Group #28378836 — subgroups

Subgroups — #28378836

Order 13952.

232 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 90 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×#16812770	C₂	1	1	yes	maximal, characteristic
2	#20295991	C₂	1	2	yes	maximal
3	C₂×C₈×#89137	C₂	1	1	yes	maximal, characteristic
4	#20295896	C₂	1	2	yes	maximal
5	#20295063	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₂²	1	2	yes	—
7	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
9	#16812770	C₄	1	4	yes	complemented
10	#16812754	—	2	2	no	—
11	C₄×#11724786	—	2	2	no	—
12	C₈×#89137	C₄	1	4	yes	complemented
13	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	#16812657	C₂²	1	2	yes	—
15	#16812633	C₄	1	1	yes	characteristic, complemented
16	#16812630	C₄	1	2	yes	complemented
17	#107×C₁₀₉	C₄	1	1	yes	characteristic, complemented
18	C₂²×#89137	—	2	1	no	—
19	#13182039	—	2	2	no	—
20	#13182039	C₂×C₄	1	4	yes	—
21	D₁₀₉⋊C₈	C₂×C₄	1	2	yes	—
22	D₁₀₉⋊C₈	Q₈	1	2	yes	—
23	C₂×#11724786	—	2	2	no	—
24	C₂×#11724786	D₄	1	1	yes	characteristic
25	C₂×#11724786	D₄	1	1	yes	characteristic
26	C₄×#89137	—	2	1	no	—
27	C₄×#89137	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
30	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
31	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
32	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
33	#13182006	—	2	2	no	—
34	C₄×Dic₁₀₉	—	2	2	no	—
35	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
37	C₂²×D₁₀₉	#41	1	1	yes	characteristic
38	C₂×#89137	—	2	2	no	—
39	C₂×#89137	—	2	2	no	—
40	C₂×#89137	—	2	2	no	—
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	—	2	2	no	—
44	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₁₀₉	#41	1	1	yes	characteristic
49	#11724786	—	2	4	no	—
50	#11724786	C₄⋊C₄	1	2	yes	—
51	#11724786	C₄⋊C₄	1	2	yes	—
52	C₈×C₁₀₉	C₄²	1	2	yes	—
53	#11724784	C₄²	1	2	yes	—
54	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
55	C₂×D₁₀₉	—	2	1	no	—
56	C₂×D₁₀₉	—	2	1	no	—
57	C₂×D₁₀₉	#117	1	1	yes	characteristic
58	C₂×D₁₀₉	C₄×Q₈	1	1	yes	characteristic
59	#89137	—	2	4	no	—
60	C₄×C₁₀₉	—	4	2	no	—
61	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
62	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
63	Dic₁₀₉	—	4	2	no	—
64	Dic₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
66	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
67	C₂×C₁₀₉	#443	1	1	yes	characteristic
68	C₂×C₁₀₉	#443	1	1	yes	characteristic
69	D₁₀₉	—	2	1	no	—
70	D₁₀₉	—	2	1	no	—
71	#1772	—	109	1	no	maximal, Sylow
72	C₁₀₉	#1772	1	1	yes	characteristic, complemented, Sylow
73	C₂×#108	—	109	1	no	—
74	#427	—	109	1	no	—
75	#405	—	109	2	no	—
76	C₂×C₄×C₈	—	109	1	no	—
77	#339	—	109	2	no	—
78	C₂×M₁₆	—	109	2	no	—
79	C₂²×C₈	—	109	1	no	—
80	C₂²×C₈	—	109	1	no	—
81	#117	—	109	2	no	—
82	C₂×C₄²	—	109	1	no	—
83	#108	—	109	4	no	—
84	#107	—	109	1	no	—
85	#107	#89137	1	1	yes	characteristic, complemented
86	#106	—	109	2	no	—
87	#97	—	218	2	no	—
88	C₄×C₈	—	109	4	no	—
89	C₄×C₈	—	218	2	no	—
90	C₂²×C₄	—	109	1	no	—
91	C₂²×C₄	—	218	1	no	—
92	M₁₆	—	109	4	no	—
93	M₁₆	—	218	2	no	—
94	C₂×C₈	—	109	1	no	—
95	C₂×C₈	—	109	1	no	—
96	C₂×C₈	—	109	2	no	—
97	C₂×C₈	—	109	2	no	—
98	C₂×C₈	—	109	1	no	—
99	C₂×C₈	—	109	2	no	—
100	C₂×C₈	—	109	2	no	—
101	C₂×C₈	—	218	2	no	—
102	C₂×C₈	C₂×#89137	1	1	yes	characteristic
103	C₄⋊C₄	—	109	2	no	—
104	C₄⋊C₄	C₂×#89137	1	2	yes	—
105	C₂²⋊C₄	—	218	2	no	—
106	C₄²	—	109	2	no	—
107	C₄²	—	218	2	no	—
108	C₄²	—	218	1	no	—
109	C₂³	—	109	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	109	1	no	—
112	C₂×C₄	—	109	1	no	—
113	C₂×C₄	—	109	1	no	—
114	C₂×C₄	—	109	1	no	—
115	C₂×C₄	—	109	1	no	—
116	C₂×C₄	—	218	2	no	—
117	C₂×C₄	—	218	2	no	—
118	C₂×C₄	—	218	2	no	—
119	C₂×C₄	—	218	2	no	—
120	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
121	C₈	—	109	2	no	—
122	C₈	—	109	2	no	—
123	C₈	—	109	2	no	—
124	C₈	—	218	4	no	—
125	C₈	C₄×#89137	1	2	yes	—
126	C₂²	—	109	1	no	—
127	C₂²	—	109	1	no	—
128	C₂²	—	218	1	no	—
129	C₂²	—	218	1	no	—
130	C₂²	D₄×#89137	1	1	yes	characteristic
131	C₄	—	4	2	no	—
132	C₄	—	109	1	no	—
133	C₄	—	109	1	no	—
134	C₄	—	436	2	no	—
135	C₄	—	218	4	no	—
136	C₄	C₂×C₄×#89137	1	1	yes	characteristic
137	C₄	Q₈×#89137	1	1	yes	characteristic
138	C₂	—	218	1	no	—
139	C₂	—	218	1	no	—
140	C₂	C₄⋊C₄×#89137	1	1	yes	characteristic
141	C₂	#20296068	1	1	yes	characteristic
142	C₂	#20296070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812770

C₂

yes

maximal, characteristic

#20295991

C₂

yes

maximal

C₂×C₈×#89137

C₂

yes

maximal, characteristic

#20295896

C₂

yes

maximal

#20295063

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×D₁₀₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×#89137

C₂²

yes

characteristic

#16812770

C₄

yes

—

—

C₄

yes

complemented

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812657

C₂²

yes

—

#16812633

C₄

yes

characteristic, complemented

#16812630

C₄

yes

complemented

#107×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

#13182039

—

#13182039

C₂×C₄

yes

—

D₁₀₉⋊C₈

C₂×C₄

yes

—

D₁₀₉⋊C₈

Q₈

yes

—

C₂×#11724786

—

C₂×#11724786

D₄

yes

characteristic

C₂×#11724786

D₄

yes

characteristic

C₄×#89137

—

C₄×#89137

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

—

C₈×D₁₀₉

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182006

—

C₄×Dic₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

yes

—

—

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

C₂²×C₄

yes

characteristic

C₄×D₁₀₉

C₂²×C₄

yes

characteristic

C₄×D₁₀₉

C₂×Q₈

yes

C₄×D₁₀₉

yes

—

C₄⋊C₄

yes

—

#11724786

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄²

yes

—

#11724784

C₄²

yes

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#117

yes

characteristic

C₂×D₁₀₉

C₄×Q₈

yes

characteristic

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂×C₄⋊C₄

yes

characteristic

Dic₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

D₁₀₉

—

#1772

—

109

maximal, Sylow

C₁₀₉

#1772

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#427

—

109

—

#405

—

109

—

C₂×C₄×C₈

—

109

—

#339

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

C₂×C₄²

—

109

—

#108

—

109

—

#107

—

109

—

#107

#89137

yes

characteristic, complemented

#106

—

109

—

#97

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

100

C₂×C₈

—

109

—

101

C₂×C₈

—

218

—

102

C₂×C₈

C₂×#89137

yes

characteristic

103

C₄⋊C₄

—

109

—

104

C₄⋊C₄

C₂×#89137

yes

—

105

C₂²⋊C₄

—

218

—

106

C₄²

—

109

—

107

C₄²

—

218

—

108

C₄²

—

218

—

109

C₂³

—

109

—

110

C₂×C₄

—

111

C₂×C₄

—

109

—

112

C₂×C₄

—

109

—

113

C₂×C₄

—

109

—

114

C₂×C₄

—

109

—

115

C₂×C₄

—

109

—

116

C₂×C₄

—

218

—

117

C₂×C₄

—

218

—

118

C₂×C₄

—

218

—

119

C₂×C₄

—

218

—

120

C₂×C₄

C₂²×#89137

yes

characteristic

121

C₈

—

109

—

122

C₈

—

109

—

123

C₈

—

109

—

124

C₈

—

218

—

125

C₈

C₄×#89137

yes

—

126

C₂²

—

109

—

127

C₂²

—

109

—

128

C₂²

—

218

—

129

C₂²

—

218

—

130

C₂²

D₄×#89137

yes

characteristic

131

C₄

—

132

C₄

—

109

—

133

C₄

—

109

—

134

C₄

—

436

—

135

C₄

—

218

—

136

C₄

C₂×C₄×#89137

yes

characteristic

137

C₄

Q₈×#89137

yes

characteristic

138

C₂

—

218

—

139

C₂

—

218

—

140

C₂

C₄⋊C₄×#89137

yes

characteristic

141

C₂

#20296068

yes

characteristic

142

C₂

#20296070

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#89137
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	#107×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₄×D₄
prime radical	C₂²×D₁₀₉	#41
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#107	#89137
hypocenter	C₁₀₉	#1772
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#89137

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂×C₄

C₂²×#89137

Fitting

#107×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₄×D₄

prime radical

C₂²×D₁₀₉

#41

cosocle

C₂×C₄×D₁₀₉

C₂³

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
9. #16812770	C₄	semidirect
12. C₈×#89137	C₄	semidirect
15. #16812633	C₄	semidirect
16. #16812630	C₄	semidirect
17. #107×C₁₀₉	C₄	semidirect
72. C₁₀₉	#1772	semidirect
85. #107	#89137	semidirect

H ≅ G/N

type

9. #16812770

C₄

semidirect

12. C₈×#89137

C₄

semidirect

15. #16812633

C₄

semidirect

16. #16812630

C₄

semidirect

17. #107×C₁₀₉

C₄

semidirect

72. C₁₀₉

#1772

semidirect

85. #107

#89137

semidirect

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

subgroup	index
C₂×#16812770	2
#20295991	2
C₂×C₈×#89137	2
#20295896	2
#20295063	2
#1772	109

subgroup

index

C₂×C₈×#89137

109

Subgroups — #28378836

Conjugacy classes of subgroups

Complements

Maximal subgroups