Subgroups — #14530642

Order 2176.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₃₆	C₂	1	1	yes	maximal, characteristic
2	#11762903	C₂	1	2	yes	maximal
3	#11762612	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₇	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#366	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₈	C₂²	1	2	yes	—
7	#10587573	—	2	2	no	—
8	C₂×C₁₆×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#107	C₂²	1	2	yes	—
10	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
11	C₂×#63214	C₂²	1	1	yes	characteristic
12	#10587481	—	2	2	no	—
13	Dic₁₃₆	C₄	1	4	yes	complemented
14	C₂×Dic₃₄	—	2	2	no	—
15	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
16	C₄⋊C₄×C₁₇	—	2	2	no	—
17	#63225	—	4	2	no	—
18	C₄×Dic₁₇	—	4	2	no	—
19	C₄×Dic₁₇	D₄	1	1	yes	characteristic
20	C₂×#3622	D₄	1	1	yes	characteristic
21	Dic₆₈	—	2	2	no	—
22	Dic₆₈	C₂×C₄	1	4	yes	—
23	C₁₆×C₁₇	C₂×C₄	1	2	yes	—
24	#63214	—	2	1	no	—
25	C₂×C₄×C₁₇	—	4	2	no	—
26	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₁₇	—	4	2	no	—
28	C₂×Dic₁₇	D₈	1	1	yes	characteristic
29	C₂×Dic₁₇	D₈	1	1	yes	characteristic
30	Dic₃₄	—	2	4	no	—
31	Dic₃₄	—	4	2	no	—
32	C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₇	#41	1	1	yes	characteristic
34	#3622	—	2	1	no	—
35	C₄×Q₃₂	—	17	1	no	maximal, Sylow
36	C₂²×C₁₇	C₂×D₈	1	1	yes	characteristic
37	C₄×C₁₇	—	8	2	no	—
38	C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
39	C₄×C₁₇	#135	1	1	yes	characteristic
40	Dic₁₇	—	2	1	no	—
41	Dic₁₇	—	4	4	no	—
42	Dic₁₇	Q₃₂	1	2	yes	complemented
43	C₂×Q₃₂	—	17	1	no	—
44	C₄×Q₁₆	—	17	2	no	—
45	#366	D₁₇	1	1	yes	characteristic
46	#358	—	17	2	no	—
47	C₄×C₁₆	—	17	1	no	—
48	C₂×C₁₇	C₄×D₈	1	1	yes	characteristic
49	C₂×C₁₇	C₂×Q₃₂	1	1	yes	characteristic
50	C₂×C₁₇	#508	1	1	yes	characteristic
51	C₂×Q₁₆	—	17	2	no	—
52	C₄×Q₈	—	34	2	no	—
53	Q₃₂	—	17	4	no	—
54	C₂×C₁₆	—	17	1	no	—
55	C₂×C₁₆	C₂×D₁₇	1	1	yes	characteristic
56	#107	C₂×D₁₇	1	2	yes	—
57	#103	—	34	2	no	—
58	C₄×C₈	—	17	1	no	—
59	C₁₇	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
60	C₂×Q₈	—	34	2	no	—
61	Q₁₆	—	17	4	no	—
62	Q₁₆	—	34	2	no	—
63	C₂×C₈	—	17	1	no	—
64	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	68	2	no	—
67	C₄²	—	17	1	no	—
68	C₄²	—	68	2	no	—
69	C₁₆	—	34	1	no	—
70	C₁₆	C₄×D₁₇	1	2	yes	—
71	Q₈	—	34	4	no	—
72	Q₈	—	68	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	17	1	no	—
75	C₂×C₄	—	17	1	no	—
76	C₂×C₄	—	68	2	no	—
77	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
78	C₈	—	34	1	no	—
79	C₈	C₂×C₄×D₁₇	1	1	yes	characteristic
80	C₈	#63256	1	1	yes	characteristic
81	C₂²	D₈×D₁₇	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	17	2	no	—
84	C₄	—	34	1	no	—
85	C₄	—	68	4	no	—
86	C₄	#10587579	1	1	yes	characteristic
87	C₄	#10587605	1	1	yes	characteristic
88	C₂	#11762891	1	1	yes	characteristic
89	C₂	Q₃₂×D₁₇	1	1	yes	characteristic
90	C₂	#11763238	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₇
commutator	C₈×C₁₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇
Fitting	C₁₇×#366	C₂
socle, prime radical	C₂²×C₁₇	C₂×D₈
cosocle	C₂×C₈×C₁₇	C₂³
hypercenter	#366	D₁₇
hypocenter	C₁₇	C₄×Q₃₂
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
13. Dic₁₃₆	C₄	semidirect
42. Dic₁₇	Q₃₂	semidirect
59. C₁₇	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₃₆	2
#11762903	2
#11762612	2
C₁₆×Dic₁₇	2
C₁₇×#366	2
C₄×Q₃₂	17