Subgroups — #14524005

Order 2144.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₆₇	C₂	1	2	yes	maximal, complemented
3	(C₂×D₆₇)⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₄×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₆₇	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×Dic₆₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₇	C₂²	1	1	yes	characteristic
11	D₂₆₈	C₄	1	4	yes	complemented
12	C₄×D₆₇	—	2	4	no	—
13	C₂²×C₆₇	C₂³	1	1	yes	characteristic
14	C₂×D₆₇	—	2	2	no	—
15	C₂×D₆₇	—	2	2	no	—
16	C₂×D₆₇	C₂×C₄	1	4	yes	complemented
17	C₄×C₆₇	—	2	2	no	—
18	C₄×C₆₇	C₂×C₄	1	2	yes	complemented
19	Dic₆₇	—	2	1	no	—
20	Dic₆₇	D₄	1	2	yes	complemented
21	C₂×C₆₇	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₆₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₆₇	#41	1	1	yes	characteristic
24	D₆₇	—	2	4	no	—
25	C₆₇	C₄×D₄	1	1	yes	characteristic, complemented, Sylow
26	C₄×D₄	—	67	1	no	maximal, Sylow
27	C₂×D₄	—	67	1	no	—
28	C₂²×C₄	—	67	2	no	—
29	C₄⋊C₄	D₆₇	1	1	yes	characteristic, complemented
30	C₂²⋊C₄	—	67	2	no	—
31	C₄²	—	67	1	no	—
32	C₂³	—	67	2	no	—
33	D₄	—	67	4	no	—
34	C₂×C₄	—	67	1	no	—
35	C₂×C₄	—	67	1	no	—
36	C₂×C₄	—	134	4	no	—
37	C₂×C₄	C₂×D₆₇	1	2	yes	—
38	C₂×C₄	C₂×D₆₇	1	1	yes	characteristic
39	C₂²	—	67	4	no	—
40	C₂²	—	134	2	no	—
41	C₂²	—	134	2	no	—
42	C₂²	C₂²×D₆₇	1	1	yes	characteristic
43	C₄	—	2	2	no	—
44	C₄	—	67	2	no	—
45	C₄	—	134	1	no	—
46	C₄	C₄×D₆₇	1	2	yes	complemented
47	C₂	—	134	4	no	—
48	C₂	C₂×C₄×D₆₇	1	1	yes	characteristic
49	C₂	D₄×D₆₇	1	1	yes	characteristic
50	C₂	#11761629	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₆₇
commutator	C₂×C₆₇	C₂²×C₄
Fitting	C₄⋊C₄×C₆₇	C₂
socle, cosocle	C₂²×C₆₇	C₂³
prime radical	C₂×D₂₆₈	C₂
hypercenter	C₄⋊C₄	D₆₇
hypocenter	C₆₇	C₄×D₄
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₆₇	C₂	semidirect
3. (C₂×D₆₇)⋊C₄	C₂	semidirect
4. C₄×Dic₆₇	C₂	semidirect
5. C₄⋊C₄×C₆₇	C₂	semidirect
11. D₂₆₈	C₄	semidirect
16. C₂×D₆₇	C₂×C₄	semidirect
18. C₄×C₆₇	C₂×C₄	semidirect
20. Dic₆₇	D₄	semidirect
25. C₆₇	C₄×D₄	semidirect
29. C₄⋊C₄	D₆₇	semidirect
46. C₄	C₄×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₆₈	2
C₂×C₄×D₆₇	2
(C₂×D₆₇)⋊C₄	2
C₄×Dic₆₇	2
C₄⋊C₄×C₆₇	2
C₄×D₄	67