Subgroups — #22568601

Order 8576.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863287	C₂	1	7	yes	maximal
2	#16863280	C₂	1	7	yes	maximal, complemented
3	C₄³×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₆₇⋊C₄	C₂²	1	7	yes	—
5	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	7	yes	—
6	C₆₇⋊#95	C₂²	1	14	yes	—
7	C₂×C₄²×C₆₇	C₂²	1	7	yes	—
8	C₂³×D₆₇	C₂³	1	1	yes	characteristic
9	C₂²×Dic₆₇	C₂³	1	7	yes	—
10	Dic₆₇⋊C₄	—	2	14	no	—
11	(C₂×D₆₇)⋊C₄	—	2	14	no	—
12	C₂²×C₄×C₆₇	C₂³	1	7	yes	—
13	C₄²×C₆₇	—	2	14	no	—
14	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
15	C₂²×D₆₇	—	2	7	no	—
16	C₂×C₄×C₆₇	—	2	14	no	—
17	C₂×C₄×C₆₇	#41	1	14	yes	—
18	C₂×Dic₆₇	—	2	7	no	—
19	C₂×Dic₆₇	—	2	14	no	—
20	C₂²×C₆₇	C₂×#41	1	7	yes	—
21	C₂×D₆₇	—	4	7	no	—
22	C₄×C₆₇	—	2	14	no	—
23	Dic₆₇	—	4	7	no	—
24	C₂×C₆₇	#529	1	7	yes	—
25	D₆₇	—	8	1	no	—
26	#2866	—	67	1	no	maximal, Sylow
27	C₆₇	#2866	1	1	yes	characteristic, complemented, Sylow
28	#388	—	67	7	no	—
29	#387	—	67	7	no	—
30	C₄³	D₆₇	1	1	yes	characteristic, complemented
31	C₂×C₄⋊C₄	—	67	7	no	—
32	C₂×C₂²⋊C₄	—	67	7	no	—
33	C₂×C₄²	C₂×D₆₇	1	7	yes	—
34	#95	—	67	14	no	—
35	C₂⁴	—	67	1	no	—
36	C₂²×C₄	—	67	7	no	—
37	C₂²×C₄	C₂²×D₆₇	1	7	yes	—
38	C₄⋊C₄	—	134	14	no	—
39	C₂²⋊C₄	—	134	14	no	—
40	C₄²	—	2	14	no	—
41	C₂³	—	134	7	no	—
42	C₂³	C₂³×D₆₇	1	1	yes	characteristic
43	C₂×C₄	—	2	14	no	—
44	C₂×C₄	—	134	7	no	—
45	C₂×C₄	—	134	14	no	—
46	C₂×C₄	#11761628	1	14	yes	—
47	C₂²	—	268	7	no	—
48	C₂²	C₂×#11761628	1	7	yes	—
49	C₄	—	2	14	no	—
50	C₄	—	268	7	no	—
51	C₂	—	536	1	no	—
52	C₂	#16863930	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	C₄³×C₆₇	C₂
hypercenter	C₄³	D₆₇
hypocenter	C₆₇	#2866
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
2. #16863280	C₂	semidirect
3. C₄³×C₆₇	C₂	semidirect
27. C₆₇	#2866	semidirect
30. C₄³	D₆₇	semidirect