Subgroups — #29504849

Order 15296.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16838531	C₂	1	2	yes	maximal, complemented
2	C₄×D₉₅₆	C₂	1	1	yes	maximal, characteristic
3	(C₂×D₂₃₉)⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₄×C₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₉₅₆	C₄	1	1	yes	characteristic
7	C₂×C₄×D₂₃₉	C₂²	1	2	yes	—
8	#16838531	—	2	4	no	—
9	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₃₉	C₄	1	1	yes	characteristic
12	(C₂×D₂₃₉)⋊C₄	C₄	1	2	yes	—
13	C₂×C₈×C₂₃₉	C₂²	1	2	yes	—
14	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂²×D₂₃₉	C₂×C₄	1	2	yes	—
16	D₉₅₆	—	2	2	no	—
17	C₂×Dic₂₃₉	C₂×C₄	1	2	yes	—
18	C₄×D₂₃₉	—	2	2	no	—
19	C₄×D₂₃₉	—	2	2	no	—
20	C₂₃₉⋊C₈	—	2	1	no	—
21	C₂₃₉⋊C₈	D₄	1	2	yes	complemented
22	C₂×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
25	C₈×C₂₃₉	—	2	2	no	—
26	C₂²×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₃₉	—	2	2	no	—
28	C₂×D₂₃₉	—	2	2	no	—
29	C₂×D₂₃₉	—	2	2	no	—
30	C₄×C₂₃₉	—	2	1	no	—
31	C₄×C₂₃₉	M₁₆	1	2	yes	complemented
32	C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₃₉	#41	1	1	yes	characteristic
34	Dic₂₃₉	—	2	2	no	—
35	C₂×C₂₃₉	C₄×D₄	1	1	yes	characteristic
36	C₂×C₂₃₉	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₂₃₉	#131	1	1	yes	characteristic
38	D₂₃₉	—	4	2	no	—
39	C₂₃₉	#436	1	1	yes	characteristic, complemented, Sylow
40	#436	—	239	1	no	maximal, Sylow
41	C₂×M₁₆	—	239	2	no	—
42	C₄×D₄	—	239	1	no	—
43	C₄⋊C₈	D₂₃₉	1	1	yes	characteristic, complemented
44	C₂²⋊C₈	—	239	2	no	—
45	C₄×C₈	—	239	1	no	—
46	C₂×D₄	—	239	1	no	—
47	C₂²×C₄	—	239	2	no	—
48	M₁₆	—	478	4	no	—
49	C₂×C₈	—	239	1	no	—
50	C₂×C₈	—	239	1	no	—
51	C₂×C₈	C₂×D₂₃₉	1	2	yes	—
52	C₄⋊C₄	—	239	1	no	—
53	C₂²⋊C₄	—	239	2	no	—
54	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
55	C₂³	—	239	2	no	—
56	D₄	—	478	2	no	—
57	C₂×C₄	—	478	2	no	—
58	C₂×C₄	—	478	2	no	—
59	C₂×C₄	—	239	2	no	—
60	C₂×C₄	C₄×D₂₃₉	1	1	yes	characteristic
61	C₂×C₄	C₄×D₂₃₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	478	1	no	—
65	C₈	—	239	2	no	—
66	C₂²	—	478	2	no	—
67	C₂²	—	478	2	no	—
68	C₂²	—	478	2	no	—
69	C₂²	C₂×C₄×D₂₃₉	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	478	2	no	—
72	C₄	#16838531	1	2	yes	complemented
73	C₄	D₄×D₂₃₉	1	1	yes	characteristic
74	C₄	#16838549	1	1	yes	characteristic
75	C₂	—	956	2	no	—
76	C₂	#21439922	1	1	yes	characteristic
77	C₂	C₂×#16838531	1	1	yes	characteristic
78	C₂	#21439968	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃₉
commutator, socle	C₂²×C₂₃₉	C₂²×C₄
Fitting	C₄⋊C₈×C₂₃₉	C₂
prime radical	C₂×D₉₅₆	C₄
cosocle	C₂×C₄×C₂₃₉	C₂³
hypercenter	C₄⋊C₈	D₂₃₉
hypocenter	C₂₃₉	#436
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16838531	C₂	semidirect
3. (C₂×D₂₃₉)⋊C₈	C₂	semidirect
4. C₄×C₂₃₉⋊C₈	C₂	semidirect
5. C₄⋊C₈×C₂₃₉	C₂	semidirect
21. C₂₃₉⋊C₈	D₄	semidirect
31. C₄×C₂₃₉	M₁₆	semidirect
39. C₂₃₉	#436	semidirect
43. C₄⋊C₈	D₂₃₉	semidirect
72. C₄	#16838531	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16838531	2
C₄×D₉₅₆	2
(C₂×D₂₃₉)⋊C₈	2
C₄×C₂₃₉⋊C₈	2
C₄⋊C₈×C₂₃₉	2
#436	239