Group #28363773 — subgroups

Subgroups — #28363773

Order 13952.

240 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 105 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×#16812631	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
3	#20294922	C₂	1	2	yes	maximal
4	#20294788	C₂	1	2	yes	maximal
5	C₂×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
8	C₂×#13182008	—	2	2	no	—
9	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	#16812631	C₄	1	4	yes	complemented
11	#16812609	—	2	2	no	—
12	#16812606	—	2	2	no	—
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
15	#107×C₁₀₉	—	2	2	no	—
16	C₂²×Dic₁₀₉	—	2	2	no	—
17	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
18	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
19	C₂×#11724784	—	2	1	no	—
20	C₂×#11724784	—	2	1	no	—
21	C₂×#11724784	Q₈	1	2	yes	—
22	C₄⋊Dic₁₀₉	—	2	2	no	—
23	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
24	#13182008	—	4	4	no	—
25	C₄×Dic₁₀₉	—	2	1	no	—
26	C₄×Dic₁₀₉	—	2	1	no	—
27	C₂²×C₄×C₁₀₉	—	2	2	no	—
28	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
31	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
32	C₄⋊C₄×C₁₀₉	—	2	2	no	—
33	C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	—	4	2	no	—
36	C₂×C₄×C₁₀₉	—	4	2	no	—
37	C₂×C₄×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
44	C₂×Dic₁₀₉	—	2	4	no	—
45	C₂×Dic₁₀₉	—	2	1	no	—
46	C₂×Dic₁₀₉	—	2	1	no	—
47	C₂×Dic₁₀₉	—	2	2	no	—
48	C₂×Dic₁₀₉	—	2	2	no	—
49	C₂×Dic₁₀₉	—	4	2	no	—
50	C₂×Dic₁₀₉	—	4	2	no	—
51	C₈×C₁₀₉	C₄⋊C₄	1	4	yes	complemented
52	#11724784	—	2	2	no	—
53	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₁₀₉	#121	1	1	yes	characteristic
55	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
57	C₂²×C₁₀₉	#137	1	2	yes	—
58	C₄×C₁₀₉	—	8	2	no	—
59	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
61	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
62	C₄×C₁₀₉	#125	1	1	yes	characteristic
63	Dic₁₀₉	—	4	4	no	—
64	Dic₁₀₉	—	4	1	no	—
65	Dic₁₀₉	—	4	1	no	—
66	C₂×C₁₀₉	#389	1	1	yes	characteristic
67	C₂×C₁₀₉	#440	1	2	yes	—
68	C₂×C₁₀₉	#469	1	1	yes	characteristic
69	C₂×C₁₀₉	#470	1	1	yes	characteristic
70	C₂×C₁₀₉	#501	1	2	yes	—
71	#1944	—	109	1	no	maximal, Sylow
72	C₁₀₉	#1944	1	1	yes	characteristic, complemented, Sylow
73	C₂×#107	D₁₀₉	1	1	yes	characteristic
74	C₂×#106	—	109	1	no	—
75	C₂×#97	—	109	1	no	—
76	#389	—	109	2	no	—
77	#340	—	109	2	no	—
78	C₂²×C₈	—	109	1	no	—
79	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	109	2	no	—
81	C₂×C₄⋊C₄	—	218	2	no	—
82	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
83	C₂×C₄²	—	109	1	no	—
84	#107	—	2	2	no	—
85	#106	—	109	4	no	—
86	#97	—	218	2	no	—
87	#95	—	218	2	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	109	1	no	—
90	C₂²×C₄	—	109	1	no	—
91	C₂²×C₄	—	218	2	no	—
92	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
93	C₂×C₈	—	109	2	no	—
94	C₂×C₈	—	218	1	no	—
95	C₂×C₈	—	218	1	no	—
96	C₂×C₈	Dic₂₁₈	1	2	yes	—
97	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
98	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	109	4	no	—
102	C₄⋊C₄	—	436	4	no	—
103	C₄⋊C₄	—	218	2	no	—
104	C₄²	—	218	1	no	—
105	C₄²	—	218	1	no	—
106	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	436	2	no	—
111	C₂×C₄	—	436	2	no	—
112	C₂×C₄	—	218	4	no	—
113	C₂×C₄	—	218	1	no	—
114	C₂×C₄	—	218	1	no	—
115	C₂×C₄	—	218	2	no	—
116	C₂×C₄	—	218	2	no	—
117	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
119	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
121	C₂×C₄	#13182028	1	1	yes	characteristic
122	C₂×C₄	#13182030	1	1	yes	characteristic
123	C₈	—	218	2	no	—
124	C₈	#13182008	1	4	yes	complemented
125	C₂²	#16812661	1	1	yes	characteristic
126	C₂²	#16812670	1	1	yes	characteristic
127	C₂²	#16812698	1	1	yes	characteristic
128	C₂²	#16812723	1	2	yes	—
129	C₂²	#16812731	1	2	yes	—
130	C₄	—	8	2	no	—
131	C₄	—	436	1	no	—
132	C₄	—	436	1	no	—
133	C₄	—	436	4	no	—
134	C₄	C₂×#13182008	1	1	yes	characteristic
135	C₄	#16812664	1	1	yes	characteristic
136	C₄	#16812675	1	1	yes	characteristic
137	C₄	#16812690	1	1	yes	characteristic
138	C₂	#20294922	1	1	yes	characteristic
139	C₂	#20295125	1	2	yes	—
140	C₂	#20295276	1	1	yes	characteristic
141	C₂	#20295281	1	1	yes	characteristic
142	C₂	#20295434	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812631

C₂

yes

maximal, characteristic

C₂×#16812609

C₂

yes

maximal, characteristic

#20294922

C₂

yes

maximal

#20294788

C₂

yes

maximal

C₂×#107×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812631

C₄

yes

complemented

#16812609

—

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

D₄

yes

—

—

Q₈

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

complemented

#11724784

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

#137

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₉

#125

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#389

yes

characteristic

C₂×C₁₀₉

#440

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

#1944

—

109

maximal, Sylow

C₁₀₉

#1944

yes

characteristic, complemented, Sylow

C₂×#107

D₁₀₉

yes

characteristic

C₂×#106

—

109

—

C₂×#97

—

109

—

#389

—

109

—

#340

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

#107

—

#106

—

109

—

#97

—

218

—

#95

—

218

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

109

—

102

C₄⋊C₄

—

436

—

103

C₄⋊C₄

—

218

—

104

C₄²

—

218

—

105

C₄²

—

218

—

106

C₂³

D₄×D₁₀₉

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

436

—

111

C₂×C₄

—

436

—

112

C₂×C₄

—

218

—

113

C₂×C₄

—

218

—

114

C₂×C₄

—

218

—

115

C₂×C₄

—

218

—

116

C₂×C₄

—

218

—

117

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

119

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

120

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

121

C₂×C₄

#13182028

yes

characteristic

122

C₂×C₄

#13182030

yes

characteristic

123

C₈

—

218

—

124

C₈

#13182008

yes

complemented

125

C₂²

#16812661

yes

characteristic

126

C₂²

#16812670

yes

characteristic

127

C₂²

#16812698

yes

characteristic

128

C₂²

#16812723

yes

—

129

C₂²

#16812731

yes

—

130

C₄

—

131

C₄

—

436

—

132

C₄

—

436

—

133

C₄

—

436

—

134

C₄

C₂×#13182008

yes

characteristic

135

C₄

#16812664

yes

characteristic

136

C₄

#16812675

yes

characteristic

137

C₄

#16812690

yes

characteristic

138

C₂

#20294922

yes

characteristic

139

C₂

#20295125

yes

—

140

C₂

#20295276

yes

characteristic

141

C₂

#20295281

yes

characteristic

142

C₂

#20295434

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#107×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#107	D₁₀₉
hypocenter	C₁₀₉	#1944
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#107×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
10. #16812631	C₄	semidirect
51. C₈×C₁₀₉	C₄⋊C₄	semidirect
72. C₁₀₉	#1944	semidirect
124. C₈	#13182008	semidirect

H ≅ G/N

type

10. #16812631

C₄

semidirect

51. C₈×C₁₀₉

C₄⋊C₄

semidirect

72. C₁₀₉

#1944

semidirect

124. C₈

#13182008

semidirect

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

subgroup	index
C₂×#16812631	2
C₂×#16812609	2
#20294922	2
#20294788	2
C₂×#107×C₁₀₉	2
#1944	109

subgroup

index

C₂×#107×C₁₀₉

109

Subgroups — #28363773

Conjugacy classes of subgroups

Complements

Maximal subgroups