Subgroups — C₂×#27212365 #41003383

Order 26944.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₈₄₂	C₂	1	2	yes	maximal
2	#27212365	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#20269750	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊C₄×C₄₂₁	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₈₄₂	—	2	8	no	—
8	C₂×Dic₈₄₂	C₂²	1	8	yes	—
9	C₂²×Dic₄₂₁	C₂²	1	2	yes	—
10	C₂²×Dic₄₂₁	C₂²	1	1	yes	characteristic
11	C₂²×Dic₄₂₁	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₄₂₁	C₂²	1	4	yes	—
13	#20269750	C₂²	1	8	yes	—
14	C₄×Dic₄₂₁	C₂²	1	4	yes	—
15	C₂²×C₄×C₄₂₁	C₂²	1	2	yes	—
16	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₄₂₁	C₂²	1	4	yes	—
18	Dic₈₄₂	—	2	16	no	—
19	C₂×Dic₄₂₁	—	2	4	no	—
20	C₂×Dic₄₂₁	D₄	1	4	yes	—
21	C₂×Dic₄₂₁	Q₈	1	4	yes	complemented
22	C₂×Dic₄₂₁	C₂³	1	4	yes	—
23	C₂×Dic₄₂₁	C₂³	1	2	yes	—
24	C₂×Dic₄₂₁	C₂³	1	2	yes	—
25	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₄₂₁	—	2	4	no	—
27	C₂×C₄×C₄₂₁	Q₈	1	4	yes	complemented
28	C₂×C₄×C₄₂₁	C₂³	1	4	yes	—
29	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
30	C₂²×C₄₂₁	C₂×D₄	1	2	yes	—
31	C₂²×C₄₂₁	C₂×Q₈	1	2	yes	—
32	C₂²×C₄₂₁	C₂×Q₈	1	2	yes	—
33	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
34	C₄×C₄₂₁	—	2	4	no	—
35	C₄×C₄₂₁	C₂×Q₈	1	4	yes	complemented
36	Dic₄₂₁	—	2	4	no	—
37	Dic₄₂₁	C₂×D₄	1	4	yes	—
38	Dic₄₂₁	C₂×Q₈	1	4	yes	complemented
39	C₂×C₄₂₁	C₄⋊Q₈	1	4	yes	complemented
40	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₄₂₁	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₄₂₁	C₂²×Q₈	1	1	yes	characteristic
43	C₄₂₁	C₂×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₄⋊Q₈	—	421	1	no	maximal, Sylow
45	C₂²×Q₈	—	421	2	no	—
46	C₄⋊Q₈	—	421	8	no	—
47	C₂×C₄⋊C₄	—	421	2	no	—
48	C₂×C₄⋊C₄	—	421	1	no	—
49	C₂×C₄⋊C₄	D₄₂₁	1	1	yes	characteristic
50	C₂×C₄²	—	421	1	no	—
51	C₂×Q₈	—	842	8	no	—
52	C₂×Q₈	—	421	8	no	—
53	C₂²×C₄	—	421	2	no	—
54	C₂²×C₄	—	421	1	no	—
55	C₂²×C₄	—	421	1	no	—
56	C₂²×C₄	C₂×D₄₂₁	1	2	yes	—
57	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
58	C₄⋊C₄	—	421	8	no	—
59	C₄⋊C₄	—	421	4	no	—
60	C₄⋊C₄	C₂×D₄₂₁	1	4	yes	—
61	C₄²	—	421	4	no	—
62	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
63	Q₈	—	842	16	no	—
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	842	4	no	—
66	C₂×C₄	—	421	4	no	—
67	C₂×C₄	—	421	4	no	—
68	C₂×C₄	—	421	4	no	—
69	C₂×C₄	—	421	2	no	—
70	C₂×C₄	—	421	2	no	—
71	C₂×C₄	Dic₈₄₂	1	4	yes	complemented
72	C₂×C₄	C₂²×D₄₂₁	1	4	yes	—
73	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
74	C₂²	D₄×D₄₂₁	1	2	yes	—
75	C₂²	Q₈×D₄₂₁	1	2	yes	—
76	C₂²	C₂×Dic₈₄₂	1	2	yes	—
77	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	842	4	no	—
80	C₄	—	421	4	no	—
81	C₄	—	421	4	no	—
82	C₄	C₂×Dic₈₄₂	1	4	yes	complemented
83	C₂	#27212365	1	4	yes	direct product factor
84	C₂	C₂×D₄×D₄₂₁	1	1	yes	characteristic
85	C₂	C₂×Q₈×D₄₂₁	1	1	yes	characteristic
86	C₂	C₂²×Dic₈₄₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₄₂₁
commutator, cosocle	C₂²×C₄₂₁	C₂⁴
Frattini	C₂²	C₂³×D₄₂₁
Fitting	C₂×C₄⋊C₄×C₄₂₁	C₂
socle, prime radical	C₂³×C₄₂₁	C₂³
hypercenter	C₂×C₄⋊C₄	D₄₂₁
hypocenter	C₄₂₁	C₂×C₄⋊Q₈
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #27212365	C₂	direct (N × H)
21. C₂×Dic₄₂₁	Q₈	semidirect
27. C₂×C₄×C₄₂₁	Q₈	semidirect
35. C₄×C₄₂₁	C₂×Q₈	semidirect
38. Dic₄₂₁	C₂×Q₈	semidirect
39. C₂×C₄₂₁	C₄⋊Q₈	semidirect
43. C₄₂₁	C₂×C₄⋊Q₈	semidirect
71. C₂×C₄	Dic₈₄₂	semidirect
82. C₄	C₂×Dic₈₄₂	semidirect
83. C₂	#27212365	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₈₄₂	2
#27212365	2
C₂×C₄⋊Dic₄₂₁	2
C₂×#20269750	2
C₂×C₄×Dic₄₂₁	2
C₂×C₄⋊C₄×C₄₂₁	2
C₂×C₄⋊Q₈	421

Subgroups — C2×#27212365 #41003383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212365 #41003383