Group #28374413 — subgroups

Subgroups — #28374413

Order 13952.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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	#20295576	C₂	1	2	yes	maximal, complemented
2	#20295317	C₂	1	4	yes	maximal, complemented
3	#20295279	C₂	1	4	yes	maximal, complemented
4	#20295222	C₂	1	4	yes	maximal, complemented
5	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182029	—	2	2	no	—
7	C₂×D₈₇₂	C₂²	1	4	yes	—
8	#16812699	C₂²	1	2	yes	—
9	#16812696	C₂²	1	2	yes	—
10	#16812687	—	2	2	no	—
11	#16812673	—	2	2	no	—
12	#16812673	C₂²	1	4	yes	—
13	C₄×D₄₃₆	—	2	2	no	—
14	C₄×D₄₃₆	C₂²	1	4	yes	—
15	#16812631	C₂²	1	4	yes	—
16	#16812621	C₂²	1	8	yes	—
17	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
20	#13182029	—	4	8	no	—
21	C₂×Dic₂₁₈	—	2	2	no	—
22	C₂×D₄₃₆	—	2	2	no	—
23	C₂×D₄₃₆	—	2	2	no	—
24	C₂×D₄₃₆	C₂³	1	4	yes	—
25	C₂×C₂²⋊D₁₀₉	—	2	4	no	—
26	C₂×C₄×D₁₀₉	—	2	4	no	—
27	D₈₇₂	—	4	4	no	—
28	C₄⋊Dic₁₀₉	—	2	2	no	—
29	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
30	#13182006	—	2	4	no	—
31	#13182006	—	2	4	no	—
32	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₀₉	—	4	2	no	—
35	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
36	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
37	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	C₂³	1	2	yes	—
39	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×D₁₀₉	—	2	4	no	—
41	C₂×C₄×C₁₀₉	—	2	4	no	—
42	C₂×C₄×C₁₀₉	—	2	1	no	—
43	C₂×C₄×C₁₀₉	—	2	1	no	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
48	C₂²⋊D₁₀₉	—	4	8	no	—
49	C₂×Dic₁₀₉	—	2	4	no	—
50	D₄₃₆	—	2	4	no	—
51	D₄₃₆	—	2	4	no	—
52	D₄₃₆	—	4	4	no	—
53	D₄₃₆	—	4	4	no	—
54	C₄×D₁₀₉	—	4	8	no	—
55	Dic₂₁₈	—	4	4	no	—
56	C₈×C₁₀₉	—	2	4	no	—
57	C₂²×C₁₀₉	—	2	1	no	—
58	C₂²×C₁₀₉	—	2	2	no	—
59	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×D₁₀₉	—	4	4	no	—
61	C₂×D₁₀₉	—	4	4	no	—
62	C₂×D₁₀₉	—	4	4	no	—
63	C₄×C₁₀₉	—	2	1	no	—
64	C₄×C₁₀₉	—	2	2	no	—
65	C₄×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
66	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
67	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₁₀₉	—	4	4	no	—
69	C₂×C₁₀₉	—	4	1	no	—
70	C₂×C₁₀₉	#537	1	1	yes	characteristic
71	C₂×C₁₀₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
72	D₁₀₉	—	8	4	no	—
73	#3253	—	109	1	no	maximal, Sylow
74	C₁₀₉	#3253	1	1	yes	characteristic, complemented, Sylow
75	#532	—	109	2	no	—
76	#476	—	109	4	no	—
77	#469	—	109	4	no	—
78	#459	—	109	4	no	—
79	#423	D₁₀₉	1	1	yes	characteristic, complemented
80	C₂×#41	—	218	2	no	—
81	C₂×D₈	—	109	4	no	—
82	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
83	C₄⋊Q₈	—	109	2	no	—
84	#127	—	109	2	no	—
85	#124	—	218	2	no	—
86	#121	—	109	4	no	—
87	#121	—	218	2	no	—
88	C₄×D₄	—	109	4	no	—
89	C₄×D₄	—	218	2	no	—
90	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
91	#106	—	109	4	no	—
92	C₄⋊C₈	C₂×D₁₀₉	1	4	yes	—
93	#102	—	109	8	no	—
94	#41	—	436	8	no	—
95	C₂×Q₈	—	218	2	no	—
96	C₂×D₄	—	109	4	no	—
97	C₂×D₄	—	218	4	no	—
98	C₂×D₄	—	218	2	no	—
99	C₂×D₄	—	218	2	no	—
100	C₂²×C₄	—	218	4	no	—
101	C₂²×C₄	D₄₃₆	1	2	yes	—
102	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
103	D₈	—	436	4	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	C₂²×D₁₀₉	1	4	yes	—
106	C₄⋊C₄	—	109	4	no	—
107	C₄⋊C₄	—	218	2	no	—
108	C₂²⋊C₄	—	218	4	no	—
109	C₂²⋊C₄	—	218	4	no	—
110	C₄²	D₄₃₆	1	1	yes	characteristic
111	C₄²	D₄₃₆	1	1	yes	characteristic
112	C₄²	C₂²×D₁₀₉	1	2	yes	—
113	C₂³	—	218	4	no	—
114	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
115	Q₈	—	436	4	no	—
116	D₄	—	436	4	no	—
117	D₄	—	436	8	no	—
118	D₄	—	436	4	no	—
119	D₄	—	218	4	no	—
120	D₄	—	218	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	436	8	no	—
125	C₂×C₄	—	218	4	no	—
126	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
127	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
128	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
129	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
130	C₈	—	2	4	no	—
131	C₂²	—	2	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	436	4	no	—
134	C₂²	—	436	4	no	—
135	C₂²	—	436	4	no	—
136	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	436	4	no	—
140	C₄	#16812727	1	4	yes	—
141	C₄	#16812744	1	1	yes	characteristic
142	C₄	#16812747	1	1	yes	characteristic
143	C₂	—	4	1	no	—
144	C₂	—	872	4	no	—
145	C₂	#20295607	1	1	yes	characteristic
146	C₂	C₂×#16812727	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295576

C₂

yes

maximal, complemented

#20295317

C₂

yes

maximal, complemented

#20295279

C₂

yes

maximal, complemented

#20295222

C₂

yes

maximal, complemented

#423×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182029

—

C₂×D₈₇₂

C₂²

yes

—

#16812699

C₂²

yes

—

#16812696

C₂²

yes

—

#16812687

—

#16812673

—

#16812673

C₂²

yes

—

C₄×D₄₃₆

—

C₄×D₄₃₆

C₂²

yes

—

#16812631

C₂²

yes

—

#16812621

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

#13182029

—

C₂×Dic₂₁₈

—

C₂×D₄₃₆

—

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂³

yes

—

C₂×C₂²⋊D₁₀₉

—

C₂×C₄×D₁₀₉

—

D₈₇₂

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182006

—

#13182006

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#537

yes

characteristic

C₂×C₁₀₉

C₂×AGL(1,ℤ/8)

yes

—

D₁₀₉

—

#3253

—

109

maximal, Sylow

C₁₀₉

#3253

yes

characteristic, complemented, Sylow

#532

—

109

—

#476

—

109

—

#469

—

109

—

#459

—

109

—

#423

D₁₀₉

yes

characteristic, complemented

C₂×#41

—

218

—

C₂×D₈

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

109

—

#127

—

109

—

#124

—

218

—

#121

—

109

—

#121

—

218

—

C₄×D₄

—

109

—

C₄×D₄

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#106

—

109

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

#102

—

109

—

#41

—

436

—

C₂×Q₈

—

218

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

100

C₂²×C₄

—

218

—

101

C₂²×C₄

D₄₃₆

yes

—

102

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

103

D₈

—

436

—

104

M₁₆

—

105

C₂×C₈

C₂²×D₁₀₉

yes

—

106

C₄⋊C₄

—

109

—

107

C₄⋊C₄

—

218

—

108

C₂²⋊C₄

—

218

—

109

C₂²⋊C₄

—

218

—

110

C₄²

D₄₃₆

yes

characteristic

111

C₄²

D₄₃₆

yes

characteristic

112

C₄²

C₂²×D₁₀₉

yes

—

113

C₂³

—

218

—

114

C₂³

C₂×D₄₃₆

yes

characteristic

115

Q₈

—

436

—

116

D₄

—

436

—

117

D₄

—

436

—

118

D₄

—

436

—

119

D₄

—

218

—

120

D₄

—

218

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

436

—

125

C₂×C₄

—

218

—

126

C₂×C₄

C₂×D₄₃₆

yes

characteristic

127

C₂×C₄

C₂×D₄₃₆

yes

—

128

C₂×C₄

C₂×D₄₃₆

yes

—

129

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

130

C₈

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

436

—

134

C₂²

—

436

—

135

C₂²

—

436

—

136

C₂²

C₂²×D₄₃₆

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

436

—

140

C₄

#16812727

yes

—

141

C₄

#16812744

yes

characteristic

142

C₄

#16812747

yes

characteristic

143

C₂

—

144

C₂

—

872

—

145

C₂

#20295607

yes

characteristic

146

C₂

C₂×#16812727

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#423×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#423	D₁₀₉
hypocenter	C₁₀₉	#3253
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₄₃₆

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#423×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
1. #20295576	C₂	semidirect
2. #20295317	C₂	semidirect
3. #20295279	C₂	semidirect
4. #20295222	C₂	semidirect
5. #423×C₁₀₉	C₂	semidirect
74. C₁₀₉	#3253	semidirect
79. #423	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295576

C₂

semidirect

2. #20295317

C₂

semidirect

3. #20295279

C₂

semidirect

4. #20295222

C₂

semidirect

5. #423×C₁₀₉

C₂

semidirect

74. C₁₀₉

#3253

semidirect

79. #423

D₁₀₉

semidirect

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

subgroup	index
#20295576	2
#20295317	2
#20295279	2
#20295222	2
#423×C₁₀₉	2
#3253	109

subgroup

index

#423×C₁₀₉

109

Subgroups — #28374413

Conjugacy classes of subgroups

Complements

Maximal subgroups