Group #28363807 — subgroups

Subgroups — #28363807

Order 13952.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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	2	yes	maximal
2	#20295050	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic
4	#20294790	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	2	yes	—
8	#16812634	—	2	4	no	—
9	C₄⋊#11724784	—	2	1	no	—
10	C₄⋊#11724784	—	2	1	no	—
11	C₄⋊#11724784	C₄	1	2	yes	—
12	C₄⋊#11724784	C₄	1	2	yes	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	#97×C₁₀₉	—	2	1	no	—
17	C₄×C₈×C₁₀₉	—	2	1	no	—
18	#13182013	—	2	2	no	—
19	#13182013	—	2	2	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×#11724784	—	2	2	no	—
23	C₂×#11724784	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₀₉	—	2	1	no	—
28	M₁₆×C₁₀₉	—	2	1	no	—
29	M₁₆×C₁₀₉	D₄	1	2	yes	—
30	M₁₆×C₁₀₉	Q₈	1	2	yes	—
31	C₂×C₈×C₁₀₉	—	2	1	no	—
32	C₂×C₈×C₁₀₉	—	2	1	no	—
33	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
36	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
37	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	—	4	1	no	—
40	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
50	C₈×C₁₀₉	—	2	2	no	—
51	C₈×C₁₀₉	—	2	2	no	—
52	#11724784	—	2	4	no	—
53	#11724784	—	4	2	no	—
54	C₂²×C₁₀₉	—	2	1	no	—
55	C₂²×C₁₀₉	—	2	1	no	—
56	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₀₉	#125	1	1	yes	characteristic
58	C₂²×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
59	C₄×C₁₀₉	—	2	1	no	—
60	C₄×C₁₀₉	—	2	1	no	—
61	C₄×C₁₀₉	—	2	1	no	—
62	C₄×C₁₀₉	#108	1	2	yes	complemented
63	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
64	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
65	C₄×C₁₀₉	#121	1	1	yes	characteristic
66	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₁₀₉	—	2	1	no	—
68	C₂×C₁₀₉	—	2	1	no	—
69	C₂×C₁₀₉	#389	1	1	yes	characteristic
70	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
71	C₂×C₁₀₉	#430	1	1	yes	characteristic
72	#1950	—	109	1	no	maximal, Sylow
73	C₁₀₉	#1950	1	1	yes	characteristic, complemented, Sylow
74	C₂×#108	—	109	2	no	—
75	#423	—	109	1	no	—
76	C₂×C₄⋊C₈	—	109	1	no	—
77	C₄×M₁₆	D₁₀₉	1	1	yes	characteristic
78	#341	—	109	2	no	—
79	C₂×M₁₆	—	109	2	no	—
80	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
81	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
82	C₂²×C₈	—	109	2	no	—
83	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
84	#108	—	218	4	no	—
85	C₄⋊C₈	—	109	2	no	—
86	C₄⋊C₈	—	109	2	no	—
87	C₄⋊C₈	—	218	1	no	—
88	C₄⋊C₈	—	218	1	no	—
89	#97	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
92	C₂²×C₄	Dic₂₁₈	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
94	M₁₆	—	2	1	no	—
95	M₁₆	—	2	1	no	—
96	M₁₆	—	218	2	no	—
97	M₁₆	—	218	2	no	—
98	M₁₆	Dic₂₁₈	1	2	yes	—
99	M₁₆	D₄₃₆	1	2	yes	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	109	4	no	—
103	C₂×C₈	—	218	2	no	—
104	C₂×C₈	—	218	2	no	—
105	C₂×C₈	—	218	2	no	—
106	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
107	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
108	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
109	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
110	C₂³	C₂×Dic₂₁₈	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	#13182008	1	1	yes	characteristic
114	C₂×C₄	#13182008	1	1	yes	characteristic
115	C₂×C₄	#13182008	1	1	yes	characteristic
116	C₂×C₄	#13182008	1	1	yes	characteristic
117	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
121	C₂×C₄	#13182029	1	1	yes	characteristic
122	C₂×C₄	#13182029	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	436	2	no	—
126	C₈	—	218	4	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×#13182008	1	1	yes	characteristic
130	C₂²	#16812691	1	1	yes	characteristic
131	C₂²	#16812699	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	#16812634	1	2	yes	complemented
136	C₄	C₄×D₄₃₆	1	1	yes	characteristic
137	C₄	C₄×Dic₂₁₈	1	1	yes	characteristic
138	C₄	#16812673	1	1	yes	characteristic
139	C₄	#16812678	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	2	1	no	—
142	C₂	#20294924	1	1	yes	characteristic
143	C₂	C₂×#16812634	1	1	yes	characteristic
144	C₂	#20295074	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812634

C₂

yes

maximal

#20295050

C₂

yes

maximal, characteristic

C₂×C₄⋊#11724784

C₂

yes

maximal, characteristic

#20294790

C₂

yes

maximal

C₄×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

—

—

—

C₄

yes

—

C₄⋊#11724784

C₄

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

—

—

—

—

—

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

D₄

yes

—

M₁₆×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

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

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#125

yes

characteristic

C₂²×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#108

yes

complemented

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1950

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#423

—

109

—

C₂×C₄⋊C₈

—

109

—

C₄×M₁₆

D₁₀₉

yes

characteristic

#341

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#108

—

218

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

#97

—

C₄×C₈

—

C₂²×C₄

Dic₂₁₈

yes

characteristic

C₂²×C₄

Dic₂₁₈

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

Dic₂₁₈

yes

—

M₁₆

D₄₃₆

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

109

—

103

C₂×C₈

—

218

—

104

C₂×C₈

—

218

—

105

C₂×C₈

—

218

—

106

C₄²

C₄×D₁₀₉

yes

characteristic

107

C₄²

C₄×D₁₀₉

yes

characteristic

108

C₄²

C₂²⋊D₁₀₉

yes

characteristic

109

C₄²

C₂²⋊D₁₀₉

yes

characteristic

110

C₂³

C₂×Dic₂₁₈

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#13182008

yes

characteristic

114

C₂×C₄

#13182008

yes

characteristic

115

C₂×C₄

#13182008

yes

characteristic

116

C₂×C₄

#13182008

yes

characteristic

117

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

119

C₂×C₄

C₂×D₄₃₆

yes

characteristic

120

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

121

C₂×C₄

#13182029

yes

characteristic

122

C₂×C₄

#13182029

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

436

—

126

C₈

—

218

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×#13182008

yes

characteristic

130

C₂²

#16812691

yes

characteristic

131

C₂²

#16812699

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#16812634

yes

complemented

136

C₄

C₄×D₄₃₆

yes

characteristic

137

C₄

C₄×Dic₂₁₈

yes

characteristic

138

C₄

#16812673

yes

characteristic

139

C₄

#16812678

yes

characteristic

140

C₂

—

141

C₂

—

142

C₂

#20294924

yes

characteristic

143

C₂

C₂×#16812634

yes

characteristic

144

C₂

#20295074

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₄×M₁₆×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₄×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#1950
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₄×M₁₆×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₄×M₁₆

D₁₀₉

hypocenter

C₁₀₉

#1950

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
62. C₄×C₁₀₉	#108	semidirect
73. C₁₀₉	#1950	semidirect
135. C₄	#16812634	semidirect

H ≅ G/N

type

62. C₄×C₁₀₉

#108

semidirect

73. C₁₀₉

#1950

semidirect

135. C₄

#16812634

semidirect

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

subgroup	index
C₂×#16812634	2
#20295050	2
C₂×C₄⋊#11724784	2
#20294790	2
C₄×M₁₆×C₁₀₉	2
#1950	109

subgroup

index

C₄×M₁₆×C₁₀₉

109

Subgroups — #28363807

Conjugacy classes of subgroups

Complements

Maximal subgroups