Group #28378791 — subgroups

Subgroups — #28378791

Order 13952.

284 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 86 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	C₂×#16812787	C₂	1	1	yes	maximal, characteristic
2	#20296007	C₂	1	1	yes	maximal, characteristic
3	#20295992	C₂	1	2	yes	maximal, complemented
4	#20295909	C₂	1	2	yes	maximal, complemented
5	#20295009	C₂	1	1	yes	maximal, characteristic
6	#16812787	—	2	2	no	—
7	#16812787	—	2	2	no	—
8	C₂×#13182039	C₄	1	2	yes	complemented
9	C₂×#13182039	C₂²	1	1	yes	characteristic
10	C₂×D₁₀₉⋊C₈	C₄	1	2	yes	complemented
11	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
12	#16812778	C₂²	1	2	yes	—
13	#16812758	C₄	1	4	yes	—
14	#16812753	—	2	2	no	—
15	C₈×#89137	—	2	2	no	—
16	C₂×#13182030	C₂²	1	1	yes	characteristic
17	M₁₆×D₁₀₉	C₂²	1	2	yes	—
18	#16812618	C₄	1	1	yes	characteristic, complemented
19	#16812615	C₄	1	2	yes	complemented
20	#101×C₁₀₉	C₄	1	1	yes	characteristic, complemented
21	#13182039	—	2	2	no	—
22	#13182039	—	2	4	no	—
23	D₁₀₉⋊C₈	—	2	2	no	—
24	D₁₀₉⋊C₈	—	2	2	no	—
25	D₁₀₉⋊C₈	D₄	1	2	yes	—
26	D₁₀₉⋊C₈	D₄	1	2	yes	—
27	C₂×#11724786	C₂×C₄	1	2	yes	—
28	C₂×#11724786	C₂×C₄	1	1	yes	characteristic
29	C₂×#11724786	C₂×C₄	1	1	yes	characteristic
30	C₄⋊#89137	—	2	2	no	—
31	C₂²⋊#89137	C₂×C₄	1	4	yes	—
32	C₄×#89137	—	2	2	no	—
33	#13182030	—	2	2	no	—
34	#13182030	—	2	2	no	—
35	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
36	C₂×D₄₃₆	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
38	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
39	#13182013	C₂×C₄	1	2	yes	—
40	#13182012	—	2	2	no	—
41	C₈×D₁₀₉	—	2	2	no	—
42	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
44	C₂²×D₁₀₉	C₄²	1	2	yes	—
45	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×#89137	—	2	4	no	—
47	Q₈×C₁₀₉	—	2	2	no	—
48	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	D₄₃₆	—	2	4	no	—
52	D₄₃₆	—	2	2	no	—
53	C₄×D₁₀₉	—	2	2	no	—
54	C₄×D₁₀₉	—	2	2	no	—
55	C₄×D₁₀₉	C₂×D₄	1	2	yes	—
56	C₄×D₁₀₉	#41	1	2	yes	—
57	#11724786	—	2	2	no	—
58	#11724786	C₂²⋊C₄	1	1	yes	characteristic
59	#11724786	C₂²⋊C₄	1	1	yes	characteristic
60	#11724786	C₂²⋊C₄	1	2	yes	—
61	C₈×C₁₀₉	—	2	2	no	—
62	#11724784	—	2	2	no	—
63	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
64	C₂×D₁₀₉	—	2	2	no	—
65	C₂×D₁₀₉	—	4	1	no	—
66	C₂×D₁₀₉	—	4	2	no	—
67	C₂×D₁₀₉	C₄×D₄	1	2	yes	—
68	#89137	—	4	4	no	—
69	C₄×C₁₀₉	—	2	2	no	—
70	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
71	Dic₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	Dic₁₀₉	#117	1	1	yes	characteristic
73	C₂×C₁₀₉	—	2	1	no	—
74	C₂×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
75	D₁₀₉	—	2	2	no	—
76	D₁₀₉	—	4	2	no	—
77	#1757	—	109	1	no	maximal, Sylow
78	C₁₀₉	#1757	1	1	yes	characteristic, complemented, Sylow
79	C₂×#131	—	109	1	no	—
80	#413	—	109	1	no	—
81	#410	—	109	1	no	—
82	#405	—	109	2	no	—
83	#344	—	109	2	no	—
84	C₂×#41	—	109	1	no	—
85	#131	—	218	2	no	—
86	#131	—	218	2	no	—
87	C₂×M₁₆	—	109	1	no	—
88	C₂×M₁₆	—	109	2	no	—
89	C₂×M₁₆	—	109	2	no	—
90	C₂²×C₈	—	109	1	no	—
91	C₂²×C₈	—	109	2	no	—
92	#117	—	109	2	no	—
93	#101	—	109	1	no	—
94	#101	#89137	1	1	yes	characteristic, complemented
95	#100	—	109	2	no	—
96	#99	—	109	4	no	—
97	#97	—	218	2	no	—
98	C₄×C₈	—	218	2	no	—
99	#41	—	218	2	no	—
100	#41	—	218	2	no	—
101	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
102	C₂×D₄	—	109	2	no	—
103	C₂×D₄	—	109	1	no	—
104	C₂²×C₄	—	109	1	no	—
105	C₂²×C₄	—	109	2	no	—
106	M₁₆	—	109	2	no	—
107	M₁₆	—	218	2	no	—
108	M₁₆	—	218	4	no	—
109	M₁₆	—	218	2	no	—
110	M₁₆	C₂×#89137	1	2	yes	—
111	C₂×C₈	—	109	1	no	—
112	C₂×C₈	—	109	1	no	—
113	C₂×C₈	—	109	2	no	—
114	C₂×C₈	—	109	2	no	—
115	C₂×C₈	—	109	2	no	—
116	C₂×C₈	—	218	2	no	—
117	C₂×C₈	—	218	2	no	—
118	C₂×C₈	—	218	2	no	—
119	C₄⋊C₄	—	218	2	no	—
120	C₂²⋊C₄	—	109	4	no	—
121	C₄²	—	218	2	no	—
122	C₂³	—	109	1	no	—
123	C₂³	—	109	2	no	—
124	Q₈	—	2	2	no	—
125	D₄	—	218	4	no	—
126	D₄	—	218	2	no	—
127	C₂×C₄	—	109	1	no	—
128	C₂×C₄	—	109	2	no	—
129	C₂×C₄	—	109	2	no	—
130	C₂×C₄	—	218	2	no	—
131	C₂×C₄	—	218	2	no	—
132	C₂×C₄	—	218	4	no	—
133	C₂×C₄	C₄×#89137	1	2	yes	—
134	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	109	1	no	—
137	C₈	—	109	1	no	—
138	C₈	—	109	2	no	—
139	C₈	—	218	2	no	—
140	C₈	—	218	2	no	—
141	C₂²	—	109	2	no	—
142	C₂²	—	436	2	no	—
143	C₂²	—	436	1	no	—
144	C₂²	—	218	2	no	—
145	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	109	1	no	—
148	C₄	—	109	1	no	—
149	C₄	—	436	4	no	—
150	C₄	D₄×#89137	1	2	yes	—
151	C₂	—	2	1	no	—
152	C₂	—	436	2	no	—
153	C₂	—	218	2	no	—
154	C₂	C₂²⋊C₄×#89137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812787

C₂

yes

maximal, characteristic

#20296007

C₂

yes

maximal, characteristic

#20295992

C₂

yes

maximal, complemented

#20295909

C₂

yes

maximal, complemented

#20295009

C₂

yes

maximal, characteristic

#16812787

—

#16812787

—

C₂×#13182039

C₄

yes

complemented

C₂×#13182039

C₂²

yes

characteristic

C₂×D₁₀₉⋊C₈

C₄

yes

complemented

C₂×D₁₀₉⋊C₈

C₂²

yes

characteristic

#16812778

C₂²

yes

—

#16812758

C₄

yes

—

#16812753

—

C₈×#89137

—

C₂×#13182030

C₂²

yes

characteristic

M₁₆×D₁₀₉

C₂²

yes

—

#16812618

C₄

yes

characteristic, complemented

#16812615

C₄

yes

complemented

#101×C₁₀₉

C₄

yes

characteristic, complemented

#13182039

—

#13182039

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

D₄

yes

—

D₁₀₉⋊C₈

D₄

yes

—

C₂×#11724786

C₂×C₄

yes

—

C₂×#11724786

C₂×C₄

yes

characteristic

C₂×#11724786

C₂×C₄

yes

characteristic

C₄⋊#89137

—

C₂²⋊#89137

C₂×C₄

yes

—

C₄×#89137

—

#13182030

—

#13182030

—

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×D₄₃₆

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

#13182013

C₂×C₄

yes

—

#13182012

—

C₈×D₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

M₁₆×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

C₄²

yes

—

C₂²×D₁₀₉

C₂²×C₄

yes

characteristic

C₂×#89137

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×D₄

yes

—

C₄×D₁₀₉

#41

yes

—

#11724786

—

#11724786

C₂²⋊C₄

yes

characteristic

#11724786

C₂²⋊C₄

yes

characteristic

#11724786

C₂²⋊C₄

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄×D₄

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

Dic₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

Dic₁₀₉

#117

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×C₂²⋊C₄

yes

characteristic

D₁₀₉

—

D₁₀₉

—

#1757

—

109

maximal, Sylow

C₁₀₉

#1757

yes

characteristic, complemented, Sylow

C₂×#131

—

109

—

#413

—

109

—

#410

—

109

—

#405

—

109

—

#344

—

109

—

C₂×#41

—

109

—

#131

—

218

—

#131

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

#101

—

109

—

#101

#89137

yes

characteristic, complemented

#100

—

109

—

#99

—

109

—

#97

—

218

—

C₄×C₈

—

218

—

#41

—

218

—

100

#41

—

218

—

101

C₂×Q₈

C₂×#89137

yes

characteristic

102

C₂×D₄

—

109

—

103

C₂×D₄

—

109

—

104

C₂²×C₄

—

109

—

105

C₂²×C₄

—

109

—

106

M₁₆

—

109

—

107

M₁₆

—

218

—

108

M₁₆

—

218

—

109

M₁₆

—

218

—

110

M₁₆

C₂×#89137

yes

—

111

C₂×C₈

—

109

—

112

C₂×C₈

—

109

—

113

C₂×C₈

—

109

—

114

C₂×C₈

—

109

—

115

C₂×C₈

—

109

—

116

C₂×C₈

—

218

—

117

C₂×C₈

—

218

—

118

C₂×C₈

—

218

—

119

C₄⋊C₄

—

218

—

120

C₂²⋊C₄

—

109

—

121

C₄²

—

218

—

122

C₂³

—

109

—

123

C₂³

—

109

—

124

Q₈

—

125

D₄

—

218

—

126

D₄

—

218

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

109

—

130

C₂×C₄

—

218

—

131

C₂×C₄

—

218

—

132

C₂×C₄

—

218

—

133

C₂×C₄

C₄×#89137

yes

—

134

C₂×C₄

C₂²×#89137

yes

characteristic

135

C₈

—

136

C₈

—

109

—

137

C₈

—

109

—

138

C₈

—

109

—

139

C₈

—

218

—

140

C₈

—

218

—

141

C₂²

—

109

—

142

C₂²

—

436

—

143

C₂²

—

436

—

144

C₂²

—

218

—

145

C₂²

C₂×C₄×#89137

yes

characteristic

146

C₄

—

147

C₄

—

109

—

148

C₄

—

109

—

149

C₄

—

436

—

150

C₄

D₄×#89137

yes

—

151

C₂

—

152

C₂

—

436

—

153

C₂

—

218

—

154

C₂

C₂²⋊C₄×#89137

yes

characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×#89137
commutator	C₂²×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	#101×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₄×C₂²⋊C₄
prime radical	C₂×#13182030	C₂²
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#101	#89137
hypocenter	C₁₀₉	#1757
LCS	G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

C₂²⋊C₄×#89137

commutator

C₂²×C₁₀₉

C₂×C₄²

Frattini

C₂×C₄

C₂²×#89137

Fitting

#101×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₄×C₂²⋊C₄

prime radical

C₂×#13182030

C₂²

cosocle

C₂×C₄×D₁₀₉

C₂³

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
3. #20295992	C₂	semidirect
4. #20295909	C₂	semidirect
8. C₂×#13182039	C₄	semidirect
10. C₂×D₁₀₉⋊C₈	C₄	semidirect
18. #16812618	C₄	semidirect
19. #16812615	C₄	semidirect
20. #101×C₁₀₉	C₄	semidirect
78. C₁₀₉	#1757	semidirect
94. #101	#89137	semidirect

H ≅ G/N

type

3. #20295992

C₂

semidirect

4. #20295909

C₂

semidirect

8. C₂×#13182039

C₄

semidirect

10. C₂×D₁₀₉⋊C₈

C₄

semidirect

18. #16812618

C₄

semidirect

19. #16812615

C₄

semidirect

20. #101×C₁₀₉

C₄

semidirect

78. C₁₀₉

#1757

semidirect

94. #101

#89137

semidirect

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

subgroup	index
C₂×#16812787	2
#20296007	2
#20295992	2
#20295909	2
#20295009	2
#1757	109

subgroup

index

109

Subgroups — #28378791

Conjugacy classes of subgroups

Complements

Maximal subgroups