Subgroups — #41003159

Order 26944.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212340	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212298	C₂	1	2	yes	maximal, complemented
4	#27212288	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₄₂₁	—	2	1	no	—
8	C₂²×Dic₄₂₁	—	2	1	no	—
9	C₄⋊Dic₄₂₁	—	2	1	no	—
10	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
11	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
12	#20269748	—	2	1	no	—
13	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
15	C₂²×D₄₂₁	—	2	1	no	—
16	D₁₆₈₄	—	2	2	no	—
17	C₂²⋊D₄₂₁	—	4	2	no	—
18	C₂×Dic₄₂₁	—	2	1	no	—
19	C₂×Dic₄₂₁	—	2	2	no	—
20	C₂×Dic₄₂₁	—	4	2	no	—
21	C₂³×C₄₂₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₂₁	—	2	2	no	—
23	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₈×C₄₂₁	—	2	2	no	—
26	C₂²×C₄₂₁	—	2	1	no	—
27	C₂²×C₄₂₁	—	2	1	no	—
28	C₂²×C₄₂₁	SD₁₆	1	2	yes	—
29	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
30	C₂×D₄₂₁	—	4	1	no	—
31	C₂×D₄₂₁	—	4	1	no	—
32	C₂×D₄₂₁	—	4	1	no	—
33	C₄×C₄₂₁	—	2	1	no	—
34	C₄×C₄₂₁	#41	1	2	yes	—
35	Dic₄₂₁	—	4	1	no	—
36	Dic₄₂₁	—	4	2	no	—
37	C₂×C₄₂₁	—	2	2	no	—
38	C₂×C₄₂₁	#123	1	1	yes	characteristic
39	C₂×C₄₂₁	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
41	D₄₂₁	—	8	1	no	—
42	C₄₂₁	#481	1	1	yes	characteristic, complemented, Sylow
43	#481	—	421	1	no	maximal, Sylow
44	#121	—	421	1	no	—
45	C₂×C₄⋊C₄	—	421	1	no	—
46	#106	—	421	2	no	—
47	#102	—	421	2	no	—
48	C₂²⋊C₈	D₄₂₁	1	1	yes	characteristic, complemented
49	C₂×D₄	—	842	1	no	—
50	C₂×D₄	—	421	1	no	—
51	C₂²×C₄	—	842	1	no	—
52	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
53	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
54	C₄⋊C₄	—	842	1	no	—
55	C₄⋊C₄	—	421	2	no	—
56	C₄⋊C₄	—	421	1	no	—
57	C₂²⋊C₄	—	842	1	no	—
58	C₂³	—	842	1	no	—
59	C₂³	D₁₆₈₄	1	1	yes	characteristic
60	D₄	—	842	2	no	—
61	D₄	—	1684	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	842	1	no	—
64	C₂×C₄	—	842	2	no	—
65	C₂×C₄	—	1684	2	no	—
66	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	1684	1	no	—
72	C₂²	—	1684	1	no	—
73	C₂²	—	1684	1	no	—
74	C₂²	#20269760	1	2	yes	—
75	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	1684	1	no	—
78	C₄	—	1684	2	no	—
79	C₄	#20269770	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	3368	1	no	—
82	C₂	#27212350	1	1	yes	characteristic
83	C₂	C₂×#20269760	1	1	yes	characteristic
84	C₂	#27212394	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₆₈₄
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂²⋊C₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	#27212340	C₂
hypercenter	C₂²⋊C₈	D₄₂₁
hypocenter	C₄₂₁	#481
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₄₂₁	C₂	semidirect
3. #27212298	C₂	semidirect
4. #27212288	C₂	semidirect
5. C₂²⋊C₈×C₄₂₁	C₂	semidirect
42. C₄₂₁	#481	semidirect
48. C₂²⋊C₈	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212340	2
C₂×C₄⋊Dic₄₂₁	2
#27212298	2
#27212288	2
C₂²⋊C₈×C₄₂₁	2
#481	421