Subgroups — #42122129

Order 27712.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28357194	C₂	1	1	yes	maximal, characteristic
2	#28357151	C₂	1	2	yes	maximal, complemented
3	#28357140	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#16812358	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₄₃₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₇₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₄₃₃	—	2	1	no	—
8	C₂×#16812358	C₂²	1	2	yes	—
9	C₄⋊Dic₄₃₃	C₂²	1	1	yes	characteristic
10	#20294074	—	2	1	no	—
11	C₂²×C₄×C₄₃₃	—	2	1	no	—
12	C₂²×C₄×C₄₃₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₃₃	—	2	1	no	—
14	C₄⋊C₄×C₄₃₃	C₂²	1	2	yes	—
15	C₂²×D₄₃₃	—	2	1	no	—
16	D₁₇₃₂	—	2	2	no	—
17	C₂²⋊D₄₃₃	—	4	2	no	—
18	C₂×Dic₄₃₃	—	2	1	no	—
19	#16812358	—	2	2	no	—
20	C₂³×C₄₃₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₃₃	—	2	2	no	—
22	C₂×C₄×C₄₃₃	—	2	2	no	—
23	C₂×C₄×C₄₃₃	—	4	2	no	—
24	C₂×C₄×C₄₃₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₄₃₃	C₂³	1	1	yes	characteristic
26	C₂²×C₄₃₃	—	2	1	no	—
27	C₂²×C₄₃₃	—	2	1	no	—
28	C₂²×C₄₃₃	SD₁₆	1	2	yes	—
29	C₂²×C₄₃₃	C₂×D₄	1	1	yes	characteristic
30	C₂×D₄₃₃	—	4	1	no	—
31	C₂×D₄₃₃	—	4	1	no	—
32	C₂×D₄₃₃	—	4	1	no	—
33	C₄×C₄₃₃	—	2	1	no	—
34	C₄×C₄₃₃	—	4	2	no	—
35	C₄×C₄₃₃	#41	1	2	yes	—
36	Dic₄₃₃	—	4	1	no	—
37	C₂×C₄₃₃	—	2	2	no	—
38	C₂×C₄₃₃	#123	1	1	yes	characteristic
39	C₂×C₄₃₃	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₄₃₃	AGL(1,ℤ/8)	1	1	yes	characteristic
41	D₄₃₃	—	8	1	no	—
42	C₄₃₃	#481	1	1	yes	characteristic, complemented, Sylow
43	#481	—	433	1	no	maximal, Sylow
44	#121	—	433	1	no	—
45	C₂×C₄⋊C₄	D₄₃₃	1	1	yes	characteristic, complemented
46	#106	—	433	2	no	—
47	#102	—	433	2	no	—
48	C₂²⋊C₈	—	433	1	no	—
49	C₂×D₄	—	866	1	no	—
50	C₂×D₄	—	433	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	C₂×D₄₃₃	1	1	yes	characteristic
53	C₂×C₈	—	433	2	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	433	1	no	—
56	C₄⋊C₄	C₂×D₄₃₃	1	2	yes	—
57	C₂²⋊C₄	—	866	1	no	—
58	C₂³	—	866	1	no	—
59	C₂³	C₂²⋊D₄₃₃	1	1	yes	characteristic
60	D₄	—	866	2	no	—
61	D₄	—	1732	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	866	1	no	—
66	C₂×C₄	C₂²⋊D₄₃₃	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₄₃₃	1	1	yes	characteristic
68	C₈	—	866	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	1732	1	no	—
72	C₂²	—	1732	1	no	—
73	C₂²	—	1732	1	no	—
74	C₂²	#20294084	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₄₃₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	1732	1	no	—
79	C₄	#20294097	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	3464	1	no	—
82	C₂	#28357203	1	1	yes	characteristic
83	C₂	C₂×#20294084	1	1	yes	characteristic
84	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	C₂×C₄⋊C₄×C₄₃₃	C₂
socle	C₂²×C₄₃₃	C₂×D₄
prime radical	#28357194	C₂
hypercenter	C₂×C₄⋊C₄	D₄₃₃
hypocenter	C₄₃₃	#481
LCS	G ▷ C₂×C₄×C₄₃₃ ▷ C₂×C₄₃₃ ▷ C₄₃₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #28357151	C₂	semidirect
3. #28357140	C₂	semidirect
4. C₂²⋊#16812358	C₂	semidirect
5. C₂×C₄⋊C₄×C₄₃₃	C₂	semidirect
42. C₄₃₃	#481	semidirect
45. C₂×C₄⋊C₄	D₄₃₃	semidirect

Maximal subgroups

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

subgroup	index
#28357194	2
#28357151	2
#28357140	2
C₂²⋊#16812358	2
C₂×C₄⋊C₄×C₄₃₃	2
#481	433