Group #28378355 — subgroups

Subgroups — #28378355

Order 13952.

218 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 64 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	#20296045	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182039	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11724786	C₂	1	4	yes	maximal, complemented
4	#20295581	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	—	2	1	no	—
6	C₄⋊#11724786	C₂²	1	2	yes	—
7	#16812756	—	2	1	no	—
8	#16812754	—	2	1	no	—
9	C₄×#11724786	—	2	1	no	—
10	C₄×#11724786	C₂²	1	2	yes	—
11	#16812699	C₄	1	1	yes	characteristic
12	#16812690	—	2	1	no	—
13	#16812679	—	2	1	no	—
14	#16812675	C₄	1	2	yes	—
15	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
16	#16812664	—	2	1	no	—
17	#16812657	—	2	1	no	—
18	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
20	#13182039	—	2	4	no	—
21	C₂×#11724786	—	2	2	no	—
22	C₂×#11724786	—	2	2	no	—
23	C₂×Dic₂₁₈	—	2	1	no	—
24	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
27	C₄⋊Dic₁₀₉	—	2	1	no	—
28	C₄⋊Dic₁₀₉	—	2	2	no	—
29	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
30	#13182008	—	2	2	no	—
31	#13182006	—	2	2	no	—
32	C₄×Dic₁₀₉	—	2	2	no	—
33	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
34	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
35	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
36	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
37	C₄⋊C₄×C₁₀₉	—	2	1	no	—
38	C₄⋊C₄×C₁₀₉	—	2	1	no	—
39	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₁₀₉	—	2	4	no	—
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₁₀₉	—	2	2	no	—
47	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₄×D₁₀₉	—	2	2	no	—
51	C₄×D₁₀₉	—	2	2	no	—
52	C₄×D₁₀₉	—	2	2	no	—
53	Dic₂₁₈	—	4	2	no	—
54	#11724786	—	2	4	no	—
55	#11724786	—	4	2	no	—
56	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂×D₁₀₉	—	2	1	no	—
58	C₂×D₁₀₉	—	2	1	no	—
59	C₂×D₁₀₉	—	2	1	no	—
60	C₄×C₁₀₉	—	2	1	no	—
61	C₄×C₁₀₉	—	4	2	no	—
62	C₄×C₁₀₉	#104	1	2	yes	—
63	C₄×C₁₀₉	#131	1	2	yes	—
64	Dic₁₀₉	—	2	1	no	—
65	Dic₁₀₉	—	2	1	no	—
66	Dic₁₀₉	—	4	2	no	—
67	Dic₁₀₉	#137	1	2	yes	—
68	C₂×C₁₀₉	#408	1	1	yes	characteristic
69	C₂×C₁₀₉	#419	1	1	yes	characteristic
70	C₂×C₁₀₉	C₂×#104	1	1	yes	characteristic
71	D₁₀₉	—	4	1	no	—
72	#1485	—	109	1	no	maximal, Sylow
73	C₁₀₉	#1485	1	1	yes	characteristic, complemented, Sylow
74	#533	—	109	1	no	—
75	#431	—	109	1	no	—
76	C₄×M₁₆	—	109	1	no	—
77	Q₈⋊C₈	—	109	4	no	—
78	C₂×M₁₆	—	218	1	no	—
79	C₄⋊Q₈	—	109	1	no	—
80	C₄⋊Q₈	#89137	1	1	yes	characteristic
81	#125	—	218	1	no	—
82	C₂²⋊Q₈	—	109	2	no	—
83	C₂²⋊Q₈	—	218	1	no	—
84	C₄×Q₈	—	109	2	no	—
85	C₄×Q₈	—	218	1	no	—
86	#117	—	218	1	no	—
87	C₂×C₄²	—	109	1	no	—
88	C₄⋊C₈	—	109	2	no	—
89	C₂²⋊C₈	—	218	1	no	—
90	#97	—	218	1	no	—
91	C₄×C₈	—	109	2	no	—
92	C₄×C₈	—	218	1	no	—
93	C₂×Q₈	—	218	1	no	—
94	C₂×Q₈	C₂×#89137	1	2	yes	—
95	C₂²×C₄	—	109	1	no	—
96	C₂²×C₄	—	109	1	no	—
97	C₂²×C₄	—	109	1	no	—
98	M₁₆	—	218	4	no	—
99	C₂×C₈	—	218	2	no	—
100	C₂×C₈	—	218	2	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	109	2	no	—
104	C₄⋊C₄	—	218	2	no	—
105	C₄⋊C₄	—	218	2	no	—
106	C₄⋊C₄	—	218	1	no	—
107	C₂²⋊C₄	—	218	2	no	—
108	C₄²	—	109	1	no	—
109	C₄²	—	109	1	no	—
110	C₄²	—	109	1	no	—
111	C₄²	—	218	2	no	—
112	C₄²	C₂×#89137	1	1	yes	characteristic
113	C₂³	—	109	1	no	—
114	Q₈	—	2	4	no	—
115	Q₈	—	436	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	109	1	no	—
118	C₂×C₄	—	109	1	no	—
119	C₂×C₄	—	109	1	no	—
120	C₂×C₄	—	218	2	no	—
121	C₂×C₄	—	218	2	no	—
122	C₂×C₄	—	218	2	no	—
123	C₂×C₄	—	218	2	no	—
124	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
126	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
127	C₈	—	436	2	no	—
128	C₈	—	218	4	no	—
129	C₂²	—	218	1	no	—
130	C₂²	—	218	1	no	—
131	C₂²	—	218	1	no	—
132	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	109	2	no	—
136	C₄	—	436	2	no	—
137	C₄	—	218	1	no	—
138	C₄	—	218	1	no	—
139	C₄	#16812764	1	2	yes	—
140	C₄	#16812786	1	2	yes	—
141	C₂	—	436	1	no	—
142	C₂	#20296000	1	1	yes	characteristic
143	C₂	#20296019	1	1	yes	characteristic
144	C₂	C₂×#16812764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296045

C₂

yes

maximal, characteristic

C₄×#13182039

C₂

yes

maximal, characteristic

Q₈⋊#11724786

C₂

yes

maximal, complemented

#20295581

C₂

yes

maximal, characteristic

C₂×#13182039

—

C₄⋊#11724786

C₂²

yes

—

—

—

—

C₂²

yes

—

#16812699

C₄

yes

—

—

C₄

yes

—

Q₈×Dic₁₀₉

C₂²

yes

—

#16812664

—

#16812657

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₄

yes

—

—

—

C₂×Dic₂₁₈

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

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

—

Dic₂₁₈

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#104

yes

—

C₄×C₁₀₉

#131

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#137

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1485

—

109

maximal, Sylow

C₁₀₉

#1485

yes

characteristic, complemented, Sylow

#533

—

109

—

#431

—

109

—

C₄×M₁₆

—

109

—

Q₈⋊C₈

—

109

—

C₂×M₁₆

—

218

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

#89137

yes

characteristic

#125

—

218

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

#117

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

218

—

#97

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×#89137

yes

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

218

—

100

C₂×C₈

—

218

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

109

—

104

C₄⋊C₄

—

218

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

—

218

—

107

C₂²⋊C₄

—

218

—

108

C₄²

—

109

—

109

C₄²

—

109

—

110

C₄²

—

109

—

111

C₄²

—

218

—

112

C₄²

C₂×#89137

yes

characteristic

113

C₂³

—

109

—

114

Q₈

—

115

Q₈

—

436

—

116

C₂×C₄

—

117

C₂×C₄

—

109

—

118

C₂×C₄

—

109

—

119

C₂×C₄

—

109

—

120

C₂×C₄

—

218

—

121

C₂×C₄

—

218

—

122

C₂×C₄

—

218

—

123

C₂×C₄

—

218

—

124

C₂×C₄

C₂²⋊#89137

yes

characteristic

125

C₂×C₄

C₂²⋊#89137

yes

characteristic

126

C₂×C₄

C₂²×#89137

yes

characteristic

127

C₈

—

436

—

128

C₈

—

218

—

129

C₂²

—

218

—

130

C₂²

—

218

—

131

C₂²

—

218

—

132

C₂²

C₂×C₂²⋊#89137

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

109

—

136

C₄

—

436

—

137

C₄

—

218

—

138

C₄

—

218

—

139

C₄

#16812764

yes

—

140

C₄

#16812786

yes

—

141

C₂

—

436

—

142

C₂

#20296000

yes

characteristic

143

C₂

#20296019

yes

characteristic

144

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	C₂²⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1485
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

C₂²⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₄⋊Q₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. Q₈⋊#11724786	C₂	semidirect
73. C₁₀₉	#1485	semidirect

H ≅ G/N

type

3. Q₈⋊#11724786

C₂

semidirect

73. C₁₀₉

#1485

semidirect

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

subgroup	index
#20296045	2
C₄×#13182039	2
Q₈⋊#11724786	2
#20295581	2
#1485	109

subgroup

index

109

Subgroups — #28378355

Conjugacy classes of subgroups

Complements

Maximal subgroups