Subgroups — #84128

Order 384.

3422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#953	C₂	1	7	yes	maximal, complemented
3	#5247	C₂	1	7	yes	maximal, complemented
4	#2846	—	3	1	no	maximal, Sylow
5	C₂⁵×C₃	—	2	28	no	—
6	C₂⁵×C₃	C₂²	1	7	yes	—
7	C₂²×#67	C₂²	1	7	yes	complemented
8	#953	—	2	28	no	—
9	C₂×#217	C₂²	1	21	yes	complemented
10	C₂⁶	D₃	1	1	yes	characteristic, complemented
11	C₂×#120	—	3	7	no	—
12	#379	—	3	7	no	—
13	C₂⁴×C₃	—	2	224	no	—
14	C₂⁴×C₃	—	2	84	no	—
15	C₂⁴×C₃	D₄	1	28	yes	complemented
16	C₂⁴×C₃	C₂³	1	7	yes	—
17	C₂³×D₃	C₂³	1	1	yes	characteristic, complemented
18	C₂×#67	—	2	42	no	—
19	C₂²×Dic₃	C₂³	1	7	yes	complemented
20	#217	—	2	42	no	—
21	C₂⁵	—	2	28	no	—
22	C₂⁵	C₂×D₃	1	7	yes	—
23	C₂²×D₄	—	3	7	no	—
24	#120	—	6	28	no	—
25	C₂×C₂²⋊C₄	—	3	21	no	—
26	C₂³×C₃	—	2	256	no	—
27	C₂³×C₃	—	2	224	no	—
28	C₂³×C₃	—	2	168	no	—
29	C₂³×C₃	—	2	28	no	—
30	C₂³×C₃	C₂×D₄	1	42	yes	—
31	C₂³×C₃	C₂⁴	1	1	yes	characteristic
32	C₂²×D₃	—	2	7	no	—
33	#67	—	4	28	no	—
34	C₂×Dic₃	—	2	21	no	—
35	C₂⁴	—	2	224	no	—
36	C₂⁴	—	2	84	no	—
37	C₂⁴	—	3	1	no	—
38	C₂⁴	#67	1	28	yes	complemented
39	C₂⁴	C₂²×D₃	1	7	yes	—
40	C₂×D₄	—	6	42	no	—
41	C₂²×C₄	—	3	7	no	—
42	C₂²⋊C₄	—	6	42	no	—
43	C₂²×C₃	—	2	224	no	—
44	C₂²×C₃	—	2	84	no	—
45	C₂²×C₃	#120	1	28	yes	complemented
46	C₂²×C₃	C₂²×D₄	1	7	yes	—
47	C₂×D₃	—	4	7	no	—
48	Dic₃	—	4	7	no	—
49	C₂³	—	2	256	no	—
50	C₂³	—	2	224	no	—
51	C₂³	—	2	168	no	—
52	C₂³	—	2	28	no	—
53	C₂³	—	6	7	no	—
54	C₂³	C₂×#67	1	42	yes	—
55	C₂³	C₂³×D₃	1	1	yes	characteristic
56	D₄	—	12	28	no	—
57	C₂×C₄	—	6	21	no	—
58	C₂×C₃	—	2	28	no	—
59	C₂×C₃	C₂×#120	1	7	yes	—
60	D₃	—	8	1	no	—
61	C₂²	—	2	224	no	—
62	C₂²	—	2	84	no	—
63	C₂²	—	12	7	no	—
64	C₂²	#953	1	28	yes	complemented
65	C₂²	C₂²×#67	1	7	yes	—
66	C₄	—	12	7	no	—
67	C₃	#2846	1	1	yes	characteristic, complemented, Sylow
68	C₂	—	2	28	no	—
69	C₂	—	24	1	no	—
70	C₂	C₂×#953	1	7	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁴
Fitting	C₂⁶×C₃	C₂
hypercenter	C₂⁶	D₃
hypocenter	C₃	#2846
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
1. C₂⁶×C₃	C₂	semidirect
2. C₂×#953	C₂	semidirect
3. #5247	C₂	semidirect
7. C₂²×#67	C₂²	semidirect
9. C₂×#217	C₂²	semidirect
10. C₂⁶	D₃	semidirect
15. C₂⁴×C₃	D₄	semidirect
17. C₂³×D₃	C₂³	semidirect
19. C₂²×Dic₃	C₂³	semidirect
38. C₂⁴	#67	semidirect
45. C₂²×C₃	#120	semidirect
64. C₂²	#953	semidirect
67. C₃	#2846	semidirect

Maximal subgroups

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

subgroup	index
C₂⁶×C₃	2
C₂×#953	2
#5247	2
#2846	3