Subgroups — #41002707

Order 26944.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
2	#27212273	C₂	1	2	yes	maximal
3	#27212273	C₂	1	2	yes	maximal
4	#27212273	C₂	1	1	yes	maximal, characteristic
5	#95×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₁	C₂²	1	2	yes	—
7	C₂²×Dic₄₂₁	C₂²	1	1	yes	characteristic
8	C₂²×Dic₄₂₁	C₂²	1	1	yes	characteristic
9	C₄×Dic₄₂₁	C₄	1	4	yes	complemented
10	C₂²×C₄×C₄₂₁	C₂²	1	2	yes	—
11	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₂₁	—	2	1	no	—
13	C₂×Dic₄₂₁	—	2	2	no	—
14	C₂×Dic₄₂₁	—	2	1	no	—
15	C₂×Dic₄₂₁	—	2	2	no	—
16	C₂×Dic₄₂₁	—	2	2	no	—
17	C₂×Dic₄₂₁	—	2	2	no	—
18	C₂×Dic₄₂₁	C₂×C₄	1	2	yes	—
19	C₂×Dic₄₂₁	C₂×C₄	1	2	yes	—
20	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₄₂₁	—	2	2	no	—
22	C₂×C₄×C₄₂₁	—	2	2	no	—
23	C₂×C₄×C₄₂₁	—	2	2	no	—
24	C₂×C₄×C₄₂₁	—	2	2	no	—
25	C₂×C₄×C₄₂₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄₂₁	#41	1	2	yes	—
28	C₂²×C₄₂₁	#41	1	1	yes	characteristic
29	C₂²×C₄₂₁	#41	1	2	yes	—
30	C₂²×C₄₂₁	#41	1	1	yes	characteristic
31	C₄×C₄₂₁	—	2	2	no	—
32	C₄×C₄₂₁	—	4	2	no	—
33	Dic₄₂₁	—	2	2	no	—
34	Dic₄₂₁	—	2	2	no	—
35	Dic₄₂₁	—	4	2	no	—
36	C₂×C₄₂₁	#117	1	1	yes	characteristic
37	C₂×C₄₂₁	#117	1	1	yes	characteristic
38	C₂×C₄₂₁	#117	1	1	yes	characteristic
39	C₂×C₄₂₁	#126	1	2	yes	—
40	C₂×C₄₂₁	#126	1	2	yes	—
41	C₄₂₁	#383	1	1	yes	characteristic, complemented, Sylow
42	#383	—	421	1	no	maximal, Sylow
43	C₂×C₄²	—	421	1	no	—
44	#95	—	421	1	no	—
45	#95	—	421	2	no	—
46	#95	—	421	2	no	—
47	#95	D₄₂₁	1	1	yes	characteristic
48	C₂²×C₄	—	421	2	no	—
49	C₂²×C₄	—	421	1	no	—
50	C₂²×C₄	—	421	1	no	—
51	C₂²×C₄	C₂×D₄₂₁	1	2	yes	—
52	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
53	C₄²	—	421	4	no	—
54	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	842	1	no	—
60	C₂×C₄	—	842	2	no	—
61	C₂×C₄	—	842	1	no	—
62	C₂×C₄	—	842	2	no	—
63	C₂×C₄	—	842	2	no	—
64	C₂×C₄	—	842	2	no	—
65	C₂×C₄	—	421	2	no	—
66	C₂×C₄	—	421	2	no	—
67	C₂×C₄	C₄×D₄₂₁	1	2	yes	—
68	C₂²	C₂×C₄×D₄₂₁	1	1	yes	characteristic
69	C₂²	#20269770	1	2	yes	—
70	C₂²	#20269770	1	1	yes	characteristic
71	C₂²	#20269771	1	2	yes	—
72	C₂²	#20269772	1	1	yes	characteristic
73	C₄	—	2	2	no	—
74	C₄	—	4	2	no	—
75	C₄	—	842	2	no	—
76	C₄	—	842	2	no	—
77	C₄	—	1684	2	no	—
78	C₂	#27212322	1	1	yes	characteristic
79	C₂	#27212323	1	1	yes	characteristic
80	C₂	#27212324	1	1	yes	characteristic
81	C₂	#27212359	1	2	yes	—
82	C₂	#27212361	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₄₂₁
commutator	C₂²×C₄₂₁	C₂²×C₄
Fitting	#95×C₄₂₁	C₂
socle, prime radical, cosocle	C₂³×C₄₂₁	C₂³
hypercenter	#95	D₄₂₁
hypocenter	C₄₂₁	#383
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂³ ◁ #95

Complements

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

N	H ≅ G/N	type
9. C₄×Dic₄₂₁	C₄	semidirect
41. C₄₂₁	#383	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×Dic₄₂₁	2
#27212273	2
#95×C₄₂₁	2
#383	421