Group #28378641 — subgroups

Subgroups — #28378641

Order 13952.

166 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 58 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×#11724786	C₂	1	1	yes	maximal, characteristic, complemented
2	#20296061	C₂	1	1	yes	maximal, characteristic
3	#20295888	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#11724786	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#11724786	C₂	1	1	yes	maximal, characteristic
6	#11724786⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182038	—	2	1	no	—
9	C₄⋊#11724786	—	2	1	no	—
10	C₄⋊#11724786	C₂²	1	1	yes	characteristic
11	C₄⋊#11724786	C₂²	1	1	yes	characteristic
12	C₂²⋊#11724786	—	2	1	no	—
13	C₄×#11724786	—	2	1	no	—
14	C₄×#11724786	C₂²	1	1	yes	characteristic
15	C₄×#11724786	C₂²	1	1	yes	characteristic
16	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
17	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	#16812631	C₄	1	1	yes	characteristic
19	#16812623	C₄	1	1	yes	characteristic
20	#16812620	C₄	1	1	yes	characteristic
21	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
22	C₂×SD₁₆×C₁₀₉	C₄	1	1	yes	characteristic
23	#13182038	—	4	2	no	—
24	C₂×#11724786	—	2	1	no	—
25	C₂×#11724786	—	2	1	no	—
26	C₂×#11724786	—	2	1	no	—
27	C₂×#11724786	D₄	1	1	yes	characteristic
28	C₂×#11724786	D₄	1	1	yes	characteristic
29	C₂²×Dic₁₀₉	—	2	1	no	—
30	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Dic₁₀₉	—	2	1	no	—
32	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂²⋊Dic₁₀₉	—	2	1	no	—
35	C₄×Dic₁₀₉	—	2	1	no	—
36	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	SD₁₆×C₁₀₉	—	2	2	no	—
40	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₁₀₉	—	2	1	no	—
42	Q₈×C₁₀₉	—	2	1	no	—
43	Q₈×C₁₀₉	M₁₆	1	2	yes	complemented
44	D₄×C₁₀₉	—	2	1	no	—
45	D₄×C₁₀₉	—	2	1	no	—
46	C₂×C₄×C₁₀₉	—	2	1	no	—
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×Dic₁₀₉	—	2	1	no	—
49	C₂×Dic₁₀₉	—	2	1	no	—
50	C₂×Dic₁₀₉	—	4	1	no	—
51	C₂×Dic₁₀₉	—	4	1	no	—
52	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
54	#11724786	—	2	1	no	—
55	#11724786	—	4	1	no	—
56	#11724786	—	4	1	no	—
57	#11724786	—	4	1	no	—
58	#11724786	SD₁₆	1	2	yes	complemented
59	C₈×C₁₀₉	—	2	1	no	—
60	#11724784	—	2	1	no	—
61	C₂²×C₁₀₉	—	4	1	no	—
62	C₂²×C₁₀₉	—	4	1	no	—
63	C₂²×C₁₀₉	—	4	1	no	—
64	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
65	C₄×C₁₀₉	—	2	2	no	—
66	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₁₀₉	#131	1	1	yes	characteristic
68	Dic₁₀₉	—	2	1	no	—
69	Dic₁₀₉	—	4	1	no	—
70	Dic₁₀₉	—	4	1	no	—
71	Dic₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
72	Dic₁₀₉	#135	1	1	yes	characteristic
73	C₂×C₁₀₉	—	8	1	no	—
74	C₂×C₁₀₉	#435	1	1	yes	characteristic
75	C₂×C₁₀₉	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₁₀₉	#444	1	1	yes	characteristic
77	#1583	—	109	1	no	maximal, Sylow
78	C₁₀₉	#1583	1	1	yes	characteristic, complemented, Sylow
79	C₈×Q₈	—	109	1	no	—
80	C₄×SD₁₆	—	109	1	no	—
81	#436	—	109	1	no	—
82	#334	—	109	1	no	—
83	Q₈⋊C₈	—	109	1	no	—
84	D₄⋊C₈	—	109	1	no	—
85	#322	—	109	1	no	—
86	C₂×SD₁₆	#89137	1	1	yes	characteristic
87	C₂×M₁₆	—	218	1	no	—
88	C₄×Q₈	—	109	1	no	—
89	C₄×D₄	—	109	1	no	—
90	#106	—	109	1	no	—
91	C₄⋊C₈	—	109	1	no	—
92	C₄⋊C₈	—	109	1	no	—
93	C₄⋊C₈	—	218	1	no	—
94	#103	—	109	1	no	—
95	#102	—	109	1	no	—
96	C₂²⋊C₈	—	218	1	no	—
97	C₄×C₈	—	109	1	no	—
98	C₄×C₈	—	109	1	no	—
99	C₄×C₈	—	109	1	no	—
100	C₄×C₈	—	218	1	no	—
101	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
102	C₂×D₄	C₂×#89137	1	1	yes	characteristic
103	C₂²×C₄	—	218	1	no	—
104	SD₁₆	—	2	2	no	—
105	M₁₆	—	436	2	no	—
106	C₂×C₈	—	109	1	no	—
107	C₂×C₈	—	109	1	no	—
108	C₂×C₈	—	109	1	no	—
109	C₂×C₈	—	218	1	no	—
110	C₂×C₈	—	218	1	no	—
111	C₂×C₈	—	218	1	no	—
112	C₂×C₈	C₂×#89137	1	1	yes	characteristic
113	C₄⋊C₄	—	109	1	no	—
114	C₄⋊C₄	—	109	1	no	—
115	C₄⋊C₄	—	218	1	no	—
116	C₂²⋊C₄	—	218	1	no	—
117	C₄²	—	109	1	no	—
118	C₄²	—	218	1	no	—
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	#13182038	1	2	yes	complemented
122	D₄	—	2	1	no	—
123	D₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	109	1	no	—
126	C₂×C₄	—	109	1	no	—
127	C₂×C₄	—	436	1	no	—
128	C₂×C₄	—	436	1	no	—
129	C₂×C₄	—	218	1	no	—
130	C₂×C₄	—	218	1	no	—
131	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	109	2	no	—
134	C₈	—	436	1	no	—
135	C₈	—	436	1	no	—
136	C₈	—	436	1	no	—
137	C₈	—	218	1	no	—
138	C₈	—	218	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	D₄×#89137	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	109	1	no	—
145	C₄	—	109	1	no	—
146	C₄	—	436	1	no	—
147	C₄	—	436	1	no	—
148	C₄	—	218	1	no	—
149	C₄	C₂×#13182038	1	1	yes	characteristic
150	C₄	#16812786	1	1	yes	characteristic
151	C₂	—	8	1	no	—
152	C₂	#20296057	1	1	yes	characteristic
153	C₂	SD₁₆×#89137	1	1	yes	characteristic
154	C₂	#20296071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11724786

C₂

yes

maximal, characteristic, complemented

#20296061

C₂

yes

maximal, characteristic

#20295888

C₂

yes

maximal, characteristic, complemented

Q₈⋊#11724786

C₂

yes

maximal, characteristic, complemented

D₄⋊#11724786

C₂

yes

maximal, characteristic

#11724786⋊C₈

C₂

yes

maximal, characteristic, complemented

SD₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182038

—

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

characteristic

C₄⋊#11724786

C₂²

yes

—

—

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

characteristic

D₄×Dic₁₀₉

C₂²

yes

characteristic

#16812631

C₄

yes

characteristic

#16812623

C₄

yes

characteristic

#16812620

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×SD₁₆×C₁₀₉

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₂×#11724786

D₄

yes

characteristic

C₂²×Dic₁₀₉

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

characteristic

C₄⋊Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₉

C₂×C₄

yes

characteristic

SD₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

M₁₆

yes

complemented

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

C₂×Dic₁₀₉

yes

—

—

—

—

SD₁₆

yes

complemented

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

#135

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

#435

yes

characteristic

C₂×C₁₀₉

C₄×SD₁₆

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1583

yes

characteristic, complemented, Sylow

C₈×Q₈

—

109

—

C₄×SD₁₆

—

109

—

#436

—

109

—

#334

—

109

—

Q₈⋊C₈

—

109

—

D₄⋊C₈

—

109

—

#322

—

109

—

C₂×SD₁₆

#89137

yes

characteristic

C₂×M₁₆

—

218

—

C₄×Q₈

—

109

—

C₄×D₄

—

109

—

#106

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

#103

—

109

—

#102

—

109

—

C₂²⋊C₈

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

100

C₄×C₈

—

218

—

101

C₂×Q₈

C₂×#89137

yes

characteristic

102

C₂×D₄

C₂×#89137

yes

characteristic

103

C₂²×C₄

—

218

—

104

SD₁₆

—

105

M₁₆

—

436

—

106

C₂×C₈

—

109

—

107

C₂×C₈

—

109

—

108

C₂×C₈

—

109

—

109

C₂×C₈

—

218

—

110

C₂×C₈

—

218

—

111

C₂×C₈

—

218

—

112

C₂×C₈

C₂×#89137

yes

characteristic

113

C₄⋊C₄

—

109

—

114

C₄⋊C₄

—

109

—

115

C₄⋊C₄

—

218

—

116

C₂²⋊C₄

—

218

—

117

C₄²

—

109

—

118

C₄²

—

218

—

119

C₂³

—

120

Q₈

—

121

Q₈

#13182038

yes

complemented

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

109

—

127

C₂×C₄

—

436

—

128

C₂×C₄

—

436

—

129

C₂×C₄

—

218

—

130

C₂×C₄

—

218

—

131

C₂×C₄

C₂²×#89137

yes

characteristic

132

C₈

—

133

C₈

—

109

—

134

C₈

—

436

—

135

C₈

—

436

—

136

C₈

—

436

—

137

C₈

—

218

—

138

C₈

—

218

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×#89137

yes

characteristic

143

C₄

—

144

C₄

—

109

—

145

C₄

—

109

—

146

C₄

—

436

—

147

C₄

—

436

—

148

C₄

—

218

—

149

C₄

C₂×#13182038

yes

characteristic

150

C₄

#16812786

yes

characteristic

151

C₂

—

152

C₂

#20296057

yes

characteristic

153

C₂

SD₁₆×#89137

yes

characteristic

154

C₂

#20296071

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×SD₁₆×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₄×D₄
prime radical	C₂×D₄×C₁₀₉	C₂×C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂×SD₁₆	#89137
hypocenter	C₁₀₉	#1583
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂×SD₁₆×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₄×D₄

prime radical

C₂×D₄×C₁₀₉

C₂×C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₂×SD₁₆

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. Q₈×#11724786	C₂	semidirect
3. #20295888	C₂	semidirect
4. Q₈⋊#11724786	C₂	semidirect
6. #11724786⋊C₈	C₂	semidirect
43. Q₈×C₁₀₉	M₁₆	semidirect
58. #11724786	SD₁₆	semidirect
78. C₁₀₉	#1583	semidirect
121. Q₈	#13182038	semidirect

H ≅ G/N

type

1. Q₈×#11724786

C₂

semidirect

3. #20295888

C₂

semidirect

4. Q₈⋊#11724786

C₂

semidirect

6. #11724786⋊C₈

C₂

semidirect

43. Q₈×C₁₀₉

M₁₆

semidirect

58. #11724786

SD₁₆

semidirect

78. C₁₀₉

#1583

semidirect

121. Q₈

#13182038

semidirect

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

subgroup	index
Q₈×#11724786	2
#20296061	2
#20295888	2
Q₈⋊#11724786	2
D₄⋊#11724786	2
#11724786⋊C₈	2
SD₁₆×Dic₁₀₉	2
#1583	109

subgroup

index

SD₁₆×Dic₁₀₉

109

Subgroups — #28378641

Conjugacy classes of subgroups

Complements

Maximal subgroups