Subgroups — #28417288

Order 14912.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21418592	C₂	1	1	yes	maximal, characteristic
2	(C₂×#11745400)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	#21418515	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11745400	—	2	1	no	—
5	C₂×#14272345	—	2	1	no	—
6	C₂×Dic₄₆₆	C₄	1	1	yes	characteristic, complemented
7	C₂×C₄×D₂₃₃	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₃₃	—	2	1	no	—
9	#16836571	—	2	1	no	—
10	C₄⋊C₄×C₂₃₃	C₄	1	1	yes	characteristic, complemented
11	C₂×#11745400	—	2	2	no	—
12	C₂×#11745400	—	4	1	no	—
13	C₂×#11745400	—	4	1	no	—
14	#14272345	—	4	1	no	—
15	C₂²×D₂₃₃	D₄	1	1	yes	characteristic
16	Dic₄₆₆	—	4	1	no	—
17	C₂×Dic₂₃₃	—	2	1	no	—
18	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
19	C₄×D₂₃₃	—	4	1	no	—
20	C₂×C₄×C₂₃₃	—	2	1	no	—
21	C₂×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×D₂₃₃	—	2	1	no	—
24	C₂×D₂₃₃	—	2	1	no	—
25	C₂×D₂₃₃	SD₁₆	1	1	yes	characteristic
26	C₂×D₂₃₃	Q₁₆	1	1	yes	characteristic
27	#11745400	—	4	2	no	—
28	C₄×C₂₃₃	—	2	1	no	—
29	C₄×C₂₃₃	—	4	1	no	—
30	Dic₂₃₃	—	2	1	no	—
31	Dic₂₃₃	—	4	1	no	—
32	C₂×C₂₃₃	#99	1	1	yes	characteristic
33	C₂×C₂₃₃	#103	1	1	yes	characteristic
34	C₂×C₂₃₃	#104	1	1	yes	characteristic
35	D₂₃₃	—	2	1	no	—
36	D₂₃₃	—	2	1	no	—
37	C₂₃₃	#328	1	1	yes	characteristic, complemented, Sylow
38	#328	—	233	1	no	maximal, Sylow
39	C₂²⋊Q₈	—	233	1	no	—
40	C₂²⋊C₈	—	233	1	no	—
41	#95	—	233	1	no	—
42	C₂×Q₈	—	233	1	no	—
43	C₂²×C₄	—	466	1	no	—
44	C₂²×C₄	—	233	1	no	—
45	C₂×C₈	—	466	1	no	—
46	C₄⋊C₄	—	466	1	no	—
47	C₄⋊C₄	#11745400	1	1	yes	characteristic, complemented
48	C₂²⋊C₄	—	466	1	no	—
49	C₂³	—	233	1	no	—
50	Q₈	—	932	1	no	—
51	C₂×C₄	—	2	1	no	—
52	C₂×C₄	—	932	1	no	—
53	C₂×C₄	—	932	1	no	—
54	C₂×C₄	—	932	1	no	—
55	C₂×C₄	—	466	2	no	—
56	C₂×C₄	—	466	1	no	—
57	C₂×C₄	—	233	1	no	—
58	C₂×C₄	C₂×#11745400	1	1	yes	characteristic
59	C₈	—	932	1	no	—
60	C₂²	—	466	1	no	—
61	C₂²	—	466	1	no	—
62	C₂²	—	233	1	no	—
63	C₂²	—	233	1	no	—
64	C₂²	C₂²⋊#11745400	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	4	1	no	—
67	C₄	—	932	2	no	—
68	C₄	—	932	1	no	—
69	C₄	—	466	1	no	—
70	C₂	—	466	1	no	—
71	C₂	—	466	1	no	—
72	C₂	#21418594	1	1	yes	characteristic
73	C₂	#21418598	1	1	yes	characteristic
74	C₂	#21418599	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11745400
commutator	C₂×C₄×C₂₃₃	C₂×C₄
Frattini	C₂×C₄	C₂×#11745400
Fitting	C₄⋊C₄×C₂₃₃	C₄
socle	C₂²×C₂₃₃	C₂²⋊C₄
prime radical	C₂²×D₂₃₃	D₄
cosocle	C₂×C₄×D₂₃₃	C₂²
hypercenter	C₄⋊C₄	#11745400
hypocenter	C₂₃₃	#328
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂²×C₂₃₃ ▷ C₂₃₃
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×Dic₄₆₆	C₄	semidirect
10. C₄⋊C₄×C₂₃₃	C₄	semidirect
37. C₂₃₃	#328	semidirect
47. C₄⋊C₄	#11745400	semidirect

Maximal subgroups

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

subgroup	index
#21418592	2
(C₂×#11745400)⋊C₄	2
#21418515	2
#328	233