Subgroups — #29505063

Order 15296.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439960	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₉₅₆	C₂	1	1	yes	maximal, characteristic
3	#21439892	C₂	1	2	yes	maximal, complemented
4	#21439882	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₇₈	—	2	1	no	—
7	C₂×D₉₅₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₃₉	—	2	1	no	—
9	C₄⋊Dic₂₃₉	—	2	1	no	—
10	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
11	C₄⋊Dic₂₃₉	C₂²	1	1	yes	characteristic
12	(C₂×D₂₃₉)⋊C₄	—	2	1	no	—
13	C₂×C₈×C₂₃₉	C₂²	1	2	yes	—
14	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂²×D₂₃₉	—	2	1	no	—
16	Dic₄₇₈	—	4	2	no	—
17	D₉₅₆	—	2	1	no	—
18	D₉₅₆	Q₈	1	2	yes	complemented
19	C₂×Dic₂₃₉	—	2	1	no	—
20	C₂×Dic₂₃₉	—	2	2	no	—
21	C₄×D₂₃₉	—	4	2	no	—
22	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
25	C₈×C₂₃₉	—	2	2	no	—
26	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
27	C₂×D₂₃₉	—	2	2	no	—
28	C₂×D₂₃₉	—	4	1	no	—
29	C₂×D₂₃₉	—	4	1	no	—
30	C₄×C₂₃₉	—	2	1	no	—
31	C₄×C₂₃₉	SD₁₆	1	2	yes	—
32	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₂₃₉	#41	1	1	yes	characteristic
34	Dic₂₃₉	—	4	1	no	—
35	Dic₂₃₉	—	4	2	no	—
36	C₂×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₂₃₉	C₂×SD₁₆	1	1	yes	characteristic
38	C₂×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
39	D₂₃₉	—	4	2	no	—
40	C₂₃₉	#476	1	1	yes	characteristic, complemented, Sylow
41	#476	—	239	1	no	maximal, Sylow
42	C₄⋊Q₈	—	239	1	no	—
43	C₄×D₄	—	239	1	no	—
44	#106	—	239	2	no	—
45	C₄⋊C₈	D₂₃₉	1	1	yes	characteristic, complemented
46	#102	—	239	2	no	—
47	C₂×Q₈	—	478	1	no	—
48	C₂×D₄	—	239	1	no	—
49	C₂²×C₄	—	478	1	no	—
50	C₂×C₈	C₂×D₂₃₉	1	2	yes	—
51	C₄⋊C₄	—	478	1	no	—
52	C₄⋊C₄	—	239	2	no	—
53	C₄⋊C₄	—	239	1	no	—
54	C₂²⋊C₄	—	478	1	no	—
55	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
56	C₂³	—	478	1	no	—
57	Q₈	—	956	2	no	—
58	D₄	—	478	1	no	—
59	D₄	—	239	2	no	—
60	C₂×C₄	—	956	2	no	—
61	C₂×C₄	—	478	1	no	—
62	C₂×C₄	—	478	2	no	—
63	C₂×C₄	D₉₅₆	1	1	yes	characteristic
64	C₂×C₄	D₉₅₆	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₂²	—	956	1	no	—
68	C₂²	—	956	1	no	—
69	C₂²	—	478	2	no	—
70	C₂²	C₂×D₉₅₆	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	956	1	no	—
73	C₄	—	956	2	no	—
74	C₄	#16838537	1	2	yes	—
75	C₄	Q₈×D₂₃₉	1	1	yes	characteristic
76	C₄	#16838547	1	1	yes	characteristic
77	C₂	—	956	2	no	—
78	C₂	#21439940	1	1	yes	characteristic
79	C₂	C₂×#16838537	1	1	yes	characteristic
80	C₂	#21439988	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₉₅₆
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	C₄⋊C₈×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₄
prime radical	C₂×D₉₅₆	C₂²
hypercenter	C₄⋊C₈	D₂₃₉
hypocenter	C₂₃₉	#476
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #21439960	C₂	semidirect
3. #21439892	C₂	semidirect
5. C₄⋊C₈×C₂₃₉	C₂	semidirect
18. D₉₅₆	Q₈	semidirect
40. C₂₃₉	#476	semidirect
45. C₄⋊C₈	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#21439960	2
C₄×D₉₅₆	2
#21439892	2
#21439882	2
C₄⋊C₈×C₂₃₉	2
#476	239