Subgroups — #42122097

Order 27712.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357220	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₈₆₆	C₂	1	1	yes	maximal, characteristic
3	#28357152	C₂	1	2	yes	maximal
4	#28357145	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₄₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₆₆	—	2	1	no	—
7	C₂×Dic₈₆₆	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₄₃₃	—	2	1	no	—
9	C₄⋊Dic₄₃₃	C₂²	1	2	yes	—
10	C₄⋊Dic₄₃₃	C₂²	1	1	yes	characteristic
11	#20294076	—	2	1	no	—
12	C₄×Dic₄₃₃	—	2	1	no	—
13	C₂×C₈×C₄₃₃	C₂²	1	2	yes	—
14	C₄²×C₄₃₃	C₂²	1	1	yes	characteristic
15	Dic₈₆₆	—	2	1	no	—
16	Dic₈₆₆	—	4	2	no	—
17	Dic₈₆₆	Q₈	1	2	yes	complemented
18	C₂×Dic₄₃₃	—	2	1	no	—
19	C₂×Dic₄₃₃	—	2	1	no	—
20	C₂×Dic₄₃₃	—	2	2	no	—
21	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
24	C₈×C₄₃₃	—	2	2	no	—
25	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
26	C₄×C₄₃₃	—	2	1	no	—
27	C₄×C₄₃₃	SD₁₆	1	2	yes	—
28	C₄×C₄₃₃	C₂×Q₈	1	1	yes	characteristic
29	C₄×C₄₃₃	#41	1	1	yes	characteristic
30	Dic₄₃₃	—	2	2	no	—
31	Dic₄₃₃	—	4	2	no	—
32	Dic₄₃₃	—	4	1	no	—
33	C₂×C₄₃₃	C₂²⋊Q₈	1	1	yes	characteristic
34	C₂×C₄₃₃	C₂×SD₁₆	1	1	yes	characteristic
35	C₂×C₄₃₃	#137	1	1	yes	characteristic
36	C₄₃₃	#475	1	1	yes	characteristic, complemented, Sylow
37	#475	—	433	1	no	maximal, Sylow
38	C₄⋊Q₈	—	433	1	no	—
39	C₄×Q₈	—	433	1	no	—
40	#106	—	433	2	no	—
41	C₄⋊C₈	D₄₃₃	1	1	yes	characteristic
42	#103	—	433	2	no	—
43	C₂×Q₈	—	866	1	no	—
44	C₂×Q₈	—	433	1	no	—
45	C₂×C₈	C₂×D₄₃₃	1	2	yes	—
46	C₄⋊C₄	—	866	1	no	—
47	C₄⋊C₄	—	866	1	no	—
48	C₄⋊C₄	—	433	2	no	—
49	C₄⋊C₄	—	433	1	no	—
50	C₄²	—	866	1	no	—
51	C₄²	C₂×D₄₃₃	1	1	yes	characteristic
52	Q₈	—	866	1	no	—
53	Q₈	—	433	2	no	—
54	Q₈	—	1732	2	no	—
55	C₂×C₄	—	866	1	no	—
56	C₂×C₄	—	866	1	no	—
57	C₂×C₄	—	866	2	no	—
58	C₂×C₄	D₁₇₃₂	1	1	yes	characteristic
59	C₂×C₄	D₁₇₃₂	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₂²	C₂×D₁₇₃₂	1	1	yes	characteristic
63	C₄	—	2	1	no	—
64	C₄	—	866	2	no	—
65	C₄	—	1732	2	no	—
66	C₄	—	1732	1	no	—
67	C₄	#20294086	1	2	yes	—
68	C₄	Q₈×D₄₃₃	1	1	yes	characteristic
69	C₄	#20294096	1	1	yes	characteristic
70	C₂	#28357200	1	1	yes	characteristic
71	C₂	C₂×#20294086	1	1	yes	characteristic
72	C₂	#28357253	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₇₃₂
commutator, cosocle	C₂×C₄×C₄₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₃
Fitting	C₄⋊C₈×C₄₃₃	C₂
socle, prime radical	C₂²×C₄₃₃	C₂×D₄
hypercenter	C₄⋊C₈	D₄₃₃
hypocenter	C₄₃₃	#475
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
17. Dic₈₆₆	Q₈	semidirect
36. C₄₃₃	#475	semidirect

Maximal subgroups

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

subgroup	index
#28357220	2
C₄×Dic₈₆₆	2
#28357152	2
#28357145	2
C₄⋊C₈×C₄₃₃	2
#475	433