Group #28379220 — subgroups

Subgroups — #28379220

Order 13952.

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

normal subgroups only

This group has 244 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	#20296031	C₂	1	1	yes	maximal, characteristic
2	#20295962	C₂	1	1	yes	maximal, characteristic
3	#20295896	C₂	1	4	yes	maximal
4	#20295581	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	C₂²	1	2	yes	—
6	C₂×C₄×#89137	C₂²	1	2	yes	—
7	#16812766	—	2	1	no	—
8	#16812766	—	2	1	no	—
9	(C₂×#89137)⋊C₄	—	2	1	no	—
10	(C₂×#89137)⋊C₄	—	2	1	no	—
11	#16812699	C₄	1	1	yes	characteristic, complemented
12	#16812690	C₄	1	2	yes	complemented
13	#16812679	—	2	1	no	—
14	#16812675	—	2	1	no	—
15	Q₈×Dic₁₀₉	—	2	1	no	—
16	#16812664	—	2	1	no	—
17	#16812657	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	#13182039	—	2	4	no	—
22	C₂×#11724786	—	2	2	no	—
23	C₄×#89137	—	2	2	no	—
24	C₄×#89137	Q₈	1	4	yes	complemented
25	C₂×Dic₂₁₈	—	2	1	no	—
26	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
27	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
28	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₄⋊Dic₁₀₉	—	2	2	no	—
31	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
32	#13182008	—	2	2	no	—
33	#13182006	—	2	2	no	—
34	C₄×Dic₁₀₉	—	2	2	no	—
35	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
37	C₄×Dic₁₀₉	D₄	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	C₂×D₄	1	1	yes	characteristic
43	C₂×#89137	—	2	4	no	—
44	C₂×#89137	—	4	2	no	—
45	C₂×#89137	—	4	2	no	—
46	Q₈×C₁₀₉	—	4	2	no	—
47	C₂×C₄×C₁₀₉	—	2	2	no	—
48	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₁₀₉	—	2	2	no	—
52	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₄×D₁₀₉	—	4	1	no	—
56	C₄×D₁₀₉	—	4	1	no	—
57	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
58	C₄×D₁₀₉	#41	1	2	yes	—
59	Dic₂₁₈	—	4	2	no	—
60	#11724786	—	4	2	no	—
61	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×D₁₀₉	—	2	1	no	—
63	C₂×D₁₀₉	—	2	1	no	—
64	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
65	#89137	—	4	4	no	—
66	C₄×C₁₀₉	—	2	1	no	—
67	C₄×C₁₀₉	—	2	1	no	—
68	C₄×C₁₀₉	—	4	2	no	—
69	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
70	Dic₁₀₉	—	2	1	no	—
71	Dic₁₀₉	—	2	1	no	—
72	Dic₁₀₉	—	4	2	no	—
73	Dic₁₀₉	#124	1	1	yes	characteristic
74	Dic₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₁₀₉	#391	1	1	yes	characteristic
76	C₂×C₁₀₉	#421	1	1	yes	characteristic
77	C₂×C₁₀₉	#421	1	1	yes	characteristic
78	D₁₀₉	—	2	2	no	—
79	#1995	—	109	1	no	maximal, Sylow
80	C₁₀₉	#1995	1	1	yes	characteristic, complemented, Sylow
81	#533	—	109	1	no	—
82	#423	—	109	1	no	—
83	#376	—	109	1	no	—
84	#339	—	109	4	no	—
85	C₂×M₁₆	—	109	2	no	—
86	C₄⋊Q₈	—	109	1	no	—
87	C₄⋊Q₈	#89137	1	1	yes	characteristic, complemented
88	#125	—	109	2	no	—
89	C₂²⋊Q₈	—	218	1	no	—
90	C₂²⋊Q₈	—	218	1	no	—
91	C₄×Q₈	—	218	1	no	—
92	C₄×Q₈	—	218	1	no	—
93	#117	—	109	2	no	—
94	C₂×C₄²	—	109	1	no	—
95	C₂×C₄²	—	109	2	no	—
96	C₄⋊C₈	—	218	1	no	—
97	C₄⋊C₈	—	218	1	no	—
98	#95	—	218	1	no	—
99	#95	—	218	1	no	—
100	C₂×Q₈	—	2	1	no	—
101	C₂×Q₈	—	218	1	no	—
102	C₂²×C₄	—	109	1	no	—
103	C₂²×C₄	—	109	1	no	—
104	C₂²×C₄	—	109	1	no	—
105	C₂²×C₄	—	218	2	no	—
106	M₁₆	—	218	4	no	—
107	C₂×C₈	—	218	2	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	109	2	no	—
110	C₄⋊C₄	—	218	2	no	—
111	C₄⋊C₄	—	218	2	no	—
112	C₄⋊C₄	—	218	1	no	—
113	C₄⋊C₄	C₂×#89137	1	2	yes	—
114	C₂²⋊C₄	—	218	2	no	—
115	C₄²	—	109	1	no	—
116	C₄²	—	109	1	no	—
117	C₄²	—	109	1	no	—
118	C₄²	—	109	4	no	—
119	C₄²	—	218	2	no	—
120	C₄²	—	218	2	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	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	109	1	no	—
127	C₂×C₄	—	109	2	no	—
128	C₂×C₄	—	109	2	no	—
129	C₂×C₄	—	109	1	no	—
130	C₂×C₄	—	109	1	no	—
131	C₂×C₄	—	436	1	no	—
132	C₂×C₄	—	436	1	no	—
133	C₂×C₄	—	436	2	no	—
134	C₂×C₄	—	436	2	no	—
135	C₂×C₄	—	218	4	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	C₂²	—	109	2	no	—
142	C₂²	—	218	1	no	—
143	C₂²	—	218	1	no	—
144	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	4	2	no	—
148	C₄	—	109	1	no	—
149	C₄	—	109	1	no	—
150	C₄	—	436	4	no	—
151	C₄	—	436	2	no	—
152	C₄	—	218	1	no	—
153	C₄	—	218	1	no	—
154	C₄	Q₈×#89137	1	2	yes	—
155	C₂	—	218	2	no	—
156	C₂	(C₂×#89137)⋊Q₈	1	1	yes	characteristic
157	C₂	#20296024	1	1	yes	characteristic
158	C₂	#20296025	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296031

C₂

yes

maximal, characteristic

#20295962

C₂

yes

maximal, characteristic

#20295896

C₂

yes

maximal

#20295581

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×C₄×#89137

C₂²

yes

—

—

—

—

—

C₄

yes

characteristic, complemented

#16812690

C₄

yes

complemented

#16812679

—

#16812675

—

Q₈×Dic₁₀₉

—

#16812664

—

#16812657

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

—

—

—

Q₈

yes

complemented

C₂×Dic₂₁₈

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

—

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

C₂×Q₈

yes

—

C₄×D₁₀₉

#41

yes

—

Dic₂₁₈

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊Q₈

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#124

yes

characteristic

Dic₁₀₉

C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1995

—

109

maximal, Sylow

C₁₀₉

#1995

yes

characteristic, complemented, Sylow

#533

—

109

—

#423

—

109

—

#376

—

109

—

#339

—

109

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

#89137

yes

characteristic, complemented

#125

—

109

—

C₂²⋊Q₈

—

218

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

218

—

C₄×Q₈

—

218

—

#117

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

#95

—

218

—

#95

—

218

—

100

C₂×Q₈

—

101

C₂×Q₈

—

218

—

102

C₂²×C₄

—

109

—

103

C₂²×C₄

—

109

—

104

C₂²×C₄

—

109

—

105

C₂²×C₄

—

218

—

106

M₁₆

—

218

—

107

C₂×C₈

—

218

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

109

—

110

C₄⋊C₄

—

218

—

111

C₄⋊C₄

—

218

—

112

C₄⋊C₄

—

218

—

113

C₄⋊C₄

C₂×#89137

yes

—

114

C₂²⋊C₄

—

218

—

115

C₄²

—

109

—

116

C₄²

—

109

—

117

C₄²

—

109

—

118

C₄²

—

109

—

119

C₄²

—

218

—

120

C₄²

—

218

—

121

C₄²

C₂×#89137

yes

characteristic

122

C₂³

—

109

—

123

Q₈

—

124

Q₈

—

436

—

125

C₂×C₄

—

126

C₂×C₄

—

109

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

109

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

436

—

132

C₂×C₄

—

436

—

133

C₂×C₄

—

436

—

134

C₂×C₄

—

436

—

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

C₂²

—

109

—

142

C₂²

—

218

—

143

C₂²

—

218

—

144

C₂²

C₂×C₂²⋊#89137

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

109

—

149

C₄

—

109

—

150

C₄

—

436

—

151

C₄

—

436

—

152

C₄

—

218

—

153

C₄

—

218

—

154

C₄

Q₈×#89137

yes

—

155

C₂

—

218

—

156

C₂

(C₂×#89137)⋊Q₈

yes

characteristic

157

C₂

#20296024

yes

characteristic

158

C₂

#20296025

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	#1995
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
11. #16812699	C₄	semidirect
12. #16812690	C₄	semidirect
19. C₄⋊Q₈×C₁₀₉	C₄	semidirect
24. C₄×#89137	Q₈	semidirect
80. C₁₀₉	#1995	semidirect
87. C₄⋊Q₈	#89137	semidirect

H ≅ G/N

type

11. #16812699

C₄

semidirect

12. #16812690

C₄

semidirect

19. C₄⋊Q₈×C₁₀₉

C₄

semidirect

24. C₄×#89137

Q₈

semidirect

80. C₁₀₉

#1995

semidirect

87. C₄⋊Q₈

#89137

semidirect

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

subgroup	index
#20296031	2
#20295962	2
#20295896	2
#20295581	2
#1995	109

subgroup

index

109

Subgroups — #28379220

Conjugacy classes of subgroups

Complements

Maximal subgroups