Subgroups — #29504900

Order 15296.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₄₇₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₂₃₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	Dic₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	Dic₂₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#21439870	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
7	#97×C₂₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₇₈	C₄	1	1	yes	characteristic
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	Dic₂₃₉⋊C₄	C₄	1	1	yes	characteristic
13	Dic₂₃₉⋊C₄	C₄	1	1	yes	characteristic
14	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
18	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
19	Dic₄₇₈	—	2	2	no	—
20	C₂×Dic₂₃₉	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₃₉	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₂₃₉	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₂₃₉	C₂×C₄	1	1	yes	characteristic
24	C₂₃₉⋊C₈	—	2	1	no	—
25	C₂₃₉⋊C₈	—	2	1	no	—
26	C₂×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
29	C₈×C₂₃₉	—	2	1	no	—
30	C₈×C₂₃₉	Q₈	1	2	yes	complemented
31	C₂²×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₃₉	—	2	1	no	—
33	C₄×C₂₃₉	—	2	1	no	—
34	C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₂₃₉	#41	1	1	yes	characteristic
36	Dic₂₃₉	—	2	1	no	—
37	Dic₂₃₉	—	2	1	no	—
38	Dic₂₃₉	—	2	1	no	—
39	Dic₂₃₉	M₁₆	1	2	yes	—
40	C₂×C₂₃₉	C₄×Q₈	1	1	yes	characteristic
41	C₂×C₂₃₉	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₂₃₉	#131	1	1	yes	characteristic
43	C₂₃₉	#446	1	1	yes	characteristic, complemented, Sylow
44	#446	—	239	1	no	maximal, Sylow
45	C₄×Q₈	—	239	1	no	—
46	C₄⋊C₈	—	239	1	no	—
47	C₄⋊C₈	—	239	1	no	—
48	C₄⋊C₈	—	239	1	no	—
49	#97	—	239	1	no	—
50	#97	D₂₃₉	1	1	yes	characteristic
51	C₄×C₈	—	239	1	no	—
52	C₂×Q₈	—	239	1	no	—
53	C₂×C₈	—	239	1	no	—
54	C₂×C₈	—	239	1	no	—
55	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
56	C₂×C₈	C₂×D₂₃₉	1	1	yes	characteristic
57	C₄⋊C₄	—	239	1	no	—
58	C₄⋊C₄	—	239	1	no	—
59	C₄⋊C₄	—	239	1	no	—
60	C₄²	—	239	1	no	—
61	C₄²	—	239	1	no	—
62	C₄²	C₂×D₂₃₉	1	1	yes	characteristic
63	Q₈	—	478	2	no	—
64	C₂×C₄	—	239	1	no	—
65	C₂×C₄	—	239	1	no	—
66	C₂×C₄	—	239	1	no	—
67	C₂×C₄	—	239	1	no	—
68	C₂×C₄	C₄×D₂₃₉	1	1	yes	characteristic
69	C₂×C₄	C₄×D₂₃₉	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
71	C₈	—	2	1	no	—
72	C₈	—	478	1	no	—
73	C₈	—	478	1	no	—
74	C₈	Dic₄₇₈	1	2	yes	complemented
75	C₂²	C₂×C₄×D₂₃₉	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	1	no	—
78	C₄	—	478	1	no	—
79	C₄	—	478	1	no	—
80	C₄	—	478	1	no	—
81	C₄	—	239	2	no	—
82	C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
83	C₄	#16838548	1	1	yes	characteristic
84	C₂	C₄×Dic₄₇₈	1	1	yes	characteristic
85	C₂	M₁₆×D₂₃₉	1	1	yes	characteristic
86	C₂	#21439968	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₃₉
commutator, socle, prime radical	C₂²×C₂₃₉	C₂²×C₄
Fitting	#97×C₂₃₉	C₂
cosocle	C₂×C₄×C₂₃₉	C₂³
hypercenter	#97	D₂₃₉
hypocenter	C₂₃₉	#446
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
30. C₈×C₂₃₉	Q₈	semidirect
43. C₂₃₉	#446	semidirect
74. C₈	Dic₄₇₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₄₇₈	2
C₄⋊(C₂₃₉⋊C₈)	2
Dic₂₃₉⋊C₈	2
#21439870	2
C₈×Dic₂₃₉	2
#97×C₂₃₉	2
#446	239