Subgroups — #14530693

Order 2176.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#63214	C₂	1	2	yes	maximal
2	C₄⋊#63214	C₂	1	1	yes	maximal, characteristic
3	#11762585	C₂	1	2	yes	maximal
4	C₄×#63214	C₂	1	1	yes	maximal, characteristic
5	C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
7	C₁₇×C₄⋊C₈	C₄	1	2	yes	—
8	C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
9	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₇	C₂²	1	2	yes	—
11	C₂×#63214	C₂²	1	1	yes	characteristic
12	C₂×#63214	C₂²	1	2	yes	—
13	C₂×#63214	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₁₇	C₈	1	1	yes	characteristic
15	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₁₇	C₈	1	2	yes	—
19	C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
20	C₄²×C₁₇	C₂×C₄	1	2	yes	—
21	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	#63214	—	2	1	no	—
23	#63214	—	2	2	no	—
24	#63214	Q₈	1	2	yes	complemented
25	Q₈×C₁₇	—	2	2	no	—
26	C₂×C₄×C₁₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₁₇	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₁₇	—	2	1	no	—
32	C₈×C₁₇	—	2	2	no	—
33	C₈×C₁₇	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₁₇	#41	1	1	yes	characteristic
35	#2183	—	17	1	no	maximal, Sylow
36	C₂²×C₁₇	C₂²×C₈	1	1	yes	characteristic
37	C₄×C₁₇	—	2	2	no	—
38	C₄×C₁₇	—	2	2	no	—
39	C₄×C₁₇	—	2	1	no	—
40	C₄×C₁₇	M₃₂	1	2	yes	—
41	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
42	C₄×C₁₇	#131	1	1	yes	characteristic
43	C₈×Q₈	D₁₇	1	1	yes	characteristic
44	C₄⋊C₁₆	—	17	1	no	—
45	C₄⋊C₁₆	—	17	2	no	—
46	#346	—	17	2	no	—
47	C₄×C₁₆	—	17	1	no	—
48	C₂×C₁₇	C₈×Q₈	1	1	yes	characteristic
49	C₂×C₁₇	C₂×M₃₂	1	1	yes	characteristic
50	C₂×C₁₇	#504	1	1	yes	characteristic
51	C₄×Q₈	Dic₁₇	1	1	yes	characteristic
52	C₂×C₁₆	—	17	1	no	—
53	C₂×C₁₆	—	17	1	no	—
54	C₂×C₁₆	—	17	2	no	—
55	C₄⋊C₈	Dic₁₇	1	2	yes	—
56	C₄⋊C₈	Dic₁₇	1	1	yes	characteristic
57	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
58	C₄×C₈	C₂×D₁₇	1	2	yes	—
59	C₁₇	#2183	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₈	#3622	1	1	yes	characteristic
61	C₂×C₈	C₂×Dic₁₇	1	1	yes	characteristic
62	C₂×C₈	C₂×Dic₁₇	1	2	yes	—
63	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
64	C₄⋊C₄	#3622	1	2	yes	—
65	C₄⋊C₄	#3622	1	1	yes	characteristic
66	C₄²	C₂×Dic₁₇	1	2	yes	—
67	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
68	C₁₆	—	17	2	no	—
69	C₁₆	—	34	1	no	—
70	C₁₆	—	34	2	no	—
71	Q₈	—	2	2	no	—
72	C₂×C₄	C₂×#3622	1	2	yes	—
73	C₂×C₄	C₂×#3622	1	1	yes	characteristic
74	C₂×C₄	C₂×#3622	1	2	yes	—
75	C₂×C₄	C₂×#3622	1	1	yes	characteristic
76	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	2	no	—
79	C₈	Q₈×D₁₇	1	1	yes	characteristic
80	C₈	#63256	1	1	yes	characteristic
81	C₂²	C₂²×#3622	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	#10587483	1	2	yes	—
86	C₄	Q₈×Dic₁₇	1	1	yes	characteristic
87	C₄	#10587634	1	1	yes	characteristic
88	C₂	Q₈×#3622	1	1	yes	characteristic
89	C₂	C₂×#10587483	1	1	yes	characteristic
90	C₂	#11763226	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₇
commutator, socle, prime radical	C₂²×C₁₇	C₂²×C₈
Fitting	C₈×Q₈×C₁₇	C₂
cosocle	C₂×C₈×C₁₇	C₂³
hypercenter	C₈×Q₈	D₁₇
hypocenter	C₁₇	#2183
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
24. #63214	Q₈	semidirect
59. C₁₇	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#63214	2
#11762585	2
C₄×#63214	2
C₈×Q₈×C₁₇	2
#2183	17