Subgroups — #29505148

Order 15296.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838533	C₂	1	4	yes	maximal, complemented
2	#21439957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#127×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₉₅₆	—	2	2	no	—
6	C₂×D₉₅₆	C₂²	1	2	yes	—
7	#16838533	—	2	8	no	—
8	C₂×C₂₃₉⋊C₈	C₂²	1	2	yes	—
9	C₂×D₄×C₂₃₉	—	2	2	no	—
10	C₂×D₄×C₂₃₉	C₂²	1	2	yes	—
11	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₂²×D₂₃₉	—	2	2	no	—
13	D₉₅₆	—	2	4	no	—
14	D₉₅₆	—	4	4	no	—
15	C₂₃₉⋊C₈	D₄	1	4	yes	complemented
16	C₂³×C₂₃₉	—	2	2	no	—
17	D₄×C₂₃₉	—	2	4	no	—
18	D₄×C₂₃₉	—	4	4	no	—
19	C₂×C₄×C₂₃₉	D₄	1	2	yes	—
20	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
21	C₂²×C₂₃₉	—	4	2	no	—
22	C₂²×C₂₃₉	—	4	4	no	—
23	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₃₉	—	4	2	no	—
25	C₂×D₂₃₉	—	4	4	no	—
26	C₄×C₂₃₉	D₈	1	4	yes	complemented
27	C₄×C₂₃₉	C₂×D₄	1	2	yes	—
28	C₂×C₂₃₉	—	8	2	no	—
29	C₂×C₂₃₉	#127	1	1	yes	characteristic
30	C₂×C₂₃₉	C₂×D₈	1	2	yes	—
31	D₂₃₉	—	8	2	no	—
32	C₂₃₉	#493	1	1	yes	characteristic, complemented, Sylow
33	#493	—	239	1	no	maximal, Sylow
34	C₂×D₈	—	239	4	no	—
35	#127	—	239	1	no	—
36	#127	D₂₃₉	1	1	yes	characteristic, complemented
37	C₄×C₈	—	239	1	no	—
38	C₂×D₄	—	2	2	no	—
39	C₂×D₄	—	478	2	no	—
40	C₂×D₄	—	239	2	no	—
41	C₂×D₄	C₂×D₂₃₉	1	2	yes	—
42	D₈	—	478	8	no	—
43	C₂×C₈	—	239	2	no	—
44	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
45	C₂³	—	2	2	no	—
46	C₂³	—	478	2	no	—
47	D₄	—	2	4	no	—
48	D₄	—	4	4	no	—
49	D₄	—	956	4	no	—
50	D₄	—	478	4	no	—
51	C₂×C₄	C₂²⋊D₂₃₉	1	2	yes	—
52	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
53	C₈	—	239	4	no	—
54	C₂²	—	4	2	no	—
55	C₂²	—	4	4	no	—
56	C₂²	—	956	2	no	—
57	C₂²	—	956	4	no	—
58	C₂²	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
59	C₄	#16838533	1	4	yes	complemented
60	C₄	D₄×D₂₃₉	1	2	yes	—
61	C₂	—	8	2	no	—
62	C₂	—	1912	2	no	—
63	C₂	#21439956	1	1	yes	characteristic
64	C₂	C₂×#16838533	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃₉
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	#127×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₄
hypercenter	#127	D₂₃₉
hypocenter	C₂₃₉	#493
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#16838533	C₂	semidirect
2. #21439957	C₂	semidirect
3. C₄×C₂₃₉⋊C₈	C₂	semidirect
4. #127×C₂₃₉	C₂	semidirect
15. C₂₃₉⋊C₈	D₄	semidirect
26. C₄×C₂₃₉	D₈	semidirect
32. C₂₃₉	#493	semidirect
36. #127	D₂₃₉	semidirect
59. C₄	#16838533	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16838533	2
#21439957	2
C₄×C₂₃₉⋊C₈	2
#127×C₂₃₉	2
#493	239