Subgroups — #14537865

Order 2176.

396 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#3622	C₂	1	8	yes	maximal, complemented
2	#11762862	C₂	1	6	yes	maximal
3	C₂×C₄×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₁₇	C₄	1	1	yes	characteristic
5	C₄×Q₈×C₁₇	C₄	1	4	yes	—
6	C₄×Q₈×C₁₇	C₂²	1	4	yes	—
7	C₂×C₄⋊C₄×C₁₇	C₄	1	3	yes	—
8	C₂×C₄²×C₁₇	C₂²	1	3	yes	—
9	C₂²⋊#3622	C₂²	1	4	yes	—
10	C₄⋊#3622	C₂²	1	12	yes	—
11	C₄×#3622	C₂²	1	12	yes	—
12	C₂×Q₈×C₁₇	C₈	1	8	yes	complemented
13	C₂×Q₈×C₁₇	C₂×C₄	1	4	yes	—
14	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
15	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
16	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₁₇	C₂×C₄	1	12	yes	—
18	C₄²×C₁₇	C₂×C₄	1	6	yes	—
19	C₄²×C₁₇	C₂³	1	6	yes	—
20	C₂×#3622	C₂³	1	8	yes	—
21	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
22	Q₈×C₁₇	C₂×C₈	1	16	yes	—
23	C₂×C₄×C₁₇	—	2	2	no	—
24	C₂×C₄×C₁₇	—	4	3	no	—
25	C₂×C₄×C₁₇	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₇	C₂²×C₄	1	6	yes	—
28	C₂×C₄×C₁₇	C₂²×C₄	1	6	yes	—
29	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
30	#3622	—	2	8	no	—
31	#2982	—	17	1	no	maximal, Sylow
32	C₂²×C₁₇	—	2	1	no	—
33	C₂²×C₁₇	—	2	1	no	—
34	C₂²×C₁₇	C₂²×C₈	1	2	yes	—
35	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
36	C₄×C₁₇	—	2	1	no	—
37	C₄×C₁₇	—	2	6	no	—
38	C₄×C₁₇	C₂²×C₈	1	12	yes	—
39	C₄×C₁₇	2₋¹⁺⁴	1	2	yes	—
40	C₂×C₄×Q₈	D₁₇	1	1	yes	characteristic
41	C₈×Q₈	—	17	8	no	—
42	#431	—	17	6	no	—
43	C₂×C₁₇	—	2	2	no	—
44	C₂×C₁₇	#519	1	1	yes	characteristic
45	C₂×C₁₇	C₂³×C₈	1	1	yes	characteristic
46	C₂×C₁₇	#568	1	1	yes	characteristic
47	C₂²×Q₈	Dic₁₇	1	1	yes	characteristic
48	C₄×Q₈	Dic₁₇	1	4	yes	—
49	C₄×Q₈	C₂×D₁₇	1	4	yes	—
50	C₂×C₄⋊C₄	Dic₁₇	1	3	yes	—
51	C₂×C₄²	C₂×D₁₇	1	3	yes	—
52	C₄⋊C₈	—	17	12	no	—
53	C₂²⋊C₈	—	17	4	no	—
54	C₄×C₈	—	17	12	no	—
55	C₁₇	#2982	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₈	#3622	1	8	yes	complemented
57	C₂×Q₈	C₂×Dic₁₇	1	4	yes	—
58	C₂²×C₄	C₂×Dic₁₇	1	3	yes	—
59	C₂²×C₄	C₂×Dic₁₇	1	3	yes	—
60	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
61	C₂×C₈	—	17	8	no	—
62	C₄⋊C₄	C₂×Dic₁₇	1	12	yes	—
63	C₄²	C₂×Dic₁₇	1	6	yes	—
64	C₄²	C₂²×D₁₇	1	6	yes	—
65	C₂³	C₂²×Dic₁₇	1	1	yes	characteristic
66	Q₈	C₂×#3622	1	16	yes	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	4	3	no	—
69	C₂×C₄	C₂×#3622	1	12	yes	—
70	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
71	C₂×C₄	C₂²×Dic₁₇	1	6	yes	—
72	C₂×C₄	C₂²×Dic₁₇	1	6	yes	—
73	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
74	C₈	—	34	8	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	C₂²×#3622	1	2	yes	—
78	C₂²	C₂³×Dic₁₇	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	2	6	no	—
81	C₄	C₂²×#3622	1	12	yes	—
82	C₄	#10587694	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#11763287	1	1	yes	characteristic
85	C₂	C₂³×#3622	1	1	yes	characteristic
86	C₂	#11763576	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator	C₂×C₁₇	C₂³×C₈
Fitting	C₂×C₄×Q₈×C₁₇	C₂
socle	C₂²×C₁₇	C₂³×C₄
prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₇
hypocenter	C₁₇	#2982
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×#3622	C₂	semidirect
12. C₂×Q₈×C₁₇	C₈	semidirect
55. C₁₇	#2982	semidirect
56. C₂×Q₈	#3622	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#3622	2
#11762862	2
C₂×C₄×Q₈×C₁₇	2
#2982	17