Subgroups — #28417294

Order 14912.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21418602	C₂	1	1	yes	maximal, characteristic
2	#21418602	C₂	1	1	yes	maximal, characteristic
3	#21418534	C₂	1	1	yes	maximal, characteristic
4	C₂×#14272345	—	2	1	no	—
5	C₂×#14272345	—	2	1	no	—
6	C₂×Dic₄₆₆	—	2	1	no	—
7	C₄⋊Dic₂₃₃	C₄	1	1	yes	characteristic
8	#16836573	—	2	1	no	—
9	C₄×Dic₂₃₃	C₂²	1	1	yes	characteristic
10	C₄⋊C₄×C₂₃₃	C₄	1	1	yes	characteristic
11	#14272345	—	4	1	no	—
12	#14272345	—	4	1	no	—
13	Dic₄₆₆	—	4	1	no	—
14	Dic₄₆₆	—	4	1	no	—
15	C₂×Dic₂₃₃	—	2	1	no	—
16	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
17	C₂×Dic₂₃₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₃₃	—	2	1	no	—
19	C₂×C₄×C₂₃₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₂₃₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₃₃	—	2	1	no	—
22	C₄×C₂₃₃	—	4	1	no	—
23	Dic₂₃₃	—	4	1	no	—
24	Dic₂₃₃	D₈	1	1	yes	characteristic
25	Dic₂₃₃	SD₁₆	1	1	yes	characteristic
26	Dic₂₃₃	SD₁₆	1	1	yes	characteristic
27	Dic₂₃₃	Q₁₆	1	1	yes	characteristic
28	C₂×C₂₃₃	#101	1	1	yes	characteristic
29	C₂×C₂₃₃	#102	1	1	yes	characteristic
30	C₂×C₂₃₃	#103	1	1	yes	characteristic
31	C₂₃₃	#332	1	1	yes	characteristic, complemented, Sylow
32	#332	—	233	1	no	maximal, Sylow
33	C₄⋊Q₈	—	233	1	no	—
34	C₄⋊C₈	—	233	1	no	—
35	C₄⋊C₈	—	233	1	no	—
36	C₂×Q₈	—	466	1	no	—
37	C₂×C₈	—	466	1	no	—
38	C₂×C₈	—	466	1	no	—
39	C₄⋊C₄	—	466	1	no	—
40	C₄⋊C₄	—	233	1	no	—
41	C₄⋊C₄	#11745400	1	1	yes	characteristic
42	C₄²	—	233	1	no	—
43	Q₈	—	932	1	no	—
44	Q₈	—	932	1	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	466	1	no	—
47	C₂×C₄	—	233	1	no	—
48	C₂×C₄	—	233	1	no	—
49	C₂×C₄	C₂×#11745400	1	1	yes	characteristic
50	C₈	—	932	1	no	—
51	C₈	—	932	1	no	—
52	C₂²	C₂²⋊#11745400	1	1	yes	characteristic
53	C₄	—	2	1	no	—
54	C₄	—	4	1	no	—
55	C₄	—	932	1	no	—
56	C₄	—	233	1	no	—
57	C₄	—	233	1	no	—
58	C₄	—	233	1	no	—
59	C₄	—	233	1	no	—
60	C₂	#21418596	1	1	yes	characteristic
61	C₂	#21418597	1	1	yes	characteristic
62	C₂	#21418598	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, prime radical	C₂²×C₂₃₃	C₂²⋊C₄
cosocle	C₄×Dic₂₃₃	C₂²
hypercenter	C₄⋊C₄	#11745400
hypocenter	C₂₃₃	#332
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
31. C₂₃₃	#332	semidirect

Maximal subgroups

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

subgroup	index
#21418602	2
#21418534	2
#332	233