Subgroups — #29504641

Order 15296.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₃₉⋊C₄	C₂	1	4	yes	maximal
3	C₂×C₄×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₄×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₃₉	C₂²	1	2	yes	—
6	C₂²×Dic₂₃₉	C₂²	1	2	yes	—
7	C₄⋊Dic₂₃₉	—	2	2	no	—
8	Dic₂₃₉⋊C₄	—	2	8	no	—
9	C₄×Dic₂₃₉	C₄	1	4	yes	complemented
10	C₂²×C₄×C₂₃₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₃₉	—	2	2	no	—
13	C₂×Dic₂₃₉	—	2	4	no	—
14	C₂×Dic₂₃₉	—	2	2	no	—
15	C₂×Dic₂₃₉	C₂×C₄	1	4	yes	—
16	C₂×Dic₂₃₉	D₄	1	4	yes	—
17	C₂×Dic₂₃₉	Q₈	1	4	yes	—
18	C₂³×C₂₃₉	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₂₃₉	—	2	2	no	—
20	C₂×C₄×C₂₃₉	—	2	4	no	—
21	C₂×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₃₉	D₄	1	2	yes	—
23	C₂×C₄×C₂₃₉	Q₈	1	2	yes	—
24	C₂²×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₂₃₉	C₂×D₄	1	2	yes	—
26	C₂²×C₂₃₉	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₂₃₉	C₂×Q₈	1	2	yes	—
28	C₂²×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₂₃₉	—	4	2	no	—
30	C₄×C₂₃₉	C₄⋊C₄	1	4	yes	complemented
31	Dic₂₃₉	—	4	2	no	—
32	Dic₂₃₉	C₄⋊C₄	1	8	yes	complemented
33	C₂×C₂₃₉	C₂×C₄⋊C₄	1	2	yes	—
34	C₂×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₂₃₉	#127	1	1	yes	characteristic
36	C₂×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
37	C₂×C₂₃₉	C₄⋊Q₈	1	2	yes	—
38	C₂₃₉	#384	1	1	yes	characteristic, complemented, Sylow
39	#384	—	239	1	no	maximal, Sylow
40	C₂×C₄⋊C₄	—	239	4	no	—
41	C₂×C₄⋊C₄	—	239	1	no	—
42	C₂×C₄⋊C₄	D₂₃₉	1	1	yes	characteristic
43	C₂×C₄²	—	239	1	no	—
44	C₂²×C₄	—	239	2	no	—
45	C₂²×C₄	—	239	2	no	—
46	C₂²×C₄	C₂×D₂₃₉	1	2	yes	—
47	C₂²×C₄	C₂×D₂₃₉	1	1	yes	characteristic
48	C₄⋊C₄	—	2	2	no	—
49	C₄⋊C₄	—	478	8	no	—
50	C₄⋊C₄	—	478	2	no	—
51	C₄²	—	239	4	no	—
52	C₂³	C₂²×D₂₃₉	1	1	yes	characteristic
53	C₂×C₄	—	2	4	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	478	4	no	—
56	C₂×C₄	—	478	2	no	—
57	C₂×C₄	—	239	4	no	—
58	C₂×C₄	—	239	4	no	—
59	C₂×C₄	—	239	4	no	—
60	C₂×C₄	C₄×D₂₃₉	1	2	yes	—
61	C₂×C₄	C₂²⋊D₂₃₉	1	2	yes	—
62	C₂×C₄	Dic₄₇₈	1	2	yes	—
63	C₂²	C₂×C₄×D₂₃₉	1	1	yes	characteristic
64	C₂²	D₄×D₂₃₉	1	2	yes	—
65	C₂²	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
66	C₂²	Q₈×D₂₃₉	1	2	yes	—
67	C₂²	C₂×Dic₄₇₈	1	1	yes	characteristic
68	C₄	—	4	2	no	—
69	C₄	—	956	2	no	—
70	C₄	—	239	8	no	—
71	C₄	Dic₂₃₉⋊C₄	1	4	yes	complemented
72	C₂	C₄⋊C₄×D₂₃₉	1	2	yes	—
73	C₂	C₂×Dic₂₃₉⋊C₄	1	1	yes	characteristic
74	C₂	#21439956	1	1	yes	characteristic
75	C₂	#21439958	1	1	yes	characteristic
76	C₂	#21439959	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₃₉
commutator	C₂²×C₂₃₉	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₂₃₉	C₂
socle, prime radical, cosocle	C₂³×C₂₃₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₂₃₉
hypocenter	C₂₃₉	#384
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. C₄×Dic₂₃₉	C₄	semidirect
30. C₄×C₂₃₉	C₄⋊C₄	semidirect
32. Dic₂₃₉	C₄⋊C₄	semidirect
38. C₂₃₉	#384	semidirect
71. C₄	Dic₂₃₉⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₃₉	2
C₂×Dic₂₃₉⋊C₄	2
C₂×C₄×Dic₂₃₉	2
C₂×C₄⋊C₄×C₂₃₉	2
#384	239