Group #28362670 — subgroups

Subgroups — #28362670

Order 13952.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20295398	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295377	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295372	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295372	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295372	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294748	C₂	1	1	yes	maximal, characteristic, complemented
7	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182017	—	2	2	no	—
9	#16812699	C₂²	1	1	yes	characteristic
10	#16812699	C₂²	1	1	yes	characteristic
11	#16812624	—	2	2	no	—
12	#16812620	—	2	2	no	—
13	#16812620	—	2	2	no	—
14	#16812620	—	2	2	no	—
15	C₄×#11724784	C₂²	1	1	yes	characteristic
16	C₄×#11724784	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₁₀₉	—	2	2	no	—
18	#127×C₁₀₉	C₂²	1	1	yes	characteristic
19	#127×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₁₈	—	2	1	no	—
22	C₂×Dic₂₁₈	—	2	1	no	—
23	#13182017	—	4	4	no	—
24	C₂×#11724784	D₄	1	2	yes	—
25	C₂×#11724784	D₄	1	2	yes	—
26	C₄⋊Dic₁₀₉	—	2	1	no	—
27	C₄⋊Dic₁₀₉	—	2	1	no	—
28	C₄⋊Dic₁₀₉	—	2	1	no	—
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₂×D₄×C₁₀₉	—	2	1	no	—
31	C₂×D₄×C₁₀₉	—	2	1	no	—
32	C₂×D₄×C₁₀₉	—	2	1	no	—
33	C₂×D₄×C₁₀₉	—	2	1	no	—
34	C₂×D₄×C₁₀₉	—	2	1	no	—
35	C₂×D₄×C₁₀₉	—	2	1	no	—
36	D₈×C₁₀₉	—	4	4	no	—
37	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
38	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂³×C₁₀₉	—	4	1	no	—
40	C₂³×C₁₀₉	—	4	1	no	—
41	D₄×C₁₀₉	—	4	2	no	—
42	D₄×C₁₀₉	—	4	2	no	—
43	D₄×C₁₀₉	—	4	2	no	—
44	D₄×C₁₀₉	—	4	2	no	—
45	D₄×C₁₀₉	—	4	2	no	—
46	D₄×C₁₀₉	—	4	2	no	—
47	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	4	1	no	—
51	C₂×Dic₁₀₉	—	4	1	no	—
52	Dic₂₁₈	—	4	2	no	—
53	Dic₂₁₈	—	4	2	no	—
54	C₈×C₁₀₉	—	2	2	no	—
55	#11724784	—	2	2	no	—
56	#11724784	—	2	2	no	—
57	C₂²×C₁₀₉	—	8	1	no	—
58	C₂²×C₁₀₉	—	8	1	no	—
59	C₂²×C₁₀₉	—	8	1	no	—
60	C₂²×C₁₀₉	—	8	1	no	—
61	C₂²×C₁₀₉	—	8	1	no	—
62	C₂²×C₁₀₉	—	8	1	no	—
63	C₂²×C₁₀₉	#127	1	1	yes	characteristic
64	C₄×C₁₀₉	#135	1	2	yes	—
65	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
66	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
67	Dic₁₀₉	—	8	1	no	—
68	Dic₁₀₉	—	8	1	no	—
69	C₂×C₁₀₉	—	16	1	no	—
70	C₂×C₁₀₉	—	16	1	no	—
71	C₂×C₁₀₉	#495	1	1	yes	characteristic
72	C₂×C₁₀₉	#496	1	1	yes	characteristic
73	C₂×C₁₀₉	#496	1	1	yes	characteristic
74	#1717	—	109	1	no	maximal, Sylow
75	C₁₀₉	#1717	1	1	yes	characteristic, complemented, Sylow
76	#493	D₁₀₉	1	1	yes	characteristic
77	#492	—	109	1	no	—
78	#487	—	109	1	no	—
79	#486	—	109	1	no	—
80	#486	—	109	1	no	—
81	#486	—	109	1	no	—
82	#322	—	109	1	no	—
83	C₂×SD₁₆	—	218	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	109	1	no	—
86	C₄⋊Q₈	—	109	1	no	—
87	#127	C₂×D₁₀₉	1	1	yes	characteristic
88	#127	C₂×D₁₀₉	1	1	yes	characteristic
89	#103	—	218	2	no	—
90	#102	—	218	2	no	—
91	#102	—	218	2	no	—
92	#102	—	218	2	no	—
93	C₄×C₈	—	109	1	no	—
94	C₄×C₈	—	109	1	no	—
95	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
96	C₂×Q₈	—	218	1	no	—
97	C₂×Q₈	—	218	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	SD₁₆	—	436	4	no	—
105	D₈	—	4	4	no	—
106	C₂×C₈	—	109	2	no	—
107	C₂×C₈	—	109	2	no	—
108	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
109	C₄⋊C₄	—	218	1	no	—
110	C₄⋊C₄	—	218	1	no	—
111	C₄⋊C₄	—	218	1	no	—
112	C₄⋊C₄	—	218	1	no	—
113	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
114	C₂³	—	4	1	no	—
115	C₂³	—	4	1	no	—
116	Q₈	—	436	2	no	—
117	Q₈	—	436	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	4	2	no	—
124	C₂×C₄	—	436	1	no	—
125	C₂×C₄	—	436	1	no	—
126	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
127	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	218	2	no	—
131	C₈	—	218	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	#16812695	1	1	yes	characteristic
139	C₄	—	872	1	no	—
140	C₄	—	872	1	no	—
141	C₄	#16812718	1	2	yes	—
142	C₄	#16812723	1	2	yes	—
143	C₄	#16812725	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	16	1	no	—
146	C₂	#20295409	1	1	yes	characteristic
147	C₂	#20295413	1	1	yes	characteristic
148	C₂	#20295414	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295398

C₂

yes

maximal, characteristic, complemented

#20295377

C₂

yes

maximal, characteristic, complemented

#20295372

C₂

yes

maximal, characteristic, complemented

#20295372

C₂

yes

maximal, characteristic, complemented

#20295372

C₂

yes

maximal, characteristic, complemented

#20294748

C₂

yes

maximal, characteristic, complemented

#493×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182017

—

#16812699

C₂²

yes

characteristic

#16812699

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₂×D₈×C₁₀₉

—

#127×C₁₀₉

C₂²

yes

characteristic

#127×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

#13182017

—

C₂×#11724784

D₄

yes

—

C₂×#11724784

D₄

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#127

yes

characteristic

C₄×C₁₀₉

#135

yes

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1717

yes

characteristic, complemented, Sylow

#493

D₁₀₉

yes

characteristic

#492

—

109

—

#487

—

109

—

#486

—

109

—

#486

—

109

—

#486

—

109

—

#322

—

109

—

C₂×SD₁₆

—

218

—

C₂×D₈

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

#127

C₂×D₁₀₉

yes

characteristic

#127

C₂×D₁₀₉

yes

characteristic

#103

—

218

—

#102

—

218

—

#102

—

218

—

#102

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

SD₁₆

—

436

—

105

D₈

—

106

C₂×C₈

—

109

—

107

C₂×C₈

—

109

—

108

C₂×C₈

C₂²⋊D₁₀₉

yes

—

109

C₄⋊C₄

—

218

—

110

C₄⋊C₄

—

218

—

111

C₄⋊C₄

—

218

—

112

C₄⋊C₄

—

218

—

113

C₄²

C₂²×D₁₀₉

yes

characteristic

114

C₂³

—

115

C₂³

—

116

Q₈

—

436

—

117

Q₈

—

436

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

436

—

125

C₂×C₄

—

436

—

126

C₂×C₄

D₄×D₁₀₉

yes

characteristic

127

C₂×C₄

D₄×D₁₀₉

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

129

C₈

—

130

C₈

—

218

—

131

C₈

—

218

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#16812695

yes

characteristic

139

C₄

—

872

—

140

C₄

—

872

—

141

C₄

#16812718

yes

—

142

C₄

#16812723

yes

—

143

C₄

#16812725

yes

—

144

C₂

—

145

C₂

—

146

C₂

#20295409

yes

characteristic

147

C₂

#20295413

yes

characteristic

148

C₂

#20295414

yes

characteristic

	subgroup	G/H
center	C₂²	#16812695
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting, prime radical	#493×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#127
hypercenter	#493	D₁₀₉
hypocenter	C₁₀₉	#1717
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#16812695

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting, prime radical

#493×C₁₀₉

C₂

C₂²×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #20295398	C₂	semidirect
2. #20295377	C₂	semidirect
3. #20295372	C₂	semidirect
4. #20295372	C₂	semidirect
5. #20295372	C₂	semidirect
6. #20294748	C₂	semidirect
75. C₁₀₉	#1717	semidirect

H ≅ G/N

type

1. #20295398

C₂

semidirect

2. #20295377

C₂

semidirect

3. #20295372

C₂

semidirect

4. #20295372

C₂

semidirect

5. #20295372

C₂

semidirect

6. #20294748

C₂

semidirect

75. C₁₀₉

#1717

semidirect

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

subgroup	index
#20295398	2
#20295377	2
#20295372	2
#20294748	2
#493×C₁₀₉	2
#1717	109

subgroup

index

#493×C₁₀₉

109

Subgroups — #28362670

Conjugacy classes of subgroups

Complements

Maximal subgroups