Subgroups — #42122011

Order 27712.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20294084	C₂	1	2	yes	maximal, complemented
2	#28357217	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#16812358	C₂	1	1	yes	maximal, characteristic, complemented
4	#28357140	C₂	1	2	yes	maximal, complemented
5	C₄×Q₈×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₇₃₂	—	2	1	no	—
7	C₂×D₁₇₃₂	—	2	1	no	—
8	C₂×D₁₇₃₂	C₂²	1	2	yes	—
9	#20294084	—	2	4	no	—
10	C₂×#16812358	C₂²	1	2	yes	—
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	C₂²×D₄₃₃	—	2	2	no	—
17	D₁₇₃₂	—	2	2	no	—
18	D₁₇₃₂	—	2	2	no	—
19	D₁₇₃₂	—	4	2	no	—
20	D₁₇₃₂	—	4	2	no	—
21	#16812358	—	2	2	no	—
22	Q₈×C₄₃₃	—	2	1	no	—
23	Q₈×C₄₃₃	D₄	1	2	yes	complemented
24	C₂×C₄×C₄₃₃	—	2	1	no	—
25	C₂×C₄×C₄₃₃	—	2	1	no	—
26	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
28	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
30	C₂×D₄₃₃	—	4	2	no	—
31	C₂×D₄₃₃	—	4	2	no	—
32	C₂×D₄₃₃	—	4	2	no	—
33	C₄×C₄₃₃	—	2	2	no	—
34	C₄×C₄₃₃	—	2	1	no	—
35	C₄×C₄₃₃	—	4	1	no	—
36	C₄×C₄₃₃	SD₁₆	1	2	yes	complemented
37	C₄×C₄₃₃	C₂×D₄	1	1	yes	characteristic
38	C₄×C₄₃₃	#41	1	1	yes	characteristic
39	C₂×C₄₃₃	#121	1	1	yes	characteristic
40	C₂×C₄₃₃	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₄₃₃	AGL(1,ℤ/8)	1	1	yes	characteristic
42	D₄₃₃	—	8	2	no	—
43	C₄₃₃	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	433	1	no	maximal, Sylow
45	C₂×SD₁₆	—	433	2	no	—
46	#127	—	433	1	no	—
47	C₄×Q₈	D₄₃₃	1	1	yes	characteristic, complemented
48	C₄⋊C₈	—	433	1	no	—
49	#102	—	433	2	no	—
50	C₂×Q₈	C₂×D₄₃₃	1	1	yes	characteristic
51	C₂×D₄	—	866	1	no	—
52	C₂×D₄	—	866	1	no	—
53	C₂×D₄	—	433	2	no	—
54	SD₁₆	—	866	4	no	—
55	C₂×C₈	—	433	2	no	—
56	C₄⋊C₄	—	2	1	no	—
57	C₄⋊C₄	C₂×D₄₃₃	1	1	yes	characteristic
58	C₄²	—	2	1	no	—
59	C₄²	C₂×D₄₃₃	1	1	yes	characteristic
60	C₂³	—	866	2	no	—
61	Q₈	—	2	1	no	—
62	Q₈	D₁₇₃₂	1	2	yes	complemented
63	D₄	—	866	2	no	—
64	D₄	—	866	2	no	—
65	D₄	—	1732	2	no	—
66	D₄	—	1732	2	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂²⋊D₄₃₃	1	1	yes	characteristic
70	C₂×C₄	C₂²⋊D₄₃₃	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
72	C₈	—	866	2	no	—
73	C₂²	—	1732	2	no	—
74	C₂²	—	1732	2	no	—
75	C₂²	—	1732	2	no	—
76	C₂²	C₂×C₂²⋊D₄₃₃	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	#20294084	1	2	yes	complemented
81	C₄	C₂×D₁₇₃₂	1	1	yes	characteristic
82	C₄	#20294097	1	1	yes	characteristic
83	C₂	—	3464	2	no	—
84	C₂	#28357194	1	1	yes	characteristic
85	C₂	C₂×#20294084	1	1	yes	characteristic
86	C₂	#28357245	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	C₂²×C₄₃₃	C₂×D₄
prime radical	#28357217	C₂
hypercenter	C₄×Q₈	D₄₃₃
hypocenter	C₄₃₃	#460
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
1. C₂×#20294084	C₂	semidirect
3. C₄⋊#16812358	C₂	semidirect
4. #28357140	C₂	semidirect
5. C₄×Q₈×C₄₃₃	C₂	semidirect
23. Q₈×C₄₃₃	D₄	semidirect
36. C₄×C₄₃₃	SD₁₆	semidirect
43. C₄₃₃	#460	semidirect
47. C₄×Q₈	D₄₃₃	semidirect
62. Q₈	D₁₇₃₂	semidirect
80. C₄	#20294084	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20294084	2
#28357217	2
C₄⋊#16812358	2
#28357140	2
C₄×Q₈×C₄₃₃	2
#460	433