Group #28378466 — subgroups

Subgroups — #28378466

Order 13952.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20295970	C₂	1	1	yes	maximal, characteristic
3	#20295880	C₂	1	4	yes	maximal
4	#20295581	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#89137	—	2	1	no	—
6	#16812766	—	2	1	no	—
7	#16812756	C₂²	1	2	yes	—
8	C₄×#11724786	—	2	1	no	—
9	(C₂×#89137)⋊C₄	—	2	1	no	—
10	(C₂×#89137)⋊C₄	C₂²	1	2	yes	—
11	#16812699	C₄	1	1	yes	characteristic, complemented
12	#16812690	—	2	1	no	—
13	#16812679	—	2	1	no	—
14	#16812675	C₂²	1	2	yes	—
15	Q₈×Dic₁₀₉	C₄	1	2	yes	complemented
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, complemented
20	C₂²×#89137	—	2	2	no	—
21	C₂×#11724786	—	2	2	no	—
22	C₄⋊#89137	—	4	2	no	—
23	C₂×Dic₂₁₈	—	2	1	no	—
24	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
25	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₀₉	C₂³	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	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
35	C₄×Dic₁₀₉	D₄	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	C₂×D₄	1	1	yes	characteristic
41	C₂×#89137	—	4	2	no	—
42	C₂×#89137	—	4	2	no	—
43	C₂×#89137	—	4	2	no	—
44	Q₈×C₁₀₉	—	4	2	no	—
45	C₂×C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₁₀₉	—	2	2	no	—
50	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₄×D₁₀₉	—	4	1	no	—
54	C₄×D₁₀₉	—	4	1	no	—
55	C₄×D₁₀₉	SD₁₆	1	2	yes	—
56	C₄×D₁₀₉	Q₁₆	1	2	yes	—
57	Dic₂₁₈	—	4	2	no	—
58	#11724786	—	4	2	no	—
59	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×D₁₀₉	—	2	1	no	—
61	C₂×D₁₀₉	—	2	1	no	—
62	C₂×D₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×D₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
64	#89137	—	8	2	no	—
65	C₄×C₁₀₉	—	2	1	no	—
66	C₄×C₁₀₉	—	2	1	no	—
67	C₄×C₁₀₉	—	4	2	no	—
68	C₄×C₁₀₉	#103	1	2	yes	—
69	Dic₁₀₉	—	2	1	no	—
70	Dic₁₀₉	—	2	1	no	—
71	Dic₁₀₉	—	4	2	no	—
72	Dic₁₀₉	#103	1	2	yes	—
73	C₂×C₁₀₉	#410	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
75	C₂×C₁₀₉	#421	1	1	yes	characteristic
76	D₁₀₉	—	2	1	no	—
77	D₁₀₉	—	2	1	no	—
78	#1518	—	109	1	no	maximal, Sylow
79	C₁₀₉	#1518	1	1	yes	characteristic, complemented, Sylow
80	#533	—	109	1	no	—
81	#431	—	109	1	no	—
82	#382	—	109	1	no	—
83	#328	—	109	4	no	—
84	C₄⋊Q₈	—	109	1	no	—
85	C₄⋊Q₈	#89137	1	1	yes	characteristic, complemented
86	#125	—	218	1	no	—
87	C₂²⋊Q₈	—	109	2	no	—
88	C₂²⋊Q₈	—	218	1	no	—
89	C₄×Q₈	—	109	2	no	—
90	C₄×Q₈	—	218	1	no	—
91	#117	—	218	1	no	—
92	C₂×C₄⋊C₄	—	218	1	no	—
93	C₂×C₄²	—	109	1	no	—
94	C₄⋊C₈	—	218	1	no	—
95	C₂²⋊C₈	—	109	2	no	—
96	C₄×C₈	—	218	1	no	—
97	#95	—	109	2	no	—
98	#95	—	218	1	no	—
99	C₂×Q₈	—	218	1	no	—
100	C₂×Q₈	C₂×#89137	1	2	yes	—
101	C₂²×C₄	—	109	1	no	—
102	C₂²×C₄	—	109	1	no	—
103	C₂²×C₄	—	109	1	no	—
104	C₂²×C₄	—	218	2	no	—
105	C₂×C₈	—	218	2	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	109	2	no	—
109	C₄⋊C₄	—	436	2	no	—
110	C₄⋊C₄	—	218	2	no	—
111	C₄⋊C₄	—	218	2	no	—
112	C₄⋊C₄	—	218	1	no	—
113	C₂²⋊C₄	—	218	2	no	—
114	C₄²	—	109	1	no	—
115	C₄²	—	109	1	no	—
116	C₄²	—	109	1	no	—
117	C₄²	—	218	2	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	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	109	1	no	—
124	C₂×C₄	—	109	2	no	—
125	C₂×C₄	—	109	2	no	—
126	C₂×C₄	—	109	1	no	—
127	C₂×C₄	—	109	1	no	—
128	C₂×C₄	—	436	1	no	—
129	C₂×C₄	—	436	1	no	—
130	C₂×C₄	—	436	2	no	—
131	C₂×C₄	—	436	2	no	—
132	C₂×C₄	—	436	2	no	—
133	C₂×C₄	—	218	2	no	—
134	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
135	C₂×C₄	C₂²⋊#89137	1	1	yes	characteristic
136	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
137	C₈	—	436	2	no	—
138	C₂²	—	109	1	no	—
139	C₂²	—	109	1	no	—
140	C₂²	—	218	1	no	—
141	C₂²	—	218	1	no	—
142	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	2	no	—
146	C₄	—	872	2	no	—
147	C₄	—	109	2	no	—
148	C₄	—	436	2	no	—
149	C₄	—	218	1	no	—
150	C₄	—	218	1	no	—
151	C₄	#16812762	1	2	yes	—
152	C₂	—	218	1	no	—
153	C₂	—	218	1	no	—
154	C₂	#20296006	1	1	yes	characteristic
155	C₂	C₂×#16812762	1	1	yes	characteristic
156	C₂	#20296025	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296044

C₂

yes

maximal, characteristic

#20295970

C₂

yes

maximal, characteristic

#20295880

C₂

yes

maximal

#20295581

C₂

yes

maximal, characteristic

C₂×C₄⋊#89137

—

#16812766

—

#16812756

C₂²

yes

—

C₄×#11724786

—

(C₂×#89137)⋊C₄

—

(C₂×#89137)⋊C₄

C₂²

yes

—

#16812699

C₄

yes

characteristic, complemented

#16812690

—

#16812679

—

#16812675

C₂²

yes

—

Q₈×Dic₁₀₉

C₄

yes

complemented

#16812664

—

#16812657

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

C₂×#11724786

—

C₄⋊#89137

—

C₂×Dic₂₁₈

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×D₄

yes

—

—

—

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

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

SD₁₆

yes

—

C₄×D₁₀₉

Q₁₆

yes

—

Dic₂₁₈

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×D₁₀₉

C₂×Q₁₆

yes

characteristic

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#103

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#103

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

D₁₀₉

—

#1518

—

109

maximal, Sylow

C₁₀₉

#1518

yes

characteristic, complemented, Sylow

#533

—

109

—

#431

—

109

—

#382

—

109

—

#328

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

#89137

yes

characteristic, complemented

#125

—

218

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

#117

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

109

—

C₄×C₈

—

218

—

#95

—

109

—

#95

—

218

—

C₂×Q₈

—

218

—

100

C₂×Q₈

C₂×#89137

yes

—

101

C₂²×C₄

—

109

—

102

C₂²×C₄

—

109

—

103

C₂²×C₄

—

109

—

104

C₂²×C₄

—

218

—

105

C₂×C₈

—

218

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

—

109

C₄⋊C₄

—

436

—

110

C₄⋊C₄

—

218

—

111

C₄⋊C₄

—

218

—

112

C₄⋊C₄

—

218

—

113

C₂²⋊C₄

—

218

—

114

C₄²

—

109

—

115

C₄²

—

109

—

116

C₄²

—

109

—

117

C₄²

—

218

—

118

C₄²

C₂×#89137

yes

characteristic

119

C₂³

—

109

—

120

Q₈

—

121

Q₈

—

436

—

122

C₂×C₄

—

123

C₂×C₄

—

109

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

109

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

436

—

129

C₂×C₄

—

436

—

130

C₂×C₄

—

436

—

131

C₂×C₄

—

436

—

132

C₂×C₄

—

436

—

133

C₂×C₄

—

218

—

134

C₂×C₄

C₂²⋊#89137

yes

characteristic

135

C₂×C₄

C₂²⋊#89137

yes

characteristic

136

C₂×C₄

C₂²×#89137

yes

characteristic

137

C₈

—

436

—

138

C₂²

—

109

—

139

C₂²

—

109

—

140

C₂²

—

218

—

141

C₂²

—

218

—

142

C₂²

C₂×C₂²⋊#89137

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

872

—

147

C₄

—

109

—

148

C₄

—

436

—

149

C₄

—

218

—

150

C₄

—

218

—

151

C₄

#16812762

yes

—

152

C₂

—

218

—

153

C₂

—

218

—

154

C₂

#20296006

yes

characteristic

155

C₂

C₂×#16812762

yes

characteristic

156

C₂

#20296025

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	C₂×D₄
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1518
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

C₂×D₄

cosocle

C₂×C₄×D₁₀₉

C₂³

C₄⋊Q₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #16812699	C₄	semidirect
15. Q₈×Dic₁₀₉	C₄	semidirect
19. C₄⋊Q₈×C₁₀₉	C₄	semidirect
79. C₁₀₉	#1518	semidirect
85. C₄⋊Q₈	#89137	semidirect

H ≅ G/N

type

11. #16812699

C₄

semidirect

15. Q₈×Dic₁₀₉

C₄

semidirect

19. C₄⋊Q₈×C₁₀₉

C₄

semidirect

79. C₁₀₉

#1518

semidirect

85. C₄⋊Q₈

#89137

semidirect

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

subgroup	index
#20296044	2
#20295970	2
#20295880	2
#20295581	2
#1518	109

subgroup

index

109

Subgroups — #28378466

Conjugacy classes of subgroups

Complements

Maximal subgroups