Group #28364020 — subgroups

Subgroups — #28364020

Order 13952.

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

normal subgroups only

This group has 260 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×#16812699	C₂	1	1	yes	maximal, characteristic
2	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
3	#20294794	C₂	1	4	yes	maximal
4	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₈	C₄	1	2	yes	—
6	C₂×#13182013	C₂²	1	2	yes	—
7	#16812699	—	2	1	no	—
8	#16812699	—	2	1	no	—
9	#16812699	—	2	1	no	—
10	#16812699	—	2	1	no	—
11	C₂×C₄⋊Dic₁₀₉	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
13	#16812610	—	2	1	no	—
14	C₄×#11724784	—	2	1	no	—
15	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₀₉	—	2	1	no	—
18	C₄⋊C₈×C₁₀₉	—	2	1	no	—
19	C₂×Dic₂₁₈	—	2	2	no	—
20	C₂×Dic₂₁₈	—	2	2	no	—
21	C₂×Dic₂₁₈	—	4	4	no	—
22	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
23	#13182013	—	2	2	no	—
24	#13182013	Q₈	1	4	yes	complemented
25	C₂×#11724784	—	2	2	no	—
26	C₄⋊Dic₁₀₉	—	2	2	no	—
27	C₄⋊Dic₁₀₉	—	2	2	no	—
28	C₄⋊Dic₁₀₉	—	2	2	no	—
29	C₄⋊Dic₁₀₉	—	2	2	no	—
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₀₉	—	2	4	no	—
34	C₂×C₈×C₁₀₉	—	2	2	no	—
35	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
36	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
37	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
39	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₀₉	#41	1	2	yes	—
50	C₂×Dic₁₀₉	—	2	4	no	—
51	C₂×Dic₁₀₉	—	4	4	no	—
52	Dic₂₁₈	—	4	4	no	—
53	Dic₂₁₈	—	4	4	no	—
54	C₈×C₁₀₉	—	4	2	no	—
55	#11724784	—	2	4	no	—
56	C₂²×C₁₀₉	—	2	1	no	—
57	C₂²×C₁₀₉	—	2	1	no	—
58	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
60	C₄×C₁₀₉	—	2	1	no	—
61	C₄×C₁₀₉	—	2	1	no	—
62	C₄×C₁₀₉	#100	1	2	yes	—
63	C₄×C₁₀₉	#101	1	2	yes	—
64	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₀₉	#124	1	1	yes	characteristic
66	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₀₉	—	4	4	no	—
68	C₂×C₁₀₉	—	2	2	no	—
69	C₂×C₁₀₉	#391	1	1	yes	characteristic
70	C₂×C₁₀₉	C₂×#100	1	1	yes	characteristic
71	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
72	#1997	—	109	1	no	maximal, Sylow
73	C₁₀₉	#1997	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	109	1	no	—
75	#423	D₁₀₉	1	1	yes	characteristic
76	C₄×M₁₆	—	109	1	no	—
77	#343	—	109	4	no	—
78	C₂²×Q₈	—	109	2	no	—
79	C₂×M₁₆	—	109	2	no	—
80	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
81	C₄⋊Q₈	—	218	1	no	—
82	C₄⋊Q₈	—	218	1	no	—
83	C₄⋊Q₈	—	218	1	no	—
84	C₄⋊Q₈	—	218	1	no	—
85	C₂×C₄⋊C₄	—	109	2	no	—
86	C₂×C₄⋊C₄	—	109	2	no	—
87	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	218	1	no	—
91	C₄×C₈	—	218	1	no	—
92	C₂×Q₈	—	436	4	no	—
93	C₂×Q₈	—	218	2	no	—
94	C₂×Q₈	—	218	2	no	—
95	C₂²×C₄	—	109	4	no	—
96	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	109	4	no	—
101	M₁₆	—	218	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	218	2	no	—
104	C₄⋊C₄	—	218	2	no	—
105	C₄⋊C₄	—	218	2	no	—
106	C₄⋊C₄	—	218	2	no	—
107	C₄⋊C₄	—	218	2	no	—
108	C₄²	D₄₃₆	1	1	yes	characteristic
109	C₄²	D₄₃₆	1	1	yes	characteristic
110	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
111	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
113	Q₈	—	436	4	no	—
114	Q₈	—	436	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	436	4	no	—
118	C₂×C₄	—	218	4	no	—
119	C₂×C₄	#13182006	1	1	yes	characteristic
120	C₂×C₄	#13182006	1	1	yes	characteristic
121	C₂×C₄	#13182006	1	1	yes	characteristic
122	C₂×C₄	#13182006	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
124	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
126	C₂×C₄	#13182028	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₈	—	218	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×#13182006	1	1	yes	characteristic
132	C₂²	#16812664	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	436	4	no	—
136	C₄	#16812615	1	2	yes	—
137	C₄	#16812617	1	2	yes	—
138	C₄	#16812679	1	2	yes	—
139	C₄	#16812686	1	1	yes	characteristic
140	C₄	#16812697	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#20294931	1	1	yes	characteristic
143	C₂	C₂×#16812615	1	1	yes	characteristic
144	C₂	C₂×#16812617	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

C₄×#13182013

C₂

yes

maximal, characteristic

#20294794

C₂

yes

maximal

#423×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₄

yes

—

C₂×#13182013

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₁₀₉

C₄

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812610

—

C₄×#11724784

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

#13182013

—

#13182013

Q₈

yes

complemented

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊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

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#100

yes

—

C₄×C₁₀₉

#101

yes

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1997

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

#423

D₁₀₉

yes

characteristic

C₄×M₁₆

—

109

—

#343

—

109

—

C₂²×Q₈

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

218

—

C₄×C₈

—

218

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

100

M₁₆

—

109

—

101

M₁₆

—

218

—

102

C₂×C₈

—

103

C₂×C₈

—

218

—

104

C₄⋊C₄

—

218

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

—

218

—

107

C₄⋊C₄

—

218

—

108

C₄²

D₄₃₆

yes

characteristic

109

C₄²

D₄₃₆

yes

characteristic

110

C₄²

C₂²⋊D₁₀₉

yes

characteristic

111

C₄²

C₂²⋊D₁₀₉

yes

characteristic

112

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

113

Q₈

—

436

—

114

Q₈

—

436

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

436

—

118

C₂×C₄

—

218

—

119

C₂×C₄

#13182006

yes

characteristic

120

C₂×C₄

#13182006

yes

characteristic

121

C₂×C₄

#13182006

yes

characteristic

122

C₂×C₄

#13182006

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

124

C₂×C₄

C₂×D₄₃₆

yes

characteristic

125

C₂×C₄

Q₈×D₁₀₉

yes

—

126

C₂×C₄

#13182028

yes

—

127

C₈

—

128

C₈

—

218

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×#13182006

yes

characteristic

132

C₂²

#16812664

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

436

—

136

C₄

#16812615

yes

—

137

C₄

#16812617

yes

—

138

C₄

#16812679

yes

—

139

C₄

#16812686

yes

characteristic

140

C₄

#16812697

yes

characteristic

141

C₂

—

142

C₂

#20294931

yes

characteristic

143

C₂

C₂×#16812615

yes

characteristic

144

C₂

C₂×#16812617

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13182006
commutator, prime radical	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#423×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#423	D₁₀₉
hypocenter	C₁₀₉	#1997
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#13182006

commutator, prime radical

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#423×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
24. #13182013	Q₈	semidirect
73. C₁₀₉	#1997	semidirect

H ≅ G/N

type

24. #13182013

Q₈

semidirect

73. C₁₀₉

#1997

semidirect

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

subgroup	index
C₂×#16812699	2
C₄×#13182013	2
#20294794	2
#423×C₁₀₉	2
#1997	109

subgroup

index

#423×C₁₀₉

109

Subgroups — #28364020

Conjugacy classes of subgroups

Complements

Maximal subgroups