Subgroups — #30770546

Order 17152.

310 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562462	C₂	1	4	yes	maximal
2	#22559827	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₃₆	—	2	8	no	—
5	#16863801	—	2	4	no	—
6	#16863776	C₂²	1	4	yes	—
7	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
8	C₈²×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₆₈	—	2	8	no	—
10	#14524043	—	2	4	no	—
11	Dic₅₃₆	—	4	16	no	—
12	#14523977	—	4	8	no	—
13	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
14	C₄×C₈×C₆₇	D₄	1	2	yes	—
15	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
16	C₂×Dic₁₃₄	—	4	4	no	—
17	Dic₂₆₈	—	4	16	no	—
18	C₄⋊Dic₆₇	—	4	8	no	—
19	C₂×C₈×C₆₇	D₈	1	4	yes	—
20	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
21	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
22	C₁₆×C₆₇	D₈	1	8	yes	—
23	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
24	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
25	C₂×Dic₆₇	—	8	4	no	—
26	Dic₁₃₄	—	8	8	no	—
27	C₈×C₆₇	Q₃₂	1	8	yes	complemented
28	C₈×C₆₇	C₂×D₈	1	4	yes	—
29	C₂²×C₆₇	#493	1	1	yes	characteristic
30	C₄×C₆₇	#493	1	2	yes	—
31	C₄×C₆₇	C₂×Q₃₂	1	4	yes	—
32	Dic₆₇	—	16	4	no	—
33	#12516	—	67	1	no	maximal, Sylow
34	C₂×C₆₇	#1712	1	1	yes	characteristic
35	C₂×C₆₇	#2247	1	2	yes	—
36	#2247	—	67	4	no	—
37	#1713	—	67	2	no	—
38	C₈×C₁₆	D₆₇	1	1	yes	characteristic
39	C₆₇	#12516	1	1	yes	characteristic, complemented, Sylow
40	C₂×Q₃₂	—	134	8	no	—
41	#500	—	134	4	no	—
42	#494	—	67	4	no	—
43	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
44	C₈²	C₂×D₆₇	1	1	yes	characteristic
45	C₂×Q₁₆	—	134	8	no	—
46	C₄⋊Q₈	—	134	4	no	—
47	Q₃₂	—	268	16	no	—
48	C₂×C₁₆	D₂₆₈	1	4	yes	—
49	#107	—	268	8	no	—
50	C₄×C₈	D₂₆₈	1	2	yes	—
51	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
52	C₂×Q₈	—	268	4	no	—
53	Q₁₆	—	268	16	no	—
54	C₂×C₈	D₅₃₆	1	4	yes	—
55	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
56	C₄⋊C₄	—	268	8	no	—
57	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
58	C₁₆	D₅₃₆	1	8	yes	—
59	Q₈	—	536	8	no	—
60	C₂×C₄	—	536	4	no	—
61	C₂×C₄	#14524040	1	1	yes	characteristic
62	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
63	C₈	Dic₅₃₆	1	8	yes	complemented
64	C₈	C₂×D₅₃₆	1	4	yes	—
65	C₂²	#16863773	1	1	yes	characteristic
66	C₄	—	1072	4	no	—
67	C₄	#16863773	1	2	yes	—
68	C₄	C₂×Dic₅₃₆	1	4	yes	—
69	C₂	#22559824	1	1	yes	characteristic
70	C₂	#22562462	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₈×C₁₆×C₆₇	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	C₈×C₁₆	D₆₇
hypocenter	C₆₇	#12516
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
27. C₈×C₆₇	Q₃₂	semidirect
39. C₆₇	#12516	semidirect
63. C₈	Dic₅₃₆	semidirect

Maximal subgroups

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

subgroup	index
#22562462	2
#22559827	2
C₈×C₁₆×C₆₇	2
#12516	67