Subgroups — #14543746

Order 2176.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763731	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#3624	C₂	1	1	yes	maximal, characteristic, complemented
3	#11762926	C₂	1	1	yes	maximal, characteristic
4	C₁₇×#135	C₄	1	1	yes	characteristic
5	#10587634	—	2	1	no	—
6	C₄×#3624	—	2	1	no	—
7	#10587538	—	2	1	no	—
8	#10587508	—	2	1	no	—
9	#10587490	C₄	1	1	yes	characteristic
10	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	#41×C₁₇	—	2	1	no	—
12	C₂×#3624	—	2	2	no	—
13	Q₁₆×C₁₇	C₈	1	1	yes	characteristic, complemented
14	SD₁₆×C₁₇	—	2	1	no	—
15	D₈×C₁₇	C₈	1	1	yes	characteristic, complemented
16	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₄×Dic₁₇	D₄	1	1	yes	characteristic
18	#63223	—	2	1	no	—
19	#63223	—	4	1	no	—
20	C₂×#3622	—	4	1	no	—
21	C₂×#3622	D₄	1	1	yes	characteristic
22	#63216	—	4	1	no	—
23	Q₈×C₁₇	—	2	1	no	—
24	D₄×C₁₇	—	2	1	no	—
25	D₄×C₁₇	—	4	1	no	—
26	C₂×C₄×C₁₇	—	4	1	no	—
27	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×Dic₁₇	—	2	1	no	—
29	#3624	—	2	4	no	—
30	C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₇	M₁₆	1	1	yes	characteristic
32	#3622	—	4	1	no	—
33	#3622	D₈	1	1	yes	characteristic
34	#3622	SD₁₆	1	1	yes	characteristic
35	#1335	—	17	1	no	maximal, Sylow
36	C₂²×C₁₇	—	4	1	no	—
37	C₂²×C₁₇	#104	1	1	yes	characteristic
38	C₄×C₁₇	—	4	1	no	—
39	C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₁₇	#102	1	1	yes	characteristic
41	Dic₁₇	—	2	2	no	—
42	#443	—	17	1	no	—
43	#364	—	17	1	no	—
44	C₈²	—	17	1	no	—
45	C₂×C₁₇	—	2	1	no	—
46	C₂×C₁₇	—	8	1	no	—
47	C₂×C₁₇	D₄⋊C₈	1	1	yes	characteristic
48	#135	#595	1	1	yes	characteristic
49	#131	—	34	1	no	—
50	M₃₂	—	34	1	no	—
51	#108	—	17	1	no	—
52	#104	—	34	1	no	—
53	C₄×C₈	—	17	1	no	—
54	C₄×C₈	—	34	1	no	—
55	C₁₇	#1335	1	1	yes	characteristic, complemented, Sylow
56	#41	—	2	1	no	—
57	Q₁₆	#3624	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	#3624	1	1	yes	characteristic, complemented
60	M₁₆	—	34	1	no	—
61	M₁₆	—	68	1	no	—
62	C₂×C₈	—	17	1	no	—
63	C₂×C₈	—	34	2	no	—
64	C₂×C₈	—	68	1	no	—
65	C₂×C₈	C₂×#595	1	1	yes	characteristic
66	C₄²	—	17	1	no	—
67	C₁₆	—	68	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	34	1	no	—
73	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
74	C₈	—	17	1	no	—
75	C₈	—	17	1	no	—
76	C₈	—	34	4	no	—
77	C₈	—	68	1	no	—
78	C₈	C₂×#3624	1	1	yes	characteristic
79	C₈	#63247	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#10587548	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	34	2	no	—
84	C₄	#10587547	1	1	yes	characteristic
85	C₄	C₂²⋊#3624	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊#3624	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10587547
commutator	C₈×C₁₇	C₂×C₈
Frattini	C₂×C₈	C₂×#595
Fitting, prime radical	C₁₇×#135	C₄
socle	C₂×C₁₇	D₄⋊C₈
cosocle	C₈×Dic₁₇	C₂²
hypercenter	#135	#595
hypocenter	C₁₇	#1335
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #11763731	C₂	semidirect
2. C₈×#3624	C₂	semidirect
13. Q₁₆×C₁₇	C₈	semidirect
15. D₈×C₁₇	C₈	semidirect
55. C₁₇	#1335	semidirect
57. Q₁₆	#3624	semidirect
59. D₈	#3624	semidirect

Maximal subgroups

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

subgroup	index
#11763731	2
C₈×#3624	2
#11762926	2
#1335	17