Group #28361703 — subgroups

Subgroups — #28361703

Order 13952.

240 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20295050	C₂	1	1	yes	maximal, characteristic
3	#20294771	C₂	1	4	yes	maximal, complemented
4	#431×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₈	—	2	1	no	—
6	C₂×#13182013	—	2	1	no	—
7	#16812699	—	2	1	no	—
8	#16812699	—	2	1	no	—
9	#16812699	C₄	1	2	yes	—
10	#16812699	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₀₉	—	2	1	no	—
12	C₂×C₄⋊Dic₁₀₉	C₄	1	2	yes	—
13	C₄⋊#11724784	—	2	1	no	—
14	C₄⋊#11724784	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
17	C₂²⋊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	2	no	—
22	C₂×Dic₂₁₈	—	4	2	no	—
23	C₂²×Dic₁₀₉	—	2	2	no	—
24	#13182013	—	4	2	no	—
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	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₀₉	—	2	2	no	—
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	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	D₈	1	2	yes	—
45	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₁₀₉	—	2	4	no	—
50	C₂×Dic₁₀₉	—	4	4	no	—
51	Dic₂₁₈	—	4	4	no	—
52	Dic₂₁₈	—	4	2	no	—
53	Dic₂₁₈	—	4	2	no	—
54	C₈×C₁₀₉	—	4	2	no	—
55	#11724784	—	4	2	no	—
56	C₂²×C₁₀₉	—	2	1	no	—
57	C₂²×C₁₀₉	—	2	1	no	—
58	C₂²×C₁₀₉	#102	1	2	yes	—
59	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₀₉	—	2	1	no	—
61	C₄×C₁₀₉	—	2	1	no	—
62	C₄×C₁₀₉	#101	1	2	yes	—
63	C₄×C₁₀₉	#102	1	2	yes	—
64	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
65	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₁₀₉	#137	1	2	yes	—
67	Dic₁₀₉	—	4	4	no	—
68	C₂×C₁₀₉	—	2	2	no	—
69	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
70	C₂×C₁₀₉	C₂×#102	1	1	yes	characteristic
71	C₂×C₁₀₉	#419	1	1	yes	characteristic
72	#1546	—	109	1	no	maximal, Sylow
73	C₁₀₉	#1546	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	109	1	no	—
75	#431	D₁₀₉	1	1	yes	characteristic
76	#423	—	109	1	no	—
77	#332	—	109	4	no	—
78	C₂²×Q₈	—	218	1	no	—
79	C₂×M₁₆	—	218	1	no	—
80	C₄⋊Q₈	—	109	2	no	—
81	C₄⋊Q₈	—	109	2	no	—
82	C₄⋊Q₈	—	218	1	no	—
83	C₄⋊Q₈	—	218	1	no	—
84	C₂×C₄⋊C₄	—	109	2	no	—
85	C₂×C₄⋊C₄	—	218	1	no	—
86	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
87	C₄⋊C₈	—	109	2	no	—
88	C₄⋊C₈	—	218	1	no	—
89	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₂×Q₈	—	436	2	no	—
93	C₂×Q₈	—	436	2	no	—
94	C₂×Q₈	—	218	2	no	—
95	C₂×Q₈	—	218	2	no	—
96	C₂²×C₄	—	218	2	no	—
97	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
98	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
100	M₁₆	—	436	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	218	2	no	—
103	C₄⋊C₄	—	109	4	no	—
104	C₄⋊C₄	—	218	2	no	—
105	C₄⋊C₄	—	218	2	no	—
106	C₄⋊C₄	—	218	2	no	—
107	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
108	C₄²	D₄₃₆	1	1	yes	characteristic
109	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
110	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
111	C₂³	#13182006	1	1	yes	characteristic
112	Q₈	—	436	4	no	—
113	Q₈	—	436	2	no	—
114	Q₈	—	436	2	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	D₈₇₂	1	2	yes	—
123	C₂×C₄	#13182018	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
126	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	436	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#16812621	1	2	yes	—
132	C₂²	C₂×#13182006	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	436	4	no	—
136	C₄	#16812617	1	2	yes	—
137	C₄	#16812621	1	2	yes	—
138	C₄	C₂×D₈₇₂	1	1	yes	characteristic
139	C₄	C₂×#13182018	1	1	yes	characteristic
140	C₄	#16812729	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#16812617	1	1	yes	characteristic
143	C₂	C₂×#16812621	1	1	yes	characteristic
144	C₂	#20295034	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812699

C₂

yes

maximal, characteristic

#20295050

C₂

yes

maximal, characteristic

#20294771

C₂

yes

maximal, complemented

#431×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

—

—

—

—

C₄

yes

—

#16812699

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

—

C₂×C₄⋊Dic₁₀₉

C₄

yes

—

C₄⋊#11724784

—

C₄⋊#11724784

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

C₂²⋊C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

—

#13182013

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

C₂³

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

D₈

yes

—

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#102

yes

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#101

yes

—

C₄×C₁₀₉

#102

yes

—

C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₀₉

#137

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1546

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

#431

D₁₀₉

yes

characteristic

#423

—

109

—

#332

—

109

—

C₂²×Q₈

—

218

—

C₂×M₁₆

—

218

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

436

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

D₄₃₆

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

100

M₁₆

—

436

—

101

C₂×C₈

—

102

C₂×C₈

—

218

—

103

C₄⋊C₄

—

109

—

104

C₄⋊C₄

—

218

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

—

218

—

107

C₄²

C₄×D₁₀₉

yes

characteristic

108

C₄²

D₄₃₆

yes

characteristic

109

C₄²

C₂²⋊D₁₀₉

yes

characteristic

110

C₄²

C₂²×D₁₀₉

yes

characteristic

111

C₂³

#13182006

yes

characteristic

112

Q₈

—

436

—

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

D₈₇₂

yes

—

123

C₂×C₄

#13182018

yes

—

124

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

126

C₂×C₄

C₂×D₄₃₆

yes

characteristic

127

C₈

—

128

C₈

—

436

—

129

C₂²

—

130

C₂²

—

131

C₂²

#16812621

yes

—

132

C₂²

C₂×#13182006

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

436

—

136

C₄

#16812617

yes

—

137

C₄

#16812621

yes

—

138

C₄

C₂×D₈₇₂

yes

characteristic

139

C₄

C₂×#13182018

yes

characteristic

140

C₄

#16812729

yes

—

141

C₂

—

142

C₂

C₂×#16812617

yes

characteristic

143

C₂

C₂×#16812621

yes

characteristic

144

C₂

#20295034

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13182006
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#431×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#431	D₁₀₉
hypocenter	C₁₀₉	#1546
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×#13182006

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#431×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₉

C₂²⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #20294771	C₂	semidirect
73. C₁₀₉	#1546	semidirect

H ≅ G/N

type

3. #20294771

C₂

semidirect

73. C₁₀₉

#1546

semidirect

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

subgroup	index
C₂×#16812699	2
#20295050	2
#20294771	2
#431×C₁₀₉	2
#1546	109

subgroup

index

#431×C₁₀₉

109

Subgroups — #28361703

Conjugacy classes of subgroups

Complements

Maximal subgroups