Group #28375674 — subgroups

Subgroups — #28375674

Order 13952.

370 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 141 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	#20295755	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295378	C₂	1	3	yes	maximal, complemented
3	#20295308	C₂	1	3	yes	maximal, complemented
4	#20295231	C₂	1	3	yes	maximal
5	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13182017	C₂	1	3	yes	maximal
7	D₄×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182017	C₂²	1	3	yes	—
9	#16812699	C₂²	1	3	yes	—
10	#16812690	—	2	3	no	—
11	Q₈×Dic₁₀₉	—	2	1	no	—
12	C₄×Dic₂₁₈	C₂²	1	3	yes	—
13	#16812630	C₂²	1	3	yes	—
14	C₄⋊#11724784	C₂²	1	3	yes	—
15	#16812622	C₂²	1	6	yes	—
16	#16812622	C₂²	1	3	yes	—
17	#16812620	C₂²	1	1	yes	characteristic
18	C₄×#11724784	C₂²	1	3	yes	—
19	C₂²×Q₈×C₁₀₉	—	2	1	no	—
20	C₄⋊Q₈×C₁₀₉	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₁₀₉	—	2	3	no	—
22	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₀₉	C₂²	1	3	yes	—
24	C₂×Dic₂₁₈	C₂³	1	3	yes	—
25	#13182017	—	2	6	no	—
26	C₂×#11724784	C₂³	1	1	yes	characteristic
27	C₂×#11724784	C₂³	1	3	yes	—
28	C₄⋊Dic₁₀₉	—	2	3	no	—
29	C₄⋊Dic₁₀₉	—	2	3	no	—
30	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
31	#13182008	—	2	3	no	—
32	C₄×Dic₁₀₉	—	2	3	no	—
33	C₂×Q₈×C₁₀₉	—	2	3	no	—
34	C₂×Q₈×C₁₀₉	—	4	4	no	—
35	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₀₉	—	2	3	no	—
38	C₄⋊C₄×C₁₀₉	—	2	3	no	—
39	C₄⋊C₄×C₁₀₉	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₀₉	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₁₀₉	—	2	3	no	—
42	C₄²×C₁₀₉	C₂³	1	3	yes	—
43	C₂³×C₁₀₉	—	2	1	no	—
44	Q₈×C₁₀₉	—	4	6	no	—
45	Q₈×C₁₀₉	SD₁₆	1	4	yes	complemented
46	D₄×C₁₀₉	—	2	1	no	—
47	D₄×C₁₀₉	#41	1	2	yes	—
48	C₂×C₄×C₁₀₉	—	2	3	no	—
49	C₂×C₄×C₁₀₉	—	4	6	no	—
50	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₁₀₉	—	2	1	no	—
54	C₂×Dic₁₀₉	—	2	3	no	—
55	Dic₂₁₈	—	2	3	no	—
56	Dic₂₁₈	—	2	3	no	—
57	#11724784	—	2	3	no	—
58	#11724784	—	2	1	no	—
59	C₂²×C₁₀₉	—	2	2	no	—
60	C₂²×C₁₀₉	—	4	1	no	—
61	C₂²×C₁₀₉	—	4	1	no	—
62	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₀₉	—	2	3	no	—
64	C₄×C₁₀₉	—	4	3	no	—
65	C₄×C₁₀₉	C₂×SD₁₆	1	6	yes	—
66	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
67	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₁₀₉	—	4	3	no	—
69	Dic₁₀₉	—	4	1	no	—
70	C₂×C₁₀₉	—	4	2	no	—
71	C₂×C₁₀₉	#548	1	1	yes	characteristic
72	C₂×C₁₀₉	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₁₀₉	#578	1	1	yes	characteristic
74	#3359	—	109	1	no	maximal, Sylow
75	C₁₀₉	#3359	1	1	yes	characteristic, complemented, Sylow
76	#557	—	109	1	no	—
77	D₄×Q₈	D₁₀₉	1	1	yes	characteristic
78	#487	—	109	3	no	—
79	#475	—	109	3	no	—
80	#461	—	109	3	no	—
81	C₈×Q₈	—	109	1	no	—
82	C₄×SD₁₆	—	109	3	no	—
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	—	109	3	no	—
85	C₄⋊Q₈	—	109	3	no	—
86	C₄⋊Q₈	C₂×D₁₀₉	1	3	yes	—
87	#125	—	218	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	109	3	no	—
90	C₄×Q₈	—	218	1	no	—
91	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
92	C₄×D₄	C₂×D₁₀₉	1	3	yes	—
93	#106	—	109	3	no	—
94	C₄⋊C₈	—	109	3	no	—
95	#103	—	109	3	no	—
96	#103	—	109	6	no	—
97	#102	—	109	1	no	—
98	C₄×C₈	—	109	3	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	109	3	no	—
102	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
103	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	218	6	no	—
106	C₂×C₈	—	109	1	no	—
107	C₂×C₈	—	109	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	109	1	no	—
110	C₄⋊C₄	—	218	3	no	—
111	C₄⋊C₄	—	218	3	no	—
112	C₄⋊C₄	—	218	3	no	—
113	C₄⋊C₄	C₂²⋊D₁₀₉	1	3	yes	—
114	C₄⋊C₄	C₂²×D₁₀₉	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	218	3	no	—
117	C₄²	C₂²×D₁₀₉	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	4	6	no	—
120	Q₈	—	218	3	no	—
121	Q₈	—	218	3	no	—
122	Q₈	#13182017	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#13182030	1	2	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	218	1	no	—
128	C₂×C₄	—	218	3	no	—
129	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	3	yes	—
131	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
132	C₈	—	218	3	no	—
133	C₈	—	218	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	436	3	no	—
141	C₄	—	436	1	no	—
142	C₄	C₂×#13182017	1	6	yes	—
143	C₄	C₂×#13182030	1	1	yes	characteristic
144	C₄	#16812747	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#20295687	1	1	yes	characteristic
147	C₂	C₂²×#13182017	1	1	yes	characteristic
148	C₂	#20295841	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295755

C₂

yes

maximal, characteristic, complemented

#20295378

C₂

yes

maximal, complemented

#20295308

C₂

yes

maximal, complemented

#20295231

C₂

yes

maximal

Q₈×#11724784

C₂

yes

maximal, characteristic, complemented

C₄×#13182017

C₂

yes

maximal

D₄×Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182017

C₂²

yes

—

#16812699

C₂²

yes

—

#16812690

—

Q₈×Dic₁₀₉

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812630

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

#16812622

C₂²

yes

—

#16812622

C₂²

yes

—

#16812620

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

—

C₂²×Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×D₄×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

C₂³

yes

—

#13182017

—

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×D₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

SD₁₆

yes

complemented

D₄×C₁₀₉

—

D₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

C₂×#41

yes

characteristic

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#548

yes

characteristic

C₂×C₁₀₉

C₂²×SD₁₆

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#3359

yes

characteristic, complemented, Sylow

#557

—

109

—

D₄×Q₈

D₁₀₉

yes

characteristic

#487

—

109

—

#475

—

109

—

#461

—

109

—

C₈×Q₈

—

109

—

C₄×SD₁₆

—

109

—

C₂²×Q₈

—

C₂×SD₁₆

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#125

—

218

—

C₂²⋊Q₈

—

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

C₄×D₄

C₂×D₁₀₉

yes

—

#106

—

109

—

C₄⋊C₈

—

109

—

#103

—

109

—

#103

—

109

—

#102

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

109

—

102

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

103

C₂×D₄

C₂²×D₁₀₉

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

218

—

106

C₂×C₈

—

109

—

107

C₂×C₈

—

109

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

109

—

110

C₄⋊C₄

—

218

—

111

C₄⋊C₄

—

218

—

112

C₄⋊C₄

—

218

—

113

C₄⋊C₄

C₂²⋊D₁₀₉

yes

—

114

C₄⋊C₄

C₂²×D₁₀₉

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

218

—

117

C₄²

C₂²×D₁₀₉

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

218

—

121

Q₈

—

218

—

122

Q₈

#13182017

yes

complemented

123

D₄

—

124

D₄

#13182030

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

218

—

128

C₂×C₄

—

218

—

129

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

131

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

132

C₈

—

218

—

133

C₈

—

218

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

436

—

141

C₄

—

436

—

142

C₄

C₂×#13182017

yes

—

143

C₄

C₂×#13182030

yes

characteristic

144

C₄

#16812747

yes

characteristic

145

C₂

—

146

C₂

#20295687

yes

characteristic

147

C₂

C₂²×#13182017

yes

characteristic

148

C₂

#20295841

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	D₄×Q₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂×D₄×C₁₀₉	C₂³
hypercenter	D₄×Q₈	D₁₀₉
hypocenter	C₁₀₉	#3359
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

D₄×Q₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂×D₄×C₁₀₉

C₂³

hypercenter

D₄×Q₈

D₁₀₉

hypocenter

C₁₀₉

#3359

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #20295755	C₂	semidirect
2. #20295378	C₂	semidirect
3. #20295308	C₂	semidirect
5. Q₈×#11724784	C₂	semidirect
45. Q₈×C₁₀₉	SD₁₆	semidirect
75. C₁₀₉	#3359	semidirect
122. Q₈	#13182017	semidirect

H ≅ G/N

type

1. #20295755

C₂

semidirect

2. #20295378

C₂

semidirect

3. #20295308

C₂

semidirect

5. Q₈×#11724784

C₂

semidirect

45. Q₈×C₁₀₉

SD₁₆

semidirect

75. C₁₀₉

#3359

semidirect

122. Q₈

#13182017

semidirect

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

subgroup	index
#20295755	2
#20295378	2
#20295308	2
#20295231	2
Q₈×#11724784	2
C₄×#13182017	2
D₄×Q₈×C₁₀₉	2
#3359	109

subgroup

index

D₄×Q₈×C₁₀₉

109

Subgroups — #28375674

Conjugacy classes of subgroups

Complements

Maximal subgroups