Subgroups — #41003046

Order 26944.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269760	C₂	1	2	yes	maximal
2	#27212366	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₆₈₄	C₂	1	1	yes	maximal, characteristic
4	#27212291	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₈₄₂	C₂²	1	2	yes	—
7	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄₂₁	—	2	1	no	—
9	#20269760	—	2	4	no	—
10	C₄⋊Dic₄₂₁	—	2	1	no	—
11	C₄⋊Dic₄₂₁	—	2	1	no	—
12	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
13	#20269748	—	2	1	no	—
14	C₂×C₈×C₄₂₁	C₂²	1	2	yes	—
15	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
16	C₂²×D₄₂₁	—	2	1	no	—
17	Dic₈₄₂	—	2	2	no	—
18	Dic₈₄₂	—	2	2	no	—
19	D₁₆₈₄	—	2	1	no	—
20	D₁₆₈₄	D₄	1	2	yes	—
21	C₂×Dic₄₂₁	—	2	1	no	—
22	C₂×Dic₄₂₁	—	2	2	no	—
23	C₄×D₄₂₁	—	4	2	no	—
24	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₄₂₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
27	C₈×C₄₂₁	—	2	2	no	—
28	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
29	C₂×D₄₂₁	—	2	2	no	—
30	C₂×D₄₂₁	—	4	1	no	—
31	C₂×D₄₂₁	—	4	1	no	—
32	C₄×C₄₂₁	—	2	1	no	—
33	C₄×C₄₂₁	SD₁₆	1	2	yes	complemented
34	C₄×C₄₂₁	C₂×D₄	1	1	yes	characteristic
35	C₄×C₄₂₁	#41	1	1	yes	characteristic
36	Dic₄₂₁	—	4	2	no	—
37	Dic₄₂₁	—	4	1	no	—
38	C₂×C₄₂₁	#121	1	1	yes	characteristic
39	C₂×C₄₂₁	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₄₂₁	#137	1	1	yes	characteristic
41	D₄₂₁	—	4	2	no	—
42	C₄₂₁	#461	1	1	yes	characteristic, complemented, Sylow
43	#461	—	421	1	no	maximal, Sylow
44	C₂×SD₁₆	—	421	2	no	—
45	C₄⋊Q₈	—	421	1	no	—
46	C₄×D₄	—	421	1	no	—
47	C₄⋊C₈	D₄₂₁	1	1	yes	characteristic, complemented
48	#103	—	421	2	no	—
49	C₂×Q₈	—	421	2	no	—
50	C₂×D₄	—	421	1	no	—
51	C₂²×C₄	—	842	1	no	—
52	SD₁₆	—	842	4	no	—
53	C₂×C₈	C₂×D₄₂₁	1	2	yes	—
54	C₄⋊C₄	—	842	1	no	—
55	C₄⋊C₄	—	842	1	no	—
56	C₄⋊C₄	—	421	1	no	—
57	C₂²⋊C₄	—	842	1	no	—
58	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
59	C₂³	—	842	1	no	—
60	Q₈	—	842	2	no	—
61	Q₈	—	842	2	no	—
62	D₄	—	842	1	no	—
63	D₄	—	421	2	no	—
64	C₂×C₄	—	842	1	no	—
65	C₂×C₄	—	842	2	no	—
66	C₂×C₄	—	1684	2	no	—
67	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
68	C₂×C₄	D₁₆₈₄	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₂²	—	842	2	no	—
72	C₂²	—	1684	1	no	—
73	C₂²	—	1684	1	no	—
74	C₂²	C₂×D₁₆₈₄	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	1684	2	no	—
77	C₄	—	1684	1	no	—
78	C₄	#20269760	1	2	yes	complemented
79	C₄	D₄×D₄₂₁	1	1	yes	characteristic
80	C₄	#20269772	1	1	yes	characteristic
81	C₂	—	1684	2	no	—
82	C₂	#27212341	1	1	yes	characteristic
83	C₂	C₂×#20269760	1	1	yes	characteristic
84	C₂	#27212399	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	C₂×D₁₆₈₄	C₂²
hypercenter	C₄⋊C₈	D₄₂₁
hypocenter	C₄₂₁	#461
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
2. #27212366	C₂	semidirect
4. #27212291	C₂	semidirect
5. C₄⋊C₈×C₄₂₁	C₂	semidirect
33. C₄×C₄₂₁	SD₁₆	semidirect
42. C₄₂₁	#461	semidirect
47. C₄⋊C₈	D₄₂₁	semidirect
78. 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
#27212366	2
C₄×D₁₆₈₄	2
#27212291	2
C₄⋊C₈×C₄₂₁	2
#461	421