Subgroups — #42122021

Order 27712.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20294087	C₂	1	2	yes	maximal
2	#28357220	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#16812358	C₂	1	1	yes	maximal, characteristic
4	#28357143	C₂	1	2	yes	maximal
5	C₄×Q₈×C₄₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₆₆	C₂²	1	2	yes	—
7	#20294087	—	2	4	no	—
8	C₂×#16812358	C₂²	1	2	yes	—
9	C₄⋊Dic₄₃₃	—	2	1	no	—
10	C₄⋊Dic₄₃₃	—	2	1	no	—
11	C₂×Q₈×C₄₃₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₄₃₃	—	2	1	no	—
13	C₄⋊C₄×C₄₃₃	C₂²	1	1	yes	characteristic
14	C₄²×C₄₃₃	—	2	1	no	—
15	C₄²×C₄₃₃	C₂²	1	1	yes	characteristic
16	Dic₈₆₆	—	2	2	no	—
17	Dic₈₆₆	—	2	2	no	—
18	C₂×Dic₄₃₃	—	2	2	no	—
19	#16812358	—	2	2	no	—
20	Q₈×C₄₃₃	—	2	1	no	—
21	Q₈×C₄₃₃	D₄	1	2	yes	—
22	C₂×C₄×C₄₃₃	—	2	1	no	—
23	C₂×C₄×C₄₃₃	—	2	1	no	—
24	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₄₃₃	—	2	2	no	—
29	C₄×C₄₃₃	—	2	1	no	—
30	C₄×C₄₃₃	—	4	1	no	—
31	C₄×C₄₃₃	Q₁₆	1	2	yes	complemented
32	C₄×C₄₃₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₄₃₃	#41	1	1	yes	characteristic
34	Dic₄₃₃	—	4	2	no	—
35	C₂×C₄₃₃	#121	1	1	yes	characteristic
36	C₂×C₄₃₃	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₄₃₃	#137	1	1	yes	characteristic
38	C₄₃₃	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	433	1	no	maximal, Sylow
40	C₂×Q₁₆	—	433	2	no	—
41	C₄⋊Q₈	—	433	1	no	—
42	C₄×Q₈	D₄₃₃	1	1	yes	characteristic
43	C₄⋊C₈	—	433	1	no	—
44	#103	—	433	2	no	—
45	C₂×Q₈	—	433	2	no	—
46	C₂×Q₈	C₂×D₄₃₃	1	1	yes	characteristic
47	Q₁₆	—	866	4	no	—
48	C₂×C₈	—	433	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	866	1	no	—
51	C₄⋊C₄	—	866	1	no	—
52	C₄⋊C₄	C₂×D₄₃₃	1	1	yes	characteristic
53	C₄²	—	2	1	no	—
54	C₄²	C₂×D₄₃₃	1	1	yes	characteristic
55	Q₈	—	2	1	no	—
56	Q₈	—	866	2	no	—
57	Q₈	—	866	2	no	—
58	Q₈	D₁₇₃₂	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	866	2	no	—
62	C₂×C₄	C₂²⋊D₄₃₃	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₄₃₃	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
65	C₈	—	866	2	no	—
66	C₂²	C₂×C₂²⋊D₄₃₃	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	1732	2	no	—
71	C₄	#20294087	1	2	yes	complemented
72	C₄	C₂×D₁₇₃₂	1	1	yes	characteristic
73	C₄	#20294097	1	1	yes	characteristic
74	C₂	#28357194	1	1	yes	characteristic
75	C₂	C₂×#20294087	1	1	yes	characteristic
76	C₂	#28357250	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₃₃
commutator, cosocle	C₂×C₄×C₄₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₃
Fitting	C₄×Q₈×C₄₃₃	C₂
socle, prime radical	C₂²×C₄₃₃	C₂×D₄
hypercenter	C₄×Q₈	D₄₃₃
hypocenter	C₄₃₃	#462
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
31. C₄×C₄₃₃	Q₁₆	semidirect
38. C₄₃₃	#462	semidirect
71. C₄	#20294087	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20294087	2
#28357220	2
C₄⋊#16812358	2
#28357143	2
C₄×Q₈×C₄₃₃	2
#462	433