Subgroups — #30787037

Order 17152.

2286 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863266	C₂	1	14	yes	maximal
2	C₂×C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊Dic₆₇	C₂²	1	28	yes	—
4	#16863266	—	2	56	no	—
5	C₂²×C₄²×C₆₇	C₂²	1	7	yes	—
6	C₄³×C₆₇	C₄	1	8	yes	complemented
7	C₂³×Dic₆₇	C₂³	1	8	yes	—
8	C₂×C₄⋊Dic₆₇	—	2	168	no	—
9	C₂³×C₄×C₆₇	C₂³	1	7	yes	—
10	C₂×C₄²×C₆₇	C₂×C₄	1	28	yes	—
11	C₂×C₄²×C₆₇	D₄	1	28	yes	—
12	C₂×C₄²×C₆₇	Q₈	1	28	yes	—
13	C₂²×Dic₆₇	—	2	56	no	—
14	C₄⋊Dic₆₇	—	4	112	no	—
15	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
16	C₂²×C₄×C₆₇	C₂²×C₄	1	14	yes	—
17	C₂²×C₄×C₆₇	C₂×D₄	1	42	yes	—
18	C₂²×C₄×C₆₇	C₂×Q₈	1	42	yes	—
19	C₄²×C₆₇	C₄⋊C₄	1	112	yes	complemented
20	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
21	C₂³×C₆₇	C₂²×D₄	1	7	yes	—
22	C₂³×C₆₇	C₂²×Q₈	1	7	yes	—
23	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	84	yes	—
24	C₂×C₄×C₆₇	#127	1	28	yes	—
25	C₂×C₄×C₆₇	C₄⋊Q₈	1	84	yes	—
26	C₂×Dic₆₇	—	4	56	no	—
27	C₂²×C₆₇	C₂²×C₄⋊C₄	1	7	yes	—
28	C₂²×C₆₇	C₂×#127	1	7	yes	—
29	C₂²×C₆₇	C₂×C₄⋊Q₈	1	21	yes	—
30	C₄×C₆₇	#384	1	56	yes	complemented
31	Dic₆₇	—	8	8	no	—
32	#14692	—	67	1	no	maximal, Sylow
33	C₂×C₆₇	C₂×#384	1	7	yes	—
34	C₂×C₆₇	#2867	1	1	yes	characteristic
35	C₂×C₆₇	#2868	1	7	yes	—
36	C₂×#384	—	67	14	no	—
37	C₂×C₄³	D₆₇	1	1	yes	characteristic
38	C₆₇	#14692	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₄⋊C₄	—	67	28	no	—
40	C₂²×C₄²	C₂×D₆₇	1	7	yes	—
41	#384	—	134	56	no	—
42	C₄³	Dic₆₇	1	8	yes	complemented
43	C₂³×C₄	—	67	8	no	—
44	C₂³×C₄	C₂²×D₆₇	1	7	yes	—
45	C₂×C₄⋊C₄	—	134	168	no	—
46	C₂×C₄²	Dic₁₃₄	1	28	yes	—
47	C₂×C₄²	D₂₆₈	1	28	yes	—
48	C₂×C₄²	C₂×Dic₆₇	1	28	yes	—
49	C₂⁴	C₂³×D₆₇	1	1	yes	characteristic
50	C₂²×C₄	—	134	56	no	—
51	C₂²×C₄	C₂²×Dic₆₇	1	14	yes	—
52	C₂²×C₄	C₂×D₂₆₈	1	42	yes	—
53	C₂²×C₄	C₂×Dic₁₃₄	1	42	yes	—
54	C₄⋊C₄	—	268	112	no	—
55	C₄²	C₄⋊Dic₆₇	1	112	yes	complemented
56	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
57	C₂³	C₂²×D₂₆₈	1	7	yes	—
58	C₂³	C₂²×Dic₁₃₄	1	7	yes	—
59	C₂×C₄	—	268	56	no	—
60	C₂×C₄	C₂×C₄⋊Dic₆₇	1	84	yes	—
61	C₂×C₄	#14524040	1	28	yes	—
62	C₂×C₄	#14524043	1	84	yes	—
63	C₂²	C₂²×C₄⋊Dic₆₇	1	7	yes	—
64	C₂²	C₂×#14524040	1	7	yes	—
65	C₂²	C₂×#14524043	1	21	yes	—
66	C₄	—	536	8	no	—
67	C₄	#16863266	1	56	yes	complemented
68	C₂	C₂×#16863266	1	7	yes	—
69	C₂	#22568604	1	1	yes	characteristic
70	C₂	#22568607	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₆₇
commutator	C₂³×C₆₇	C₂³×C₄
Fitting	C₂×C₄³×C₆₇	C₂
socle, prime radical, cosocle	C₂⁴×C₆₇	C₂⁴
hypercenter	C₂×C₄³	D₆₇
hypocenter	C₆₇	#14692
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
6. C₄³×C₆₇	C₄	semidirect
19. C₄²×C₆₇	C₄⋊C₄	semidirect
30. C₄×C₆₇	#384	semidirect
38. C₆₇	#14692	semidirect
42. C₄³	Dic₆₇	semidirect
55. C₄²	C₄⋊Dic₆₇	semidirect
67. C₄	#16863266	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863266	2
C₂×C₄³×C₆₇	2
#14692	67