Group #28363782 — subgroups

Subgroups — #28363782

Order 13952.

200 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16812634	C₂	1	1	yes	maximal, characteristic
2	#20295062	C₂	1	1	yes	maximal, characteristic
3	#20295052	C₂	1	1	yes	maximal, characteristic
4	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
5	#20294787	C₂	1	1	yes	maximal, characteristic
6	#20294785	C₂	1	1	yes	maximal, characteristic
7	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182013	C₂²	1	1	yes	characteristic
9	C₂²×#11724784	C₂²	1	1	yes	characteristic
10	#16812658	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	#16812634	—	2	2	no	—
13	#16812632	C₄	1	2	yes	complemented
14	#16812630	C₄	1	2	yes	complemented
15	#16812628	—	2	2	no	—
16	#16812609	—	2	2	no	—
17	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	#117×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₁₀₉	—	2	2	no	—
21	C₂²×Dic₁₀₉	—	2	1	no	—
22	#13182013	—	2	1	no	—
23	#13182013	—	2	1	no	—
24	C₂×#11724784	—	2	1	no	—
25	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
26	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
27	C₂×#11724784	D₄	1	2	yes	—
28	C₂×#11724784	Q₈	1	2	yes	—
29	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
30	C₂²⋊Dic₁₀₉	—	2	1	no	—
31	C₄×Dic₁₀₉	—	2	1	no	—
32	C₄×Dic₁₀₉	—	2	1	no	—
33	C₄×Dic₁₀₉	Q₈	1	2	yes	—
34	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	M₁₆×C₁₀₉	—	2	1	no	—
36	M₁₆×C₁₀₉	—	2	1	no	—
37	C₂×C₈×C₁₀₉	—	2	1	no	—
38	C₂×C₈×C₁₀₉	—	2	1	no	—
39	C₂×C₈×C₁₀₉	—	2	1	no	—
40	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
41	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
42	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
43	C₄²×C₁₀₉	—	2	1	no	—
44	C₂³×C₁₀₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	—	2	2	no	—
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
52	C₂×Dic₁₀₉	—	2	2	no	—
53	C₂×Dic₁₀₉	—	2	2	no	—
54	C₂×Dic₁₀₉	—	4	2	no	—
55	C₈×C₁₀₉	—	2	2	no	—
56	C₈×C₁₀₉	—	4	1	no	—
57	#11724784	—	4	1	no	—
58	#11724784	C₄⋊C₄	1	2	yes	—
59	#11724784	C₄⋊C₄	1	2	yes	—
60	C₂²×C₁₀₉	—	2	2	no	—
61	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
62	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₁₀₉	#125	1	1	yes	characteristic
64	C₄×C₁₀₉	—	4	2	no	—
65	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
67	C₄×C₁₀₉	#121	1	1	yes	characteristic
68	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
69	Dic₁₀₉	—	4	2	no	—
70	Dic₁₀₉	—	4	2	no	—
71	C₂×C₁₀₉	—	2	1	no	—
72	C₂×C₁₀₉	—	2	1	no	—
73	C₂×C₁₀₉	#389	1	1	yes	characteristic
74	C₂×C₁₀₉	#444	1	2	yes	—
75	#1946	—	109	1	no	maximal, Sylow
76	C₁₀₉	#1946	1	1	yes	characteristic, complemented, Sylow
77	C₂×#108	—	109	1	no	—
78	#427	—	109	1	no	—
79	#424	—	109	1	no	—
80	#424	D₁₀₉	1	1	yes	characteristic
81	C₂×#97	—	109	1	no	—
82	#339	—	109	1	no	—
83	#339	—	109	1	no	—
84	C₂×M₁₆	—	109	1	no	—
85	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
86	C₂²×C₈	—	109	1	no	—
87	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
88	#117	—	109	1	no	—
89	#117	C₂×D₁₀₉	1	1	yes	characteristic
90	C₂×C₄²	—	109	1	no	—
91	#108	—	218	2	no	—
92	#107	—	109	2	no	—
93	#106	—	109	2	no	—
94	C₄⋊C₈	—	2	2	no	—
95	C₄⋊C₈	—	218	2	no	—
96	#97	—	218	2	no	—
97	C₂²×C₄	—	218	1	no	—
98	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
99	M₁₆	—	2	1	no	—
100	M₁₆	—	2	1	no	—
101	M₁₆	—	218	1	no	—
102	M₁₆	—	218	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	109	1	no	—
107	C₂×C₈	—	109	1	no	—
108	C₂×C₈	—	109	2	no	—
109	C₂×C₈	—	109	2	no	—
110	C₂×C₈	—	218	1	no	—
111	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
112	C₄⋊C₄	—	109	2	no	—
113	C₄⋊C₄	C₄×D₁₀₉	1	2	yes	—
114	C₂²⋊C₄	—	2	1	no	—
115	C₂²⋊C₄	—	218	1	no	—
116	C₄²	—	2	1	no	—
117	C₄²	—	109	2	no	—
118	C₄²	—	218	1	no	—
119	C₄²	—	218	1	no	—
120	C₂³	#13182029	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	436	2	no	—
123	C₂×C₄	—	218	2	no	—
124	C₂×C₄	—	218	2	no	—
125	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
126	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
128	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
129	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
130	C₂×C₄	#13182028	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	4	1	no	—
133	C₈	—	109	2	no	—
134	C₈	—	109	2	no	—
135	C₈	—	436	1	no	—
136	C₂²	—	2	2	no	—
137	C₂²	C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
138	C₂²	#16812677	1	1	yes	characteristic
139	C₂²	#16812689	1	1	yes	characteristic
140	C₄	—	4	2	no	—
141	C₄	—	436	2	no	—
142	C₄	—	436	2	no	—
143	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
144	C₄	#16812664	1	1	yes	characteristic
145	C₄	#16812671	1	1	yes	characteristic
146	C₄	#16812697	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#20294921	1	1	yes	characteristic
150	C₂	#20295146	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812634

C₂

yes

maximal, characteristic

#20295062

C₂

yes

maximal, characteristic

#20295052

C₂

yes

maximal, characteristic

C₂×#16812609

C₂

yes

maximal, characteristic

#20294787

C₂

yes

maximal, characteristic

#20294785

C₂

yes

maximal, characteristic

#424×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

#16812658

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812634

—

#16812632

C₄

yes

complemented

#16812630

C₄

yes

complemented

#16812628

—

#16812609

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂²×Dic₁₀₉

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

D₄

yes

—

C₂×#11724784

Q₈

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

C₄⋊C₄

yes

—

#11724784

C₄⋊C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#125

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#389

yes

characteristic

C₂×C₁₀₉

#444

yes

—

#1946

—

109

maximal, Sylow

C₁₀₉

#1946

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#427

—

109

—

#424

—

109

—

#424

D₁₀₉

yes

characteristic

C₂×#97

—

109

—

#339

—

109

—

#339

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

#117

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

#108

—

218

—

#107

—

109

—

#106

—

109

—

C₄⋊C₈

—

C₄⋊C₈

—

218

—

#97

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

218

—

102

M₁₆

—

218

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

109

—

107

C₂×C₈

—

109

—

108

C₂×C₈

—

109

—

109

C₂×C₈

—

109

—

110

C₂×C₈

—

218

—

111

C₂×C₈

C₂²⋊D₁₀₉

yes

—

112

C₄⋊C₄

—

109

—

113

C₄⋊C₄

C₄×D₁₀₉

yes

—

114

C₂²⋊C₄

—

115

C₂²⋊C₄

—

218

—

116

C₄²

—

117

C₄²

—

109

—

118

C₄²

—

218

—

119

C₄²

—

218

—

120

C₂³

#13182029

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

436

—

123

C₂×C₄

—

218

—

124

C₂×C₄

—

218

—

125

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

126

C₂×C₄

D₄×D₁₀₉

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

128

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

129

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

130

C₂×C₄

#13182028

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

109

—

134

C₈

—

109

—

135

C₈

—

436

—

136

C₂²

—

137

C₂²

C₄×C₂²⋊D₁₀₉

yes

characteristic

138

C₂²

#16812677

yes

characteristic

139

C₂²

#16812689

yes

characteristic

140

C₄

—

141

C₄

—

436

—

142

C₄

—

436

—

143

C₄

C₄⋊C₄×D₁₀₉

yes

characteristic

144

C₄

#16812664

yes

characteristic

145

C₄

#16812671

yes

characteristic

146

C₄

#16812697

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#20294921

yes

characteristic

150

C₂

#20295146

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#424×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₄×D₄
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#424	D₁₀₉
hypocenter	C₁₀₉	#1946
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#424×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₄×D₄

prime radical

C₂³×C₁₀₉

#41

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

N	H ≅ G/N	type
13. #16812632	C₄	semidirect
14. #16812630	C₄	semidirect
76. C₁₀₉	#1946	semidirect

H ≅ G/N

type

13. #16812632

C₄

semidirect

14. #16812630

C₄

semidirect

76. C₁₀₉

#1946

semidirect

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

subgroup	index
C₂×#16812634	2
#20295062	2
#20295052	2
C₂×#16812609	2
#20294787	2
#20294785	2
#424×C₁₀₉	2
#1946	109

subgroup

index

#424×C₁₀₉

109

Subgroups — #28363782

Conjugacy classes of subgroups

Complements

Maximal subgroups