Group #41003584 — subgroups

Subgroups — #41003584

Order 26944.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#27212358	C₂	1	1	yes	maximal, characteristic, complemented
2	#27212354	C₂	1	1	yes	maximal, characteristic, complemented
3	#27212350	C₂	1	2	yes	maximal, complemented
4	#27212348	C₂	1	2	yes	maximal, complemented
5	#27212343	C₂	1	2	yes	maximal, complemented
6	#27212342	C₂	1	2	yes	maximal, complemented
7	#27212339	C₂	1	2	yes	maximal, complemented
8	#27212338	C₂	1	2	yes	maximal, complemented
9	#124×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
11	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₄₂₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₄₂₁	C₂²	1	2	yes	—
14	C₂²×Dic₄₂₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄₂₁	C₂²	1	2	yes	—
16	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
17	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
19	#20269750	C₂²	1	2	yes	—
20	#20269750	C₂²	1	2	yes	—
21	#20269750	C₂²	1	2	yes	—
22	C₂²⋊Dic₄₂₁	C₂²	1	2	yes	—
23	#20269748	C₂²	1	2	yes	—
24	#20269748	C₂²	1	2	yes	—
25	#20269748	C₂²	1	2	yes	—
26	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₄₂₁	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₄₂₁	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₄₂₁	C₂²	1	2	yes	—
30	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
31	C₂²×D₄₂₁	C₂³	1	2	yes	—
32	Dic₈₄₂	—	4	1	no	—
33	D₁₆₈₄	—	4	1	no	—
34	C₂²⋊D₄₂₁	—	4	2	no	—
35	C₂²⋊D₄₂₁	—	4	2	no	—
36	C₂×Dic₄₂₁	—	4	2	no	—
37	C₂×Dic₄₂₁	C₂³	1	2	yes	—
38	C₂×Dic₄₂₁	C₂³	1	2	yes	—
39	C₂×Dic₄₂₁	C₂³	1	2	yes	—
40	C₄×D₄₂₁	—	4	2	no	—
41	C₂³×C₄₂₁	C₂³	1	2	yes	—
42	Q₈×C₄₂₁	—	4	1	no	—
43	D₄×C₄₂₁	—	4	1	no	—
44	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
48	C₂²×C₄₂₁	—	2	2	no	—
49	C₂²×C₄₂₁	—	2	2	no	—
50	C₂²×C₄₂₁	—	2	2	no	—
51	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₄₂₁	—	2	2	no	—
53	C₂×D₄₂₁	—	2	2	no	—
54	C₂×D₄₂₁	—	2	2	no	—
55	C₄×C₄₂₁	—	2	1	no	—
56	C₄×C₄₂₁	—	2	2	no	—
57	C₄×C₄₂₁	—	2	1	no	—
58	C₄×C₄₂₁	—	2	1	no	—
59	Dic₄₂₁	—	2	2	no	—
60	Dic₄₂₁	—	2	2	no	—
61	Dic₄₂₁	—	2	2	no	—
62	C₂×C₄₂₁	—	4	2	no	—
63	C₂×C₄₂₁	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₄₂₁	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₄₂₁	2₋¹⁺⁴	1	1	yes	characteristic
66	D₄₂₁	—	4	2	no	—
67	C₄₂₁	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	421	1	no	maximal, Sylow
69	#125	—	421	1	no	—
70	#124	—	421	1	no	—
71	#124	D₄₂₁	1	1	yes	characteristic, complemented
72	#123	—	421	2	no	—
73	#123	—	421	2	no	—
74	C₂²⋊Q₈	—	421	2	no	—
75	C₂²⋊Q₈	—	421	2	no	—
76	#121	—	421	2	no	—
77	#121	—	421	2	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	—	421	2	no	—
82	C₂×D₄	—	421	1	no	—
83	C₂×D₄	C₂×D₄₂₁	1	1	yes	characteristic
84	C₂²×C₄	—	421	2	no	—
85	C₂²×C₄	—	421	2	no	—
86	C₄⋊C₄	—	421	2	no	—
87	C₄⋊C₄	—	421	2	no	—
88	C₄⋊C₄	—	421	2	no	—
89	C₄⋊C₄	—	421	2	no	—
90	C₄⋊C₄	—	421	1	no	—
91	C₄⋊C₄	—	421	1	no	—
92	C₂²⋊C₄	—	421	2	no	—
93	C₂²⋊C₄	—	421	2	no	—
94	C₂²⋊C₄	—	421	2	no	—
95	C₂²⋊C₄	—	421	2	no	—
96	C₂²⋊C₄	C₂×D₄₂₁	1	2	yes	—
97	C₂²⋊C₄	C₂×D₄₂₁	1	2	yes	—
98	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
99	C₂³	—	421	2	no	—
100	C₂³	C₂²×D₄₂₁	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	1684	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	1684	1	no	—
105	D₄	—	1684	2	no	—
106	D₄	—	1684	2	no	—
107	C₂×C₄	—	1684	2	no	—
108	C₂×C₄	—	1684	2	no	—
109	C₂×C₄	—	421	2	no	—
110	C₂×C₄	—	421	2	no	—
111	C₂×C₄	—	421	2	no	—
112	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	842	2	no	—
120	C₂²	—	842	2	no	—
121	C₂²	—	842	2	no	—
122	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	842	2	no	—
128	C₄	—	842	2	no	—
129	C₄	—	842	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	1684	2	no	—
132	C₂	#27212411	1	1	yes	characteristic
133	C₂	#27212412	1	1	yes	characteristic
134	C₂	#27212413	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27212358

C₂

yes

maximal, characteristic, complemented

#27212354

C₂

yes

maximal, characteristic, complemented

#27212350

C₂

yes

maximal, complemented

#27212348

C₂

yes

maximal, complemented

#27212343

C₂

yes

maximal, complemented

#27212342

C₂

yes

maximal, complemented

#27212339

C₂

yes

maximal, complemented

#27212338

C₂

yes

maximal, complemented

#124×C₄₂₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₈₄₂

C₂²

yes

characteristic

C₂×D₁₆₈₄

C₂²

yes

characteristic

C₂×C₂²⋊D₄₂₁

C₂²

yes

—

C₂×C₂²⋊D₄₂₁

C₂²

yes

—

C₂²×Dic₄₂₁

C₂²

yes

—

C₂×C₄×D₄₂₁

C₂²

yes

—

C₄⋊Dic₄₂₁

C₂²

yes

—

C₄⋊Dic₄₂₁

C₂²

yes

characteristic

C₄⋊Dic₄₂₁

C₂²

yes

characteristic

#20269750

C₂²

yes

—

#20269750

C₂²

yes

—

#20269750

C₂²

yes

—

C₂²⋊Dic₄₂₁

C₂²

yes

—

#20269748

C₂²

yes

—

#20269748

C₂²

yes

—

#20269748

C₂²

yes

—

C₂×Q₈×C₄₂₁

C₂²

yes

characteristic

C₂×D₄×C₄₂₁

C₂²

yes

characteristic

C₂²⋊C₄×C₄₂₁

C₂²

yes

—

C₂²⋊C₄×C₄₂₁

C₂²

yes

—

C₄²×C₄₂₁

C₂²

yes

characteristic

C₂²×D₄₂₁

C₂³

yes

—

Dic₈₄₂

—

D₁₆₈₄

—

C₂²⋊D₄₂₁

—

C₂²⋊D₄₂₁

—

C₂×Dic₄₂₁

—

C₂×Dic₄₂₁

C₂³

yes

—

C₂×Dic₄₂₁

C₂³

yes

—

C₂×Dic₄₂₁

C₂³

yes

—

C₄×D₄₂₁

—

C₂³×C₄₂₁

C₂³

yes

—

Q₈×C₄₂₁

—

D₄×C₄₂₁

—

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂×C₄×C₄₂₁

C₂³

yes

—

C₂²×C₄₂₁

—

C₂²×C₄₂₁

—

C₂²×C₄₂₁

—

C₂²×C₄₂₁

C₂⁴

yes

characteristic

C₂×D₄₂₁

—

C₂×D₄₂₁

—

C₂×D₄₂₁

—

C₄×C₄₂₁

—

C₄×C₄₂₁

—

C₄×C₄₂₁

—

C₄×C₄₂₁

—

Dic₄₂₁

—

Dic₄₂₁

—

Dic₄₂₁

—

C₂×C₄₂₁

—

C₂×C₄₂₁

yes

C₂×C₄₂₁

yes

C₂×C₄₂₁

2₋¹⁺⁴

yes

characteristic

D₄₂₁

—

C₄₂₁

#562

yes

characteristic, complemented, Sylow

#562

—

421

maximal, Sylow

#125

—

421

—

#124

—

421

—

#124

D₄₂₁

yes

characteristic, complemented

#123

—

421

—

#123

—

421

—

C₂²⋊Q₈

—

421

—

C₂²⋊Q₈

—

421

—

#121

—

421

—

#121

—

421

—

C₂×Q₈

—

421

—

C₂×Q₈

C₂×D₄₂₁

yes

characteristic

C₂×D₄

—

421

—

C₂×D₄

—

421

—

C₂×D₄

—

421

—

C₂×D₄

C₂×D₄₂₁

yes

characteristic

C₂²×C₄

—

421

—

C₂²×C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

C₂×D₄₂₁

yes

—

C₂²⋊C₄

C₂×D₄₂₁

yes

—

C₄²

C₂×D₄₂₁

yes

characteristic

C₂³

—

421

—

100

C₂³

C₂²×D₄₂₁

yes

—

101

Q₈

—

102

Q₈

—

1684

—

103

D₄

—

104

D₄

—

1684

—

105

D₄

—

1684

—

106

D₄

—

1684

—

107

C₂×C₄

—

1684

—

108

C₂×C₄

—

1684

—

109

C₂×C₄

—

421

—

110

C₂×C₄

—

421

—

111

C₂×C₄

—

421

—

112

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

113

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

114

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

115

C₂×C₄

C₂²×D₄₂₁

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

842

—

120

C₂²

—

842

—

121

C₂²

—

842

—

122

C₂²

C₂³×D₄₂₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

842

—

128

C₄

—

842

—

129

C₄

—

842

—

130

C₂

—

131

C₂

—

1684

—

132

C₂

#27212411

yes

characteristic

133

C₂

#27212412

yes

characteristic

134

C₂

#27212413

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	#124×C₄₂₁	C₂
hypercenter	#124	D₄₂₁
hypocenter	C₄₂₁	#562
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

D₄₂₁

C₄₂₁

G ▷ C₂²×C₄₂₁ ▷ C₄₂₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #27212358	C₂	semidirect
2. #27212354	C₂	semidirect
3. #27212350	C₂	semidirect
4. #27212348	C₂	semidirect
5. #27212343	C₂	semidirect
6. #27212342	C₂	semidirect
7. #27212339	C₂	semidirect
8. #27212338	C₂	semidirect
9. #124×C₄₂₁	C₂	semidirect
67. C₄₂₁	#562	semidirect
71. #124	D₄₂₁	semidirect

H ≅ G/N

type

1. #27212358

C₂

semidirect

2. #27212354

C₂

semidirect

3. #27212350

C₂

semidirect

4. #27212348

C₂

semidirect

5. #27212343

C₂

semidirect

6. #27212342

C₂

semidirect

7. #27212339

C₂

semidirect

8. #27212338

C₂

semidirect

9. #124×C₄₂₁

C₂

semidirect

67. C₄₂₁

#562

semidirect

71. #124

D₄₂₁

semidirect

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

subgroup	index
#27212358	2
#27212354	2
#27212350	2
#27212348	2
#27212343	2
#27212342	2
#27212339	2
#27212338	2
#124×C₄₂₁	2
#562	421

subgroup

index

#124×C₄₂₁

421

Subgroups — #41003584

Conjugacy classes of subgroups

Complements

Maximal subgroups