Group #28379217 — subgroups

Subgroups — #28379217

Order 13952.

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

normal subgroups only

This group has 284 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	#20296030	C₂	1	1	yes	maximal, characteristic
2	C₄²×#89137	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	—	2	1	no	—
7	C₂×C₄×#89137	—	2	1	no	—
8	C₂×C₄×#89137	C₂²	1	2	yes	—
9	C₄⋊#11724786	—	2	1	no	—
10	C₄⋊#11724786	—	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	4	no	—
24	C₄×#89137	—	2	4	no	—
25	C₄×#89137	—	2	2	no	—
26	C₄×#89137	Q₈	1	4	yes	complemented
27	C₂×Dic₂₁₈	—	2	1	no	—
28	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
29	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₁₀₉	—	2	1	no	—
32	C₄⋊Dic₁₀₉	—	2	2	no	—
33	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
34	#13182008	—	2	2	no	—
35	#13182006	—	2	2	no	—
36	C₄×Dic₁₀₉	—	2	2	no	—
37	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
39	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
40	C₂×Q₈×C₁₀₉	—	2	1	no	—
41	C₄⋊C₄×C₁₀₉	—	2	1	no	—
42	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×#89137	—	2	4	no	—
46	C₂×#89137	—	2	4	no	—
47	C₂×#89137	—	2	4	no	—
48	Q₈×C₁₀₉	—	4	2	no	—
49	C₂×C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₁₀₉	—	2	2	no	—
54	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₄×D₁₀₉	—	2	2	no	—
58	C₄×D₁₀₉	—	2	2	no	—
59	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
60	C₄×D₁₀₉	#41	1	2	yes	—
61	Dic₂₁₈	—	4	2	no	—
62	#11724786	—	4	2	no	—
63	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₁₀₉	—	2	1	no	—
65	C₂×D₁₀₉	—	2	1	no	—
66	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
67	#89137	—	2	8	no	—
68	C₄×C₁₀₉	—	4	2	no	—
69	C₄×C₁₀₉	#104	1	2	yes	—
70	C₄×C₁₀₉	#104	1	2	yes	—
71	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
72	Dic₁₀₉	—	2	1	no	—
73	Dic₁₀₉	—	2	1	no	—
74	Dic₁₀₉	—	4	2	no	—
75	Dic₁₀₉	#124	1	1	yes	characteristic
76	Dic₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₁₀₉	#391	1	1	yes	characteristic
78	C₂×C₁₀₉	C₂×#104	1	1	yes	characteristic
79	C₂×C₁₀₉	C₂×#104	1	1	yes	characteristic
80	D₁₀₉	—	2	2	no	—
81	#1994	—	109	1	no	maximal, Sylow
82	C₁₀₉	#1994	1	1	yes	characteristic, complemented, Sylow
83	#533	—	109	1	no	—
84	#423	—	109	1	no	—
85	C₄³	—	109	1	no	—
86	#339	—	109	4	no	—
87	C₂×M₁₆	—	109	2	no	—
88	C₄⋊Q₈	—	109	1	no	—
89	C₄⋊Q₈	#89137	1	1	yes	characteristic, complemented
90	#125	—	109	2	no	—
91	C₂²⋊Q₈	—	218	1	no	—
92	C₂²⋊Q₈	—	218	1	no	—
93	C₄×Q₈	—	218	1	no	—
94	C₄×Q₈	—	218	1	no	—
95	#117	—	109	2	no	—
96	C₂×C₄²	—	109	1	no	—
97	C₂×C₄²	—	109	2	no	—
98	C₂×C₄²	—	218	1	no	—
99	C₂×C₄²	—	218	1	no	—
100	C₄⋊C₈	—	218	1	no	—
101	C₄⋊C₈	—	218	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	218	1	no	—
104	C₂²×C₄	—	109	1	no	—
105	C₂²×C₄	—	109	1	no	—
106	C₂²×C₄	—	109	1	no	—
107	C₂²×C₄	—	218	2	no	—
108	M₁₆	—	218	4	no	—
109	C₂×C₈	—	218	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	109	2	no	—
112	C₄⋊C₄	—	218	2	no	—
113	C₄⋊C₄	—	218	2	no	—
114	C₄⋊C₄	—	218	1	no	—
115	C₄⋊C₄	C₂×#89137	1	2	yes	—
116	C₂²⋊C₄	—	218	2	no	—
117	C₄²	—	109	1	no	—
118	C₄²	—	109	1	no	—
119	C₄²	—	109	1	no	—
120	C₄²	—	109	4	no	—
121	C₄²	—	218	2	no	—
122	C₄²	—	218	2	no	—
123	C₄²	—	218	4	no	—
124	C₄²	—	218	4	no	—
125	C₄²	C₂×#89137	1	1	yes	characteristic
126	C₂³	—	109	1	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	436	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	109	1	no	—
131	C₂×C₄	—	109	2	no	—
132	C₂×C₄	—	109	2	no	—
133	C₂×C₄	—	109	1	no	—
134	C₂×C₄	—	109	1	no	—
135	C₂×C₄	—	218	2	no	—
136	C₂×C₄	—	218	4	no	—
137	C₂×C₄	—	218	4	no	—
138	C₂×C₄	—	218	4	no	—
139	C₂×C₄	—	218	2	no	—
140	C₂×C₄	—	218	2	no	—
141	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
143	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
144	C₈	—	436	2	no	—
145	C₂²	—	109	2	no	—
146	C₂²	—	218	1	no	—
147	C₂²	—	218	1	no	—
148	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	109	1	no	—
151	C₄	—	109	1	no	—
152	C₄	—	436	2	no	—
153	C₄	—	218	8	no	—
154	C₄	—	218	1	no	—
155	C₄	—	218	1	no	—
156	C₄	#16812763	1	2	yes	—
157	C₄	#16812764	1	2	yes	—
158	C₄	Q₈×#89137	1	2	yes	—
159	C₂	—	218	2	no	—
160	C₂	(C₂×#89137)⋊Q₈	1	1	yes	characteristic
161	C₂	C₂×#16812763	1	1	yes	characteristic
162	C₂	C₂×#16812764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296030

C₂

yes

maximal, characteristic

C₄²×#89137

C₂

yes

maximal, characteristic

#20295896

C₂

yes

maximal

#20295581

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×C₄×#89137

—

C₂×C₄×#89137

—

C₂×C₄×#89137

C₂²

yes

—

C₄⋊#11724786

—

C₄⋊#11724786

—

#16812699

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

#104

yes

—

C₄×C₁₀₉

#104

yes

—

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

—

#1994

—

109

maximal, Sylow

C₁₀₉

#1994

yes

characteristic, complemented, Sylow

#533

—

109

—

#423

—

109

—

C₄³

—

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

—

C₂×C₄²

—

218

—

C₂×C₄²

—

218

—

100

C₄⋊C₈

—

218

—

101

C₄⋊C₈

—

218

—

102

C₂×Q₈

—

103

C₂×Q₈

—

218

—

104

C₂²×C₄

—

109

—

105

C₂²×C₄

—

109

—

106

C₂²×C₄

—

109

—

107

C₂²×C₄

—

218

—

108

M₁₆

—

218

—

109

C₂×C₈

—

218

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

109

—

112

C₄⋊C₄

—

218

—

113

C₄⋊C₄

—

218

—

114

C₄⋊C₄

—

218

—

115

C₄⋊C₄

C₂×#89137

yes

—

116

C₂²⋊C₄

—

218

—

117

C₄²

—

109

—

118

C₄²

—

109

—

119

C₄²

—

109

—

120

C₄²

—

109

—

121

C₄²

—

218

—

122

C₄²

—

218

—

123

C₄²

—

218

—

124

C₄²

—

218

—

125

C₄²

C₂×#89137

yes

characteristic

126

C₂³

—

109

—

127

Q₈

—

128

Q₈

—

436

—

129

C₂×C₄

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

109

—

132

C₂×C₄

—

109

—

133

C₂×C₄

—

109

—

134

C₂×C₄

—

109

—

135

C₂×C₄

—

218

—

136

C₂×C₄

—

218

—

137

C₂×C₄

—

218

—

138

C₂×C₄

—

218

—

139

C₂×C₄

—

218

—

140

C₂×C₄

—

218

—

141

C₂×C₄

C₂²⋊#89137

yes

characteristic

142

C₂×C₄

C₂²⋊#89137

yes

characteristic

143

C₂×C₄

C₂²×#89137

yes

characteristic

144

C₈

—

436

—

145

C₂²

—

109

—

146

C₂²

—

218

—

147

C₂²

—

218

—

148

C₂²

C₂×C₂²⋊#89137

yes

characteristic

149

C₄

—

150

C₄

—

109

—

151

C₄

—

109

—

152

C₄

—

436

—

153

C₄

—

218

—

154

C₄

—

218

—

155

C₄

—

218

—

156

C₄

#16812763

yes

—

157

C₄

#16812764

yes

—

158

C₄

Q₈×#89137

yes

—

159

C₂

—

218

—

160

C₂

(C₂×#89137)⋊Q₈

yes

characteristic

161

C₂

C₂×#16812763

yes

characteristic

162

C₂

C₂×#16812764

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	#1994
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
26. C₄×#89137	Q₈	semidirect
82. C₁₀₉	#1994	semidirect
89. C₄⋊Q₈	#89137	semidirect

H ≅ G/N

type

11. #16812699

C₄

semidirect

12. #16812690

C₄

semidirect

19. C₄⋊Q₈×C₁₀₉

C₄

semidirect

26. C₄×#89137

Q₈

semidirect

82. C₁₀₉

#1994

semidirect

89. C₄⋊Q₈

#89137

semidirect

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

subgroup	index
#20296030	2
C₄²×#89137	2
#20295896	2
#20295581	2
#1994	109

subgroup

index

C₄²×#89137

109

Subgroups — #28379217

Conjugacy classes of subgroups

Complements

Maximal subgroups