Subgroups — #30770575

Order 17152.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562491	C₂	1	4	yes	maximal
2	#22559827	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863801	—	2	2	no	—
5	#16863801	C₂²	1	4	yes	—
6	#16863776	—	2	2	no	—
7	#16863224	—	2	8	no	—
8	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
9	C₈²×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₂₆₈	—	4	4	no	—
11	#14524043	—	2	4	no	—
12	#14523977	—	2	8	no	—
13	#14523977	—	4	4	no	—
14	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
15	C₄×C₈×C₆₇	D₄	1	2	yes	—
16	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₁₃₄	—	4	4	no	—
18	Dic₂₆₈	—	8	8	no	—
19	C₄⋊Dic₆₇	—	4	4	no	—
20	C₄⋊Dic₆₇	—	4	4	no	—
21	C₂×C₈×C₆₇	D₈	1	4	yes	—
22	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
23	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
24	C₁₆×C₆₇	Q₁₆	1	8	yes	complemented
25	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
26	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
27	C₂×Dic₆₇	—	8	4	no	—
28	Dic₁₃₄	—	8	8	no	—
29	C₈×C₆₇	SD₃₂	1	8	yes	—
30	C₈×C₆₇	C₂×Q₁₆	1	4	yes	—
31	C₂²×C₆₇	#493	1	1	yes	characteristic
32	C₄×C₆₇	#494	1	2	yes	—
33	C₄×C₆₇	C₂×SD₃₂	1	4	yes	—
34	Dic₆₇	—	16	4	no	—
35	#12523	—	67	1	no	maximal, Sylow
36	C₂×C₆₇	#1713	1	1	yes	characteristic
37	C₂×C₆₇	#2248	1	2	yes	—
38	#2254	—	67	4	no	—
39	#1713	—	67	2	no	—
40	C₈×C₁₆	D₆₇	1	1	yes	characteristic
41	C₆₇	#12523	1	1	yes	characteristic, complemented, Sylow
42	#500	—	67	4	no	—
43	#500	—	134	2	no	—
44	#494	—	134	2	no	—
45	#367	—	134	8	no	—
46	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
47	C₈²	C₂×D₆₇	1	1	yes	characteristic
48	C₂×Q₁₆	—	268	4	no	—
49	C₄⋊Q₈	—	134	4	no	—
50	C₂×C₁₆	D₂₆₈	1	4	yes	—
51	#107	—	268	4	no	—
52	#107	—	134	8	no	—
53	C₄×C₈	D₂₆₈	1	2	yes	—
54	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
55	C₂×Q₈	—	268	4	no	—
56	Q₁₆	—	536	8	no	—
57	C₂×C₈	D₅₃₆	1	4	yes	—
58	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
59	C₄⋊C₄	—	268	4	no	—
60	C₄⋊C₄	—	268	4	no	—
61	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
62	C₁₆	Dic₂₆₈	1	8	yes	complemented
63	Q₈	—	536	8	no	—
64	C₂×C₄	—	536	4	no	—
65	C₂×C₄	#14524040	1	1	yes	characteristic
66	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
67	C₈	#14523988	1	8	yes	—
68	C₈	C₂×Dic₂₆₈	1	4	yes	—
69	C₂²	#16863773	1	1	yes	characteristic
70	C₄	—	1072	4	no	—
71	C₄	#16863776	1	2	yes	—
72	C₄	C₂×#14523988	1	4	yes	—
73	C₂	#22559827	1	1	yes	characteristic
74	C₂	#22562465	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	#12523
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
24. C₁₆×C₆₇	Q₁₆	semidirect
41. C₆₇	#12523	semidirect
62. C₁₆	Dic₂₆₈	semidirect

Maximal subgroups

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

subgroup	index
#22562491	2
#22559827	2
C₈×C₁₆×C₆₇	2
#12523	67