Subgroups — #42122008

Order 27712.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20294085	C₂	1	2	yes	maximal, complemented
2	C₄×Dic₈₆₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#16812358	C₂	1	1	yes	maximal, characteristic, complemented
4	#28357141	C₂	1	2	yes	maximal, complemented
5	#127×C₄₃₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₆₆	C₂²	1	1	yes	characteristic
7	#20294085	—	2	4	no	—
8	C₂×#16812358	C₂²	1	2	yes	—
9	C₄⋊Dic₄₃₃	C₂²	1	1	yes	characteristic
10	#20294076	—	2	1	no	—
11	C₄×Dic₄₃₃	—	2	1	no	—
12	C₂×D₄×C₄₃₃	—	2	1	no	—
13	C₂×D₄×C₄₃₃	—	2	1	no	—
14	C₂×D₄×C₄₃₃	C₂²	1	2	yes	—
15	C₄²×C₄₃₃	C₂²	1	1	yes	characteristic
16	Dic₈₆₆	—	2	1	no	—
17	Dic₈₆₆	D₄	1	2	yes	complemented
18	C₂×Dic₄₃₃	—	2	1	no	—
19	C₂×Dic₄₃₃	—	2	1	no	—
20	#16812358	—	2	2	no	—
21	C₂³×C₄₃₃	—	2	2	no	—
22	D₄×C₄₃₃	—	2	2	no	—
23	D₄×C₄₃₃	—	2	2	no	—
24	D₄×C₄₃₃	—	4	2	no	—
25	D₄×C₄₃₃	—	4	2	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	—	4	2	no	—
30	C₂²×C₄₃₃	—	4	2	no	—
31	C₂²×C₄₃₃	—	4	2	no	—
32	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₄₃₃	—	2	1	no	—
34	C₄×C₄₃₃	SD₁₆	1	2	yes	complemented
35	C₄×C₄₃₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₃₃	#41	1	1	yes	characteristic
37	Dic₄₃₃	—	2	2	no	—
38	Dic₄₃₃	—	4	1	no	—
39	C₂×C₄₃₃	—	8	2	no	—
40	C₂×C₄₃₃	#121	1	1	yes	characteristic
41	C₂×C₄₃₃	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₄₃₃	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₄₃₃	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	433	1	no	maximal, Sylow
45	C₂×SD₁₆	—	433	2	no	—
46	#127	D₄₃₃	1	1	yes	characteristic
47	C₄×Q₈	—	433	1	no	—
48	C₄⋊C₈	—	433	1	no	—
49	#102	—	433	2	no	—
50	C₂×Q₈	—	433	1	no	—
51	C₂×D₄	—	2	1	no	—
52	C₂×D₄	—	2	1	no	—
53	C₂×D₄	C₂×D₄₃₃	1	2	yes	—
54	SD₁₆	—	866	4	no	—
55	C₂×C₈	—	433	2	no	—
56	C₄⋊C₄	—	866	1	no	—
57	C₄⋊C₄	—	433	1	no	—
58	C₄²	—	866	1	no	—
59	C₄²	C₂×D₄₃₃	1	1	yes	characteristic
60	C₂³	—	2	2	no	—
61	Q₈	—	866	1	no	—
62	Q₈	—	433	2	no	—
63	D₄	—	2	2	no	—
64	D₄	—	2	2	no	—
65	D₄	—	4	2	no	—
66	D₄	—	4	2	no	—
67	C₂×C₄	—	866	1	no	—
68	C₂×C₄	—	866	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	—	4	2	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	4	2	no	—
76	C₂²	C₂×C₂²⋊D₄₃₃	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	866	2	no	—
79	C₄	—	1732	1	no	—
80	C₄	#20294085	1	2	yes	complemented
81	C₄	D₄×D₄₃₃	1	1	yes	characteristic
82	C₄	#20294096	1	1	yes	characteristic
83	C₂	—	8	2	no	—
84	C₂	#28357191	1	1	yes	characteristic
85	C₂	C₂×#20294085	1	1	yes	characteristic
86	C₂	#28357246	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₃₃
commutator, cosocle	C₂×C₄×C₄₃₃	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₃
Fitting, prime radical	#127×C₄₃₃	C₂
socle	C₂²×C₄₃₃	C₂×D₄
hypercenter	#127	D₄₃₃
hypocenter	C₄₃₃	#460
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#20294085	C₂	semidirect
2. C₄×Dic₈₆₆	C₂	semidirect
3. C₄⋊#16812358	C₂	semidirect
4. #28357141	C₂	semidirect
17. Dic₈₆₆	D₄	semidirect
34. C₄×C₄₃₃	SD₁₆	semidirect
43. C₄₃₃	#460	semidirect
80. C₄	#20294085	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20294085	2
C₄×Dic₈₆₆	2
C₄⋊#16812358	2
#28357141	2
#127×C₄₃₃	2
#460	433