Subgroups — C₂×#27212285 #41002820

Order 26944.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269755	C₂	1	2	yes	maximal
2	#27212285	C₂	1	4	yes	maximal, direct product factor
3	C₂²×Q₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
4	#20269755	—	2	2	no	—
5	#20269755	C₂²	1	4	yes	—
6	C₂×#16809890	—	2	2	no	—
7	C₂×Q₈×C₄₂₁	—	2	2	no	—
8	C₂×Q₈×C₄₂₁	—	2	2	no	—
9	C₂×Q₈×C₄₂₁	C₄	1	2	yes	—
10	C₂×Q₈×C₄₂₁	C₂²	1	2	yes	—
11	C₂²×C₄×C₄₂₁	C₄	1	2	yes	—
12	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
13	#16809890	—	2	4	no	—
14	C₂³×C₄₂₁	C₂×C₄	1	1	yes	characteristic
15	Q₈×C₄₂₁	—	2	4	no	—
16	Q₈×C₄₂₁	—	2	4	no	—
17	C₂×C₄×C₄₂₁	—	2	2	no	—
18	C₂×C₄×C₄₂₁	—	2	2	no	—
19	C₂×C₄×C₄₂₁	C₂×C₄	1	4	yes	—
20	C₂×C₄×C₄₂₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
22	C₂×C₄×C₄₂₁	D₄	1	2	yes	—
23	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
24	C₂²×C₄₂₁	—	2	2	no	—
25	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₄₂₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₄₂₁	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₄₂₁	—	2	4	no	—
29	C₄×C₄₂₁	C₂²⋊C₄	1	2	yes	—
30	C₄×C₄₂₁	C₂×D₄	1	2	yes	—
31	C₂×C₄₂₁	—	2	1	no	—
32	C₂×C₄₂₁	—	2	1	no	—
33	C₂×C₄₂₁	#101	1	2	yes	complemented
34	C₂×C₄₂₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
35	C₄₂₁	C₂×#101	1	1	yes	characteristic, complemented, Sylow
36	C₂×#101	—	421	1	no	maximal, Sylow
37	C₂²×Q₈	D₄₂₁	1	1	yes	characteristic
38	C₂×M₁₆	—	421	2	no	—
39	#101	—	421	4	no	—
40	C₂×Q₈	—	2	2	no	—
41	C₂×Q₈	—	2	2	no	—
42	C₂×Q₈	Dic₄₂₁	1	2	yes	—
43	C₂×Q₈	C₂×D₄₂₁	1	2	yes	—
44	C₂²×C₄	Dic₄₂₁	1	2	yes	—
45	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
46	M₁₆	—	842	2	no	—
47	M₁₆	—	421	4	no	—
48	C₂×C₈	—	842	2	no	—
49	C₂³	C₂×Dic₄₂₁	1	1	yes	characteristic
50	Q₈	—	2	4	no	—
51	Q₈	—	2	4	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	C₂×Dic₄₂₁	1	4	yes	—
55	C₂×C₄	C₂×Dic₄₂₁	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₄₂₁	1	2	yes	—
57	C₂×C₄	C₂²⋊D₄₂₁	1	2	yes	—
58	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
59	C₈	—	842	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	C₂²⋊Dic₄₂₁	1	1	yes	characteristic
62	C₂²	C₂²⋊Dic₄₂₁	1	1	yes	characteristic
63	C₂²	C₂²×Dic₄₂₁	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	C₂²⋊Dic₄₂₁	1	2	yes	—
66	C₄	C₂×C₂²⋊D₄₂₁	1	2	yes	—
67	C₂	—	2	1	no	—
68	C₂	—	2	1	no	—
69	C₂	#27212285	1	2	yes	direct product factor
70	C₂	C₂×C₂²⋊Dic₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊Dic₄₂₁
commutator	C₂²×C₄₂₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₄₂₁
Fitting	C₂²×Q₈×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂²⋊C₄
prime radical	C₂³×C₄₂₁	C₂×C₄
cosocle	C₂×C₄×C₄₂₁	C₂³
hypercenter	C₂²×Q₈	D₄₂₁
hypocenter	C₄₂₁	C₂×#101
LCS	G ▷ C₂²×C₄₂₁ ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. #27212285	C₂	direct (N × H)
33. C₂×C₄₂₁	#101	semidirect
35. C₄₂₁	C₂×#101	semidirect
69. C₂	#27212285	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#20269755	2
#27212285	2
C₂²×Q₈×C₄₂₁	2
C₂×#101	421

Subgroups — C2×#27212285 #41002820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212285 #41002820