Group #28378562 — subgroups

Subgroups — #28378562

Order 13952.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#20296049	C₂	1	1	yes	maximal, characteristic
2	#20296049	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	#16812766	C₂²	1	2	yes	—
6	#16812766	C₂²	1	2	yes	—
7	#16812756	—	2	1	no	—
8	#16812756	—	2	1	no	—
9	#16812754	—	2	1	no	—
10	#16812754	—	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	C₂×#11724786	—	2	2	no	—
21	C₂×#11724786	—	2	2	no	—
22	C₂×Dic₂₁₈	—	2	1	no	—
23	C₂×Dic₂₁₈	—	2	2	no	—
24	Q₈×D₁₀₉	—	4	4	no	—
25	C₂×C₄×D₁₀₉	—	2	2	no	—
26	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
28	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
31	#13182008	—	2	2	no	—
32	#13182008	—	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	—	2	1	no	—
37	C₄⋊C₄×C₁₀₉	—	2	1	no	—
38	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
39	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₁₀₉	—	4	2	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	C₄×D₁₀₉	—	4	4	no	—
54	Dic₂₁₈	—	4	2	no	—
55	Dic₂₁₈	—	4	4	no	—
56	#11724786	—	4	2	no	—
57	#11724786	—	4	2	no	—
58	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₁₀₉	—	2	1	no	—
60	C₂×D₁₀₉	—	2	1	no	—
61	C₂×D₁₀₉	#101	1	2	yes	—
62	C₄×C₁₀₉	—	2	1	no	—
63	C₄×C₁₀₉	—	2	1	no	—
64	C₄×C₁₀₉	—	2	1	no	—
65	C₄×C₁₀₉	—	4	2	no	—
66	Dic₁₀₉	—	2	1	no	—
67	Dic₁₀₉	—	4	2	no	—
68	Dic₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₁₀₉	#137	1	2	yes	—
70	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
71	C₂×C₁₀₉	#418	1	1	yes	characteristic
72	C₂×C₁₀₉	#419	1	1	yes	characteristic
73	D₁₀₉	—	2	2	no	—
74	#1556	—	109	1	no	maximal, Sylow
75	C₁₀₉	#1556	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊Q₈	—	109	1	no	—
77	#432	—	109	1	no	—
78	#432	—	109	1	no	—
79	#332	—	109	4	no	—
80	C₂²×Q₈	—	218	1	no	—
81	C₄⋊Q₈	—	109	2	no	—
82	C₄⋊Q₈	—	109	1	no	—
83	C₄⋊Q₈	—	218	1	no	—
84	C₄⋊Q₈	—	218	1	no	—
85	C₄⋊Q₈	#89137	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	109	2	no	—
87	C₂×C₄⋊C₄	—	218	1	no	—
88	C₂×C₄²	—	109	1	no	—
89	C₄⋊C₈	—	109	2	no	—
90	C₄⋊C₈	—	109	2	no	—
91	C₂²⋊C₈	—	218	1	no	—
92	C₂²⋊C₈	—	218	1	no	—
93	#97	—	218	1	no	—
94	#97	—	218	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	436	4	no	—
97	C₂×Q₈	—	218	2	no	—
98	C₂×Q₈	—	218	1	no	—
99	C₂²×C₄	—	109	1	no	—
100	C₂²×C₄	—	109	1	no	—
101	C₂²×C₄	—	109	1	no	—
102	C₂²×C₄	—	218	2	no	—
103	C₂×C₈	—	218	2	no	—
104	C₂×C₈	—	218	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	109	2	no	—
107	C₄⋊C₄	—	218	2	no	—
108	C₄⋊C₄	—	218	2	no	—
109	C₄⋊C₄	—	218	1	no	—
110	C₄⋊C₄	C₂×#89137	1	2	yes	—
111	C₄²	—	109	1	no	—
112	C₄²	—	109	1	no	—
113	C₄²	—	109	1	no	—
114	C₄²	C₂×#89137	1	1	yes	characteristic
115	C₂³	—	109	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	436	2	no	—
118	Q₈	—	436	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	109	1	no	—
121	C₂×C₄	—	109	1	no	—
122	C₂×C₄	—	109	1	no	—
123	C₂×C₄	—	436	4	no	—
124	C₂×C₄	—	218	2	no	—
125	C₂×C₄	—	218	2	no	—
126	C₂×C₄	—	218	2	no	—
127	C₂×C₄	—	218	2	no	—
128	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
130	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
131	C₈	—	436	2	no	—
132	C₈	—	436	2	no	—
133	C₂²	—	109	2	no	—
134	C₂²	—	218	1	no	—
135	C₂²	—	218	1	no	—
136	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	—	109	2	no	—
142	C₄	—	109	2	no	—
143	C₄	—	436	2	no	—
144	C₄	—	218	1	no	—
145	C₂	—	218	2	no	—
146	C₂	D₁₀₉⋊#101	1	1	yes	characteristic
147	C₂	#20296018	1	1	yes	characteristic
148	C₂	#20296019	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296049

C₂

yes

maximal, characteristic

#20296049

C₂

yes

maximal, characteristic

#20295885

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₀₉

C₂

yes

maximal, characteristic

#16812766

C₂²

yes

—

#16812766

C₂²

yes

—

—

—

—

—

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

characteristic

C₂×#11724786

—

C₂×#11724786

—

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

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#101

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

AGL(1,ℤ/8)

yes

—

Dic₁₀₉

#137

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#1556

—

109

maximal, Sylow

C₁₀₉

#1556

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

#432

—

109

—

#432

—

109

—

#332

—

109

—

C₂²×Q₈

—

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

—

C₂²⋊C₈

—

218

—

C₂²⋊C₈

—

218

—

#97

—

218

—

#97

—

218

—

C₂×Q₈

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

—

109

—

102

C₂²×C₄

—

218

—

103

C₂×C₈

—

218

—

104

C₂×C₈

—

218

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

109

—

107

C₄⋊C₄

—

218

—

108

C₄⋊C₄

—

218

—

109

C₄⋊C₄

—

218

—

110

C₄⋊C₄

C₂×#89137

yes

—

111

C₄²

—

109

—

112

C₄²

—

109

—

113

C₄²

—

109

—

114

C₄²

C₂×#89137

yes

characteristic

115

C₂³

—

109

—

116

Q₈

—

117

Q₈

—

436

—

118

Q₈

—

436

—

119

C₂×C₄

—

120

C₂×C₄

—

109

—

121

C₂×C₄

—

109

—

122

C₂×C₄

—

109

—

123

C₂×C₄

—

436

—

124

C₂×C₄

—

218

—

125

C₂×C₄

—

218

—

126

C₂×C₄

—

218

—

127

C₂×C₄

—

218

—

128

C₂×C₄

C₂²⋊#89137

yes

characteristic

129

C₂×C₄

C₂²⋊#89137

yes

characteristic

130

C₂×C₄

C₂²×#89137

yes

characteristic

131

C₈

—

436

—

132

C₈

—

436

—

133

C₂²

—

109

—

134

C₂²

—

218

—

135

C₂²

—

218

—

136

C₂²

C₂×C₂²⋊#89137

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

109

—

142

C₄

—

109

—

143

C₄

—

436

—

144

C₄

—

218

—

145

C₂

—

218

—

146

C₂

D₁₀₉⋊#101

yes

characteristic

147

C₂

#20296018

yes

characteristic

148

C₂

#20296019

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	#1556
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
75. C₁₀₉	#1556	semidirect

H ≅ G/N

type

3. #20295885

C₂

semidirect

75. C₁₀₉

#1556

semidirect

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

subgroup	index
#20296049	2
#20295885	2
C₄⋊Q₈×D₁₀₉	2
#1556	109

subgroup

index

#20296049

#20295885

C₄⋊Q₈×D₁₀₉

#1556

109

Subgroups — #28378562

Conjugacy classes of subgroups

Complements

Maximal subgroups