Group #14581080 — subgroups

Subgroups — #14581080

Order 2624.

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	#13050156	C₂	1	1	yes	maximal, characteristic, complemented
2	#13050152	C₂	1	1	yes	maximal, characteristic, complemented
3	#13050148	C₂	1	2	yes	maximal, complemented
4	#13050146	C₂	1	2	yes	maximal, complemented
5	#13050141	C₂	1	2	yes	maximal, complemented
6	#13050140	C₂	1	2	yes	maximal, complemented
7	#13050137	C₂	1	2	yes	maximal, complemented
8	#13050136	C₂	1	2	yes	maximal, complemented
9	#124×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
14	C₂²×Dic₄₁	C₂²	1	2	yes	—
15	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄₁	C₂²	1	2	yes	—
17	C₂×Dic₈₂	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
19	C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
20	C₄²×C₄₁	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
22	#10620281	C₂²	1	2	yes	—
23	#10620281	C₂²	1	2	yes	—
24	#10620281	C₂²	1	2	yes	—
25	C₄⋊Dic₄₁	C₂²	1	2	yes	—
26	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
28	#10620279	C₂²	1	2	yes	—
29	#10620279	C₂²	1	2	yes	—
30	#10620279	C₂²	1	2	yes	—
31	C₂³×C₄₁	C₂³	1	2	yes	—
32	C₂²×D₄₁	C₂³	1	2	yes	—
33	Q₈×C₄₁	—	4	1	no	—
34	D₄×C₄₁	—	4	1	no	—
35	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₄₁	C₂³	1	2	yes	—
38	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
39	C₂²⋊D₄₁	—	4	2	no	—
40	C₂²⋊D₄₁	—	4	2	no	—
41	C₂×Dic₄₁	—	4	2	no	—
42	C₂×Dic₄₁	C₂³	1	2	yes	—
43	C₂×Dic₄₁	C₂³	1	2	yes	—
44	C₂×Dic₄₁	C₂³	1	2	yes	—
45	D₁₆₄	—	4	1	no	—
46	C₄×D₄₁	—	4	2	no	—
47	Dic₈₂	—	4	1	no	—
48	C₂²×C₄₁	—	2	2	no	—
49	C₂²×C₄₁	—	2	2	no	—
50	C₂²×C₄₁	—	2	2	no	—
51	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₄₁	—	2	2	no	—
53	C₂×D₄₁	—	2	2	no	—
54	C₂×D₄₁	—	2	2	no	—
55	C₄×C₄₁	—	2	1	no	—
56	C₄×C₄₁	—	2	2	no	—
57	C₄×C₄₁	—	2	1	no	—
58	C₄×C₄₁	—	2	1	no	—
59	Dic₄₁	—	2	2	no	—
60	Dic₄₁	—	2	2	no	—
61	Dic₄₁	—	2	2	no	—
62	C₂×C₄₁	—	4	2	no	—
63	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
66	D₄₁	—	4	2	no	—
67	#562	—	41	1	no	maximal, Sylow
68	C₄₁	#562	1	1	yes	characteristic, complemented, Sylow
69	#125	—	41	1	no	—
70	#124	—	41	1	no	—
71	#124	D₄₁	1	1	yes	characteristic, complemented
72	#123	—	41	2	no	—
73	#123	—	41	2	no	—
74	C₂²⋊Q₈	—	41	2	no	—
75	C₂²⋊Q₈	—	41	2	no	—
76	#121	—	41	2	no	—
77	#121	—	41	2	no	—
78	C₂×Q₈	—	41	1	no	—
79	C₂×Q₈	C₂×D₄₁	1	1	yes	characteristic
80	C₂×D₄	—	41	2	no	—
81	C₂×D₄	—	41	2	no	—
82	C₂×D₄	—	41	1	no	—
83	C₂×D₄	C₂×D₄₁	1	1	yes	characteristic
84	C₂²×C₄	—	41	2	no	—
85	C₂²×C₄	—	41	2	no	—
86	C₄⋊C₄	—	41	2	no	—
87	C₄⋊C₄	—	41	2	no	—
88	C₄⋊C₄	—	41	2	no	—
89	C₄⋊C₄	—	41	2	no	—
90	C₄⋊C₄	—	41	1	no	—
91	C₄⋊C₄	—	41	1	no	—
92	C₂²⋊C₄	—	41	2	no	—
93	C₂²⋊C₄	—	41	2	no	—
94	C₂²⋊C₄	—	41	2	no	—
95	C₂²⋊C₄	—	41	2	no	—
96	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
97	C₂²⋊C₄	C₂×D₄₁	1	2	yes	—
98	C₄²	C₂×D₄₁	1	1	yes	characteristic
99	C₂³	—	41	2	no	—
100	C₂³	C₂²×D₄₁	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	164	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	164	1	no	—
105	D₄	—	164	2	no	—
106	D₄	—	164	2	no	—
107	C₂×C₄	—	41	2	no	—
108	C₂×C₄	—	41	2	no	—
109	C₂×C₄	—	41	2	no	—
110	C₂×C₄	—	164	2	no	—
111	C₂×C₄	—	164	2	no	—
112	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₄₁	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	82	2	no	—
120	C₂²	—	82	2	no	—
121	C₂²	—	82	2	no	—
122	C₂²	C₂³×D₄₁	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	—	82	2	no	—
128	C₄	—	82	2	no	—
129	C₄	—	82	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	164	2	no	—
132	C₂	#13050209	1	1	yes	characteristic
133	C₂	#13050210	1	1	yes	characteristic
134	C₂	#13050211	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13050156

C₂

yes

maximal, characteristic, complemented

#13050152

C₂

yes

maximal, characteristic, complemented

#13050148

C₂

yes

maximal, complemented

#13050146

C₂

yes

maximal, complemented

#13050141

C₂

yes

maximal, complemented

#13050140

C₂

yes

maximal, complemented

#13050137

C₂

yes

maximal, complemented

#13050136

C₂

yes

maximal, complemented

#124×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₁

C₂²

yes

characteristic

C₂×D₄×C₄₁

C₂²

yes

characteristic

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂×C₂²⋊D₄₁

C₂²

yes

—

C₂²×Dic₄₁

C₂²

yes

—

C₂×D₁₆₄

C₂²

yes

characteristic

C₂×C₄×D₄₁

C₂²

yes

—

C₂×Dic₈₂

C₂²

yes

characteristic

C₂²⋊C₄×C₄₁

C₂²

yes

—

C₂²⋊C₄×C₄₁

C₂²

yes

—

C₄²×C₄₁

C₂²

yes

characteristic

C₂²⋊Dic₄₁

C₂²

yes

—

#10620281

C₂²

yes

—

#10620281

C₂²

yes

—

#10620281

C₂²

yes

—

C₄⋊Dic₄₁

C₂²

yes

—

C₄⋊Dic₄₁

C₂²

yes

characteristic

C₄⋊Dic₄₁

C₂²

yes

characteristic

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

#10620279

C₂²

yes

—

C₂³×C₄₁

C₂³

yes

—

C₂²×D₄₁

C₂³

yes

—

Q₈×C₄₁

—

D₄×C₄₁

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂²⋊D₄₁

—

C₂²⋊D₄₁

—

C₂×Dic₄₁

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

D₁₆₄

—

C₄×D₄₁

—

Dic₈₂

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

—

C₂²×C₄₁

C₂⁴

yes

characteristic

C₂×D₄₁

—

C₂×D₄₁

—

C₂×D₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

C₄×C₄₁

—

Dic₄₁

—

Dic₄₁

—

Dic₄₁

—

C₂×C₄₁

—

C₂×C₄₁

yes

C₂×C₄₁

yes

C₂×C₄₁

2₋¹⁺⁴

yes

characteristic

D₄₁

—

#562

—

maximal, Sylow

C₄₁

#562

yes

characteristic, complemented, Sylow

#125

—

#124

—

#124

D₄₁

yes

characteristic, complemented

#123

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₁

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₁

yes

—

C₂²⋊C₄

C₂×D₄₁

yes

—

C₄²

C₂×D₄₁

yes

characteristic

C₂³

—

100

C₂³

C₂²×D₄₁

yes

—

101

Q₈

—

102

Q₈

—

164

—

103

D₄

—

104

D₄

—

164

—

105

D₄

—

164

—

106

D₄

—

164

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

164

—

111

C₂×C₄

—

164

—

112

C₂×C₄

C₂²×D₄₁

yes

characteristic

113

C₂×C₄

C₂²×D₄₁

yes

characteristic

114

C₂×C₄

C₂²×D₄₁

yes

characteristic

115

C₂×C₄

C₂²×D₄₁

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂³×D₄₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₂

—

131

C₂

—

164

—

132

C₂

#13050209

yes

characteristic

133

C₂

#13050210

yes

characteristic

134

C₂

#13050211

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	#124×C₄₁	C₂
hypercenter	#124	D₄₁
hypocenter	C₄₁	#562
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₁

commutator, socle, cosocle

C₂²×C₄₁

C₂⁴

Fitting

#124×C₄₁

C₂

D₄₁

C₄₁

G ▷ C₂²×C₄₁ ▷ C₄₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #13050156	C₂	semidirect
2. #13050152	C₂	semidirect
3. #13050148	C₂	semidirect
4. #13050146	C₂	semidirect
5. #13050141	C₂	semidirect
6. #13050140	C₂	semidirect
7. #13050137	C₂	semidirect
8. #13050136	C₂	semidirect
9. #124×C₄₁	C₂	semidirect
68. C₄₁	#562	semidirect
71. #124	D₄₁	semidirect

H ≅ G/N

type

1. #13050156

C₂

semidirect

2. #13050152

C₂

semidirect

3. #13050148

C₂

semidirect

4. #13050146

C₂

semidirect

5. #13050141

C₂

semidirect

6. #13050140

C₂

semidirect

7. #13050137

C₂

semidirect

8. #13050136

C₂

semidirect

9. #124×C₄₁

C₂

semidirect

68. C₄₁

#562

semidirect

71. #124

D₄₁

semidirect

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

subgroup	index
#13050156	2
#13050152	2
#13050148	2
#13050146	2
#13050141	2
#13050140	2
#13050137	2
#13050136	2
#124×C₄₁	2
#562	41

subgroup

index

#124×C₄₁

Subgroups — #14581080

Conjugacy classes of subgroups

Complements

Maximal subgroups