Group #28378548 — subgroups

Subgroups — #28378548

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	4	no	—
26	C₂×C₄×D₁₀₉	—	2	2	no	—
27	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
29	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
30	C₄⋊Dic₁₀₉	—	2	1	no	—
31	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
32	#13182008	—	2	2	no	—
33	#13182008	—	2	2	no	—
34	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
35	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
36	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₀₉	—	2	1	no	—
38	C₄⋊C₄×C₁₀₉	—	2	1	no	—
39	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
40	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₁₀₉	—	4	2	no	—
43	C₂×C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₁₀₉	—	2	2	no	—
48	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₄×D₁₀₉	—	2	2	no	—
52	C₄×D₁₀₉	—	2	2	no	—
53	C₄×D₁₀₉	—	4	4	no	—
54	C₄×D₁₀₉	SD₁₆	1	2	yes	—
55	C₄×D₁₀₉	Q₁₆	1	2	yes	—
56	Dic₂₁₈	—	4	2	no	—
57	Dic₂₁₈	—	4	4	no	—
58	#11724786	—	4	2	no	—
59	#11724786	—	4	2	no	—
60	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×D₁₀₉	—	2	1	no	—
62	C₂×D₁₀₉	—	2	1	no	—
63	C₂×D₁₀₉	#103	1	2	yes	—
64	C₄×C₁₀₉	—	2	1	no	—
65	C₄×C₁₀₉	—	2	1	no	—
66	C₄×C₁₀₉	—	4	2	no	—
67	C₄×C₁₀₉	#103	1	2	yes	—
68	Dic₁₀₉	—	4	2	no	—
69	Dic₁₀₉	#101	1	2	yes	—
70	Dic₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
71	Dic₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
72	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
75	C₂×C₁₀₉	#418	1	1	yes	characteristic
76	D₁₀₉	—	2	2	no	—
77	#1549	—	109	1	no	maximal, Sylow
78	C₁₀₉	#1549	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₄⋊Q₈	—	109	1	no	—
80	#431	—	109	1	no	—
81	#423	—	109	1	no	—
82	#332	—	109	4	no	—
83	C₂²×Q₈	—	218	1	no	—
84	C₂×M₁₆	—	218	1	no	—
85	C₄⋊Q₈	—	109	2	no	—
86	C₄⋊Q₈	—	109	1	no	—
87	C₄⋊Q₈	—	218	1	no	—
88	C₄⋊Q₈	—	218	1	no	—
89	C₄⋊Q₈	#89137	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	109	2	no	—
91	C₂×C₄⋊C₄	—	218	1	no	—
92	C₂×C₄²	—	109	1	no	—
93	C₄⋊C₈	—	109	2	no	—
94	C₄⋊C₈	—	109	2	no	—
95	C₄⋊C₈	—	218	1	no	—
96	C₂²⋊C₈	—	218	1	no	—
97	C₄×C₈	—	218	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	436	4	no	—
100	C₂×Q₈	—	218	2	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	—	436	2	no	—
107	C₂×C₈	—	218	2	no	—
108	C₂×C₈	—	218	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	109	2	no	—
111	C₄⋊C₄	—	218	2	no	—
112	C₄⋊C₄	—	218	2	no	—
113	C₄⋊C₄	—	218	1	no	—
114	C₄⋊C₄	C₂×#89137	1	2	yes	—
115	C₄²	—	109	1	no	—
116	C₄²	—	109	1	no	—
117	C₄²	—	109	1	no	—
118	C₄²	C₂×#89137	1	1	yes	characteristic
119	C₂³	—	109	1	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	436	2	no	—
122	Q₈	—	436	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	109	1	no	—
125	C₂×C₄	—	109	1	no	—
126	C₂×C₄	—	109	1	no	—
127	C₂×C₄	—	109	2	no	—
128	C₂×C₄	—	109	2	no	—
129	C₂×C₄	—	436	4	no	—
130	C₂×C₄	—	218	2	no	—
131	C₂×C₄	—	218	2	no	—
132	C₂×C₄	—	218	2	no	—
133	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
135	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
136	C₈	—	436	2	no	—
137	C₈	—	436	2	no	—
138	C₂²	—	109	2	no	—
139	C₂²	—	218	1	no	—
140	C₂²	—	218	1	no	—
141	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	—	109	2	no	—
146	C₄	—	109	1	no	—
147	C₄	—	109	1	no	—
148	C₄	—	109	2	no	—
149	C₄	—	436	2	no	—
150	C₄	#16812762	1	2	yes	—
151	C₂	—	218	2	no	—
152	C₂	D₁₀₉⋊#101	1	1	yes	characteristic
153	C₂	C₂×#16812762	1	1	yes	characteristic
154	C₂	#20296018	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

—

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

SD₁₆

yes

—

C₄×D₁₀₉

Q₁₆

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#103

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#103

yes

—

Dic₁₀₉

—

Dic₁₀₉

#101

yes

—

Dic₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

C₂×Q₁₆

yes

characteristic

Dic₁₀₉

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1549

—

109

maximal, Sylow

C₁₀₉

#1549

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

—

C₂×Q₈

—

C₂×Q₈

—

436

—

100

C₂×Q₈

—

218

—

101

C₂×Q₈

—

218

—

102

C₂²×C₄

—

109

—

103

C₂²×C₄

—

109

—

104

C₂²×C₄

—

109

—

105

C₂²×C₄

—

218

—

106

M₁₆

—

436

—

107

C₂×C₈

—

218

—

108

C₂×C₈

—

218

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

109

—

111

C₄⋊C₄

—

218

—

112

C₄⋊C₄

—

218

—

113

C₄⋊C₄

—

218

—

114

C₄⋊C₄

C₂×#89137

yes

—

115

C₄²

—

109

—

116

C₄²

—

109

—

117

C₄²

—

109

—

118

C₄²

C₂×#89137

yes

characteristic

119

C₂³

—

109

—

120

Q₈

—

121

Q₈

—

436

—

122

Q₈

—

436

—

123

C₂×C₄

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

109

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

436

—

130

C₂×C₄

—

218

—

131

C₂×C₄

—

218

—

132

C₂×C₄

—

218

—

133

C₂×C₄

C₂²⋊#89137

yes

characteristic

134

C₂×C₄

C₂²⋊#89137

yes

characteristic

135

C₂×C₄

C₂²×#89137

yes

characteristic

136

C₈

—

436

—

137

C₈

—

436

—

138

C₂²

—

109

—

139

C₂²

—

218

—

140

C₂²

—

218

—

141

C₂²

C₂×C₂²⋊#89137

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

109

—

146

C₄

—

109

—

147

C₄

—

109

—

148

C₄

—

109

—

149

C₄

—

436

—

150

C₄

#16812762

yes

—

151

C₂

—

218

—

152

C₂

D₁₀₉⋊#101

yes

characteristic

153

C₂

C₂×#16812762

yes

characteristic

154

C₂

#20296018

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	#1549
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
78. C₁₀₉	#1549	semidirect

H ≅ G/N

type

3. #20295885

C₂

semidirect

78. C₁₀₉

#1549

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
#1549	109

subgroup

index

C₄⋊Q₈×D₁₀₉

109

Subgroups — #28378548

Conjugacy classes of subgroups

Complements

Maximal subgroups