Subgroups — #41003193

Order 26944.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212357	C₂	1	1	yes	maximal, characteristic, complemented
2	#27212353	C₂	1	1	yes	maximal, characteristic
3	#27212291	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212289	C₂	1	1	yes	maximal, characteristic, complemented
5	#27212287	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₄₂₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
9	C₂×#16809890	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₂₁	—	2	1	no	—
11	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
12	#20269750	—	2	1	no	—
13	C₂²⋊Dic₄₂₁	—	2	1	no	—
14	C₂²⋊Dic₄₂₁	—	2	1	no	—
15	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₄₂₁	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
19	Dic₈₄₂	—	2	1	no	—
20	Dic₈₄₂	—	2	1	no	—
21	C₂×Dic₄₂₁	—	2	1	no	—
22	C₂×Dic₄₂₁	—	2	1	no	—
23	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
24	C₂×Dic₄₂₁	D₄	1	1	yes	characteristic
25	#16809890	—	2	1	no	—
26	C₂³×C₄₂₁	—	2	1	no	—
27	D₄×C₄₂₁	—	2	1	no	—
28	D₄×C₄₂₁	—	2	1	no	—
29	C₂×C₄×C₄₂₁	—	2	1	no	—
30	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
31	C₈×C₄₂₁	—	2	1	no	—
32	C₂²×C₄₂₁	—	4	1	no	—
33	C₂²×C₄₂₁	—	4	1	no	—
34	C₂²×C₄₂₁	—	4	1	no	—
35	C₂²×C₄₂₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₂₁	—	4	1	no	—
37	C₄×C₄₂₁	#41	1	1	yes	characteristic
38	C₄×C₄₂₁	#41	1	1	yes	characteristic
39	Dic₄₂₁	—	2	1	no	—
40	Dic₄₂₁	—	2	1	no	—
41	Dic₄₂₁	—	4	1	no	—
42	Dic₄₂₁	—	4	1	no	—
43	C₂×C₄₂₁	—	8	1	no	—
44	C₂×C₄₂₁	#124	1	1	yes	characteristic
45	C₂×C₄₂₁	#135	1	1	yes	characteristic
46	C₂×C₄₂₁	#135	1	1	yes	characteristic
47	C₄₂₁	#488	1	1	yes	characteristic, complemented, Sylow
48	#488	—	421	1	no	maximal, Sylow
49	#125	—	421	1	no	—
50	#124	—	421	1	no	—
51	#103	—	421	1	no	—
52	#103	—	421	1	no	—
53	#102	—	421	1	no	—
54	#102	D₄₂₁	1	1	yes	characteristic
55	C₄×C₈	—	421	1	no	—
56	C₂×Q₈	—	421	1	no	—
57	C₂×D₄	C₂×D₄₂₁	1	1	yes	characteristic
58	C₂×C₈	—	421	1	no	—
59	C₂×C₈	C₂×D₄₂₁	1	1	yes	characteristic
60	C₄⋊C₄	—	842	1	no	—
61	C₄⋊C₄	—	842	1	no	—
62	C₄⋊C₄	—	421	1	no	—
63	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
64	C₂²⋊C₄	—	842	1	no	—
65	C₂²⋊C₄	—	842	1	no	—
66	C₄²	—	421	1	no	—
67	C₂³	—	2	1	no	—
68	Q₈	—	842	1	no	—
69	Q₈	—	842	1	no	—
70	D₄	—	2	1	no	—
71	D₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	842	1	no	—
74	C₂×C₄	—	842	1	no	—
75	C₂×C₄	—	421	1	no	—
76	C₂×C₄	—	421	1	no	—
77	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
78	C₈	—	2	1	no	—
79	C₈	—	842	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	4	1	no	—
82	C₂²	—	4	1	no	—
83	C₂²	D₄×D₄₂₁	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	842	1	no	—
86	C₄	—	842	1	no	—
87	C₄	—	1684	1	no	—
88	C₄	—	1684	1	no	—
89	C₄	#20269770	1	1	yes	characteristic
90	C₄	#20269771	1	1	yes	characteristic
91	C₂	—	8	1	no	—
92	C₂	#27212352	1	1	yes	characteristic
93	C₂	#27212385	1	1	yes	characteristic
94	C₂	#27212387	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₄₂₁
commutator, cosocle	C₂×C₄×C₄₂₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	#102×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂×D₄
prime radical	C₂×D₄×C₄₂₁	C₂²
hypercenter	#102	D₄₂₁
hypocenter	C₄₂₁	#488
LCS	G ▷ C₂×C₄×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

Complements

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

N	H ≅ G/N	type
1. #27212357	C₂	semidirect
3. #27212291	C₂	semidirect
4. #27212289	C₂	semidirect
6. C₈×Dic₄₂₁	C₂	semidirect
47. C₄₂₁	#488	semidirect

Maximal subgroups

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

subgroup	index
#27212357	2
#27212353	2
#27212291	2
#27212289	2
#27212287	2
C₈×Dic₄₂₁	2
#102×C₄₂₁	2
#488	421