Group #28377866 — subgroups

Subgroups — #28377866

Order 13952.

208 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₄×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#89137	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×D₁₀₉⋊C₈	—	2	1	no	—
5	C₂×C₄⋊#89137	C₄	1	2	yes	—
6	C₂×C₄×#89137	—	2	1	no	—
7	C₄×#11724786	—	2	1	no	—
8	C₄×#11724786	C₄	1	2	yes	complemented
9	(C₂×#89137)⋊C₄	—	2	1	no	—
10	C₂×C₈×D₁₀₉	—	2	1	no	—
11	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊#11724784	C₄	1	1	yes	characteristic, complemented
13	#16812628	—	2	1	no	—
14	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×#89137	—	2	2	no	—
16	D₁₀₉⋊C₈	—	2	4	no	—
17	C₂×#11724786	—	2	2	no	—
18	C₄⋊#89137	—	2	2	no	—
19	C₄⋊#89137	C₈	1	4	yes	complemented
20	C₄×#89137	—	4	1	no	—
21	C₄×#89137	—	4	1	no	—
22	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
25	C₂×#11724784	—	2	1	no	—
26	C₈×D₁₀₉	—	4	1	no	—
27	C₈×D₁₀₉	—	4	1	no	—
28	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
30	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	—	2	1	no	—
32	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×#89137	—	2	4	no	—
35	C₂×#89137	—	4	2	no	—
36	C₂×#89137	—	4	2	no	—
37	C₂×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
43	C₄×D₁₀₉	—	2	2	no	—
44	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
45	C₄×D₁₀₉	M₁₆	1	2	yes	—
46	C₄×D₁₀₉	D₈	1	1	yes	characteristic
47	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
48	C₄×D₁₀₉	SD₁₆	1	1	yes	characteristic
49	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
50	#11724786	—	2	4	no	—
51	C₈×C₁₀₉	—	4	1	no	—
52	#11724784	—	4	1	no	—
53	C₂²×C₁₀₉	#95	1	1	yes	characteristic
54	C₂×D₁₀₉	C₂²⋊C₈	1	2	yes	—
55	C₂×D₁₀₉	#102	1	1	yes	characteristic
56	C₂×D₁₀₉	#103	1	1	yes	characteristic
57	C₂×D₁₀₉	#104	1	2	yes	—
58	#89137	—	4	4	no	—
59	C₄×C₁₀₉	—	2	1	no	—
60	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
61	C₄×C₁₀₉	#97	1	1	yes	characteristic
62	C₄×C₁₀₉	#102	1	1	yes	characteristic
63	C₄×C₁₀₉	#103	1	1	yes	characteristic
64	Dic₁₀₉	—	2	1	no	—
65	Dic₁₀₉	C₄⋊C₈	1	2	yes	—
66	Dic₁₀₉	#106	1	1	yes	characteristic
67	Dic₁₀₉	#107	1	1	yes	characteristic
68	C₂×C₁₀₉	#336	1	1	yes	characteristic
69	C₂×C₁₀₉	#339	1	1	yes	characteristic
70	C₂×C₁₀₉	#340	1	1	yes	characteristic
71	D₁₀₉	D₄⋊C₈	1	2	yes	—
72	D₁₀₉	Q₈⋊C₈	1	2	yes	—
73	#1279	—	109	1	no	maximal, Sylow
74	C₁₀₉	#1279	1	1	yes	characteristic, complemented, Sylow
75	C₂×C₄⋊C₈	—	109	1	no	—
76	C₂×C₄×C₈	—	109	1	no	—
77	C₄×C₄⋊C₄	—	109	1	no	—
78	C₂²×C₈	—	218	1	no	—
79	C₂²×C₈	—	218	1	no	—
80	C₂×C₄⋊C₄	—	109	2	no	—
81	C₂×C₄²	—	109	1	no	—
82	C₂×C₄²	—	218	1	no	—
83	C₄⋊C₈	—	109	1	no	—
84	C₄⋊C₈	—	218	1	no	—
85	C₄⋊C₈	#89137	1	1	yes	characteristic, complemented
86	C₄×C₈	—	109	2	no	—
87	C₄×C₈	—	218	1	no	—
88	#95	—	218	1	no	—
89	C₂²×C₄	—	109	1	no	—
90	C₂²×C₄	—	109	1	no	—
91	C₂²×C₄	—	109	1	no	—
92	C₂²×C₄	—	218	2	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	436	1	no	—
95	C₂×C₈	—	436	1	no	—
96	C₂×C₈	—	218	1	no	—
97	C₂×C₈	—	218	2	no	—
98	C₂×C₈	—	218	4	no	—
99	C₄⋊C₄	—	109	4	no	—
100	C₄⋊C₄	—	218	2	no	—
101	C₄²	—	109	1	no	—
102	C₄²	—	109	1	no	—
103	C₄²	—	109	1	no	—
104	C₄²	—	436	1	no	—
105	C₄²	—	436	1	no	—
106	C₄²	C₂×#89137	1	1	yes	characteristic
107	C₂³	—	109	1	no	—
108	C₂×C₄	—	109	1	no	—
109	C₂×C₄	—	109	1	no	—
110	C₂×C₄	—	109	2	no	—
111	C₂×C₄	—	109	2	no	—
112	C₂×C₄	—	109	1	no	—
113	C₂×C₄	—	109	1	no	—
114	C₂×C₄	—	109	1	no	—
115	C₂×C₄	—	109	1	no	—
116	C₂×C₄	—	109	1	no	—
117	C₂×C₄	—	436	2	no	—
118	C₂×C₄	—	436	2	no	—
119	C₂×C₄	—	218	2	no	—
120	C₂×C₄	—	218	4	no	—
121	C₂×C₄	C₄×#89137	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
123	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
124	C₈	—	4	1	no	—
125	C₈	—	436	1	no	—
126	C₈	—	218	4	no	—
127	C₂²	—	109	2	no	—
128	C₂²	—	109	2	no	—
129	C₂²	—	109	1	no	—
130	C₂²	—	109	1	no	—
131	C₂²	(C₂×#89137)⋊C₄	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	109	2	no	—
134	C₄	—	109	1	no	—
135	C₄	—	109	1	no	—
136	C₄	—	436	4	no	—
137	C₄	—	218	1	no	—
138	C₄	C₈×#89137	1	1	yes	characteristic
139	C₄	#16812753	1	1	yes	characteristic
140	C₄	#16812761	1	1	yes	characteristic
141	C₄	#16812762	1	1	yes	characteristic
142	C₂	—	109	2	no	—
143	C₂	—	109	2	no	—
144	C₂	(C₂×#89137)⋊C₈	1	1	yes	characteristic
145	C₂	#20295898	1	1	yes	characteristic
146	C₂	#20295899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊#89137

C₂

yes

maximal, characteristic

C₄⋊C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

—

C₂×C₄⋊#89137

C₄

yes

—

C₂×C₄×#89137

—

C₄×#11724786

—

C₄×#11724786

C₄

yes

complemented

(C₂×#89137)⋊C₄

—

C₂×C₈×D₁₀₉

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic, complemented

#16812628

—

C₄⋊C₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

D₁₀₉⋊C₈

—

C₂×#11724786

—

C₄⋊#89137

—

C₄⋊#89137

C₈

yes

complemented

C₄×#89137

—

C₄×#89137

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×#11724784

—

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

—

—

—

C₂×C₄×C₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₈

yes

—

C₄×D₁₀₉

M₁₆

yes

—

C₄×D₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#95

yes

characteristic

C₂×D₁₀₉

C₂²⋊C₈

yes

—

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×C₈

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

Dic₁₀₉

C₄⋊C₈

yes

—

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

D₄⋊C₈

yes

—

D₁₀₉

Q₈⋊C₈

yes

—

#1279

—

109

maximal, Sylow

C₁₀₉

#1279

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄×C₈

—

109

—

C₄×C₄⋊C₄

—

109

—

C₂²×C₈

—

218

—

C₂²×C₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

218

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

#89137

yes

characteristic, complemented

C₄×C₈

—

109

—

C₄×C₈

—

218

—

#95

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂×C₈

—

C₂×C₈

—

436

—

C₂×C₈

—

436

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

109

—

100

C₄⋊C₄

—

218

—

101

C₄²

—

109

—

102

C₄²

—

109

—

103

C₄²

—

109

—

104

C₄²

—

436

—

105

C₄²

—

436

—

106

C₄²

C₂×#89137

yes

characteristic

107

C₂³

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

109

—

111

C₂×C₄

—

109

—

112

C₂×C₄

—

109

—

113

C₂×C₄

—

109

—

114

C₂×C₄

—

109

—

115

C₂×C₄

—

109

—

116

C₂×C₄

—

109

—

117

C₂×C₄

—

436

—

118

C₂×C₄

—

436

—

119

C₂×C₄

—

218

—

120

C₂×C₄

—

218

—

121

C₂×C₄

C₄×#89137

yes

characteristic

122

C₂×C₄

C₂²⋊#89137

yes

characteristic

123

C₂×C₄

C₄⋊#89137

yes

characteristic

124

C₈

—

125

C₈

—

436

—

126

C₈

—

218

—

127

C₂²

—

109

—

128

C₂²

—

109

—

129

C₂²

—

109

—

130

C₂²

—

109

—

131

C₂²

(C₂×#89137)⋊C₄

yes

characteristic

132

C₄

—

133

C₄

—

109

—

134

C₄

—

109

—

135

C₄

—

109

—

136

C₄

—

436

—

137

C₄

—

218

—

138

C₄

C₈×#89137

yes

characteristic

139

C₄

#16812753

yes

characteristic

140

C₄

#16812761

yes

characteristic

141

C₄

#16812762

yes

characteristic

142

C₂

—

109

—

143

C₂

—

109

—

144

C₂

(C₂×#89137)⋊C₈

yes

characteristic

145

C₂

#20295898

yes

characteristic

146

C₂

#20295899

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#89137
commutator	C₄×C₁₀₉	C₄×C₈
Frattini	C₄²	C₂×#89137
Fitting	C₄⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#95
prime radical	C₂²×D₁₀₉	C₂²⋊C₄
cosocle	C₄²×D₁₀₉	C₂²
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#1279
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊#89137

commutator

C₄×C₁₀₉

C₄×C₈

Frattini

C₄²

C₂×#89137

Fitting

C₄⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#95

prime radical

C₂²×D₁₀₉

C₂²⋊C₄

cosocle

C₄²×D₁₀₉

C₂²

C₄⋊C₈

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
8. C₄×#11724786	C₄	semidirect
12. C₄⋊#11724784	C₄	semidirect
14. C₄⋊C₈×C₁₀₉	C₄	semidirect
19. C₄⋊#89137	C₈	semidirect
74. C₁₀₉	#1279	semidirect
85. C₄⋊C₈	#89137	semidirect

H ≅ G/N

type

8. C₄×#11724786

C₄

semidirect

12. C₄⋊#11724784

C₄

semidirect

14. C₄⋊C₈×C₁₀₉

C₄

semidirect

19. C₄⋊#89137

C₈

semidirect

74. C₁₀₉

#1279

semidirect

85. C₄⋊C₈

#89137

semidirect

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

subgroup	index
C₄×D₁₀₉⋊C₈	2
C₄×C₄⋊#89137	2
C₄⋊C₈×D₁₀₉	2
#1279	109

subgroup

index

C₄×D₁₀₉⋊C₈

C₄×C₄⋊#89137

C₄⋊C₈×D₁₀₉

#1279

109

Subgroups — #28377866

Conjugacy classes of subgroups

Complements

Maximal subgroups