Subgroups — #41003041

Order 26944.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269760	C₂	1	2	yes	maximal, complemented
2	#27212363	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₈₄₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212288	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
7	C₂×D₁₆₈₄	—	2	1	no	—
8	C₂×D₁₆₈₄	—	2	1	no	—
9	C₂×D₁₆₈₄	C₂²	1	2	yes	—
10	#20269760	—	2	4	no	—
11	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
12	#20269750	—	2	1	no	—
13	C₄×Dic₄₂₁	—	2	1	no	—
14	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
15	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
16	C₂²×D₄₂₁	—	2	2	no	—
17	Dic₈₄₂	—	2	1	no	—
18	Dic₈₄₂	D₄	1	2	yes	complemented
19	D₁₆₈₄	—	2	2	no	—
20	D₁₆₈₄	—	2	2	no	—
21	D₁₆₈₄	—	4	2	no	—
22	D₁₆₈₄	—	4	2	no	—
23	C₂×Dic₄₂₁	—	2	1	no	—
24	C₂×Dic₄₂₁	—	2	1	no	—
25	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
27	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
28	C₈×C₄₂₁	—	2	2	no	—
29	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
30	C₂×D₄₂₁	—	4	2	no	—
31	C₂×D₄₂₁	—	4	2	no	—
32	C₂×D₄₂₁	—	4	2	no	—
33	C₄×C₄₂₁	—	2	1	no	—
34	C₄×C₄₂₁	SD₁₆	1	2	yes	complemented
35	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₂₁	#41	1	1	yes	characteristic
37	Dic₄₂₁	—	2	2	no	—
38	Dic₄₂₁	—	4	1	no	—
39	C₂×C₄₂₁	#121	1	1	yes	characteristic
40	C₂×C₄₂₁	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₄₂₁	AGL(1,ℤ/8)	1	1	yes	characteristic
42	D₄₂₁	—	8	2	no	—
43	C₄₂₁	#460	1	1	yes	characteristic, complemented, Sylow
44	#460	—	421	1	no	maximal, Sylow
45	C₂×SD₁₆	—	421	2	no	—
46	#127	—	421	1	no	—
47	C₄×Q₈	—	421	1	no	—
48	C₄⋊C₈	D₄₂₁	1	1	yes	characteristic, complemented
49	#102	—	421	2	no	—
50	C₂×Q₈	—	421	1	no	—
51	C₂×D₄	—	842	1	no	—
52	C₂×D₄	—	842	1	no	—
53	C₂×D₄	—	421	2	no	—
54	SD₁₆	—	842	4	no	—
55	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
56	C₄⋊C₄	—	842	1	no	—
57	C₄⋊C₄	—	421	1	no	—
58	C₄²	—	842	1	no	—
59	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
60	C₂³	—	842	2	no	—
61	Q₈	—	842	1	no	—
62	Q₈	—	421	2	no	—
63	D₄	—	842	2	no	—
64	D₄	—	842	2	no	—
65	D₄	—	1684	2	no	—
66	D₄	—	1684	2	no	—
67	C₂×C₄	—	842	1	no	—
68	C₂×C₄	—	842	1	no	—
69	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
70	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
72	C₈	—	2	2	no	—
73	C₂²	—	1684	2	no	—
74	C₂²	—	1684	2	no	—
75	C₂²	—	1684	2	no	—
76	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	842	2	no	—
79	C₄	—	1684	1	no	—
80	C₄	#20269760	1	2	yes	complemented
81	C₄	D₄×D₄₂₁	1	1	yes	characteristic
82	C₄	#20269772	1	1	yes	characteristic
83	C₂	—	3368	2	no	—
84	C₂	#27212341	1	1	yes	characteristic
85	C₂	C₂×#20269760	1	1	yes	characteristic
86	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	C₄⋊C₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	#27212363	C₂
hypercenter	C₄⋊C₈	D₄₂₁
hypocenter	C₄₂₁	#460
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#20269760	C₂	semidirect
3. C₄×Dic₈₄₂	C₂	semidirect
4. #27212288	C₂	semidirect
5. C₄⋊C₈×C₄₂₁	C₂	semidirect
18. Dic₈₄₂	D₄	semidirect
34. C₄×C₄₂₁	SD₁₆	semidirect
43. C₄₂₁	#460	semidirect
48. C₄⋊C₈	D₄₂₁	semidirect
80. C₄	#20269760	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269760	2
#27212363	2
C₄×Dic₈₄₂	2
#27212288	2
C₄⋊C₈×C₄₂₁	2
#460	421