Group #28375663 — subgroups

Subgroups — #28375663

Order 13952.

426 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×D₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295372	C₂	1	3	yes	maximal, complemented
3	#20295306	C₂	1	3	yes	maximal, complemented
4	#20295218	C₂	1	3	yes	maximal, complemented
5	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13182014	C₂	1	3	yes	maximal, complemented
7	#550×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₁₀₉	—	2	1	no	—
9	C₂×#13182014	C₂²	1	3	yes	—
10	#16812699	C₂²	1	3	yes	—
11	#16812679	—	2	3	no	—
12	C₄×D₄₃₆	C₂²	1	3	yes	—
13	#16812632	C₂²	1	3	yes	—
14	C₄⋊#11724784	C₂²	1	3	yes	—
15	#16812620	C₂²	1	6	yes	—
16	#16812620	C₂²	1	3	yes	—
17	#16812619	C₂²	1	1	yes	characteristic
18	C₄×#11724784	C₂²	1	3	yes	—
19	C₂×#41×C₁₀₉	—	2	1	no	—
20	#127×C₁₀₉	C₂²	1	3	yes	—
21	#121×C₁₀₉	—	2	3	no	—
22	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₀₉	C₂²	1	3	yes	—
24	C₂×Dic₂₁₈	—	2	3	no	—
25	Q₈×D₁₀₉	—	4	4	no	—
26	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₁₀₉	—	2	3	no	—
28	#13182014	—	2	6	no	—
29	C₂×#11724784	C₂³	1	1	yes	characteristic
30	C₂×#11724784	C₂³	1	3	yes	—
31	C₄⋊Dic₁₀₉	—	2	3	no	—
32	C₄⋊Dic₁₀₉	C₂³	1	3	yes	—
33	#13182006	—	2	3	no	—
34	#41×C₁₀₉	—	4	4	no	—
35	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₀₉	—	2	3	no	—
37	C₂×D₄×C₁₀₉	—	2	3	no	—
38	C₂×D₄×C₁₀₉	C₂³	1	3	yes	—
39	C₂²×C₄×C₁₀₉	—	2	3	no	—
40	C₄⋊C₄×C₁₀₉	D₄	1	3	yes	—
41	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂²⋊C₄×C₁₀₉	—	2	3	no	—
43	C₄²×C₁₀₉	C₂³	1	3	yes	—
44	C₂³×C₁₀₉	—	2	3	no	—
45	C₂²×D₁₀₉	—	2	1	no	—
46	Q₈×C₁₀₉	D₈	1	4	yes	complemented
47	D₄×C₁₀₉	—	2	3	no	—
48	D₄×C₁₀₉	—	2	3	no	—
49	D₄×C₁₀₉	—	4	6	no	—
50	D₄×C₁₀₉	—	8	3	no	—
51	C₂×C₄×C₁₀₉	—	2	1	no	—
52	C₂×C₄×C₁₀₉	—	4	6	no	—
53	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
54	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₁₀₉	—	2	3	no	—
57	D₄₃₆	—	2	1	no	—
58	D₄₃₆	#41	1	2	yes	complemented
59	C₄×D₁₀₉	—	4	6	no	—
60	Dic₂₁₈	—	4	6	no	—
61	#11724784	—	2	3	no	—
62	#11724784	—	2	1	no	—
63	C₂²×C₁₀₉	—	4	3	no	—
64	C₂²×C₁₀₉	—	4	3	no	—
65	C₂²×C₁₀₉	—	4	3	no	—
66	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₁₀₉	—	2	2	no	—
68	C₂×D₁₀₉	—	4	1	no	—
69	C₂×D₁₀₉	—	4	1	no	—
70	C₄×C₁₀₉	—	2	3	no	—
71	C₄×C₁₀₉	—	4	1	no	—
72	C₄×C₁₀₉	C₂×D₈	1	6	yes	—
73	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
74	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₁₀₉	—	4	3	no	—
76	C₂×C₁₀₉	—	8	3	no	—
77	C₂×C₁₀₉	#548	1	1	yes	characteristic
78	C₂×C₁₀₉	C₂²×D₈	1	1	yes	characteristic
79	C₂×C₁₀₉	#577	1	1	yes	characteristic
80	D₁₀₉	—	4	2	no	—
81	#3358	—	109	1	no	maximal, Sylow
82	C₁₀₉	#3358	1	1	yes	characteristic, complemented, Sylow
83	#550	D₁₀₉	1	1	yes	characteristic, complemented
84	D₄×Q₈	—	109	1	no	—
85	#486	—	109	3	no	—
86	#474	—	109	3	no	—
87	#459	—	109	3	no	—
88	C₈×Q₈	—	109	1	no	—
89	C₄×D₈	—	109	3	no	—
90	C₂×#41	—	2	1	no	—
91	C₂²×Q₈	—	218	1	no	—
92	C₂×D₈	—	109	3	no	—
93	C₄⋊Q₈	—	109	3	no	—
94	#127	C₂×D₁₀₉	1	3	yes	—
95	C₂²⋊Q₈	—	218	3	no	—
96	#121	—	2	3	no	—
97	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
98	C₄×D₄	—	109	3	no	—
99	C₄×D₄	C₂×D₁₀₉	1	3	yes	—
100	#107	—	109	3	no	—
101	C₄⋊C₈	—	109	3	no	—
102	#102	—	109	3	no	—
103	#102	—	109	6	no	—
104	#102	—	109	1	no	—
105	C₄×C₈	—	109	3	no	—
106	#41	—	4	4	no	—
107	C₂×Q₈	—	436	4	no	—
108	C₂×Q₈	—	218	3	no	—
109	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
110	C₂×D₄	—	2	3	no	—
111	C₂×D₄	—	2	3	no	—
112	C₂×D₄	—	109	1	no	—
113	C₂×D₄	C₂²×D₁₀₉	1	3	yes	—
114	C₂²×C₄	—	2	3	no	—
115	C₂²×C₄	—	218	3	no	—
116	D₈	—	218	6	no	—
117	C₂×C₈	—	109	1	no	—
118	C₂×C₈	—	109	3	no	—
119	C₄⋊C₄	—	109	3	no	—
120	C₄⋊C₄	—	218	3	no	—
121	C₄⋊C₄	C₂²⋊D₁₀₉	1	3	yes	—
122	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	3	no	—
124	C₂²⋊C₄	—	218	3	no	—
125	C₄²	C₂²×D₁₀₉	1	3	yes	—
126	C₂³	—	2	3	no	—
127	C₂³	—	218	1	no	—
128	Q₈	—	436	6	no	—
129	Q₈	#13182014	1	4	yes	complemented
130	D₄	—	2	3	no	—
131	D₄	—	2	3	no	—
132	D₄	—	4	6	no	—
133	D₄	—	8	3	no	—
134	D₄	—	109	2	no	—
135	D₄	—	218	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	—	436	6	no	—
139	C₂×C₄	—	218	3	no	—
140	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
141	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
142	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
143	C₈	—	218	3	no	—
144	C₈	—	218	1	no	—
145	C₂²	—	4	3	no	—
146	C₂²	—	4	3	no	—
147	C₂²	—	4	3	no	—
148	C₂²	—	436	1	no	—
149	C₂²	—	436	1	no	—
150	C₂²	—	218	2	no	—
151	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
152	C₄	—	2	3	no	—
153	C₄	—	4	1	no	—
154	C₄	—	436	3	no	—
155	C₄	C₂×#13182014	1	6	yes	—
156	C₄	#41×D₁₀₉	1	1	yes	characteristic
157	C₄	#16812746	1	1	yes	characteristic
158	C₂	—	8	3	no	—
159	C₂	—	436	2	no	—
160	C₂	#20295683	1	1	yes	characteristic
161	C₂	C₂²×#13182014	1	1	yes	characteristic
162	C₂	#20295837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₄₃₆

C₂

yes

maximal, characteristic, complemented

#20295372

C₂

yes

maximal, complemented

#20295306

C₂

yes

maximal, complemented

#20295218

C₂

yes

maximal, complemented

Q₈×#11724784

C₂

yes

maximal, characteristic, complemented

C₄×#13182014

C₂

yes

maximal, complemented

#550×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₀₉

—

C₂×#13182014

C₂²

yes

—

#16812699

C₂²

yes

—

#16812679

—

C₄×D₄₃₆

C₂²

yes

—

#16812632

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

#16812620

C₂²

yes

—

#16812620

C₂²

yes

—

#16812619

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

—

C₂×#41×C₁₀₉

—

#127×C₁₀₉

C₂²

yes

—

#121×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×D₄×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

Q₈×D₁₀₉

—

C₂×D₄₃₆

C₂³

yes

characteristic

C₂×C₄×D₁₀₉

—

#13182014

—

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182006

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

D₈

yes

complemented

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

#41

yes

complemented

C₄×D₁₀₉

—

Dic₂₁₈

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₈

yes

—

C₄×C₁₀₉

C₂×#41

yes

characteristic

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#548

yes

characteristic

C₂×C₁₀₉

C₂²×D₈

yes

characteristic

C₂×C₁₀₉

#577

yes

characteristic

D₁₀₉

—

#3358

—

109

maximal, Sylow

C₁₀₉

#3358

yes

characteristic, complemented, Sylow

#550

D₁₀₉

yes

characteristic, complemented

D₄×Q₈

—

109

—

#486

—

109

—

#474

—

109

—

#459

—

109

—

C₈×Q₈

—

109

—

C₄×D₈

—

109

—

C₂×#41

—

C₂²×Q₈

—

218

—

C₂×D₈

—

109

—

C₄⋊Q₈

—

109

—

#127

C₂×D₁₀₉

yes

—

C₂²⋊Q₈

—

218

—

#121

—

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

C₄×D₄

—

109

—

C₄×D₄

C₂×D₁₀₉

yes

—

100

#107

—

109

—

101

C₄⋊C₈

—

109

—

102

#102

—

109

—

103

#102

—

109

—

104

#102

—

109

—

105

C₄×C₈

—

109

—

106

#41

—

107

C₂×Q₈

—

436

—

108

C₂×Q₈

—

218

—

109

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

109

—

113

C₂×D₄

C₂²×D₁₀₉

yes

—

114

C₂²×C₄

—

115

C₂²×C₄

—

218

—

116

D₈

—

218

—

117

C₂×C₈

—

109

—

118

C₂×C₈

—

109

—

119

C₄⋊C₄

—

109

—

120

C₄⋊C₄

—

218

—

121

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

122

C₄⋊C₄

C₂²×D₁₀₉

yes

characteristic

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

218

—

125

C₄²

C₂²×D₁₀₉

yes

—

126

C₂³

—

127

C₂³

—

218

—

128

Q₈

—

436

—

129

Q₈

#13182014

yes

complemented

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

109

—

135

D₄

—

218

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

436

—

139

C₂×C₄

—

218

—

140

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

141

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

142

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

143

C₈

—

218

—

144

C₈

—

218

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

436

—

149

C₂²

—

436

—

150

C₂²

—

218

—

151

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

436

—

155

C₄

C₂×#13182014

yes

—

156

C₄

#41×D₁₀₉

yes

characteristic

157

C₄

#16812746

yes

characteristic

158

C₂

—

159

C₂

—

436

—

160

C₂

#20295683

yes

characteristic

161

C₂

C₂²×#13182014

yes

characteristic

162

C₂

#20295837

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#550×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#550	D₁₀₉
hypocenter	C₁₀₉	#3358
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#550×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×D₄₃₆	C₂	semidirect
2. #20295372	C₂	semidirect
3. #20295306	C₂	semidirect
4. #20295218	C₂	semidirect
5. Q₈×#11724784	C₂	semidirect
6. C₄×#13182014	C₂	semidirect
7. #550×C₁₀₉	C₂	semidirect
46. Q₈×C₁₀₉	D₈	semidirect
58. D₄₃₆	#41	semidirect
82. C₁₀₉	#3358	semidirect
83. #550	D₁₀₉	semidirect
129. Q₈	#13182014	semidirect

H ≅ G/N

type

1. Q₈×D₄₃₆

C₂

semidirect

2. #20295372

C₂

semidirect

3. #20295306

C₂

semidirect

4. #20295218

C₂

semidirect

5. Q₈×#11724784

C₂

semidirect

6. C₄×#13182014

C₂

semidirect

7. #550×C₁₀₉

C₂

semidirect

46. Q₈×C₁₀₉

D₈

semidirect

58. D₄₃₆

#41

semidirect

82. C₁₀₉

#3358

semidirect

83. #550

D₁₀₉

semidirect

129. Q₈

#13182014

semidirect

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

subgroup	index
Q₈×D₄₃₆	2
#20295372	2
#20295306	2
#20295218	2
Q₈×#11724784	2
C₄×#13182014	2
#550×C₁₀₉	2
#3358	109

subgroup

index

Q₈×D₄₃₆

#550×C₁₀₉

109

Subgroups — #28375663

Conjugacy classes of subgroups

Complements

Maximal subgroups