Subgroups — #28416717

Order 14912.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₄₆₆	C₂	1	1	yes	maximal, characteristic
2	#21418466	C₂	1	2	yes	maximal
3	C₄⋊#14272338	C₂	1	1	yes	maximal, characteristic
4	#21418458	C₂	1	2	yes	maximal
5	C₄⋊Q₈×C₂₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₆₆	C₂²	1	1	yes	characteristic
7	C₂×#14272338	C₂²	1	2	yes	—
8	C₄⋊Dic₂₃₃	C₂²	1	1	yes	characteristic
9	#16836573	—	2	1	no	—
10	C₄×Dic₂₃₃	—	2	1	no	—
11	C₂×Q₈×C₂₃₃	—	2	1	no	—
12	C₄⋊C₄×C₂₃₃	—	2	1	no	—
13	C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
14	C₄²×C₂₃₃	C₂²	1	1	yes	characteristic
15	Dic₄₆₆	—	2	1	no	—
16	Dic₄₆₆	Q₈	1	2	yes	complemented
17	C₂×Dic₂₃₃	—	2	1	no	—
18	C₂×Dic₂₃₃	—	2	1	no	—
19	#14272338	—	2	2	no	—
20	Q₈×C₂₃₃	—	4	2	no	—
21	C₂×C₄×C₂₃₃	—	2	2	no	—
22	C₂×C₄×C₂₃₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₃₃	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
25	C₂²×C₂₃₃	C₂×D₄	1	1	yes	characteristic
26	C₄×C₂₃₃	—	2	1	no	—
27	C₄×C₂₃₃	—	4	2	no	—
28	C₄×C₂₃₃	SD₁₆	1	2	yes	—
29	C₄×C₂₃₃	C₂×Q₈	1	1	yes	characteristic
30	C₄×C₂₃₃	#41	1	1	yes	characteristic
31	Dic₂₃₃	—	2	2	no	—
32	Dic₂₃₃	—	4	1	no	—
33	C₂×C₂₃₃	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₂₃₃	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₂₃₃	#137	1	1	yes	characteristic
36	C₂₃₃	#475	1	1	yes	characteristic, complemented, Sylow
37	#475	—	233	1	no	maximal, Sylow
38	C₄⋊Q₈	D₂₃₃	1	1	yes	characteristic
39	C₄×Q₈	—	233	1	no	—
40	#106	—	233	2	no	—
41	C₄⋊C₈	—	233	1	no	—
42	#103	—	233	2	no	—
43	C₂×Q₈	—	2	1	no	—
44	C₂×Q₈	—	233	1	no	—
45	C₂×C₈	—	233	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	—	466	1	no	—
48	C₄⋊C₄	—	233	1	no	—
49	C₄⋊C₄	C₂×D₂₃₃	1	2	yes	—
50	C₄²	—	466	1	no	—
51	C₄²	C₂×D₂₃₃	1	1	yes	characteristic
52	Q₈	—	4	2	no	—
53	Q₈	—	466	1	no	—
54	Q₈	—	233	2	no	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	466	1	no	—
57	C₂×C₄	—	466	1	no	—
58	C₂×C₄	C₂²⋊D₂₃₃	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₂₃₃	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
61	C₈	—	466	2	no	—
62	C₂²	C₂×C₂²⋊D₂₃₃	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	4	2	no	—
65	C₄	—	932	1	no	—
66	C₄	—	466	2	no	—
67	C₄	#16836582	1	2	yes	—
68	C₄	Q₈×D₂₃₃	1	1	yes	characteristic
69	C₄	#16836595	1	1	yes	characteristic
70	C₂	#21418511	1	1	yes	characteristic
71	C₂	C₂×#16836582	1	1	yes	characteristic
72	C₂	#21418566	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₃₃
commutator, cosocle	C₂×C₄×C₂₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₃₃
Fitting	C₄⋊Q₈×C₂₃₃	C₂
socle, prime radical	C₂²×C₂₃₃	C₂×D₄
hypercenter	C₄⋊Q₈	D₂₃₃
hypocenter	C₂₃₃	#475
LCS	G ▷ C₂×C₄×C₂₃₃ ▷ C₂×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
16. Dic₄₆₆	Q₈	semidirect
36. C₂₃₃	#475	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₄₆₆	2
#21418466	2
C₄⋊#14272338	2
#21418458	2
C₄⋊Q₈×C₂₃₃	2
#475	233