Subgroups — #14543720

Order 2176.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#63216	C₂	1	2	yes	maximal
2	C₂×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₁₇	C₄	1	1	yes	characteristic
4	C₂²×#3622	C₄	1	1	yes	characteristic
5	C₂²×#3622	C₂²	1	1	yes	characteristic
6	C₂×#63216	—	2	2	no	—
7	C₄⋊#3622	—	2	2	no	—
8	C₂²×C₄×C₁₇	C₈	1	1	yes	characteristic
9	C₂²×C₄×C₁₇	C₈	1	1	yes	characteristic
10	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
11	C₄²×C₁₇	—	2	1	no	—
12	C₄²×C₁₇	—	2	1	no	—
13	C₂×#3622	—	2	1	no	—
14	C₂×#3622	—	4	1	no	—
15	C₂×#3622	—	4	1	no	—
16	C₂×#3622	D₄	1	2	yes	—
17	#63216	—	4	2	no	—
18	C₂³×C₁₇	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄×C₁₇	—	2	1	no	—
20	C₂×C₄×C₁₇	—	2	1	no	—
21	C₂×C₄×C₁₇	—	2	1	no	—
22	C₂×C₄×C₁₇	—	2	1	no	—
23	C₂×C₄×C₁₇	—	2	1	no	—
24	C₂×C₄×C₁₇	—	4	1	no	—
25	C₂×C₄×C₁₇	C₁₆	1	2	yes	complemented
26	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₇	M₁₆	1	1	yes	characteristic
28	#3622	—	2	2	no	—
29	#3622	—	4	1	no	—
30	#1322	—	17	1	no	maximal, Sylow
31	C₂²×C₁₇	—	2	1	no	—
32	C₂²×C₁₇	—	2	1	no	—
33	C₂²×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₂²×C₁₇	C₂×C₁₆	1	1	yes	characteristic
35	C₂²×C₁₇	M₃₂	1	1	yes	characteristic
36	C₄×C₁₇	—	2	1	no	—
37	C₄×C₁₇	—	2	2	no	—
38	C₄×C₁₇	—	4	1	no	—
39	C₄×C₁₇	#99	1	1	yes	characteristic
40	C₄×C₁₇	#101	1	1	yes	characteristic
41	C₂×C₄⋊C₈	—	17	1	no	—
42	C₂²⋊C₁₆	—	17	2	no	—
43	C₂×C₁₇	—	2	1	no	—
44	C₂×C₁₇	—	2	1	no	—
45	C₂×C₁₇	#324	1	1	yes	characteristic
46	C₂×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
47	C₂×C₁₇	#349	1	1	yes	characteristic
48	C₂²×C₈	—	17	1	no	—
49	C₂²×C₈	—	17	1	no	—
50	C₂×C₄²	#595	1	1	yes	characteristic
51	C₂×C₁₆	—	34	2	no	—
52	C₄⋊C₈	—	34	2	no	—
53	C₁₇	#1322	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₄	#3624	1	1	yes	characteristic
55	C₂²×C₄	#3624	1	1	yes	characteristic
56	C₂²×C₄	C₂×#595	1	1	yes	characteristic
57	C₂×C₈	—	17	2	no	—
58	C₂×C₈	—	34	1	no	—
59	C₂×C₈	—	68	1	no	—
60	C₂×C₈	—	68	1	no	—
61	C₄²	—	2	1	no	—
62	C₄²	—	2	1	no	—
63	C₁₆	—	68	2	no	—
64	C₂³	C₂×#3624	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	#63216	1	2	yes	complemented
72	C₂×C₄	#63247	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
74	C₈	—	34	2	no	—
75	C₈	—	68	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	C₂×#63216	1	1	yes	characteristic
79	C₂²	#10587538	1	1	yes	characteristic
80	C₂²	C₂²⋊#3624	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	#10587539	1	1	yes	characteristic
85	C₄	#10587545	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#11763654	1	1	yes	characteristic
89	C₂	C₂²⋊#63216	1	1	yes	characteristic
90	C₂	#11763705	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#595
commutator	C₂²×C₁₇	C₂×C₁₆
Frattini	C₂²×C₄	C₂×#595
Fitting	C₂×C₄²×C₁₇	C₄
socle	C₂²×C₁₇	C₂²⋊C₈
prime radical	C₂³×C₁₇	C₂×C₈
cosocle	C₂²×#3622	C₂²
hypercenter	C₂×C₄²	#595
hypocenter	C₁₇	#1322
LCS	G ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
25. C₂×C₄×C₁₇	C₁₆	semidirect
53. C₁₇	#1322	semidirect
71. C₂×C₄	#63216	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#63216	2
C₂×C₄⋊#3622	2
#1322	17