Group #29508461 — subgroups

Subgroups — #29508461

Order 15424.

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	#21440634	C₂	1	1	yes	maximal, characteristic, complemented
2	#21440630	C₂	1	1	yes	maximal, characteristic, complemented
3	#21440626	C₂	1	2	yes	maximal, complemented
4	#21440624	C₂	1	2	yes	maximal, complemented
5	#21440619	C₂	1	2	yes	maximal, complemented
6	#21440618	C₂	1	2	yes	maximal, complemented
7	#21440615	C₂	1	2	yes	maximal, complemented
8	#21440614	C₂	1	2	yes	maximal, complemented
9	#124×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₄₈₂	C₂²	1	1	yes	characteristic
11	C₂×D₉₆₄	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₂₄₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₂₄₁	C₂²	1	2	yes	—
14	C₂²×Dic₂₄₁	C₂²	1	2	yes	—
15	C₂×C₄×D₂₄₁	C₂²	1	2	yes	—
16	C₄⋊Dic₂₄₁	C₂²	1	2	yes	—
17	C₄⋊Dic₂₄₁	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₂₄₁	C₂²	1	1	yes	characteristic
19	#16838749	C₂²	1	2	yes	—
20	#16838749	C₂²	1	2	yes	—
21	#16838749	C₂²	1	2	yes	—
22	C₂²⋊Dic₂₄₁	C₂²	1	2	yes	—
23	#16838747	C₂²	1	2	yes	—
24	#16838747	C₂²	1	2	yes	—
25	#16838747	C₂²	1	2	yes	—
26	C₂×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₂₄₁	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₂₄₁	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₂₄₁	C₂²	1	2	yes	—
30	C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
31	C₂²×D₂₄₁	C₂³	1	2	yes	—
32	Dic₄₈₂	—	4	1	no	—
33	D₉₆₄	—	4	1	no	—
34	C₂²⋊D₂₄₁	—	4	2	no	—
35	C₂²⋊D₂₄₁	—	4	2	no	—
36	C₂×Dic₂₄₁	—	4	2	no	—
37	C₂×Dic₂₄₁	C₂³	1	2	yes	—
38	C₂×Dic₂₄₁	C₂³	1	2	yes	—
39	C₂×Dic₂₄₁	C₂³	1	2	yes	—
40	C₄×D₂₄₁	—	4	2	no	—
41	C₂³×C₂₄₁	C₂³	1	2	yes	—
42	Q₈×C₂₄₁	—	4	1	no	—
43	D₄×C₂₄₁	—	4	1	no	—
44	C₂×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₂₄₁	C₂³	1	2	yes	—
48	C₂²×C₂₄₁	—	2	2	no	—
49	C₂²×C₂₄₁	—	2	2	no	—
50	C₂²×C₂₄₁	—	2	2	no	—
51	C₂²×C₂₄₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₂₄₁	—	2	2	no	—
53	C₂×D₂₄₁	—	2	2	no	—
54	C₂×D₂₄₁	—	2	2	no	—
55	C₄×C₂₄₁	—	2	1	no	—
56	C₄×C₂₄₁	—	2	2	no	—
57	C₄×C₂₄₁	—	2	1	no	—
58	C₄×C₂₄₁	—	2	1	no	—
59	Dic₂₄₁	—	2	2	no	—
60	Dic₂₄₁	—	2	2	no	—
61	Dic₂₄₁	—	2	2	no	—
62	C₂×C₂₄₁	—	4	2	no	—
63	C₂×C₂₄₁	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₂₄₁	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₂₄₁	2₋¹⁺⁴	1	1	yes	characteristic
66	D₂₄₁	—	4	2	no	—
67	C₂₄₁	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	241	1	no	maximal, Sylow
69	#125	—	241	1	no	—
70	#124	—	241	1	no	—
71	#124	D₂₄₁	1	1	yes	characteristic, complemented
72	#123	—	241	2	no	—
73	#123	—	241	2	no	—
74	C₂²⋊Q₈	—	241	2	no	—
75	C₂²⋊Q₈	—	241	2	no	—
76	#121	—	241	2	no	—
77	#121	—	241	2	no	—
78	C₂×Q₈	—	241	1	no	—
79	C₂×Q₈	C₂×D₂₄₁	1	1	yes	characteristic
80	C₂×D₄	—	241	2	no	—
81	C₂×D₄	—	241	2	no	—
82	C₂×D₄	—	241	1	no	—
83	C₂×D₄	C₂×D₂₄₁	1	1	yes	characteristic
84	C₂²×C₄	—	241	2	no	—
85	C₂²×C₄	—	241	2	no	—
86	C₄⋊C₄	—	241	2	no	—
87	C₄⋊C₄	—	241	2	no	—
88	C₄⋊C₄	—	241	2	no	—
89	C₄⋊C₄	—	241	2	no	—
90	C₄⋊C₄	—	241	1	no	—
91	C₄⋊C₄	—	241	1	no	—
92	C₂²⋊C₄	—	241	2	no	—
93	C₂²⋊C₄	—	241	2	no	—
94	C₂²⋊C₄	—	241	2	no	—
95	C₂²⋊C₄	—	241	2	no	—
96	C₂²⋊C₄	C₂×D₂₄₁	1	2	yes	—
97	C₂²⋊C₄	C₂×D₂₄₁	1	2	yes	—
98	C₄²	C₂×D₂₄₁	1	1	yes	characteristic
99	C₂³	—	241	2	no	—
100	C₂³	C₂²×D₂₄₁	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	964	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	964	1	no	—
105	D₄	—	964	2	no	—
106	D₄	—	964	2	no	—
107	C₂×C₄	—	964	2	no	—
108	C₂×C₄	—	964	2	no	—
109	C₂×C₄	—	241	2	no	—
110	C₂×C₄	—	241	2	no	—
111	C₂×C₄	—	241	2	no	—
112	C₂×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₂₄₁	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	482	2	no	—
120	C₂²	—	482	2	no	—
121	C₂²	—	482	2	no	—
122	C₂²	C₂³×D₂₄₁	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	—	482	2	no	—
128	C₄	—	482	2	no	—
129	C₄	—	482	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	964	2	no	—
132	C₂	#21440687	1	1	yes	characteristic
133	C₂	#21440688	1	1	yes	characteristic
134	C₂	#21440689	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21440634

C₂

yes

maximal, characteristic, complemented

#21440630

C₂

yes

maximal, characteristic, complemented

#21440626

C₂

yes

maximal, complemented

#21440624

C₂

yes

maximal, complemented

#21440619

C₂

yes

maximal, complemented

#21440618

C₂

yes

maximal, complemented

#21440615

C₂

yes

maximal, complemented

#21440614

C₂

yes

maximal, complemented

#124×C₂₄₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₈₂

C₂²

yes

characteristic

C₂×D₉₆₄

C₂²

yes

characteristic

C₂×C₂²⋊D₂₄₁

C₂²

yes

—

C₂×C₂²⋊D₂₄₁

C₂²

yes

—

C₂²×Dic₂₄₁

C₂²

yes

—

C₂×C₄×D₂₄₁

C₂²

yes

—

C₄⋊Dic₂₄₁

C₂²

yes

—

C₄⋊Dic₂₄₁

C₂²

yes

characteristic

C₄⋊Dic₂₄₁

C₂²

yes

characteristic

#16838749

C₂²

yes

—

#16838749

C₂²

yes

—

#16838749

C₂²

yes

—

C₂²⋊Dic₂₄₁

C₂²

yes

—

#16838747

C₂²

yes

—

#16838747

C₂²

yes

—

#16838747

C₂²

yes

—

C₂×Q₈×C₂₄₁

C₂²

yes

characteristic

C₂×D₄×C₂₄₁

C₂²

yes

characteristic

C₂²⋊C₄×C₂₄₁

C₂²

yes

—

C₂²⋊C₄×C₂₄₁

C₂²

yes

—

C₄²×C₂₄₁

C₂²

yes

characteristic

C₂²×D₂₄₁

C₂³

yes

—

Dic₄₈₂

—

D₉₆₄

—

C₂²⋊D₂₄₁

—

C₂²⋊D₂₄₁

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

C₂³

yes

—

C₂×Dic₂₄₁

C₂³

yes

—

C₂×Dic₂₄₁

C₂³

yes

—

C₄×D₂₄₁

—

C₂³×C₂₄₁

C₂³

yes

—

Q₈×C₂₄₁

—

D₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂³

yes

characteristic

C₂×C₄×C₂₄₁

C₂³

yes

characteristic

C₂×C₄×C₂₄₁

C₂³

yes

characteristic

C₂×C₄×C₂₄₁

C₂³

yes

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

C₂⁴

yes

characteristic

C₂×D₂₄₁

—

C₂×D₂₄₁

—

C₂×D₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

Dic₂₄₁

—

Dic₂₄₁

—

Dic₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

2₋¹⁺⁴

yes

characteristic

D₂₄₁

—

C₂₄₁

#562

yes

characteristic, complemented, Sylow

#562

—

241

maximal, Sylow

#125

—

241

—

#124

—

241

—

#124

D₂₄₁

yes

characteristic, complemented

#123

—

241

—

#123

—

241

—

C₂²⋊Q₈

—

241

—

C₂²⋊Q₈

—

241

—

#121

—

241

—

#121

—

241

—

C₂×Q₈

—

241

—

C₂×Q₈

C₂×D₂₄₁

yes

characteristic

C₂×D₄

—

241

—

C₂×D₄

—

241

—

C₂×D₄

—

241

—

C₂×D₄

C₂×D₂₄₁

yes

characteristic

C₂²×C₄

—

241

—

C₂²×C₄

—

241

—

C₄⋊C₄

—

241

—

C₄⋊C₄

—

241

—

C₄⋊C₄

—

241

—

C₄⋊C₄

—

241

—

C₄⋊C₄

—

241

—

C₄⋊C₄

—

241

—

C₂²⋊C₄

—

241

—

C₂²⋊C₄

—

241

—

C₂²⋊C₄

—

241

—

C₂²⋊C₄

—

241

—

C₂²⋊C₄

C₂×D₂₄₁

yes

—

C₂²⋊C₄

C₂×D₂₄₁

yes

—

C₄²

C₂×D₂₄₁

yes

characteristic

C₂³

—

241

—

100

C₂³

C₂²×D₂₄₁

yes

—

101

Q₈

—

102

Q₈

—

964

—

103

D₄

—

104

D₄

—

964

—

105

D₄

—

964

—

106

D₄

—

964

—

107

C₂×C₄

—

964

—

108

C₂×C₄

—

964

—

109

C₂×C₄

—

241

—

110

C₂×C₄

—

241

—

111

C₂×C₄

—

241

—

112

C₂×C₄

C₂²×D₂₄₁

yes

characteristic

113

C₂×C₄

C₂²×D₂₄₁

yes

characteristic

114

C₂×C₄

C₂²×D₂₄₁

yes

characteristic

115

C₂×C₄

C₂²×D₂₄₁

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

482

—

120

C₂²

—

482

—

121

C₂²

—

482

—

122

C₂²

C₂³×D₂₄₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

482

—

128

C₄

—

482

—

129

C₄

—

482

—

130

C₂

—

131

C₂

—

964

—

132

C₂

#21440687

yes

characteristic

133

C₂

#21440688

yes

characteristic

134

C₂

#21440689

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₄₁
commutator, socle, cosocle	C₂²×C₂₄₁	C₂⁴
Fitting	#124×C₂₄₁	C₂
hypercenter	#124	D₂₄₁
hypocenter	C₂₄₁	#562
LCS	G ▷ C₂²×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₄₁

commutator, socle, cosocle

C₂²×C₂₄₁

C₂⁴

Fitting

#124×C₂₄₁

C₂

D₂₄₁

C₂₄₁

G ▷ C₂²×C₂₄₁ ▷ C₂₄₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #21440634	C₂	semidirect
2. #21440630	C₂	semidirect
3. #21440626	C₂	semidirect
4. #21440624	C₂	semidirect
5. #21440619	C₂	semidirect
6. #21440618	C₂	semidirect
7. #21440615	C₂	semidirect
8. #21440614	C₂	semidirect
9. #124×C₂₄₁	C₂	semidirect
67. C₂₄₁	#562	semidirect
71. #124	D₂₄₁	semidirect

H ≅ G/N

type

1. #21440634

C₂

semidirect

2. #21440630

C₂

semidirect

3. #21440626

C₂

semidirect

4. #21440624

C₂

semidirect

5. #21440619

C₂

semidirect

6. #21440618

C₂

semidirect

7. #21440615

C₂

semidirect

8. #21440614

C₂

semidirect

9. #124×C₂₄₁

C₂

semidirect

67. C₂₄₁

#562

semidirect

71. #124

D₂₄₁

semidirect

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

subgroup	index
#21440634	2
#21440630	2
#21440626	2
#21440624	2
#21440619	2
#21440618	2
#21440615	2
#21440614	2
#124×C₂₄₁	2
#562	241

subgroup

index

#124×C₂₄₁

241

Subgroups — #29508461

Conjugacy classes of subgroups

Complements

Maximal subgroups