Group #28379146 — subgroups

Subgroups — #28379146

Order 13952.

200 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 74 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	2	yes	maximal
2	#20296030	C₂	1	1	yes	maximal, characteristic
3	C₄×#13182039	C₂	1	1	yes	maximal, characteristic
4	#20295902	C₂	1	2	yes	maximal
5	C₄⋊C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₂²	1	2	yes	—
7	C₂×#13182039	C₂²	1	1	yes	characteristic
8	C₂×#13182039	C₂²	1	1	yes	characteristic
9	#16812770	—	2	4	no	—
10	C₄⋊#11724786	—	2	1	no	—
11	C₄⋊#11724786	—	2	1	no	—
12	#16812754	—	2	1	no	—
13	C₄×#11724786	—	2	1	no	—
14	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
15	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊#11724784	C₄	1	1	yes	characteristic
17	#16812628	C₄	1	2	yes	—
18	C₄⋊C₈×C₁₀₉	C₄	1	1	yes	characteristic
19	#13182039	—	2	1	no	—
20	#13182039	—	2	1	no	—
21	#13182039	—	2	2	no	—
22	#13182039	—	2	2	no	—
23	#13182039	D₄	1	2	yes	—
24	#13182039	Q₈	1	2	yes	—
25	C₂×#11724786	—	2	1	no	—
26	C₂×#11724786	—	2	1	no	—
27	C₂×#11724786	—	2	2	no	—
28	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
29	C₂×C₄×D₁₀₉	Q₈	1	1	yes	characteristic
30	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×#11724784	C₂×C₄	1	2	yes	—
32	C₈×D₁₀₉	—	2	2	no	—
33	C₈×D₁₀₉	—	2	2	no	—
34	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
36	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
38	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
39	C₂²×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂×Dic₁₀₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₄×D₁₀₉	—	2	2	no	—
47	C₄×D₁₀₉	—	4	1	no	—
48	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
50	C₄×D₁₀₉	#41	1	1	yes	characteristic
51	C₄×D₁₀₉	#41	1	1	yes	characteristic
52	#11724786	—	2	2	no	—
53	#11724786	—	2	2	no	—
54	#11724786	—	4	2	no	—
55	C₈×C₁₀₉	—	2	2	no	—
56	#11724784	—	2	2	no	—
57	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×D₁₀₉	—	2	1	no	—
59	C₂×D₁₀₉	—	2	1	no	—
60	C₂×D₁₀₉	#125	1	1	yes	characteristic
61	C₂×D₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₄×C₁₀₉	—	2	1	no	—
63	C₄×C₁₀₉	#108	1	2	yes	complemented
64	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
65	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
66	Dic₁₀₉	—	2	1	no	—
67	Dic₁₀₉	—	2	1	no	—
68	Dic₁₀₉	#121	1	1	yes	characteristic
69	Dic₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂×C₁₀₉	#389	1	1	yes	characteristic
71	C₂×C₁₀₉	C₂×#108	1	1	yes	characteristic
72	C₂×C₁₀₉	#430	1	1	yes	characteristic
73	D₁₀₉	—	2	1	no	—
74	D₁₀₉	—	2	1	no	—
75	#1950	—	109	1	no	maximal, Sylow
76	C₁₀₉	#1950	1	1	yes	characteristic, complemented, Sylow
77	C₂×#108	—	109	2	no	—
78	#423	—	109	1	no	—
79	C₂×C₄⋊C₈	—	109	1	no	—
80	C₄×M₁₆	—	109	1	no	—
81	#341	—	109	2	no	—
82	C₂×M₁₆	—	109	1	no	—
83	C₂×M₁₆	—	109	1	no	—
84	C₂×M₁₆	—	109	2	no	—
85	C₂²×C₈	—	109	2	no	—
86	C₂×C₄²	—	109	1	no	—
87	#108	—	218	4	no	—
88	C₄⋊C₈	—	109	2	no	—
89	C₄⋊C₈	—	109	1	no	—
90	C₄⋊C₈	—	218	1	no	—
91	C₄⋊C₈	—	218	1	no	—
92	C₄⋊C₈	#89137	1	1	yes	characteristic
93	#97	—	218	1	no	—
94	C₄×C₈	—	218	1	no	—
95	C₂²×C₄	—	109	1	no	—
96	C₂²×C₄	—	109	1	no	—
97	C₂²×C₄	—	109	1	no	—
98	M₁₆	—	109	2	no	—
99	M₁₆	—	109	2	no	—
100	M₁₆	—	218	2	no	—
101	M₁₆	—	218	2	no	—
102	M₁₆	—	218	1	no	—
103	M₁₆	—	218	1	no	—
104	C₂×C₈	—	109	2	no	—
105	C₂×C₈	—	218	2	no	—
106	C₂×C₈	—	218	2	no	—
107	C₂×C₈	—	218	2	no	—
108	C₂×C₈	—	218	1	no	—
109	C₂×C₈	—	218	1	no	—
110	C₂×C₈	C₂×#89137	1	2	yes	—
111	C₄²	—	109	1	no	—
112	C₄²	—	109	1	no	—
113	C₄²	—	109	1	no	—
114	C₄²	C₂×#89137	1	1	yes	characteristic
115	C₂³	—	109	1	no	—
116	C₂×C₄	—	109	1	no	—
117	C₂×C₄	—	109	1	no	—
118	C₂×C₄	—	109	1	no	—
119	C₂×C₄	—	109	1	no	—
120	C₂×C₄	—	109	1	no	—
121	C₂×C₄	—	109	1	no	—
122	C₂×C₄	—	109	1	no	—
123	C₂×C₄	—	436	1	no	—
124	C₂×C₄	—	218	2	no	—
125	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
126	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
127	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	436	2	no	—
130	C₈	—	218	2	no	—
131	C₈	—	218	2	no	—
132	C₈	—	218	2	no	—
133	C₂²	—	109	1	no	—
134	C₂²	—	109	1	no	—
135	C₂²	—	218	1	no	—
136	C₂²	—	218	1	no	—
137	C₂²	C₂×C₄⋊#89137	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	109	1	no	—
140	C₄	—	109	1	no	—
141	C₄	—	218	1	no	—
142	C₄	—	218	1	no	—
143	C₄	#16812770	1	2	yes	complemented
144	C₄	D₄×#89137	1	1	yes	characteristic
145	C₄	Q₈×#89137	1	1	yes	characteristic
146	C₂	—	218	1	no	—
147	C₂	—	218	1	no	—
148	C₂	#20295977	1	1	yes	characteristic
149	C₂	C₂×#16812770	1	1	yes	characteristic
150	C₂	#20296041	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812770

C₂

yes

maximal

#20296030

C₂

yes

maximal, characteristic

C₄×#13182039

C₂

yes

maximal, characteristic

#20295902

C₂

yes

maximal

C₄⋊C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×#13182039

C₂²

yes

characteristic

C₂×#13182039

C₂²

yes

—

—

—

—

—

C₂×C₈×D₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

characteristic

#16812628

C₄

yes

—

C₄⋊C₈×C₁₀₉

C₄

yes

—

—

—

—

D₄

yes

—

#13182039

Q₈

yes

—

C₂×#11724786

—

C₂×#11724786

—

C₂×#11724786

—

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×C₄×D₁₀₉

Q₈

yes

characteristic

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₈×D₁₀₉

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

C₂×Q₈

yes

C₄×D₁₀₉

yes

C₄×D₁₀₉

yes

—

—

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

#125

yes

characteristic

C₂×D₁₀₉

C₄⋊Q₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#108

yes

complemented

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#121

yes

characteristic

Dic₁₀₉

C₂²⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

D₁₀₉

—

#1950

—

109

maximal, Sylow

C₁₀₉

#1950

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#423

—

109

—

C₂×C₄⋊C₈

—

109

—

C₄×M₁₆

—

109

—

#341

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

#108

—

218

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

—

218

—

C₄⋊C₈

#89137

yes

characteristic

#97

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

109

—

M₁₆

—

109

—

100

M₁₆

—

218

—

101

M₁₆

—

218

—

102

M₁₆

—

218

—

103

M₁₆

—

218

—

104

C₂×C₈

—

109

—

105

C₂×C₈

—

218

—

106

C₂×C₈

—

218

—

107

C₂×C₈

—

218

—

108

C₂×C₈

—

218

—

109

C₂×C₈

—

218

—

110

C₂×C₈

C₂×#89137

yes

—

111

C₄²

—

109

—

112

C₄²

—

109

—

113

C₄²

—

109

—

114

C₄²

C₂×#89137

yes

characteristic

115

C₂³

—

109

—

116

C₂×C₄

—

109

—

117

C₂×C₄

—

109

—

118

C₂×C₄

—

109

—

119

C₂×C₄

—

109

—

120

C₂×C₄

—

109

—

121

C₂×C₄

—

109

—

122

C₂×C₄

—

109

—

123

C₂×C₄

—

436

—

124

C₂×C₄

—

218

—

125

C₂×C₄

C₄⋊#89137

yes

characteristic

126

C₂×C₄

C₄⋊#89137

yes

characteristic

127

C₂×C₄

C₂²×#89137

yes

characteristic

128

C₈

—

129

C₈

—

436

—

130

C₈

—

218

—

131

C₈

—

218

—

132

C₈

—

218

—

133

C₂²

—

109

—

134

C₂²

—

109

—

135

C₂²

—

218

—

136

C₂²

—

218

—

137

C₂²

C₂×C₄⋊#89137

yes

characteristic

138

C₄

—

139

C₄

—

109

—

140

C₄

—

109

—

141

C₄

—

218

—

142

C₄

—

218

—

143

C₄

#16812770

yes

complemented

144

C₄

D₄×#89137

yes

characteristic

145

C₄

Q₈×#89137

yes

characteristic

146

C₂

—

218

—

147

C₂

—

218

—

148

C₂

#20295977

yes

characteristic

149

C₂

C₂×#16812770

yes

characteristic

150

C₂

#20296041

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄⋊C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂²×D₁₀₉	C₂×Q₈
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄⋊C₈	#89137
hypocenter	C₁₀₉	#1950
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄⋊C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂²×D₁₀₉

C₂×Q₈

cosocle

C₂×C₄×D₁₀₉

C₂³

C₄⋊C₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

N	H ≅ G/N	type
63. C₄×C₁₀₉	#108	semidirect
76. C₁₀₉	#1950	semidirect
143. C₄	#16812770	semidirect

H ≅ G/N

type

63. C₄×C₁₀₉

#108

semidirect

76. C₁₀₉

#1950

semidirect

143. C₄

#16812770

semidirect

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

subgroup	index
C₂×#16812770	2
#20296030	2
C₄×#13182039	2
#20295902	2
C₄⋊C₈×D₁₀₉	2
#1950	109

subgroup

index

C₄⋊C₈×D₁₀₉

109

Subgroups — #28379146

Conjugacy classes of subgroups

Complements

Maximal subgroups