Group #41003410 — subgroups

Subgroups — #41003410

Order 26944.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269770	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212366	C₂	1	1	yes	maximal, characteristic, complemented
4	#27212364	C₂	1	1	yes	maximal, characteristic, complemented
5	#27212353	C₂	1	1	yes	maximal, characteristic, complemented
6	#27212348	C₂	1	4	yes	maximal, complemented
7	#27212343	C₂	1	4	yes	maximal, complemented
8	#27212323	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#20269770	—	2	4	no	—
11	C₂×Dic₈₄₂	C₂²	1	2	yes	—
12	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
13	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
14	Q₈×D₄₂₁	—	2	4	no	—
15	C₂×C₂²⋊D₄₂₁	C₂²	1	2	yes	—
16	C₂²×Dic₄₂₁	C₂²	1	2	yes	—
17	C₂×C₄×D₄₂₁	C₂²	1	2	yes	—
18	C₂×C₄×D₄₂₁	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₄₂₁	C₂²	1	4	yes	—
20	#20269750	C₂²	1	2	yes	—
21	#20269750	C₂²	1	4	yes	—
22	C₂²⋊Dic₄₂₁	C₂²	1	4	yes	—
23	#20269748	C₂²	1	2	yes	—
24	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₄₂₁	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₄₂₁	C₂²	1	4	yes	—
28	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
29	C₂²×D₄₂₁	C₂³	1	1	yes	characteristic
30	Dic₈₄₂	—	2	2	no	—
31	Dic₈₄₂	—	2	2	no	—
32	Dic₈₄₂	—	2	4	no	—
33	C₂²⋊D₄₂₁	—	2	4	no	—
34	C₂×Dic₄₂₁	—	2	4	no	—
35	C₂×Dic₄₂₁	C₂³	1	2	yes	—
36	C₂×Dic₄₂₁	C₂³	1	2	yes	—
37	C₂×Dic₄₂₁	C₂³	1	2	yes	—
38	C₂×Dic₄₂₁	C₂³	1	1	yes	characteristic
39	C₄×D₄₂₁	—	2	4	no	—
40	C₄×D₄₂₁	D₄	1	4	yes	—
41	C₂³×C₄₂₁	C₂³	1	2	yes	—
42	Q₈×C₄₂₁	—	2	2	no	—
43	D₄×C₄₂₁	—	2	2	no	—
44	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
45	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
46	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
47	C₂²×C₄₂₁	—	2	2	no	—
48	C₂²×C₄₂₁	—	2	4	no	—
49	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
50	C₂×D₄₂₁	—	2	2	no	—
51	C₂×D₄₂₁	C₂×D₄	1	2	yes	—
52	C₄×C₄₂₁	—	2	2	no	—
53	C₄×C₄₂₁	—	2	2	no	—
54	C₄×C₄₂₁	C₂×D₄	1	2	yes	—
55	Dic₄₂₁	—	2	2	no	—
56	Dic₄₂₁	—	2	2	no	—
57	Dic₄₂₁	—	2	2	no	—
58	Dic₄₂₁	C₂×D₄	1	2	yes	—
59	C₂×C₄₂₁	—	4	2	no	—
60	C₂×C₄₂₁	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₄₂₁	2₋¹⁺⁴	1	2	yes	—
62	D₄₂₁	—	2	2	no	—
63	C₄₂₁	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	421	1	no	maximal, Sylow
65	C₂×#41	—	421	1	no	—
66	C₂²×Q₈	—	421	1	no	—
67	C₄⋊Q₈	—	421	1	no	—
68	C₄⋊Q₈	—	421	1	no	—
69	#124	—	421	1	no	—
70	#124	D₄₂₁	1	1	yes	characteristic, complemented
71	#123	—	421	4	no	—
72	C₂²⋊Q₈	—	421	4	no	—
73	#117	—	421	1	no	—
74	#41	—	842	4	no	—
75	C₂×Q₈	—	842	4	no	—
76	C₂×Q₈	—	421	2	no	—
77	C₂×Q₈	—	421	1	no	—
78	C₂×Q₈	—	421	1	no	—
79	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
80	C₂×D₄	—	421	2	no	—
81	C₂×D₄	C₂×D₄₂₁	1	1	yes	characteristic
82	C₂²×C₄	—	421	2	no	—
83	C₂²×C₄	—	421	2	no	—
84	C₂²×C₄	—	421	1	no	—
85	C₄⋊C₄	—	421	2	no	—
86	C₄⋊C₄	—	421	4	no	—
87	C₄⋊C₄	—	421	4	no	—
88	C₂²⋊C₄	—	421	4	no	—
89	C₂²⋊C₄	—	421	2	no	—
90	C₂²⋊C₄	C₂×D₄₂₁	1	4	yes	—
91	C₄²	—	421	1	no	—
92	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
93	C₂³	—	421	1	no	—
94	C₂³	C₂²×D₄₂₁	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	842	4	no	—
97	Q₈	—	842	2	no	—
98	Q₈	—	842	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	842	4	no	—
101	C₂×C₄	—	842	4	no	—
102	C₂×C₄	—	842	4	no	—
103	C₂×C₄	—	421	2	no	—
104	C₂×C₄	—	421	2	no	—
105	C₂×C₄	—	421	2	no	—
106	C₂×C₄	—	421	1	no	—
107	C₂×C₄	—	421	4	no	—
108	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
109	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
110	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	842	2	no	—
114	C₂²	—	421	2	no	—
115	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	842	2	no	—
119	C₄	—	842	2	no	—
120	C₄	—	842	2	no	—
121	C₄	—	421	2	no	—
122	C₄	D₄×D₄₂₁	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	842	2	no	—
125	C₂	C₂×D₄×D₄₂₁	1	1	yes	characteristic
126	C₂	#27212413	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269770

C₂

yes

maximal, characteristic

C₂×Q₈×D₄₂₁

C₂

yes

maximal, characteristic, complemented

#27212366

C₂

yes

maximal, characteristic, complemented

#27212364

C₂

yes

maximal, characteristic, complemented

#27212353

C₂

yes

maximal, characteristic, complemented

#27212348

C₂

yes

maximal, complemented

#27212343

C₂

yes

maximal, complemented

#27212323

C₂

yes

maximal, characteristic, complemented

#124×C₄₂₁

C₂

yes

maximal, characteristic, complemented

#20269770

—

C₂×Dic₈₄₂

C₂²

yes

—

C₂×Dic₈₄₂

C₂²

yes

characteristic

C₂×Dic₈₄₂

C₂²

yes

characteristic

Q₈×D₄₂₁

—

C₂×C₂²⋊D₄₂₁

C₂²

yes

—

C₂²×Dic₄₂₁

C₂²

yes

—

C₂×C₄×D₄₂₁

C₂²

yes

—

C₂×C₄×D₄₂₁

C₂²

yes

characteristic

C₄⋊Dic₄₂₁

C₂²

yes

—

#20269750

C₂²

yes

—

#20269750

C₂²

yes

—

C₂²⋊Dic₄₂₁

C₂²

yes

—

#20269748

C₂²

yes

—

C₄×Dic₄₂₁

C₂²

yes

characteristic

C₂×Q₈×C₄₂₁

C₂²

yes

characteristic

C₂×D₄×C₄₂₁

C₂²

yes

characteristic

C₂²⋊C₄×C₄₂₁

C₂²

yes

—

C₄²×C₄₂₁

C₂²

yes

characteristic

C₂²×D₄₂₁

C₂³

yes

characteristic

Dic₈₄₂

—

Dic₈₄₂

—

Dic₈₄₂

—

C₂²⋊D₄₂₁

—

C₂×Dic₄₂₁

—

C₂×Dic₄₂₁

C₂³

yes

—

C₂×Dic₄₂₁

C₂³

yes

—

C₂×Dic₄₂₁

C₂³

yes

—

C₂×Dic₄₂₁

C₂³

yes

characteristic

C₄×D₄₂₁

—

C₄×D₄₂₁

D₄

yes

—

C₂³×C₄₂₁

C₂³

yes

—

Q₈×C₄₂₁

—

D₄×C₄₂₁

—

C₂×C₄×C₄₂₁

C₂³

yes

—

C₂×C₄×C₄₂₁

C₂³

yes

—

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂²×C₄₂₁

—

C₂²×C₄₂₁

—

C₂²×C₄₂₁

C₂⁴

yes

characteristic

C₂×D₄₂₁

—

C₂×D₄₂₁

C₂×D₄

yes

—

C₄×C₄₂₁

—

C₄×C₄₂₁

—

C₄×C₄₂₁

C₂×D₄

yes

—

Dic₄₂₁

—

Dic₄₂₁

—

Dic₄₂₁

—

Dic₄₂₁

C₂×D₄

yes

—

C₂×C₄₂₁

—

C₂×C₄₂₁

C₂²×D₄

yes

characteristic

C₂×C₄₂₁

2₋¹⁺⁴

yes

—

D₄₂₁

—

C₄₂₁

#536

yes

characteristic, complemented, Sylow

#536

—

421

maximal, Sylow

C₂×#41

—

421

—

C₂²×Q₈

—

421

—

C₄⋊Q₈

—

421

—

C₄⋊Q₈

—

421

—

#124

—

421

—

#124

D₄₂₁

yes

characteristic, complemented

#123

—

421

—

C₂²⋊Q₈

—

421

—

#117

—

421

—

#41

—

842

—

C₂×Q₈

—

842

—

C₂×Q₈

—

421

—

C₂×Q₈

—

421

—

C₂×Q₈

—

421

—

C₂×Q₈

C₂×D₄₂₁

yes

characteristic

C₂×D₄

—

421

—

C₂×D₄

C₂×D₄₂₁

yes

characteristic

C₂²×C₄

—

421

—

C₂²×C₄

—

421

—

C₂²×C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

C₂×D₄₂₁

yes

—

C₄²

—

421

—

C₄²

C₂×D₄₂₁

yes

characteristic

C₂³

—

421

—

C₂³

C₂²×D₄₂₁

yes

—

Q₈

—

Q₈

—

842

—

Q₈

—

842

—

Q₈

—

842

—

D₄

—

100

D₄

—

842

—

101

C₂×C₄

—

842

—

102

C₂×C₄

—

842

—

103

C₂×C₄

—

421

—

104

C₂×C₄

—

421

—

105

C₂×C₄

—

421

—

106

C₂×C₄

—

421

—

107

C₂×C₄

—

421

—

108

C₂×C₄

C₂²×D₄₂₁

yes

—

109

C₂×C₄

C₂²×D₄₂₁

yes

—

110

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

842

—

114

C₂²

—

421

—

115

C₂²

C₂³×D₄₂₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

842

—

119

C₄

—

842

—

120

C₄

—

842

—

121

C₄

—

421

—

122

C₄

D₄×D₄₂₁

yes

—

123

C₂

—

124

C₂

—

842

—

125

C₂

C₂×D₄×D₄₂₁

yes

characteristic

126

C₂

#27212413

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	#124×C₄₂₁	C₂
prime radical	C₂×#20269770	C₂
hypercenter	#124	D₄₂₁
hypocenter	C₄₂₁	#536
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₂₁

commutator, socle, cosocle

C₂²×C₄₂₁

C₂⁴

Fitting

#124×C₄₂₁

C₂

C₂

D₄₂₁

C₄₂₁

G ▷ C₂²×C₄₂₁ ▷ C₄₂₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₄₂₁	C₂	semidirect
3. #27212366	C₂	semidirect
4. #27212364	C₂	semidirect
5. #27212353	C₂	semidirect
6. #27212348	C₂	semidirect
7. #27212343	C₂	semidirect
8. #27212323	C₂	semidirect
9. #124×C₄₂₁	C₂	semidirect
63. C₄₂₁	#536	semidirect
70. #124	D₄₂₁	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₄₂₁

C₂

semidirect

3. #27212366

C₂

semidirect

4. #27212364

C₂

semidirect

5. #27212353

C₂

semidirect

6. #27212348

C₂

semidirect

7. #27212343

C₂

semidirect

8. #27212323

C₂

semidirect

9. #124×C₄₂₁

C₂

semidirect

63. C₄₂₁

#536

semidirect

70. #124

D₄₂₁

semidirect

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

subgroup	index
C₂×#20269770	2
C₂×Q₈×D₄₂₁	2
#27212366	2
#27212364	2
#27212353	2
#27212348	2
#27212343	2
#27212323	2
#124×C₄₂₁	2
#536	421

subgroup

index

C₂×Q₈×D₄₂₁

#124×C₄₂₁

421

Subgroups — #41003410

Conjugacy classes of subgroups

Complements

Maximal subgroups