Subgroups — #41003444

Order 26944.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212358	C₂	1	1	yes	maximal, characteristic, complemented
2	#27212347	C₂	1	4	yes	maximal, complemented
3	#27212338	C₂	1	4	yes	maximal, complemented
4	#27212337	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₄₂₁	C₂	1	2	yes	maximal, complemented
6	#27212323	C₂	1	1	yes	maximal, characteristic, complemented
7	#127×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊D₄₂₁	C₂²	1	4	yes	—
9	C₂²×Dic₄₂₁	C₂²	1	4	yes	—
10	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
12	#20269750	C₂²	1	2	yes	—
13	#20269750	C₂²	1	4	yes	—
14	C₂²⋊Dic₄₂₁	C₂²	1	4	yes	—
15	C₂²⋊Dic₄₂₁	C₂²	1	4	yes	—
16	#20269748	C₂²	1	2	yes	—
17	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₄₂₁	C₂²	1	2	yes	—
19	C₂×D₄×C₄₂₁	C₂²	1	4	yes	—
20	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
21	C₂²×D₄₂₁	C₂³	1	1	yes	characteristic
22	C₂²⋊D₄₂₁	—	4	4	no	—
23	C₂×Dic₄₂₁	—	4	4	no	—
24	C₂×Dic₄₂₁	C₂³	1	2	yes	—
25	C₂×Dic₄₂₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₄₂₁	C₂³	1	4	yes	—
27	C₄×D₄₂₁	—	2	2	no	—
28	C₂³×C₄₂₁	C₂³	1	4	yes	—
29	D₄×C₄₂₁	—	2	4	no	—
30	D₄×C₄₂₁	—	4	4	no	—
31	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
32	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
33	C₂²×C₄₂₁	—	2	4	no	—
34	C₂²×C₄₂₁	—	2	4	no	—
35	C₂²×C₄₂₁	—	2	4	no	—
36	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
37	C₂×D₄₂₁	—	2	1	no	—
38	C₂×D₄₂₁	—	2	2	no	—
39	C₄×C₄₂₁	—	2	2	no	—
40	C₄×C₄₂₁	#41	1	2	yes	—
41	Dic₄₂₁	—	2	1	no	—
42	Dic₄₂₁	—	2	2	no	—
43	Dic₄₂₁	—	2	4	no	—
44	C₂×C₄₂₁	—	4	4	no	—
45	C₂×C₄₂₁	C₂×#41	1	1	yes	characteristic
46	C₂×C₄₂₁	H(5,𝔽₂)	1	2	yes	—
47	D₄₂₁	—	4	1	no	—
48	C₄₂₁	#540	1	1	yes	characteristic, complemented, Sylow
49	#540	—	421	1	no	maximal, Sylow
50	#127	D₄₂₁	1	1	yes	characteristic, complemented
51	#125	—	421	1	no	—
52	#123	—	421	4	no	—
53	#121	—	421	2	no	—
54	#121	—	421	4	no	—
55	C₄×D₄	—	421	2	no	—
56	#117	—	421	1	no	—
57	C₂×D₄	—	421	4	no	—
58	C₂×D₄	C₂×D₄₂₁	1	2	yes	—
59	C₂×D₄	C₂×D₄₂₁	1	4	yes	—
60	C₂²×C₄	—	421	1	no	—
61	C₂²×C₄	—	421	4	no	—
62	C₄⋊C₄	—	421	2	no	—
63	C₄⋊C₄	—	421	2	no	—
64	C₄⋊C₄	—	421	4	no	—
65	C₂²⋊C₄	—	421	4	no	—
66	C₂²⋊C₄	—	421	4	no	—
67	C₂²⋊C₄	—	421	2	no	—
68	C₄²	—	421	1	no	—
69	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
70	C₂³	—	421	1	no	—
71	C₂³	C₂²×D₄₂₁	1	4	yes	—
72	D₄	—	2	4	no	—
73	D₄	—	4	4	no	—
74	D₄	—	1684	4	no	—
75	C₂×C₄	—	842	2	no	—
76	C₂×C₄	—	1684	4	no	—
77	C₂×C₄	—	421	2	no	—
78	C₂×C₄	—	421	1	no	—
79	C₂×C₄	—	421	4	no	—
80	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
81	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
82	C₂²	—	2	4	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	842	1	no	—
86	C₂²	—	842	2	no	—
87	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	842	1	no	—
90	C₄	—	842	2	no	—
91	C₄	—	842	4	no	—
92	C₄	#20269770	1	2	yes	—
93	C₂	—	4	4	no	—
94	C₂	—	1684	1	no	—
95	C₂	C₂×#20269770	1	1	yes	characteristic
96	C₂	#27212411	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	#127×C₄₂₁	C₂
hypercenter	#127	D₄₂₁
hypocenter	C₄₂₁	#540
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #27212358	C₂	semidirect
2. #27212347	C₂	semidirect
3. #27212338	C₂	semidirect
4. #27212337	C₂	semidirect
5. D₄×Dic₄₂₁	C₂	semidirect
6. #27212323	C₂	semidirect
7. #127×C₄₂₁	C₂	semidirect
48. C₄₂₁	#540	semidirect
50. #127	D₄₂₁	semidirect

Maximal subgroups

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

subgroup	index
#27212358	2
#27212347	2
#27212338	2
#27212337	2
D₄×Dic₄₂₁	2
#27212323	2
#127×C₄₂₁	2
#540	421