Subgroups — #29505005

Order 15296.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#21439931	C₂	1	2	yes	maximal, complemented
3	#21439894	C₂	1	1	yes	maximal, characteristic, complemented
4	#21439881	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂²⋊D₂₃₉	—	2	2	no	—
7	C₂×C₄×D₂₃₉	C₂²	1	1	yes	characteristic
8	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₈×D₂₃₉	—	2	2	no	—
10	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
11	C₂²⋊Dic₂₃₉	—	2	2	no	—
12	C₂×D₄×C₂₃₉	C₂²	1	2	yes	—
13	D₈×C₂₃₉	—	2	2	no	—
14	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂²×D₂₃₉	D₄	1	1	yes	characteristic
16	C₂²⋊D₂₃₉	—	4	4	no	—
17	C₂×Dic₂₃₉	—	2	2	no	—
18	C₂×Dic₂₃₉	D₄	1	1	yes	characteristic
19	C₄×D₂₃₉	—	2	1	no	—
20	C₄×D₂₃₉	—	2	1	no	—
21	C₂₃₉⋊C₈	—	2	1	no	—
22	C₂³×C₂₃₉	—	2	2	no	—
23	D₄×C₂₃₉	—	2	2	no	—
24	D₄×C₂₃₉	—	2	2	no	—
25	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
26	C₈×C₂₃₉	D₄	1	2	yes	complemented
27	C₂²×C₂₃₉	—	4	2	no	—
28	C₂²×C₂₃₉	—	4	2	no	—
29	C₂²×C₂₃₉	—	4	2	no	—
30	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
31	C₂×D₂₃₉	—	2	1	no	—
32	C₂×D₂₃₉	—	2	1	no	—
33	C₂×D₂₃₉	D₈	1	2	yes	complemented
34	C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₃₉	#41	1	1	yes	characteristic
36	Dic₂₃₉	—	2	1	no	—
37	Dic₂₃₉	—	4	2	no	—
38	C₂×C₂₃₉	—	8	2	no	—
39	C₂×C₂₃₉	#121	1	1	yes	characteristic
40	C₂×C₂₃₉	C₂×D₈	1	1	yes	characteristic
41	C₂×C₂₃₉	#135	1	1	yes	characteristic
42	D₂₃₉	—	2	2	no	—
43	C₂₃₉	#466	1	1	yes	characteristic, complemented, Sylow
44	#466	—	239	1	no	maximal, Sylow
45	C₂×D₈	D₂₃₉	1	1	yes	characteristic, complemented
46	C₂²×C₈	—	239	1	no	—
47	#121	—	239	2	no	—
48	#107	—	239	1	no	—
49	#102	—	239	2	no	—
50	C₂×D₄	—	478	2	no	—
51	C₂×D₄	C₂×D₂₃₉	1	2	yes	—
52	C₂²×C₄	—	239	1	no	—
53	D₈	—	2	2	no	—
54	C₂×C₈	—	478	2	no	—
55	C₂×C₈	—	239	1	no	—
56	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
57	C₄⋊C₄	—	239	2	no	—
58	C₂²⋊C₄	—	478	2	no	—
59	C₂³	—	2	2	no	—
60	C₂³	—	239	1	no	—
61	D₄	—	2	2	no	—
62	D₄	—	2	2	no	—
63	D₄	—	956	4	no	—
64	C₂×C₄	—	478	2	no	—
65	C₂×C₄	—	478	1	no	—
66	C₂×C₄	—	478	1	no	—
67	C₂×C₄	—	239	1	no	—
68	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
69	C₈	—	478	1	no	—
70	C₈	C₂²⋊D₂₃₉	1	2	yes	complemented
71	C₂²	—	4	2	no	—
72	C₂²	—	4	2	no	—
73	C₂²	—	4	2	no	—
74	C₂²	—	478	1	no	—
75	C₂²	—	478	1	no	—
76	C₂²	—	239	2	no	—
77	C₂²	D₄×D₂₃₉	1	1	yes	characteristic
78	C₄	—	956	2	no	—
79	C₄	—	478	1	no	—
80	C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
81	C₄	#16838547	1	1	yes	characteristic
82	C₂	—	8	2	no	—
83	C₂	—	478	2	no	—
84	C₂	#21439931	1	1	yes	characteristic
85	C₂	D₈×D₂₃₉	1	1	yes	characteristic
86	C₂	#21439979	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃₉
commutator, cosocle	C₂×C₄×C₂₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₉
Fitting	C₂×D₈×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₄
hypercenter	C₂×D₈	D₂₃₉
hypocenter	C₂₃₉	#466
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×D₂₃₉	C₂	semidirect
2. #21439931	C₂	semidirect
3. #21439894	C₂	semidirect
4. #21439881	C₂	semidirect
5. C₂×D₈×C₂₃₉	C₂	semidirect
26. C₈×C₂₃₉	D₄	semidirect
33. C₂×D₂₃₉	D₈	semidirect
43. C₂₃₉	#466	semidirect
45. C₂×D₈	D₂₃₉	semidirect
70. C₈	C₂²⋊D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₃₉	2
#21439931	2
#21439894	2
#21439881	2
C₂×D₈×C₂₃₉	2
#466	239