Subgroups — #14545200

Order 2176.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763730	C₂	1	2	yes	maximal
2	C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
3	#11763698	C₂	1	2	yes	maximal
4	C₄×#63216	C₂	1	1	yes	maximal, characteristic
5	#11762933	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₃₄	C₄	1	1	yes	characteristic
7	C₂×#63216	C₂²	1	2	yes	—
8	C₂×#63216	C₂²	1	1	yes	characteristic
9	C₂×#63216	C₂²	1	1	yes	characteristic
10	C₁₇×C₄⋊C₈	C₄	1	1	yes	characteristic
11	#10587485	C₄	1	2	yes	—
12	C₈×Dic₁₇	C₂²	1	2	yes	—
13	C₄×#3622	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₄	C₈	1	1	yes	characteristic
15	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
16	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₄⋊Dic₁₇	C₈	1	1	yes	characteristic
18	#63225	C₈	1	2	yes	—
19	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
20	C₂×#3622	C₂×C₄	1	1	yes	characteristic
21	C₂×#3622	C₂³	1	1	yes	characteristic
22	#63216	—	2	2	no	—
23	#63216	—	2	1	no	—
24	#63216	Q₈	1	2	yes	complemented
25	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
26	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×Dic₁₇	C₂×C₈	1	2	yes	—
29	C₂×Dic₁₇	C₂×C₈	1	2	yes	—
30	Dic₃₄	—	2	2	no	—
31	C₈×C₁₇	—	2	2	no	—
32	#3622	—	2	1	no	—
33	#3622	C₂×Q₈	1	1	yes	characteristic
34	#3622	#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	1	no	—
38	C₄×C₁₇	M₃₂	1	2	yes	—
39	C₄×C₁₇	C₄×Q₈	1	1	yes	characteristic
40	C₄×C₁₇	#131	1	1	yes	characteristic
41	Dic₁₇	—	2	2	no	—
42	Dic₁₇	—	2	2	no	—
43	C₈×Q₈	—	17	1	no	—
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	—	17	1	no	—
52	C₂×C₁₆	—	17	1	no	—
53	C₂×C₁₆	—	17	1	no	—
54	C₂×C₁₆	—	17	2	no	—
55	C₄⋊C₈	—	17	2	no	—
56	C₄⋊C₈	#595	1	1	yes	characteristic
57	C₄×C₈	—	17	2	no	—
58	C₄×C₈	—	17	1	no	—
59	C₁₇	#2183	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₈	—	17	1	no	—
61	C₂×C₈	—	17	1	no	—
62	C₂×C₈	—	17	1	no	—
63	C₂×C₈	C₂×#595	1	2	yes	—
64	C₄⋊C₄	—	17	2	no	—
65	C₄⋊C₄	—	17	1	no	—
66	C₄²	—	17	2	no	—
67	C₄²	C₂×#595	1	1	yes	characteristic
68	C₁₆	—	17	2	no	—
69	C₁₆	—	34	1	no	—
70	C₁₆	—	34	2	no	—
71	Q₈	—	34	2	no	—
72	C₂×C₄	—	17	2	no	—
73	C₂×C₄	—	17	2	no	—
74	C₂×C₄	#63242	1	1	yes	characteristic
75	C₂×C₄	#63242	1	1	yes	characteristic
76	C₂×C₄	C₂²×#595	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₈	—	17	1	no	—
79	C₈	—	17	1	no	—
80	C₈	—	34	1	no	—
81	C₂²	C₂×#63242	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	34	2	no	—
84	C₄	—	34	2	no	—
85	C₄	#10587530	1	2	yes	—
86	C₄	#10587678	1	1	yes	characteristic
87	C₄	Q₈×#595	1	1	yes	characteristic
88	C₂	#11763858	1	1	yes	characteristic
89	C₂	C₂×#10587530	1	1	yes	characteristic
90	C₂	#11763868	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#595
commutator, socle, prime radical	C₂²×C₁₇	C₂²×C₈
Fitting	C₁₇×C₄⋊C₈	C₄
cosocle	C₂×#3622	C₂³
hypercenter	C₄⋊C₈	#595
hypocenter	C₁₇	#2183
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
24. #63216	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
#11763730	2
C₄⋊#63216	2
#11763698	2
C₄×#63216	2
#11762933	2
#2183	17