Subgroups — C₂×#27212353 #41003356

Order 26944.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₈₄₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#27212353	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₂²⋊Dic₄₂₁	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Dic₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₄₂	—	2	4	no	—
7	C₂×Dic₈₄₂	C₂²	1	4	yes	complemented
8	C₂²×Dic₄₂₁	C₂²	1	2	yes	—
9	C₂²×Dic₄₂₁	C₂²	1	2	yes	—
10	C₂²⋊Dic₄₂₁	C₂²	1	16	yes	complemented
11	C₄×Dic₄₂₁	C₂²	1	4	yes	complemented
12	C₂⁴×C₄₂₁	C₂²	1	2	yes	—
13	C₂×D₄×C₄₂₁	—	2	4	no	—
14	C₂×D₄×C₄₂₁	C₂²	1	4	yes	—
15	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
16	Dic₈₄₂	—	2	8	no	—
17	C₂×Dic₄₂₁	—	2	4	no	—
18	C₂×Dic₄₂₁	—	2	4	no	—
19	C₂×Dic₄₂₁	C₂³	1	4	yes	—
20	C₂×Dic₄₂₁	C₂³	1	4	yes	—
21	C₂³×C₄₂₁	—	2	4	no	—
22	C₂³×C₄₂₁	—	2	8	no	—
23	C₂³×C₄₂₁	C₂³	1	4	yes	—
24	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
25	D₄×C₄₂₁	—	2	8	no	—
26	C₂×C₄×C₄₂₁	D₄	1	4	yes	—
27	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
28	C₂²×C₄₂₁	—	2	4	no	—
29	C₂²×C₄₂₁	—	2	8	no	—
30	C₂²×C₄₂₁	—	4	8	no	—
31	C₂²×C₄₂₁	C₂×D₄	1	2	yes	—
32	C₂²×C₄₂₁	#41	1	4	yes	—
33	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
34	C₄×C₄₂₁	C₂×D₄	1	4	yes	—
35	Dic₄₂₁	—	2	4	no	—
36	Dic₄₂₁	—	2	4	no	—
37	C₂×C₄₂₁	—	4	4	no	—
38	C₂×C₄₂₁	#124	1	4	yes	complemented
39	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
40	C₂×C₄₂₁	C₂×#41	1	2	yes	—
41	C₄₂₁	C₂×#124	1	1	yes	characteristic, complemented, Sylow
42	C₂×#124	—	421	1	no	maximal, Sylow
43	C₂²×Q₈	—	421	1	no	—
44	C₂²×D₄	D₄₂₁	1	1	yes	characteristic
45	#124	—	421	8	no	—
46	C₂×C₂²⋊C₄	—	421	4	no	—
47	C₂×C₄²	—	421	1	no	—
48	C₂⁴	C₂×D₄₂₁	1	2	yes	—
49	C₂×Q₈	—	842	4	no	—
50	C₂×Q₈	—	421	4	no	—
51	C₂×D₄	—	2	4	no	—
52	C₂×D₄	C₂×D₄₂₁	1	4	yes	—
53	C₂²×C₄	—	421	2	no	—
54	C₂²×C₄	—	421	2	no	—
55	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
56	C₂²⋊C₄	—	421	16	no	—
57	C₄²	—	421	4	no	—
58	C₂³	—	2	4	no	—
59	C₂³	—	2	8	no	—
60	C₂³	C₂²×D₄₂₁	1	4	yes	—
61	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
62	Q₈	—	842	8	no	—
63	D₄	—	2	8	no	—
64	C₂×C₄	—	842	4	no	—
65	C₂×C₄	—	842	4	no	—
66	C₂×C₄	—	421	4	no	—
67	C₂×C₄	—	421	4	no	—
68	C₂×C₄	C₂²⋊D₄₂₁	1	4	yes	—
69	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
70	C₂²	—	2	4	no	—
71	C₂²	—	2	8	no	—
72	C₂²	—	4	8	no	—
73	C₂²	C₂×C₂²⋊D₄₂₁	1	2	yes	—
74	C₂²	#20269770	1	4	yes	—
75	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
76	C₄	—	842	4	no	—
77	C₄	—	842	4	no	—
78	C₄	C₂×C₂²⋊D₄₂₁	1	4	yes	—
79	C₂	—	4	4	no	—
80	C₂	#27212353	1	4	yes	direct product factor
81	C₂	C₂²×C₂²⋊D₄₂₁	1	1	yes	characteristic
82	C₂	C₂×#20269770	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₄₂₁
commutator, cosocle	C₂²×C₄₂₁	C₂⁴
Frattini	C₂²	C₂³×D₄₂₁
Fitting, prime radical	C₂²×D₄×C₄₂₁	C₂
socle	C₂³×C₄₂₁	C₂³
hypercenter	C₂²×D₄	D₄₂₁
hypocenter	C₄₂₁	C₂×#124
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₈₄₂	C₂	semidirect
2. #27212353	C₂	direct (N × H)
3. C₂×C₂²⋊Dic₄₂₁	C₂	semidirect
4. C₂×C₄×Dic₄₂₁	C₂	semidirect
7. C₂×Dic₈₄₂	C₂²	semidirect
10. C₂²⋊Dic₄₂₁	C₂²	semidirect
11. C₄×Dic₄₂₁	C₂²	semidirect
38. C₂×C₄₂₁	#124	semidirect
41. C₄₂₁	C₂×#124	semidirect
80. C₂	#27212353	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
#27212353	2
C₂×C₂²⋊Dic₄₂₁	2
C₂×C₄×Dic₄₂₁	2
C₂²×D₄×C₄₂₁	2
C₂×#124	421

Subgroups — C2×#27212353 #41003356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212353 #41003356