Subgroups — #14543706

Order 2176.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#63216	C₂	1	2	yes	maximal, complemented
2	#11762776	C₂	1	1	yes	maximal, characteristic
3	C₂×#41×C₁₇	C₄	1	1	yes	characteristic
4	C₂×#63223	C₄	1	1	yes	characteristic
5	C₂²×#3622	C₂²	1	1	yes	characteristic
6	C₂×#63216	—	2	2	no	—
7	C₂²⋊#3622	—	2	2	no	—
8	#41×C₁₇	—	4	1	no	—
9	#41×C₁₇	—	4	1	no	—
10	C₂×Q₈×C₁₇	C₈	1	1	yes	characteristic
11	C₂×D₄×C₁₇	—	2	1	no	—
12	C₂×D₄×C₁₇	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₁₇	—	2	1	no	—
14	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
15	#63223	—	4	1	no	—
16	C₂×#3622	—	2	1	no	—
17	C₂×#3622	—	4	1	no	—
18	C₂×#3622	D₄	1	2	yes	—
19	#63216	—	4	2	no	—
20	C₂³×C₁₇	—	2	1	no	—
21	C₂³×C₁₇	M₁₆	1	1	yes	characteristic
22	Q₈×C₁₇	—	4	1	no	—
23	D₄×C₁₇	—	4	1	no	—
24	D₄×C₁₇	—	4	2	no	—
25	C₂×C₄×C₁₇	—	2	1	no	—
26	C₂×C₄×C₁₇	—	2	1	no	—
27	C₂×C₄×C₁₇	—	2	1	no	—
28	C₂×C₄×C₁₇	—	4	1	no	—
29	C₂×C₄×C₁₇	—	4	1	no	—
30	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
32	#3622	—	2	2	no	—
33	#3622	—	4	1	no	—
34	#1315	—	17	1	no	maximal, Sylow
35	C₂²×C₁₇	—	2	1	no	—
36	C₂²×C₁₇	—	2	2	no	—
37	C₂²×C₁₇	—	4	1	no	—
38	C₂²×C₁₇	—	4	2	no	—
39	C₂²×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₁₇	—	2	1	no	—
41	C₄×C₁₇	—	4	1	no	—
42	C₄×C₁₇	#99	1	1	yes	characteristic
43	C₄×C₁₇	#100	1	1	yes	characteristic
44	#408	—	17	1	no	—
45	C₂²⋊C₁₆	—	17	2	no	—
46	C₂×C₁₇	—	4	1	no	—
47	C₂×C₁₇	—	8	1	no	—
48	C₂×C₁₇	#323	1	1	yes	characteristic
49	C₂×C₁₇	#350	1	2	yes	—
50	C₂×#41	#595	1	1	yes	characteristic
51	C₂×M₁₆	—	17	1	no	—
52	C₂²×C₈	—	17	1	no	—
53	C₂×C₁₆	—	34	2	no	—
54	C₂²⋊C₈	—	34	2	no	—
55	C₁₇	#1315	1	1	yes	characteristic, complemented, Sylow
56	#41	—	4	1	no	—
57	#41	—	4	1	no	—
58	C₂×Q₈	#3624	1	1	yes	characteristic
59	C₂×D₄	—	2	1	no	—
60	C₂×D₄	#3624	1	1	yes	characteristic
61	C₂²×C₄	—	2	1	no	—
62	C₂²×C₄	C₂×#595	1	1	yes	characteristic
63	M₁₆	—	68	1	no	—
64	C₂×C₈	—	17	2	no	—
65	C₂×C₈	—	34	1	no	—
66	C₂×C₈	—	68	1	no	—
67	C₁₆	—	68	2	no	—
68	C₂³	—	2	1	no	—
69	C₂³	#63247	1	1	yes	characteristic
70	Q₈	—	4	1	no	—
71	D₄	—	4	1	no	—
72	D₄	—	4	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	C₂×#3624	1	1	yes	characteristic
79	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
80	C₈	—	34	2	no	—
81	C₈	—	68	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	4	2	no	—
86	C₂²	C₂²⋊#3624	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	4	1	no	—
89	C₄	#10587551	1	1	yes	characteristic
90	C₄	#10587553	1	1	yes	characteristic
91	C₂	—	4	1	no	—
92	C₂	—	8	1	no	—
93	C₂	#11763652	1	1	yes	characteristic
94	C₂	#11763707	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂×C₈
Frattini	C₂²×C₄	C₂×#595
Fitting, prime radical	C₂×#41×C₁₇	C₄
socle	C₂²×C₁₇	C₂²⋊C₈
cosocle	C₂²×#3622	C₂²
hypercenter	C₂×#41	#595
hypocenter	C₁₇	#1315
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂²⋊#63216	C₂	semidirect
55. C₁₇	#1315	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
#11762776	2
#1315	17