Subgroups — #14526956

Order 2176.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763177	C₂	1	1	yes	maximal, characteristic
2	#11762608	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#356	C₂	1	1	yes	maximal, characteristic
4	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
5	#10587556	C₄	1	1	yes	characteristic
6	C₁₇×#101	—	2	1	no	—
7	#10587507	—	2	1	no	—
8	#10587506	—	2	1	no	—
9	#10587506	—	2	1	no	—
10	#10587474	C₄	1	1	yes	characteristic
11	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
12	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
13	M₁₆×C₁₇	—	4	1	no	—
14	C₄⋊C₄×C₁₇	—	4	1	no	—
15	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₄⋊Dic₁₇	—	2	1	no	—
17	#63225	—	4	1	no	—
18	#63223	—	4	1	no	—
19	C₂×#3622	—	2	1	no	—
20	Q₈×C₁₇	—	4	1	no	—
21	Q₈×C₁₇	—	4	1	no	—
22	C₂×C₄×C₁₇	—	2	1	no	—
23	C₂×C₄×C₁₇	—	2	1	no	—
24	C₂×C₄×C₁₇	—	2	1	no	—
25	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×Dic₁₇	—	4	1	no	—
27	C₈×C₁₇	—	8	1	no	—
28	#3622	—	4	1	no	—
29	#3622	—	8	1	no	—
30	#1413	—	17	1	no	maximal, Sylow
31	C₂²×C₁₇	#99	1	1	yes	characteristic
32	C₄×C₁₇	—	2	1	no	—
33	C₄×C₁₇	—	4	1	no	—
34	C₄×C₁₇	—	4	1	no	—
35	C₄×C₁₇	—	4	1	no	—
36	Dic₁₇	—	8	1	no	—
37	#491	—	17	1	no	—
38	#356	—	17	1	no	—
39	#356	D₁₇	1	1	yes	characteristic
40	C₂×C₁₇	—	2	1	no	—
41	C₂×C₁₇	#354	1	1	yes	characteristic
42	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
43	#125	—	17	1	no	—
44	#103	—	34	1	no	—
45	#103	—	34	1	no	—
46	#101	—	2	1	no	—
47	#101	—	34	1	no	—
48	#97	—	17	1	no	—
49	C₁₇	#1413	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₈	D₆₈	1	1	yes	characteristic
51	C₂×Q₈	C₂²⋊D₁₇	1	1	yes	characteristic
52	M₁₆	—	4	1	no	—
53	M₁₆	—	68	1	no	—
54	C₂×C₈	—	34	1	no	—
55	C₄⋊C₄	—	4	1	no	—
56	C₄⋊C₄	—	34	1	no	—
57	C₄⋊C₄	—	68	1	no	—
58	C₄²	C₄×D₁₇	1	1	yes	characteristic
59	Q₈	—	4	1	no	—
60	Q₈	—	4	1	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	68	1	no	—
65	C₂×C₄	#63227	1	1	yes	characteristic
66	C₈	—	8	1	no	—
67	C₈	—	68	1	no	—
68	C₈	—	136	1	no	—
69	C₂²	#10587477	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	4	1	no	—
72	C₄	—	4	1	no	—
73	C₄	—	4	1	no	—
74	C₄	—	136	1	no	—
75	C₂	—	2	1	no	—
76	C₂	#11762603	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11762603
commutator	C₄²×C₁₇	C₂×C₄
Frattini	C₄⋊Q₈	C₂×D₁₇
Fitting	C₁₇×#356	C₂
socle	C₂×C₁₇	#354
prime radical	C₂²×C₁₇	#99
cosocle	C₁₇×C₄⋊Q₈	C₂²
hypercenter	#356	D₁₇
hypocenter	C₁₇	#1413
LCS	G ▷ C₄²×C₁₇ ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊Q₈ ◁ #356

Complements

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

N	H ≅ G/N	type
49. C₁₇	#1413	semidirect

Maximal subgroups

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

subgroup	index
#11763177	2
#11762608	2
C₁₇×#356	2
#1413	17