Subgroups — C₂×#27212325 #41003296

Order 26944.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212325	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₄₂₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊Dic₄₂₁	C₂	1	2	yes	maximal
4	C₂×C₄×Dic₄₂₁	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₄₂₁	C₂²	1	4	yes	—
7	C₄⋊Dic₄₂₁	C₂²	1	8	yes	complemented
8	C₂²⋊Dic₄₂₁	C₂²	1	8	yes	—
9	C₄×Dic₄₂₁	C₂²	1	8	yes	complemented
10	C₂⁴×C₄₂₁	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₄₂₁	C₄	1	8	yes	complemented
12	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₄₂₁	C₂²	1	4	yes	—
14	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₂₁	—	2	8	no	—
16	C₂×Dic₄₂₁	C₂³	1	8	yes	—
17	C₂³×C₄₂₁	—	2	4	no	—
18	C₂³×C₄₂₁	C₂×C₄	1	4	yes	—
19	C₂³×C₄₂₁	C₂³	1	2	yes	—
20	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₄₂₁	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₄₂₁	C₂×C₄	1	4	yes	—
23	C₂×C₄×C₄₂₁	C₂×C₄	1	16	yes	complemented
24	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
25	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
26	C₂²×C₄₂₁	—	2	8	no	—
27	C₂²×C₄₂₁	—	2	4	no	—
28	C₂²×C₄₂₁	C₂²×C₄	1	4	yes	—
29	C₂²×C₄₂₁	C₂²×C₄	1	2	yes	—
30	C₂²×C₄₂₁	#41	1	4	yes	—
31	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
32	C₄×C₄₂₁	C₂²×C₄	1	4	yes	—
33	C₄×C₄₂₁	C₂²×C₄	1	4	yes	—
34	Dic₄₂₁	—	2	8	no	—
35	C₂×C₄₂₁	—	2	4	no	—
36	C₂×C₄₂₁	#117	1	4	yes	complemented
37	C₂×C₄₂₁	C₂³×C₄	1	1	yes	characteristic
38	C₂×C₄₂₁	C₂×#41	1	2	yes	—
39	C₄₂₁	C₂×#117	1	1	yes	characteristic, complemented, Sylow
40	C₂×#117	—	421	1	no	maximal, Sylow
41	C₂³×C₄	D₄₂₁	1	1	yes	characteristic
42	#117	—	421	8	no	—
43	C₂×C₄⋊C₄	—	421	2	no	—
44	C₂×C₂²⋊C₄	—	421	2	no	—
45	C₂×C₄²	—	421	2	no	—
46	C₂⁴	C₂×D₄₂₁	1	1	yes	characteristic
47	C₂²×C₄	—	421	4	no	—
48	C₂²×C₄	Dic₄₂₁	1	8	yes	complemented
49	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₄₂₁	1	4	yes	—
51	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
52	C₄⋊C₄	—	421	8	no	—
53	C₂²⋊C₄	—	421	8	no	—
54	C₄²	—	421	8	no	—
55	C₂³	—	2	4	no	—
56	C₂³	C₂×Dic₄₂₁	1	4	yes	—
57	C₂³	C₂²×D₄₂₁	1	2	yes	—
58	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
59	C₂×C₄	—	842	8	no	—
60	C₂×C₄	—	421	8	no	—
61	C₂×C₄	C₂×Dic₄₂₁	1	4	yes	—
62	C₂×C₄	C₂×Dic₄₂₁	1	4	yes	—
63	C₂×C₄	C₂×Dic₄₂₁	1	16	yes	complemented
64	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
65	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
66	C₂²	—	2	8	no	—
67	C₂²	—	2	4	no	—
68	C₂²	C₂²×Dic₄₂₁	1	4	yes	—
69	C₂²	C₂²×Dic₄₂₁	1	2	yes	—
70	C₂²	#20269771	1	4	yes	—
71	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
72	C₄	—	842	8	no	—
73	C₄	C₂²×Dic₄₂₁	1	4	yes	—
74	C₄	C₂²×Dic₄₂₁	1	4	yes	—
75	C₂	—	2	4	no	—
76	C₂	#27212325	1	4	yes	direct product factor
77	C₂	C₂³×Dic₄₂₁	1	1	yes	characteristic
78	C₂	C₂×#20269771	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₄₂₁
commutator	C₂×C₄₂₁	C₂³×C₄
Frattini	C₂²	C₂³×D₄₂₁
Fitting	C₂³×C₄×C₄₂₁	C₂
socle	C₂³×C₄₂₁	C₂³
prime radical	C₂⁴×C₄₂₁	C₂²
cosocle	C₂²×C₄₂₁	C₂⁴
hypercenter	C₂³×C₄	D₄₂₁
hypocenter	C₄₂₁	C₂×#117
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. #27212325	C₂	direct (N × H)
2. C₂×C₄⋊Dic₄₂₁	C₂	semidirect
4. C₂×C₄×Dic₄₂₁	C₂	semidirect
7. C₄⋊Dic₄₂₁	C₂²	semidirect
9. C₄×Dic₄₂₁	C₂²	semidirect
11. C₂²×C₄×C₄₂₁	C₄	semidirect
23. C₂×C₄×C₄₂₁	C₂×C₄	semidirect
36. C₂×C₄₂₁	#117	semidirect
39. C₄₂₁	C₂×#117	semidirect
48. C₂²×C₄	Dic₄₂₁	semidirect
63. C₂×C₄	C₂×Dic₄₂₁	semidirect
76. C₂	#27212325	direct (N × H)

Maximal subgroups

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

subgroup	index
#27212325	2
C₂×C₄⋊Dic₄₂₁	2
C₂×C₂²⋊Dic₄₂₁	2
C₂×C₄×Dic₄₂₁	2
C₂³×C₄×C₄₂₁	2
C₂×#117	421

Subgroups — C2×#27212325 #41003296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212325 #41003296