Subgroups — #38734127

Order 24896.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15691005	C₂	1	1	yes	maximal, characteristic
2	#27159343	C₂	1	1	yes	maximal, characteristic
3	#27159313	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₃₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691005	—	2	2	no	—
6	C₂×#15691005	C₂²	1	2	yes	—
7	C₄⋊Dic₃₈₉	C₄	1	2	yes	—
8	C₂²⋊Dic₃₈₉	C₄	1	2	yes	—
9	C₄×Dic₃₈₉	C₂²	1	2	yes	—
10	C₂²×C₄×C₃₈₉	C₂²	1	1	yes	characteristic
11	M₁₆×C₃₈₉	—	2	2	no	—
12	C₂×C₈×C₃₈₉	C₂²	1	2	yes	—
13	C₂×Dic₃₈₉	C₂×C₄	1	4	yes	—
14	#15691005	—	2	2	no	—
15	C₂³×C₃₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₃₈₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₃₈₉	D₄	1	2	yes	—
18	C₂×C₄×C₃₈₉	Q₈	1	2	yes	—
19	C₂×C₄×C₃₈₉	C₂³	1	1	yes	characteristic
20	C₈×C₃₈₉	—	2	2	no	—
21	C₂²×C₃₈₉	—	2	2	no	—
22	C₂²×C₃₈₉	C₄⋊C₄	1	2	yes	—
23	C₂²×C₃₈₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₈₉	C₄⋊C₄	1	2	yes	—
25	C₄×C₃₈₉	C₂×D₄	1	1	yes	characteristic
26	C₄×C₃₈₉	C₂×Q₈	1	1	yes	characteristic
27	Dic₃₈₉	—	2	4	no	—
28	C₂×C₃₈₉	—	2	2	no	—
29	C₂×C₃₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₃₈₉	#131	1	2	yes	—
31	C₃₈₉	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	389	1	no	maximal, Sylow
33	C₂×M₁₆	D₃₈₉	1	1	yes	characteristic
34	C₂²×C₈	—	389	1	no	—
35	#117	—	389	1	no	—
36	C₄⋊C₈	—	389	4	no	—
37	C₂²×C₄	C₂×D₃₈₉	1	1	yes	characteristic
38	M₁₆	—	2	2	no	—
39	C₂×C₈	—	778	2	no	—
40	C₂×C₈	—	389	2	no	—
41	C₂×C₈	C₂×D₃₈₉	1	2	yes	—
42	C₄⋊C₄	—	389	2	no	—
43	C₂²⋊C₄	—	389	2	no	—
44	C₄²	—	389	2	no	—
45	C₂³	C₄×D₃₈₉	1	1	yes	characteristic
46	C₂×C₄	—	389	4	no	—
47	C₂×C₄	C₄×D₃₈₉	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₃₈₉	1	2	yes	—
49	C₂×C₄	Dic₇₇₈	1	2	yes	—
50	C₂×C₄	C₂²×D₃₈₉	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	778	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	#20261945	1	2	yes	—
55	C₂²	C₂×C₄×D₃₈₉	1	1	yes	characteristic
56	C₄	—	778	4	no	—
57	C₄	#20261945	1	2	yes	—
58	C₄	C₂×C₂²⋊D₃₈₉	1	1	yes	characteristic
59	C₄	C₂×Dic₇₇₈	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×#20261945	1	1	yes	characteristic
62	C₂	#27159392	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₈₉
commutator, socle	C₂²×C₃₈₉	C₂²×C₄
Fitting	C₂×M₁₆×C₃₈₉	C₂
prime radical	C₂³×C₃₈₉	C₂×C₄
cosocle	C₂×C₄×C₃₈₉	C₂³
hypercenter	C₂×M₁₆	D₃₈₉
hypocenter	C₃₈₉	#424
LCS	G ▷ C₂²×C₃₈₉ ▷ C₃₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
3. #27159313	C₂	semidirect
31. C₃₈₉	#424	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15691005	2
#27159343	2
#27159313	2
C₂×M₁₆×C₃₈₉	2
#424	389