Group #28379216 — subgroups

Subgroups — #28379216

Order 13952.

276 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20295985	C₂	1	1	yes	maximal, characteristic
2	#20295970	C₂	1	1	yes	maximal, characteristic
3	#20295899	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#89137	C₂²	1	2	yes	—
7	#16812754	—	2	1	no	—
8	#16812754	—	2	1	no	—
9	(C₂×#89137)⋊C₄	—	2	1	no	—
10	(C₂×#89137)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₁₀₉	—	2	1	no	—
12	#16812699	C₄	1	1	yes	characteristic, complemented
13	#16812698	—	2	1	no	—
14	#16812698	C₄	1	2	yes	complemented
15	#16812697	—	2	1	no	—
16	C₄⋊C₄×D₁₀₉	—	2	1	no	—
17	C₄⋊C₄×D₁₀₉	C₂²	1	2	yes	—
18	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
20	C₂²×#89137	—	2	2	no	—
21	D₁₀₉⋊C₈	—	2	2	no	—
22	D₁₀₉⋊C₈	Q₈	1	4	yes	complemented
23	C₂×#11724786	—	2	2	no	—
24	C₄⋊#89137	—	2	4	no	—
25	C₂×Dic₂₁₈	—	2	1	no	—
26	C₂×Dic₂₁₈	—	2	2	no	—
27	Q₈×D₁₀₉	—	4	4	no	—
28	C₂×C₄×D₁₀₉	—	2	2	no	—
29	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
31	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₁₀₉	—	2	1	no	—
33	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
34	#13182008	—	2	2	no	—
35	#13182008	—	2	2	no	—
36	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
38	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
39	C₂×Q₈×C₁₀₉	—	2	1	no	—
40	C₄⋊C₄×C₁₀₉	—	2	1	no	—
41	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
42	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×#89137	—	4	2	no	—
45	C₂×#89137	—	4	2	no	—
46	C₂×#89137	—	4	2	no	—
47	Q₈×C₁₀₉	—	4	2	no	—
48	C₂×C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₁₀₉	—	2	2	no	—
53	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
56	C₄×D₁₀₉	—	2	2	no	—
57	C₄×D₁₀₉	—	2	2	no	—
58	C₄×D₁₀₉	—	4	4	no	—
59	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
60	C₄×D₁₀₉	#41	1	2	yes	—
61	Dic₂₁₈	—	4	4	no	—
62	Dic₂₁₈	—	4	2	no	—
63	#11724786	—	2	4	no	—
64	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₁₀₉	#124	1	1	yes	characteristic
66	C₂×D₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×D₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
68	C₂×D₁₀₉	#137	1	2	yes	—
69	#89137	—	8	2	no	—
70	C₄×C₁₀₉	—	2	1	no	—
71	C₄×C₁₀₉	—	2	1	no	—
72	C₄×C₁₀₉	—	4	2	no	—
73	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
74	Dic₁₀₉	—	2	1	no	—
75	Dic₁₀₉	—	2	1	no	—
76	Dic₁₀₉	—	4	2	no	—
77	Dic₁₀₉	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₁₀₉	#391	1	1	yes	characteristic
79	C₂×C₁₀₉	#418	1	1	yes	characteristic
80	C₂×C₁₀₉	#419	1	1	yes	characteristic
81	D₁₀₉	#489	1	2	yes	—
82	D₁₀₉	#501	1	2	yes	—
83	#1993	—	109	1	no	maximal, Sylow
84	C₁₀₉	#1993	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊Q₈	—	109	1	no	—
86	C₂×#97	—	109	1	no	—
87	#382	—	109	1	no	—
88	#340	—	109	4	no	—
89	C₂²×Q₈	—	218	1	no	—
90	C₂²×C₈	—	109	2	no	—
91	C₄⋊Q₈	—	109	2	no	—
92	C₄⋊Q₈	—	109	1	no	—
93	C₄⋊Q₈	—	218	1	no	—
94	C₄⋊Q₈	—	218	1	no	—
95	C₄⋊Q₈	#89137	1	1	yes	characteristic, complemented
96	C₂×C₄⋊C₄	—	109	2	no	—
97	C₂×C₄⋊C₄	—	109	2	no	—
98	C₂×C₄⋊C₄	—	218	1	no	—
99	C₂×C₄²	—	109	1	no	—
100	#97	—	218	1	no	—
101	#97	—	218	1	no	—
102	#95	—	218	1	no	—
103	#95	—	218	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	436	4	no	—
106	C₂×Q₈	—	218	2	no	—
107	C₂×Q₈	—	218	1	no	—
108	C₂²×C₄	—	109	1	no	—
109	C₂²×C₄	—	109	1	no	—
110	C₂²×C₄	—	109	1	no	—
111	C₂²×C₄	—	218	2	no	—
112	C₂²×C₄	—	218	2	no	—
113	C₂×C₈	—	109	4	no	—
114	C₂×C₈	—	218	2	no	—
115	C₂×C₈	—	218	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	109	2	no	—
118	C₄⋊C₄	—	218	2	no	—
119	C₄⋊C₄	—	218	2	no	—
120	C₄⋊C₄	—	218	4	no	—
121	C₄⋊C₄	—	218	1	no	—
122	C₄⋊C₄	C₂×#89137	1	2	yes	—
123	C₄²	—	109	1	no	—
124	C₄²	—	109	1	no	—
125	C₄²	—	109	1	no	—
126	C₄²	C₂×#89137	1	1	yes	characteristic
127	C₂³	—	109	1	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	436	2	no	—
130	Q₈	—	436	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	109	1	no	—
133	C₂×C₄	—	109	2	no	—
134	C₂×C₄	—	109	2	no	—
135	C₂×C₄	—	109	1	no	—
136	C₂×C₄	—	109	1	no	—
137	C₂×C₄	—	436	4	no	—
138	C₂×C₄	—	436	2	no	—
139	C₂×C₄	—	436	2	no	—
140	C₂×C₄	—	436	2	no	—
141	C₂×C₄	—	218	2	no	—
142	C₂×C₄	—	218	2	no	—
143	C₂×C₄	—	218	2	no	—
144	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
146	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
147	C₈	—	218	4	no	—
148	C₂²	—	109	1	no	—
149	C₂²	—	109	1	no	—
150	C₂²	—	109	2	no	—
151	C₂²	—	109	2	no	—
152	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	872	2	no	—
157	C₄	—	109	2	no	—
158	C₄	—	436	2	no	—
159	C₄	—	218	1	no	—
160	C₄	—	218	1	no	—
161	C₄	Q₈×#89137	1	2	yes	—
162	C₂	—	109	2	no	—
163	C₂	—	109	2	no	—
164	C₂	(C₂×#89137)⋊Q₈	1	1	yes	characteristic
165	C₂	#20296018	1	1	yes	characteristic
166	C₂	#20296019	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295985

C₂

yes

maximal, characteristic

#20295970

C₂

yes

maximal, characteristic

#20295899

C₂

yes

maximal

C₄⋊Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

C₂²

yes

—

C₂×C₄⋊#89137

C₂²

yes

—

—

—

—

—

C₂×Q₈×D₁₀₉

—

#16812699

C₄

yes

characteristic, complemented

#16812698

—

#16812698

C₄

yes

complemented

#16812697

—

C₄⋊C₄×D₁₀₉

—

C₄⋊C₄×D₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

Q₈

yes

complemented

C₂×#11724786

—

C₄⋊#89137

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×D₄

yes

—

—

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×Q₈

yes

—

C₄×D₁₀₉

#41

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

#124

yes

characteristic

C₂×D₁₀₉

C₄⋊Q₈

yes

characteristic

C₂×D₁₀₉

AGL(1,ℤ/8)

yes

—

C₂×D₁₀₉

#137

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊Q₈

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

yes

—

D₁₀₉

#501

yes

—

#1993

—

109

maximal, Sylow

C₁₀₉

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

C₂×#97

—

109

—

#382

—

109

—

#340

—

109

—

C₂²×Q₈

—

218

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

#89137

yes

characteristic, complemented

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

—

109

—

100

#97

—

218

—

101

#97

—

218

—

102

#95

—

218

—

103

#95

—

218

—

104

C₂×Q₈

—

105

C₂×Q₈

—

436

—

106

C₂×Q₈

—

218

—

107

C₂×Q₈

—

218

—

108

C₂²×C₄

—

109

—

109

C₂²×C₄

—

109

—

110

C₂²×C₄

—

109

—

111

C₂²×C₄

—

218

—

112

C₂²×C₄

—

218

—

113

C₂×C₈

—

109

—

114

C₂×C₈

—

218

—

115

C₂×C₈

—

218

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

109

—

118

C₄⋊C₄

—

218

—

119

C₄⋊C₄

—

218

—

120

C₄⋊C₄

—

218

—

121

C₄⋊C₄

—

218

—

122

C₄⋊C₄

C₂×#89137

yes

—

123

C₄²

—

109

—

124

C₄²

—

109

—

125

C₄²

—

109

—

126

C₄²

C₂×#89137

yes

characteristic

127

C₂³

—

109

—

128

Q₈

—

129

Q₈

—

436

—

130

Q₈

—

436

—

131

C₂×C₄

—

132

C₂×C₄

—

109

—

133

C₂×C₄

—

109

—

134

C₂×C₄

—

109

—

135

C₂×C₄

—

109

—

136

C₂×C₄

—

109

—

137

C₂×C₄

—

436

—

138

C₂×C₄

—

436

—

139

C₂×C₄

—

436

—

140

C₂×C₄

—

436

—

141

C₂×C₄

—

218

—

142

C₂×C₄

—

218

—

143

C₂×C₄

—

218

—

144

C₂×C₄

C₂²⋊#89137

yes

characteristic

145

C₂×C₄

C₂²⋊#89137

yes

characteristic

146

C₂×C₄

C₂²×#89137

yes

characteristic

147

C₈

—

218

—

148

C₂²

—

109

—

149

C₂²

—

109

—

150

C₂²

—

109

—

151

C₂²

—

109

—

152

C₂²

C₂×C₂²⋊#89137

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

872

—

157

C₄

—

109

—

158

C₄

—

436

—

159

C₄

—

218

—

160

C₄

—

218

—

161

C₄

Q₈×#89137

yes

—

162

C₂

—

109

—

163

C₂

—

109

—

164

C₂

(C₂×#89137)⋊Q₈

yes

characteristic

165

C₂

#20296018

yes

characteristic

166

C₂

#20296019

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄⋊Q₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₉	C₂×D₄
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1993
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄⋊Q₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₉

C₂×D₄

cosocle

C₂×C₄×D₁₀₉

C₂³

C₄⋊Q₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #16812699	C₄	semidirect
14. #16812698	C₄	semidirect
19. C₄⋊Q₈×C₁₀₉	C₄	semidirect
22. D₁₀₉⋊C₈	Q₈	semidirect
84. C₁₀₉	#1993	semidirect
95. C₄⋊Q₈	#89137	semidirect

H ≅ G/N

type

12. #16812699

C₄

semidirect

14. #16812698

C₄

semidirect

19. C₄⋊Q₈×C₁₀₉

C₄

semidirect

22. D₁₀₉⋊C₈

Q₈

semidirect

84. C₁₀₉

#1993

semidirect

95. C₄⋊Q₈

#89137

semidirect

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

subgroup	index
#20295985	2
#20295970	2
#20295899	2
C₄⋊Q₈×D₁₀₉	2
#1993	109

subgroup

index

C₄⋊Q₈×D₁₀₉

109

Subgroups — #28379216

Conjugacy classes of subgroups

Complements

Maximal subgroups