Subgroups — #41003402

Order 26944.

346 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂²⋊D₄₂₁	C₂	1	2	yes	maximal, complemented
2	#27212350	C₂	1	4	yes	maximal, complemented
3	#27212340	C₂	1	4	yes	maximal, complemented
4	#27212336	C₂	1	4	yes	maximal, complemented
5	C₂×C₂²⋊C₄×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄₂₁	C₂²	1	2	yes	complemented
7	C₂×D₁₆₈₄	C₂²	1	4	yes	complemented
8	C₂×C₂²⋊D₄₂₁	—	2	8	no	—
9	C₂×C₂²⋊D₄₂₁	C₂²	1	4	yes	—
10	C₂×C₂²⋊D₄₂₁	C₂²	1	4	yes	complemented
11	C₂²×Dic₄₂₁	C₂²	1	2	yes	complemented
12	C₄⋊Dic₄₂₁	C₂²	1	4	yes	complemented
13	#20269748	C₂²	1	8	yes	complemented
14	C₂⁴×C₄₂₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₄₂₁	C₂²	1	2	yes	complemented
16	C₂²⋊C₄×C₄₂₁	C₂²	1	4	yes	complemented
17	C₂²×D₄₂₁	—	2	4	no	—
18	C₂²×D₄₂₁	—	4	4	no	—
19	C₂²×D₄₂₁	C₂³	1	4	yes	—
20	D₁₆₈₄	—	4	4	no	—
21	C₂²⋊D₄₂₁	—	2	8	no	—
22	C₂²⋊D₄₂₁	—	2	8	no	—
23	C₂×Dic₄₂₁	—	2	4	no	—
24	C₂×Dic₄₂₁	C₂³	1	4	yes	—
25	C₂³×C₄₂₁	—	4	2	no	—
26	C₂³×C₄₂₁	D₄	1	4	yes	complemented
27	C₂³×C₄₂₁	C₂³	1	2	yes	—
28	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₄₂₁	—	4	2	no	—
30	C₂×C₄×C₄₂₁	C₂³	1	4	yes	—
31	C₂²×C₄₂₁	—	2	2	no	—
32	C₂²×C₄₂₁	—	2	4	no	—
33	C₂²×C₄₂₁	—	4	4	no	—
34	C₂²×C₄₂₁	C₂×D₄	1	4	yes	—
35	C₂²×C₄₂₁	C₂×D₄	1	2	yes	—
36	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
37	C₂×D₄₂₁	—	2	4	no	—
38	C₂×D₄₂₁	—	2	8	no	—
39	C₂×D₄₂₁	—	4	8	no	—
40	C₄×C₄₂₁	—	2	4	no	—
41	Dic₄₂₁	—	2	4	no	—
42	C₂×C₄₂₁	—	2	4	no	—
43	C₂×C₄₂₁	—	2	2	no	—
44	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
45	C₂×C₄₂₁	H(5,𝔽₂)	1	2	yes	—
46	D₄₂₁	—	4	4	no	—
47	C₄₂₁	#534	1	1	yes	characteristic, complemented, Sylow
48	#534	—	421	1	no	maximal, Sylow
49	C₂²×D₄	—	421	2	no	—
50	#123	—	421	4	no	—
51	#121	—	421	4	no	—
52	#120	—	421	4	no	—
53	C₂×C₂²⋊C₄	D₄₂₁	1	1	yes	characteristic, complemented
54	C₂⁴	—	421	2	no	—
55	C₂⁴	C₂×D₄₂₁	1	1	yes	characteristic
56	C₂×D₄	—	842	8	no	—
57	C₂×D₄	—	421	4	no	—
58	C₂×D₄	—	421	4	no	—
59	C₂×D₄	—	421	4	no	—
60	C₂²×C₄	—	421	2	no	—
61	C₂²×C₄	C₂×D₄₂₁	1	2	yes	complemented
62	C₄⋊C₄	—	421	4	no	—
63	C₂²⋊C₄	—	421	8	no	—
64	C₂²⋊C₄	C₂×D₄₂₁	1	4	yes	complemented
65	C₂³	—	4	2	no	—
66	C₂³	—	842	4	no	—
67	C₂³	—	1684	4	no	—
68	C₂³	—	421	4	no	—
69	C₂³	D₁₆₈₄	1	4	yes	complemented
70	C₂³	C₂²×D₄₂₁	1	2	yes	—
71	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
72	D₄	—	842	8	no	—
73	D₄	—	842	8	no	—
74	D₄	—	1684	4	no	—
75	C₂×C₄	—	4	2	no	—
76	C₂×C₄	—	842	4	no	—
77	C₂×C₄	—	421	4	no	—
78	C₂×C₄	C₂²×D₄₂₁	1	4	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	4	4	no	—
82	C₂²	—	842	4	no	—
83	C₂²	—	842	8	no	—
84	C₂²	—	1684	8	no	—
85	C₂²	C₂×D₁₆₈₄	1	4	yes	—
86	C₂²	C₂×D₁₆₈₄	1	2	yes	—
87	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	842	4	no	—
90	C₂	—	2	4	no	—
91	C₂	—	2	2	no	—
92	C₂	—	1684	4	no	—
93	C₂	C₂²×D₁₆₈₄	1	1	yes	characteristic
94	C₂	#27212411	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₄₂₁	C₂
hypercenter	C₂×C₂²⋊C₄	D₄₂₁
hypocenter	C₄₂₁	#534
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂²⋊D₄₂₁	C₂	semidirect
2. #27212350	C₂	semidirect
3. #27212340	C₂	semidirect
4. #27212336	C₂	semidirect
5. C₂×C₂²⋊C₄×C₄₂₁	C₂	semidirect
6. C₂³×D₄₂₁	C₂²	semidirect
7. C₂×D₁₆₈₄	C₂²	semidirect
10. C₂×C₂²⋊D₄₂₁	C₂²	semidirect
11. C₂²×Dic₄₂₁	C₂²	semidirect
12. C₄⋊Dic₄₂₁	C₂²	semidirect
13. #20269748	C₂²	semidirect
15. C₂²×C₄×C₄₂₁	C₂²	semidirect
16. C₂²⋊C₄×C₄₂₁	C₂²	semidirect
26. C₂³×C₄₂₁	D₄	semidirect
47. C₄₂₁	#534	semidirect
53. C₂×C₂²⋊C₄	D₄₂₁	semidirect
61. C₂²×C₄	C₂×D₄₂₁	semidirect
64. C₂²⋊C₄	C₂×D₄₂₁	semidirect
69. C₂³	D₁₆₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂²⋊D₄₂₁	2
#27212350	2
#27212340	2
#27212336	2
C₂×C₂²⋊C₄×C₄₂₁	2
#534	421