Subgroups — #28412334

Order 14656.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20301045	C₂	1	1	yes	maximal, characteristic
2	(C₂×#11745021)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#20300968	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11745021	—	2	1	no	—
5	C₂×#14270087	—	2	1	no	—
6	C₂×Dic₄₅₈	C₄	1	1	yes	characteristic, complemented
7	C₂×C₄×D₂₂₉	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₂₉	—	2	1	no	—
9	#16814643	—	2	1	no	—
10	C₄⋊C₄×C₂₂₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#11745021	—	2	2	no	—
12	C₂×#11745021	—	4	1	no	—
13	C₂×#11745021	—	4	1	no	—
14	#14270087	—	4	1	no	—
15	C₂²×D₂₂₉	D₄	1	1	yes	characteristic
16	Dic₄₅₈	—	4	1	no	—
17	C₂×Dic₂₂₉	—	2	1	no	—
18	C₂×Dic₂₂₉	D₄	1	1	yes	characteristic
19	C₄×D₂₂₉	—	4	1	no	—
20	C₂×C₄×C₂₂₉	—	2	1	no	—
21	C₂×C₄×C₂₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₂₂₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₂₂₉	—	2	1	no	—
24	C₂×D₂₂₉	—	2	1	no	—
25	C₂×D₂₂₉	SD₁₆	1	1	yes	characteristic
26	C₂×D₂₂₉	Q₁₆	1	1	yes	characteristic
27	#11745021	—	4	2	no	—
28	C₄×C₂₂₉	—	2	1	no	—
29	C₄×C₂₂₉	—	4	1	no	—
30	Dic₂₂₉	—	2	1	no	—
31	Dic₂₂₉	—	4	1	no	—
32	C₂×C₂₂₉	#99	1	1	yes	characteristic
33	C₂×C₂₂₉	#103	1	1	yes	characteristic
34	C₂×C₂₂₉	#104	1	1	yes	characteristic
35	D₂₂₉	—	2	1	no	—
36	D₂₂₉	—	2	1	no	—
37	C₂₂₉	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	229	1	no	maximal, Sylow
39	C₂²⋊Q₈	—	229	1	no	—
40	C₂²⋊C₈	—	229	1	no	—
41	#95	—	229	1	no	—
42	C₂×Q₈	—	229	1	no	—
43	C₂²×C₄	—	458	1	no	—
44	C₂²×C₄	—	229	1	no	—
45	C₂×C₈	—	458	1	no	—
46	C₄⋊C₄	—	458	1	no	—
47	C₄⋊C₄	#11745021	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	458	1	no	—
49	C₂³	—	229	1	no	—
50	Q₈	—	916	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	916	1	no	—
53	C₂×C₄	—	916	1	no	—
54	C₂×C₄	—	916	1	no	—
55	C₂×C₄	—	458	2	no	—
56	C₂×C₄	—	458	1	no	—
57	C₂×C₄	—	229	1	no	—
58	C₂×C₄	C₂×#11745021	1	1	yes	characteristic
59	C₈	—	916	1	no	—
60	C₂²	—	458	1	no	—
61	C₂²	—	458	1	no	—
62	C₂²	—	229	1	no	—
63	C₂²	—	229	1	no	—
64	C₂²	C₂²⋊#11745021	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	916	2	no	—
68	C₄	—	916	1	no	—
69	C₄	—	458	1	no	—
70	C₂	—	458	1	no	—
71	C₂	—	458	1	no	—
72	C₂	#20301047	1	1	yes	characteristic
73	C₂	#20301051	1	1	yes	characteristic
74	C₂	#20301052	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11745021
commutator	C₂×C₄×C₂₂₉	C₂×C₄
Frattini	C₂×C₄	C₂×#11745021
Fitting	C₄⋊C₄×C₂₂₉	C₄
socle	C₂²×C₂₂₉	C₂²⋊C₄
prime radical	C₂²×D₂₂₉	D₄
cosocle	C₂×C₄×D₂₂₉	C₂²
hypercenter	C₄⋊C₄	#11745021
hypocenter	C₂₂₉	#328
LCS	G ▷ C₂×C₄×C₂₂₉ ▷ C₂²×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. C₂×Dic₄₅₈	C₄	semidirect
10. C₄⋊C₄×C₂₂₉	C₄	semidirect
37. C₂₂₉	#328	semidirect
47. C₄⋊C₄	#11745021	semidirect

Maximal subgroups

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

subgroup	index
#20301045	2
(C₂×#11745021)⋊C₄	2
#20300968	2
#328	229