Subgroups — #14537178

Order 2176.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10587560	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10587626	C₂	1	12	yes	maximal, complemented
3	C₄²×Dic₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₁₇	C₂²	1	6	yes	—
6	C₂²×C₂²⋊D₁₇	C₂²	1	16	yes	—
7	C₂²×D₆₈	C₂²	1	6	yes	—
8	C₁₇×#127	—	2	4	no	—
9	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
10	#10587626	—	2	48	no	—
11	C₂×C₄×Dic₁₇	C₂²	1	6	yes	—
12	#10587560	—	2	4	no	—
13	C₂⁴×C₁₇	C₂³	1	4	yes	—
14	C₂³×D₁₇	C₂³	1	4	yes	—
15	C₂×D₄×C₁₇	—	2	12	no	—
16	C₂×D₄×C₁₇	—	2	24	no	—
17	C₂²×C₄×C₁₇	C₂³	1	3	yes	—
18	C₂×C₂²⋊D₁₇	—	2	96	no	—
19	C₂²×Dic₁₇	C₂³	1	4	yes	—
20	C₂×D₆₈	—	2	12	no	—
21	C₂×D₆₈	—	2	24	no	—
22	C₄²×C₁₇	D₄	1	4	yes	complemented
23	C₄×Dic₁₇	D₄	1	24	yes	complemented
24	C₂³×C₁₇	—	2	24	no	—
25	C₂³×C₁₇	—	2	4	no	—
26	C₂³×C₁₇	C₂⁴	1	1	yes	characteristic
27	C₂²×D₁₇	—	2	4	no	—
28	C₂²×D₁₇	—	2	24	no	—
29	D₄×C₁₇	—	4	24	no	—
30	C₂×C₄×C₁₇	C₂×D₄	1	12	yes	—
31	C₂×C₄×C₁₇	C₂×D₄	1	6	yes	—
32	C₂²⋊D₁₇	—	4	64	no	—
33	C₂×Dic₁₇	C₂×D₄	1	24	yes	—
34	D₆₈	—	4	24	no	—
35	#2867	—	17	1	no	maximal, Sylow
36	C₂²×C₁₇	—	4	16	no	—
37	C₂²×C₁₇	—	4	12	no	—
38	C₂²×C₁₇	C₂²×D₄	1	6	yes	—
39	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
40	C₂×D₁₇	—	4	12	no	—
41	C₂×D₁₇	—	4	16	no	—
42	C₄×C₁₇	#127	1	12	yes	complemented
43	Dic₁₇	#127	1	16	yes	complemented
44	C₂×#127	—	17	12	no	—
45	C₂×#127	—	17	1	no	—
46	C₂×#127	D₁₇	1	1	yes	characteristic, complemented
47	C₄³	—	17	1	no	—
48	C₂×C₁₇	—	8	4	no	—
49	C₂×C₁₇	C₂×#127	1	4	yes	—
50	C₂×C₁₇	C₂×#127	1	3	yes	—
51	D₁₇	—	8	4	no	—
52	C₂²×D₄	—	17	16	no	—
53	C₂²×D₄	—	17	6	no	—
54	C₂²×D₄	C₂×D₁₇	1	6	yes	—
55	#127	—	2	4	no	—
56	#127	—	34	48	no	—
57	#127	—	34	4	no	—
58	C₂×C₄²	—	17	6	no	—
59	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
60	C₁₇	#2867	1	1	yes	characteristic, complemented, Sylow
61	C₂⁴	—	17	4	no	—
62	C₂⁴	C₂²×D₁₇	1	4	yes	—
63	C₂×D₄	—	2	24	no	—
64	C₂×D₄	—	2	12	no	—
65	C₂×D₄	—	34	96	no	—
66	C₂×D₄	—	34	24	no	—
67	C₂×D₄	—	34	12	no	—
68	C₂²×C₄	—	17	4	no	—
69	C₂²×C₄	C₂²×D₁₇	1	3	yes	—
70	C₄²	—	17	24	no	—
71	C₄²	C₂²⋊D₁₇	1	4	yes	complemented
72	C₂³	—	2	24	no	—
73	C₂³	—	2	4	no	—
74	C₂³	—	34	24	no	—
75	C₂³	—	34	4	no	—
76	C₂³	C₂³×D₁₇	1	1	yes	characteristic
77	D₄	—	4	24	no	—
78	D₄	—	68	64	no	—
79	D₄	—	68	24	no	—
80	C₂×C₄	—	17	24	no	—
81	C₂×C₄	D₄×D₁₇	1	12	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₇	1	6	yes	—
83	C₂²	—	4	16	no	—
84	C₂²	—	4	12	no	—
85	C₂²	—	68	16	no	—
86	C₂²	—	68	12	no	—
87	C₂²	C₂×D₄×D₁₇	1	6	yes	—
88	C₂²	C₂²×C₂²⋊D₁₇	1	1	yes	characteristic
89	C₄	—	17	16	no	—
90	C₄	#10587626	1	12	yes	complemented
91	C₂	—	8	4	no	—
92	C₂	—	136	4	no	—
93	C₂	#127×D₁₇	1	4	yes	—
94	C₂	C₂×#10587626	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₇
commutator, socle, cosocle	C₂³×C₁₇	C₂⁴
Fitting	C₂×C₁₇×#127	C₂
hypercenter	C₂×#127	D₁₇
hypocenter	C₁₇	#2867
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. C₂×#10587560	C₂	semidirect
2. C₂×#10587626	C₂	semidirect
3. C₄²×Dic₁₇	C₂	semidirect
4. C₂×C₁₇×#127	C₂	semidirect
22. C₄²×C₁₇	D₄	semidirect
23. C₄×Dic₁₇	D₄	semidirect
42. C₄×C₁₇	#127	semidirect
43. Dic₁₇	#127	semidirect
46. C₂×#127	D₁₇	semidirect
60. C₁₇	#2867	semidirect
71. C₄²	C₂²⋊D₁₇	semidirect
90. C₄	#10587626	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10587560	2
C₂×#10587626	2
C₄²×Dic₁₇	2
C₂×C₁₇×#127	2
#2867	17