Subgroups — #29505384

Order 15296.

192 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	#21439953	C₂	1	4	yes	maximal
3	#21439952	C₂	1	1	yes	maximal, characteristic, complemented
4	#21439950	C₂	1	4	yes	maximal, complemented
5	#21439939	C₂	1	2	yes	maximal
6	C₄×Dic₄₇₈	C₂	1	2	yes	maximal, complemented
7	#117×C₂₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₇₈	C₂²	1	2	yes	—
9	C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
10	C₄⋊Dic₂₃₉	C₂²	1	4	yes	—
11	Dic₂₃₉⋊C₄	C₂²	1	4	yes	—
12	Dic₂₃₉⋊C₄	C₂²	1	4	yes	—
13	Dic₂₃₉⋊C₄	C₂²	1	4	yes	—
14	C₂²⋊Dic₂₃₉	C₂²	1	4	yes	—
15	C₄×Dic₂₃₉	C₂²	1	4	yes	—
16	C₂²×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₃₉	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₂₃₉	C₂²	1	2	yes	—
19	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
20	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
21	Dic₄₇₈	—	2	4	no	—
22	C₂×Dic₂₃₉	C₂³	1	4	yes	—
23	C₂×Dic₂₃₉	C₂³	1	4	yes	—
24	C₂³×C₂₃₉	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₂₃₉	—	2	2	no	—
26	C₂×C₄×C₂₃₉	C₂³	1	2	yes	—
27	C₂×C₄×C₂₃₉	C₂³	1	2	yes	—
28	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
30	C₂²×C₂₃₉	—	2	1	no	—
31	C₂²×C₂₃₉	—	2	2	no	—
32	C₂²×C₂₃₉	C₂⁴	1	1	yes	characteristic
33	C₄×C₂₃₉	—	2	1	no	—
34	C₄×C₂₃₉	—	2	2	no	—
35	C₄×C₂₃₉	—	2	2	no	—
36	C₄×C₂₃₉	#41	1	2	yes	—
37	Dic₂₃₉	—	2	4	no	—
38	Dic₂₃₉	—	2	4	no	—
39	C₂×C₂₃₉	—	4	1	no	—
40	C₂×C₂₃₉	C₂×#41	1	1	yes	characteristic
41	C₂×C₂₃₉	2₋¹⁺⁴	1	2	yes	—
42	C₂₃₉	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	239	1	no	maximal, Sylow
44	C₄⋊Q₈	—	239	1	no	—
45	#126	—	239	4	no	—
46	#125	—	239	4	no	—
47	#125	—	239	1	no	—
48	C₂²⋊Q₈	—	239	2	no	—
49	C₄×Q₈	—	239	2	no	—
50	#117	D₂₃₉	1	1	yes	characteristic
51	C₂×Q₈	—	239	2	no	—
52	C₂²×C₄	C₂×D₂₃₉	1	1	yes	characteristic
53	C₄⋊C₄	—	239	2	no	—
54	C₄⋊C₄	—	239	4	no	—
55	C₄⋊C₄	—	239	4	no	—
56	C₄⋊C₄	—	239	4	no	—
57	C₄⋊C₄	—	239	4	no	—
58	C₄⋊C₄	C₂×D₂₃₉	1	2	yes	—
59	C₂²⋊C₄	—	239	4	no	—
60	C₂²⋊C₄	C₂×D₂₃₉	1	2	yes	—
61	C₄²	—	239	4	no	—
62	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
63	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
64	C₂³	C₂²×D₂₃₉	1	1	yes	characteristic
65	Q₈	—	478	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	239	4	no	—
68	C₂×C₄	—	239	4	no	—
69	C₂×C₄	C₂²×D₂₃₉	1	2	yes	—
70	C₂×C₄	C₂²×D₂₃₉	1	2	yes	—
71	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
73	C₂²	—	2	1	no	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂³×D₂₃₉	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	2	no	—
79	C₄	—	478	4	no	—
80	C₄	—	478	4	no	—
81	C₄	#16838548	1	2	yes	—
82	C₂	—	4	1	no	—
83	C₂	C₂×#16838548	1	1	yes	characteristic
84	C₂	#21440007	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃₉
commutator, socle, cosocle	C₂²×C₂₃₉	C₂⁴
Fitting	#117×C₂₃₉	C₂
prime radical	C₂³×C₂₃₉	C₂³
hypercenter	#117	D₂₃₉
hypocenter	C₂₃₉	#541
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #21439960	C₂	semidirect
3. #21439952	C₂	semidirect
4. #21439950	C₂	semidirect
6. C₄×Dic₄₇₈	C₂	semidirect
42. C₂₃₉	#541	semidirect

Maximal subgroups

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

subgroup	index
#21439960	2
#21439953	2
#21439952	2
#21439950	2
#21439939	2
C₄×Dic₄₇₈	2
#117×C₂₃₉	2
#541	239