Subgroups — #41003421

Order 26944.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269771	C₂	1	2	yes	maximal, complemented
2	#27212346	C₂	1	4	yes	maximal, complemented
3	#27212341	C₂	1	4	yes	maximal, complemented
4	#27212340	C₂	1	4	yes	maximal, complemented
5	C₂×C₄⋊C₄×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#20269771	—	2	8	no	—
7	C₂×Dic₈₄₂	C₂²	1	2	yes	—
8	C₂×D₁₆₈₄	C₂²	1	2	yes	—
9	C₂×D₁₆₈₄	C₂²	1	4	yes	—
10	C₂×C₂²⋊D₄₂₁	C₂²	1	4	yes	—
11	C₂×C₄×D₄₂₁	C₂²	1	4	yes	—
12	C₄⋊Dic₄₂₁	C₂²	1	4	yes	—
13	#20269748	C₂²	1	8	yes	—
14	C₂²×C₄×C₄₂₁	C₂²	1	2	yes	—
15	C₂²×C₄×C₄₂₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₄₂₁	C₂²	1	4	yes	—
17	C₂²×D₄₂₁	C₂³	1	4	yes	—
18	Dic₈₄₂	—	2	4	no	—
19	D₁₆₈₄	—	2	4	no	—
20	D₁₆₈₄	—	4	4	no	—
21	C₂²⋊D₄₂₁	—	2	8	no	—
22	C₂×Dic₄₂₁	C₂³	1	4	yes	—
23	C₄×D₄₂₁	—	2	8	no	—
24	C₂³×C₄₂₁	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₄₂₁	—	4	2	no	—
26	C₂×C₄×C₄₂₁	D₄	1	4	yes	complemented
27	C₂×C₄×C₄₂₁	C₂³	1	4	yes	—
28	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
29	C₂²×C₄₂₁	—	2	1	no	—
30	C₂²×C₄₂₁	—	2	1	no	—
31	C₂²×C₄₂₁	C₂×D₄	1	2	yes	—
32	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
33	C₂×D₄₂₁	—	2	4	no	—
34	C₂×D₄₂₁	—	2	4	no	—
35	C₂×D₄₂₁	—	2	4	no	—
36	C₄×C₄₂₁	—	2	4	no	—
37	C₄×C₄₂₁	C₂×D₄	1	4	yes	—
38	Dic₄₂₁	—	2	4	no	—
39	C₂×C₄₂₁	—	2	2	no	—
40	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₄₂₁	H(5,𝔽₂)	1	1	yes	characteristic
42	C₂×C₄₂₁	2₋¹⁺⁴	1	1	yes	characteristic
43	D₄₂₁	—	4	4	no	—
44	C₄₂₁	#537	1	1	yes	characteristic, complemented, Sylow
45	#537	—	421	1	no	maximal, Sylow
46	C₂×#41	—	421	2	no	—
47	C₂²⋊Q₈	—	421	4	no	—
48	#121	—	421	4	no	—
49	#121	—	421	4	no	—
50	C₂×C₄⋊C₄	D₄₂₁	1	1	yes	characteristic, complemented
51	#41	—	842	8	no	—
52	C₂×Q₈	—	421	2	no	—
53	C₂×D₄	—	421	4	no	—
54	C₂×D₄	—	421	2	no	—
55	C₂×D₄	—	421	4	no	—
56	C₂²×C₄	—	421	4	no	—
57	C₂²×C₄	C₂×D₄₂₁	1	2	yes	—
58	C₂²×C₄	C₂×D₄₂₁	1	1	yes	characteristic
59	C₄⋊C₄	—	421	4	no	—
60	C₄⋊C₄	C₂×D₄₂₁	1	4	yes	—
61	C₂²⋊C₄	—	421	8	no	—
62	C₂³	—	421	4	no	—
63	C₂³	C₂²×D₄₂₁	1	1	yes	characteristic
64	Q₈	—	842	4	no	—
65	D₄	—	842	8	no	—
66	D₄	—	842	4	no	—
67	D₄	—	1684	4	no	—
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	—	842	8	no	—
70	C₂×C₄	—	421	4	no	—
71	C₂×C₄	D₁₆₈₄	1	4	yes	complemented
72	C₂×C₄	C₂²×D₄₂₁	1	4	yes	—
73	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	—	842	4	no	—
77	C₂²	—	842	4	no	—
78	C₂²	—	842	4	no	—
79	C₂²	C₂×D₁₆₈₄	1	2	yes	—
80	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
81	C₄	—	2	4	no	—
82	C₄	—	842	4	no	—
83	C₄	C₂×D₁₆₈₄	1	4	yes	—
84	C₂	—	2	2	no	—
85	C₂	—	1684	4	no	—
86	C₂	C₂²×D₁₆₈₄	1	1	yes	characteristic
87	C₂	#27212411	1	1	yes	characteristic
88	C₂	#27212414	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	C₂×C₄⋊C₄×C₄₂₁	C₂
hypercenter	C₂×C₄⋊C₄	D₄₂₁
hypocenter	C₄₂₁	#537
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#20269771	C₂	semidirect
2. #27212346	C₂	semidirect
3. #27212341	C₂	semidirect
4. #27212340	C₂	semidirect
5. C₂×C₄⋊C₄×C₄₂₁	C₂	semidirect
26. C₂×C₄×C₄₂₁	D₄	semidirect
44. C₄₂₁	#537	semidirect
50. C₂×C₄⋊C₄	D₄₂₁	semidirect
71. C₂×C₄	D₁₆₈₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20269771	2
#27212346	2
#27212341	2
#27212340	2
C₂×C₄⋊C₄×C₄₂₁	2
#537	421