Subgroups — #14524014

Order 2144.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊Dic₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₆₇	C₂²	1	2	yes	—
7	C₂²×D₆₇	C₂²	1	1	yes	characteristic
8	D₄×C₆₇	—	2	2	no	—
9	C₂×C₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²⋊D₆₇	—	2	4	no	—
11	C₂×Dic₆₇	C₂²	1	1	yes	characteristic
12	C₂×Dic₆₇	C₂²	1	2	yes	—
13	C₄×D₆₇	—	2	2	no	—
14	C₂²×C₆₇	—	2	2	no	—
15	C₂²×C₆₇	—	2	2	no	—
16	C₂²×C₆₇	—	2	2	no	—
17	C₂²×C₆₇	C₂³	1	1	yes	characteristic
18	C₂×D₆₇	—	2	1	no	—
19	C₂×D₆₇	—	2	1	no	—
20	C₂×D₆₇	D₄	1	2	yes	complemented
21	C₄×C₆₇	D₄	1	2	yes	complemented
22	Dic₆₇	—	2	1	no	—
23	Dic₆₇	—	2	2	no	—
24	C₂×C₆₇	—	4	2	no	—
25	C₂×C₆₇	C₂×D₄	1	1	yes	characteristic
26	C₂×C₆₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₆₇	#41	1	1	yes	characteristic
28	D₆₇	—	2	2	no	—
29	C₆₇	#121	1	1	yes	characteristic, complemented, Sylow
30	#121	—	67	1	no	maximal, Sylow
31	C₂×D₄	—	67	2	no	—
32	C₂×D₄	D₆₇	1	1	yes	characteristic, complemented
33	C₂²×C₄	—	67	1	no	—
34	C₄⋊C₄	—	67	1	no	—
35	C₂²⋊C₄	—	67	2	no	—
36	C₂³	—	67	1	no	—
37	C₂³	C₂×D₆₇	1	2	yes	—
38	D₄	—	2	2	no	—
39	D₄	—	134	4	no	—
40	C₂×C₄	—	67	1	no	—
41	C₂×C₄	—	67	2	no	—
42	C₂×C₄	—	134	2	no	—
43	C₂×C₄	C₂×D₆₇	1	1	yes	characteristic
44	C₂²	—	2	2	no	—
45	C₂²	—	2	2	no	—
46	C₂²	—	2	2	no	—
47	C₂²	—	67	2	no	—
48	C₂²	—	134	1	no	—
49	C₂²	—	134	1	no	—
50	C₂²	C₂²×D₆₇	1	1	yes	characteristic
51	C₄	—	134	2	no	—
52	C₄	—	134	1	no	—
53	C₄	C₂²⋊D₆₇	1	2	yes	complemented
54	C₂	—	4	2	no	—
55	C₂	—	134	2	no	—
56	C₂	D₄×D₆₇	1	1	yes	characteristic
57	C₂	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
58	C₂	#11761627	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₇
commutator, socle, cosocle	C₂²×C₆₇	C₂³
Fitting	C₂×D₄×C₆₇	C₂
hypercenter	C₂×D₄	D₆₇
hypocenter	C₆₇	#121
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊D₆₇	C₂	semidirect
2. C₂×C₄×D₆₇	C₂	semidirect
3. C₄⋊Dic₆₇	C₂	semidirect
4. C₂²⋊Dic₆₇	C₂	semidirect
5. C₂×D₄×C₆₇	C₂	semidirect
20. C₂×D₆₇	D₄	semidirect
21. C₄×C₆₇	D₄	semidirect
29. C₆₇	#121	semidirect
32. C₂×D₄	D₆₇	semidirect
53. C₄	C₂²⋊D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₆₇	2
C₂×C₄×D₆₇	2
C₄⋊Dic₆₇	2
C₂²⋊Dic₆₇	2
C₂×D₄×C₆₇	2
#121	67