₄×D₁₁	—	8	1	no	—
31	Dic₂₂	—	4	1	no	—
32	Dic₂₂	—	4	2	no	—
33	Dic₂₂	—	8	1	no	—
34	C₈×C₁₁	D₈	1	1	yes	characteristic
35	C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
36	#508	—	11	1	no	—
37	C₂×Q₃₂	—	11	1	no	—
38	Q₆₄	—	11	2	no	—
39	SD₆₄	—	11	2	no	—
40	M₆₄	D₁₁	1	1	yes	characteristic, complemented
41	C₂²×C₁₁	D₁₆	1	1	yes	characteristic
42	C₂×D₁₁	—	8	1	no	—
43	C₄×C₁₁	D₁₆	1	1	yes	characteristic
44	C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
45	Dic₁₁	—	8	2	no	—
46	Dic₁₁	—	8	1	no	—
47	#135	—	22	1	no	—
48	C₂×Q₁₆	—	22	1	no	—
49	Q₃₂	—	11	2	no	—
50	Q₃₂	—	11	1	no	—
51	Q₃₂	—	22	1	no	—
52	SD₃₂	—	22	1	no	—
53	D₁₆	—	11	1	no	—
54	C₂×C₁₆	C₂×D₁₁	1	1	yes	characteristic
55	C₃₂	C₂×D₁₁	1	2	yes	—
56	C₂×C₁₁	—	2	1	no	—
57	C₂×C₁₁	C₂×D₁₆	1	1	yes	characteristic
58	D₁₁	—	16	1	no	—
59	#41	—	44	1	no	—
60	C₂×Q₈	—	44	1	no	—
61	Q₁₆	—	22	1	no	—
62	Q₁₆	—	22	2	no	—
63	Q₁₆	—	44	1	no	—
64	SD₁₆	—	44	1	no	—
65	D₈	—	22	1	no	—
66	C₂×C₈	D₄₄	1	1	yes	characteristic
67	C₁₆	D₄₄	1	1	yes	characteristic
68	C₁₆	C₂²×D₁₁	1	1	yes	characteristic
69	C₁₁	#2264	1	1	yes	characteristic, complemented, Sylow
70	Q₈	—	44	1	no	—
71	Q₈	—	44	2	no	—
72	Q₈	—	88	1	no	—
73	D₄	—	44	1	no	—
74	D₄	—	88	1	no	—
75	C₂×C₄	—	88	1	no	—
76	C₂×C₄	—	88	1	no	—
77	C₂×C₄	D₈₈	1	1	yes	characteristic
78	C₈	D₈₈	1	1	yes	characteristic
79	C₈	C₂×D₄₄	1	1	yes	characteristic
80	C₂²	—	88	1	no	—
81	C₂²	D₁₇₆	1	1	yes	characteristic
82	C₄	—	88	1	no	—
83	C₄	—	88	2	no	—
84	C₄	D₁₇₆	1	1	yes	characteristic
85	C₄	C₂×D₈₈	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	176	1	no	—
88	C₂	C₂×D₁₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₇₆
commutator, cosocle	C₁₁×C₁₆	C₂³
Frattini	C₁₆	C₂²×D₁₁
Fitting	C₁₁×M₆₄	C₂
socle	C₂×C₁₁	C₂×D₁₆
prime radical	#10622517	C₂
hypercenter	M₆₄	D₁₁
hypocenter	C₁₁	#2264
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
1. C₂×Dic₈₈	C₂	semidirect
3. C₁₁×M₆₄	C₂	semidirect
4. Dic₁₇₆	C₂	semidirect
5. #10622087	C₂	semidirect
40. M₆₄	D₁₁	semidirect
69. C₁₁	#2264	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₈	2
#10622517	2
C₁₁×M₆₄	2
Dic₁₇₆	2
#10622087	2
#2264	11