Group #28378542 — subgroups

Subgroups — #28378542

Order 13952.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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	#20296044	C₂	1	1	yes	maximal, characteristic
2	#20296031	C₂	1	1	yes	maximal, characteristic
3	#20295885	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	—	2	1	no	—
6	#16812766	—	2	1	no	—
7	#16812766	C₂²	1	2	yes	—
8	#16812766	C₂²	1	2	yes	—
9	#16812756	—	2	1	no	—
10	C₄×#11724786	—	2	1	no	—
11	C₂×Q₈×D₁₀₉	—	2	1	no	—
12	#16812699	C₄	1	1	yes	characteristic
13	#16812698	—	2	1	no	—
14	#16812698	C₂²	1	2	yes	—
15	#16812697	—	2	1	no	—
16	C₄⋊C₄×D₁₀₉	—	2	1	no	—
17	C₄⋊C₄×D₁₀₉	C₄	1	2	yes	—
18	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
20	#13182039	—	4	2	no	—
21	C₂×#11724786	—	2	2	no	—
22	C₂×#11724786	—	2	2	no	—
23	C₂×Dic₂₁₈	—	2	1	no	—
24	C₂×Dic₂₁₈	—	2	2	no	—
25	Q₈×D₁₀₉	—	4	2	no	—
26	Q₈×D₁₀₉	—	4	2	no	—
27	C₂×C₄×D₁₀₉	—	2	2	no	—
28	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
31	C₄⋊Dic₁₀₉	—	2	1	no	—
32	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
33	#13182008	—	2	2	no	—
34	#13182008	—	2	2	no	—
35	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
36	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
37	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
38	C₂×Q₈×C₁₀₉	—	2	1	no	—
39	C₄⋊C₄×C₁₀₉	—	2	1	no	—
40	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
41	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
43	Q₈×C₁₀₉	—	4	2	no	—
44	C₂×C₄×C₁₀₉	—	2	2	no	—
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₀₉	—	2	2	no	—
49	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₄×D₁₀₉	—	2	2	no	—
53	C₄×D₁₀₉	—	2	2	no	—
54	C₄×D₁₀₉	—	4	4	no	—
55	C₄×D₁₀₉	D₈	1	2	yes	—
56	C₄×D₁₀₉	SD₁₆	1	2	yes	—
57	Dic₂₁₈	—	4	2	no	—
58	Dic₂₁₈	—	4	2	no	—
59	Dic₂₁₈	—	4	2	no	—
60	#11724786	—	4	2	no	—
61	#11724786	—	4	2	no	—
62	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×D₁₀₉	—	2	1	no	—
64	C₂×D₁₀₉	—	2	1	no	—
65	C₂×D₁₀₉	#102	1	2	yes	—
66	C₄×C₁₀₉	—	2	1	no	—
67	C₄×C₁₀₉	—	2	1	no	—
68	C₄×C₁₀₉	—	4	2	no	—
69	C₄×C₁₀₉	#102	1	2	yes	—
70	Dic₁₀₉	—	4	2	no	—
71	Dic₁₀₉	#101	1	2	yes	—
72	Dic₁₀₉	C₂×D₈	1	1	yes	characteristic
73	Dic₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
74	Dic₁₀₉	#137	1	2	yes	—
75	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
76	C₂×C₁₀₉	C₂×#102	1	1	yes	characteristic
77	C₂×C₁₀₉	#419	1	1	yes	characteristic
78	D₁₀₉	—	2	2	no	—
79	#1546	—	109	1	no	maximal, Sylow
80	C₁₀₉	#1546	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄⋊Q₈	—	109	1	no	—
82	#431	—	109	1	no	—
83	#423	—	109	1	no	—
84	#332	—	109	4	no	—
85	C₂²×Q₈	—	218	1	no	—
86	C₂×M₁₆	—	218	1	no	—
87	C₄⋊Q₈	—	109	2	no	—
88	C₄⋊Q₈	—	109	1	no	—
89	C₄⋊Q₈	—	218	1	no	—
90	C₄⋊Q₈	—	218	1	no	—
91	C₄⋊Q₈	#89137	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	109	2	no	—
93	C₂×C₄⋊C₄	—	218	1	no	—
94	C₂×C₄²	—	109	1	no	—
95	C₄⋊C₈	—	109	2	no	—
96	C₄⋊C₈	—	109	2	no	—
97	C₄⋊C₈	—	218	1	no	—
98	C₂²⋊C₈	—	218	1	no	—
99	C₄×C₈	—	218	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	436	2	no	—
102	C₂×Q₈	—	436	2	no	—
103	C₂×Q₈	—	218	2	no	—
104	C₂×Q₈	—	218	1	no	—
105	C₂²×C₄	—	109	1	no	—
106	C₂²×C₄	—	109	1	no	—
107	C₂²×C₄	—	109	1	no	—
108	C₂²×C₄	—	218	2	no	—
109	M₁₆	—	436	2	no	—
110	C₂×C₈	—	218	2	no	—
111	C₂×C₈	—	218	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	109	2	no	—
114	C₄⋊C₄	—	218	2	no	—
115	C₄⋊C₄	—	218	2	no	—
116	C₄⋊C₄	—	218	1	no	—
117	C₄⋊C₄	C₂×#89137	1	2	yes	—
118	C₄²	—	109	1	no	—
119	C₄²	—	109	1	no	—
120	C₄²	—	109	1	no	—
121	C₄²	C₂×#89137	1	1	yes	characteristic
122	C₂³	—	109	1	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	436	2	no	—
125	Q₈	—	436	2	no	—
126	Q₈	—	436	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	109	1	no	—
129	C₂×C₄	—	109	1	no	—
130	C₂×C₄	—	109	1	no	—
131	C₂×C₄	—	109	2	no	—
132	C₂×C₄	—	109	2	no	—
133	C₂×C₄	—	436	4	no	—
134	C₂×C₄	—	218	2	no	—
135	C₂×C₄	—	218	2	no	—
136	C₂×C₄	—	218	2	no	—
137	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
139	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
140	C₈	—	436	2	no	—
141	C₈	—	436	2	no	—
142	C₂²	—	109	2	no	—
143	C₂²	—	218	1	no	—
144	C₂²	—	218	1	no	—
145	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	2	no	—
149	C₄	—	109	2	no	—
150	C₄	—	109	1	no	—
151	C₄	—	109	1	no	—
152	C₄	—	109	2	no	—
153	C₄	—	436	2	no	—
154	C₄	#16812761	1	2	yes	—
155	C₂	—	218	2	no	—
156	C₂	D₁₀₉⋊#101	1	1	yes	characteristic
157	C₂	C₂×#16812761	1	1	yes	characteristic
158	C₂	#20296019	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296044

C₂

yes

maximal, characteristic

#20296031

C₂

yes

maximal, characteristic

#20295885

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

—

#16812766

—

#16812766

C₂²

yes

—

#16812766

C₂²

yes

—

#16812756

—

C₄×#11724786

—

C₂×Q₈×D₁₀₉

—

#16812699

C₄

yes

characteristic

#16812698

—

#16812698

C₂²

yes

—

#16812697

—

C₄⋊C₄×D₁₀₉

—

C₄⋊C₄×D₁₀₉

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₄

yes

—

—

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

Q₈×D₁₀₉

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

characteristic

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

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₈

yes

—

C₄×D₁₀₉

SD₁₆

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#102

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#102

yes

—

Dic₁₀₉

—

Dic₁₀₉

#101

yes

—

Dic₁₀₉

C₂×D₈

yes

characteristic

Dic₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

#137

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1546

—

109

maximal, Sylow

C₁₀₉

#1546

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

#431

—

109

—

#423

—

109

—

#332

—

109

—

C₂²×Q₈

—

218

—

C₂×M₁₆

—

218

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

#89137

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

218

—

100

C₂×Q₈

—

101

C₂×Q₈

—

436

—

102

C₂×Q₈

—

436

—

103

C₂×Q₈

—

218

—

104

C₂×Q₈

—

218

—

105

C₂²×C₄

—

109

—

106

C₂²×C₄

—

109

—

107

C₂²×C₄

—

109

—

108

C₂²×C₄

—

218

—

109

M₁₆

—

436

—

110

C₂×C₈

—

218

—

111

C₂×C₈

—

218

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

109

—

114

C₄⋊C₄

—

218

—

115

C₄⋊C₄

—

218

—

116

C₄⋊C₄

—

218

—

117

C₄⋊C₄

C₂×#89137

yes

—

118

C₄²

—

109

—

119

C₄²

—

109

—

120

C₄²

—

109

—

121

C₄²

C₂×#89137

yes

characteristic

122

C₂³

—

109

—

123

Q₈

—

124

Q₈

—

436

—

125

Q₈

—

436

—

126

Q₈

—

436

—

127

C₂×C₄

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

109

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

109

—

132

C₂×C₄

—

109

—

133

C₂×C₄

—

436

—

134

C₂×C₄

—

218

—

135

C₂×C₄

—

218

—

136

C₂×C₄

—

218

—

137

C₂×C₄

C₂²⋊#89137

yes

characteristic

138

C₂×C₄

C₂²⋊#89137

yes

characteristic

139

C₂×C₄

C₂²×#89137

yes

characteristic

140

C₈

—

436

—

141

C₈

—

436

—

142

C₂²

—

109

—

143

C₂²

—

218

—

144

C₂²

—

218

—

145

C₂²

C₂×C₂²⋊#89137

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

109

—

150

C₄

—

109

—

151

C₄

—

109

—

152

C₄

—

109

—

153

C₄

—

436

—

154

C₄

#16812761

yes

—

155

C₂

—

218

—

156

C₂

D₁₀₉⋊#101

yes

characteristic

157

C₂

C₂×#16812761

yes

characteristic

158

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	C₂²⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1546
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×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

C₂²⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₄⋊Q₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #20295885	C₂	semidirect
80. C₁₀₉	#1546	semidirect

H ≅ G/N

type

3. #20295885

C₂

semidirect

80. C₁₀₉

#1546

semidirect

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

subgroup	index
#20296044	2
#20296031	2
#20295885	2
C₄⋊Q₈×D₁₀₉	2
#1546	109

subgroup

index

C₄⋊Q₈×D₁₀₉

109

Subgroups — #28378542

Conjugacy classes of subgroups

Complements

Maximal subgroups