Subgroups — #14526749

Order 2176.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₈	C₂	1	1	yes	maximal, characteristic
2	#11762562	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₈	C₄	1	1	yes	characteristic
5	C₄×Dic₃₄	—	2	1	no	—
6	C₂×C₁₆×C₁₇	—	2	1	no	—
7	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
8	#10587489	C₄	1	1	yes	characteristic
9	#10587487	—	2	1	no	—
10	C₄⋊#3622	—	2	1	no	—
11	C₂×Dic₃₄	—	2	1	no	—
12	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
14	C₄²×C₁₇	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₇	—	2	1	no	—
16	#63225	—	4	1	no	—
17	C₄×Dic₁₇	—	4	1	no	—
18	C₂×#3622	—	4	1	no	—
19	Dic₆₈	—	2	1	no	—
20	Dic₆₈	C₈	1	2	yes	complemented
21	C₁₆×C₁₇	—	2	2	no	—
22	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
24	C₂×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
25	C₂×Dic₁₇	—	4	1	no	—
26	C₂×Dic₁₇	—	4	1	no	—
27	Dic₃₄	—	2	2	no	—
28	Dic₃₄	—	4	1	no	—
29	C₈×C₁₇	—	2	1	no	—
30	C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
31	C₈×C₁₇	M₁₆	1	1	yes	characteristic
32	#3622	—	8	1	no	—
33	#1332	—	17	1	no	maximal, Sylow
34	C₂²×C₁₇	#102	1	1	yes	characteristic
35	C₄×C₁₇	—	2	1	no	—
36	C₄×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₁₇	#104	1	1	yes	characteristic
38	C₄×C₁₇	SD₃₂	1	1	yes	characteristic
39	C₄×C₁₇	Q₃₂	1	1	yes	characteristic
40	Dic₁₇	—	4	2	no	—
41	Dic₁₇	—	8	1	no	—
42	C₄×Q₁₆	—	17	1	no	—
43	C₄×C₁₆	D₁₇	1	1	yes	characteristic
44	#335	—	17	1	no	—
45	C₂×C₁₇	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₁₇	#358	1	1	yes	characteristic
47	C₂×C₁₇	#359	1	1	yes	characteristic
48	C₂×Q₁₆	—	17	1	no	—
49	C₄×Q₈	—	34	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	#107	—	17	1	no	—
52	C₄⋊C₈	—	34	1	no	—
53	#103	—	34	1	no	—
54	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
55	C₁₇	#1332	1	1	yes	characteristic, complemented, Sylow
56	C₂×Q₈	—	34	1	no	—
57	Q₁₆	—	17	2	no	—
58	Q₁₆	—	34	1	no	—
59	C₂×C₈	—	68	1	no	—
60	C₂×C₈	C₄×D₁₇	1	1	yes	characteristic
61	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
62	C₄⋊C₄	—	34	1	no	—
63	C₄⋊C₄	—	68	1	no	—
64	C₄²	—	68	1	no	—
65	C₄²	D₆₈	1	1	yes	characteristic
66	C₁₆	—	2	2	no	—
67	Q₈	—	34	2	no	—
68	Q₈	—	68	1	no	—
69	C₂×C₄	—	68	1	no	—
70	C₂×C₄	—	68	1	no	—
71	C₂×C₄	#63219	1	1	yes	characteristic
72	C₂×C₄	D₁₃₆	1	1	yes	characteristic
73	C₂×C₄	#63227	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	136	1	no	—
76	C₈	C₈×D₁₇	1	1	yes	characteristic
77	C₈	#63218	1	1	yes	characteristic
78	C₂²	#10587492	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	68	2	no	—
81	C₄	—	136	1	no	—
82	C₄	#10587471	1	1	yes	characteristic
83	C₄	Dic₁₃₆	1	1	yes	characteristic
84	C₄	#10587476	1	1	yes	characteristic
85	C₄	#10587491	1	1	yes	characteristic
86	C₂	#11762538	1	1	yes	characteristic
87	C₂	#11762614	1	1	yes	characteristic
88	C₂	#11762617	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₃₆
commutator	C₈×C₁₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₇
Fitting	C₄×C₁₆×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	#102
cosocle	C₄×C₈×C₁₇	C₂²
hypercenter	C₄×C₁₆	D₁₇
hypocenter	C₁₇	#1332
LCS	G ▷ C₈×C₁₇ ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
20. Dic₆₈	C₈	semidirect
55. C₁₇	#1332	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆₈	2
#11762562	2
C₄×C₁₆×C₁₇	2
#1332	17