Subgroups — C₂×#27212413 #41003673

Order 26944.

794 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212413	C₂	1	16	yes	maximal, direct product factor
2	C₂×#20269771	C₂	1	3	yes	maximal, complemented
3	C₂×#20269770	C₂	1	6	yes	maximal, complemented
4	C₂²×Dic₈₄₂	C₂	1	3	yes	maximal, complemented
5	C₂×Q₈×D₄₂₁	C₂	1	2	yes	maximal, complemented
6	C₂×#41×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#20269771	C₂²	1	24	yes	complemented
8	#20269770	C₂²	1	48	yes	complemented
9	C₂×Dic₈₄₂	C₂²	1	6	yes	—
10	C₂×Dic₈₄₂	C₂²	1	3	yes	—
11	C₂×Dic₈₄₂	C₂²	1	24	yes	complemented
12	Q₈×D₄₂₁	C₂²	1	16	yes	complemented
13	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₄₂₁	C₂²	1	6	yes	—
15	C₂²×Dic₄₂₁	C₂²	1	6	yes	—
16	C₂×C₄×D₄₂₁	C₂²	1	6	yes	—
17	#41×C₄₂₁	C₂²	1	8	yes	complemented
18	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₄₂₁	C₂²	1	3	yes	—
20	C₂²×C₄×C₄₂₁	C₂²	1	3	yes	—
21	C₂²×D₄₂₁	C₂³	1	2	yes	—
22	Dic₈₄₂	C₂³	1	24	yes	—
23	Dic₈₄₂	C₂³	1	12	yes	—
24	D₁₆₈₄	C₂³	1	4	yes	—
25	C₂²⋊D₄₂₁	C₂³	1	24	yes	—
26	C₂×Dic₄₂₁	C₂³	1	6	yes	—
27	C₂×Dic₄₂₁	C₂³	1	24	yes	—
28	C₄×D₄₂₁	C₂³	1	24	yes	—
29	C₂³×C₄₂₁	C₂³	1	3	yes	—
30	Q₈×C₄₂₁	C₂³	1	4	yes	—
31	D₄×C₄₂₁	C₂³	1	12	yes	—
32	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₄₂₁	C₂³	1	3	yes	—
34	C₂×C₄×C₄₂₁	C₂³	1	12	yes	—
35	C₂²×C₄₂₁	—	2	6	no	—
36	C₂²×C₄₂₁	C₂⁴	1	6	yes	—
37	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
38	C₂×D₄₂₁	—	2	4	no	—
39	C₂×D₄₂₁	C₂⁴	1	4	yes	—
40	C₄×C₄₂₁	C₂⁴	1	2	yes	—
41	C₄×C₄₂₁	C₂⁴	1	6	yes	—
42	Dic₄₂₁	C₂⁴	1	12	yes	—
43	C₂×C₄₂₁	—	2	6	no	—
44	C₂×C₄₂₁	2₋¹⁺⁴	1	2	yes	complemented
45	C₂×C₄₂₁	C₂⁵	1	1	yes	characteristic
46	D₄₂₁	—	2	4	no	—
47	C₄₂₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
48	C₂×2₋¹⁺⁴	—	421	1	no	maximal, Sylow
49	2₋¹⁺⁴	—	421	16	no	—
50	C₂×#41	—	421	6	no	—
51	C₂×#41	—	421	3	no	—
52	C₂×#41	D₄₂₁	1	1	yes	characteristic, complemented
53	C₂²×Q₈	—	421	2	no	—
54	C₂²×Q₈	—	421	3	no	—
55	#41	—	421	48	no	—
56	#41	—	421	24	no	—
57	#41	C₂×D₄₂₁	1	8	yes	complemented
58	C₂×Q₈	—	421	16	no	—
59	C₂×Q₈	—	421	6	no	—
60	C₂×Q₈	—	421	3	no	—
61	C₂×Q₈	—	421	24	no	—
62	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
63	C₂×D₄	—	421	6	no	—
64	C₂×D₄	—	421	1	no	—
65	C₂×D₄	C₂×D₄₂₁	1	3	yes	—
66	C₂²×C₄	—	421	6	no	—
67	C₂²×C₄	—	421	6	no	—
68	C₂²×C₄	C₂×D₄₂₁	1	3	yes	—
69	C₂³	—	421	2	no	—
70	C₂³	C₂²×D₄₂₁	1	3	yes	—
71	Q₈	—	421	24	no	—
72	Q₈	—	421	12	no	—
73	Q₈	C₂²×D₄₂₁	1	4	yes	—
74	D₄	—	421	24	no	—
75	D₄	—	421	4	no	—
76	D₄	C₂²×D₄₂₁	1	12	yes	—
77	C₂×C₄	—	421	24	no	—
78	C₂×C₄	—	421	6	no	—
79	C₂×C₄	—	421	24	no	—
80	C₂×C₄	C₂²×D₄₂₁	1	12	yes	—
81	C₂×C₄	C₂²×D₄₂₁	1	3	yes	—
82	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
83	C₂²	—	2	6	no	—
84	C₂²	—	842	4	no	—
85	C₂²	—	421	4	no	—
86	C₂²	C₂³×D₄₂₁	1	6	yes	—
87	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
88	C₄	—	421	12	no	—
89	C₄	C₂³×D₄₂₁	1	6	yes	—
90	C₄	C₂³×D₄₂₁	1	2	yes	—
91	C₂	—	2	6	no	—
92	C₂	—	842	4	no	—
93	C₂	#27212413	1	2	yes	direct product factor
94	C₂	C₂⁴×D₄₂₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₄₂₁
commutator, cosocle	C₂×C₄₂₁	C₂⁵
Frattini	C₂	C₂⁴×D₄₂₁
Fitting	C₂×#41×C₄₂₁	C₂
socle	C₂²×C₄₂₁	C₂⁴
hypercenter	C₂×#41	D₄₂₁
hypocenter	C₄₂₁	C₂×2₋¹⁺⁴
LCS	G ▷ C₂×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #27212413	C₂	direct (N × H)
2. C₂×#20269771	C₂	semidirect
3. C₂×#20269770	C₂	semidirect
4. C₂²×Dic₈₄₂	C₂	semidirect
5. C₂×Q₈×D₄₂₁	C₂	semidirect
6. C₂×#41×C₄₂₁	C₂	semidirect
7. #20269771	C₂²	semidirect
8. #20269770	C₂²	semidirect
11. C₂×Dic₈₄₂	C₂²	semidirect
12. Q₈×D₄₂₁	C₂²	semidirect
17. #41×C₄₂₁	C₂²	semidirect
44. C₂×C₄₂₁	2₋¹⁺⁴	semidirect
47. C₄₂₁	C₂×2₋¹⁺⁴	semidirect
52. C₂×#41	D₄₂₁	semidirect
57. #41	C₂×D₄₂₁	semidirect
93. C₂	#27212413	direct (N × H)

Maximal subgroups

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

subgroup	index
#27212413	2
C₂×#20269771	2
C₂×#20269770	2
C₂²×Dic₈₄₂	2
C₂×Q₈×D₄₂₁	2
C₂×#41×C₄₂₁	2
C₂×2₋¹⁺⁴	421

Subgroups — C2×#27212413 #41003673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27212413 #41003673