Group #28361479 — subgroups

Subgroups — #28361479

Order 13952.

252 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20295083	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11724784	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	—	2	1	no	—
6	C₄⋊#11724784	C₂²	1	2	yes	—
7	C₂²⋊#11724784	—	2	1	no	—
8	#16812610	—	2	1	no	—
9	#16812610	—	2	1	no	—
10	C₄×#11724784	C₂²	1	2	yes	—
11	C₂²×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
12	C₄×Q₈×C₁₀₉	—	2	1	no	—
13	C₄×Q₈×C₁₀₉	—	2	1	no	—
14	C₄×Q₈×C₁₀₉	C₄	1	2	yes	—
15	C₄×Q₈×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₀₉	—	2	1	no	—
17	C₂×C₄⋊C₄×C₁₀₉	C₄	1	1	yes	characteristic
18	C₂×C₄²×C₁₀₉	—	2	1	no	—
19	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
20	#13182013	—	2	4	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×#11724784	—	2	2	no	—
23	C₂×Q₈×C₁₀₉	—	2	4	no	—
24	C₂×Q₈×C₁₀₉	—	2	1	no	—
25	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	—	2	1	no	—
27	C₂²×C₄×C₁₀₉	—	2	1	no	—
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₄⋊C₄×C₁₀₉	—	2	1	no	—
33	C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₄²×C₁₀₉	—	2	2	no	—
36	C₄²×C₁₀₉	—	2	2	no	—
37	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
39	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
40	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₁₀₉	—	2	4	no	—
43	Q₈×C₁₀₉	—	2	2	no	—
44	Q₈×C₁₀₉	M₁₆	1	4	yes	complemented
45	C₂×C₄×C₁₀₉	—	2	4	no	—
46	C₂×C₄×C₁₀₉	—	2	2	no	—
47	C₂×C₄×C₁₀₉	—	2	2	no	—
48	C₂×C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₄×C₁₀₉	—	2	2	no	—
50	C₂×C₄×C₁₀₉	—	2	2	no	—
51	C₂×C₄×C₁₀₉	—	4	2	no	—
52	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
58	#11724784	—	2	4	no	—
59	#11724784	—	4	2	no	—
60	C₂²×C₁₀₉	—	2	1	no	—
61	C₂²×C₁₀₉	—	2	1	no	—
62	C₂²×C₁₀₉	#104	1	2	yes	—
63	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₁₀₉	—	2	4	no	—
65	C₄×C₁₀₉	—	2	1	no	—
66	C₄×C₁₀₉	—	2	1	no	—
67	C₄×C₁₀₉	—	2	1	no	—
68	C₄×C₁₀₉	—	2	1	no	—
69	C₄×C₁₀₉	—	4	2	no	—
70	C₄×C₁₀₉	C₂×M₁₆	1	2	yes	—
71	C₄×C₁₀₉	#137	1	2	yes	—
72	C₂×C₁₀₉	—	2	2	no	—
73	C₂×C₁₀₉	#407	1	1	yes	characteristic
74	C₂×C₁₀₉	#419	1	1	yes	characteristic
75	C₂×C₁₀₉	C₂×#104	1	1	yes	characteristic
76	#1491	—	109	1	no	maximal, Sylow
77	C₁₀₉	#1491	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄×Q₈	D₁₀₉	1	1	yes	characteristic
79	#432	—	109	1	no	—
80	C₄×M₁₆	—	109	1	no	—
81	Q₈⋊C₈	—	109	4	no	—
82	C₂²×Q₈	Dic₁₀₉	1	1	yes	characteristic
83	C₂×M₁₆	—	218	1	no	—
84	C₄×Q₈	—	2	1	no	—
85	C₄×Q₈	—	2	1	no	—
86	C₄×Q₈	Dic₁₀₉	1	2	yes	—
87	C₄×Q₈	C₂×D₁₀₉	1	2	yes	—
88	C₂×C₄⋊C₄	—	2	1	no	—
89	C₂×C₄⋊C₄	Dic₁₀₉	1	1	yes	characteristic
90	C₂×C₄²	—	2	1	no	—
91	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
92	C₄⋊C₈	—	109	2	no	—
93	C₂²⋊C₈	—	218	1	no	—
94	#97	—	218	1	no	—
95	#97	—	218	1	no	—
96	C₄×C₈	—	109	2	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	C₂×Dic₁₀₉	1	2	yes	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
103	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
105	M₁₆	—	218	4	no	—
106	C₂×C₈	—	218	2	no	—
107	C₂×C₈	—	218	2	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂×Dic₁₀₉	1	2	yes	—
112	C₄²	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
115	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
116	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
117	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
118	C₂³	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	2	no	—
121	Q₈	#13182013	1	4	yes	complemented
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
131	C₂×C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
132	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
135	C₈	—	436	2	no	—
136	C₈	—	218	4	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	#16812626	1	2	yes	—
140	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	4	2	no	—
147	C₄	C₂×#13182013	1	2	yes	—
148	C₄	#16812730	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#20294990	1	1	yes	characteristic
151	C₂	#20295035	1	1	yes	characteristic
152	C₂	C₂×#16812626	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295083

C₂

yes

maximal, characteristic

C₄×#13182013

C₂

yes

maximal, characteristic

Q₈⋊#11724784

C₂

yes

maximal, complemented

C₂×C₄×Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

C₄⋊#11724784

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₄

yes

characteristic

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₄

yes

—

C₄×Q₈×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

characteristic

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

—

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

M₁₆

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×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×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#104

yes

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

C₄×C₁₀₉

#137

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1491

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₁₀₉

yes

characteristic

#432

—

109

—

C₄×M₁₆

—

109

—

Q₈⋊C₈

—

109

—

C₂²×Q₈

Dic₁₀₉

yes

characteristic

C₂×M₁₆

—

218

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

Dic₁₀₉

yes

—

C₄×Q₈

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₀₉

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

218

—

#97

—

218

—

#97

—

218

—

C₄×C₈

—

109

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₀₉

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

103

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

104

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

105

M₁₆

—

218

—

106

C₂×C₈

—

218

—

107

C₂×C₈

—

218

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×Dic₁₀₉

yes

characteristic

115

C₄²

C₂²⋊D₁₀₉

yes

characteristic

116

C₄²

C₂²⋊D₁₀₉

yes

characteristic

117

C₄²

C₂²×D₁₀₉

yes

characteristic

118

C₂³

C₂²⋊Dic₁₀₉

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

#13182013

yes

complemented

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²⋊Dic₁₀₉

yes

characteristic

130

C₂×C₄

C₂²⋊Dic₁₀₉

yes

characteristic

131

C₂×C₄

C₂²⋊Dic₁₀₉

yes

characteristic

132

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

135

C₈

—

436

—

136

C₈

—

218

—

137

C₂²

—

138

C₂²

—

139

C₂²

#16812626

yes

—

140

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂×#13182013

yes

—

148

C₄

#16812730

yes

—

149

C₂

—

150

C₂

#20294990

yes

characteristic

151

C₂

#20295035

yes

characteristic

152

C₂

C₂×#16812626

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	C₂×C₄×Q₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	C₂×C₄×Q₈	D₁₀₉
hypocenter	C₁₀₉	#1491
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

C₂×C₄×Q₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₉

C₂²⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

hypercenter

C₂×C₄×Q₈

D₁₀₉

hypocenter

C₁₀₉

#1491

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
3. Q₈⋊#11724784	C₂	semidirect
44. Q₈×C₁₀₉	M₁₆	semidirect
77. C₁₀₉	#1491	semidirect
121. Q₈	#13182013	semidirect

H ≅ G/N

type

3. Q₈⋊#11724784

C₂

semidirect

44. Q₈×C₁₀₉

M₁₆

semidirect

77. C₁₀₉

#1491

semidirect

121. Q₈

#13182013

semidirect

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

subgroup	index
#20295083	2
C₄×#13182013	2
Q₈⋊#11724784	2
C₂×C₄×Q₈×C₁₀₉	2
#1491	109

subgroup

index

C₂×C₄×Q₈×C₁₀₉

109

Subgroups — #28361479

Conjugacy classes of subgroups

Complements

Maximal subgroups