Group #28373435 — subgroups

Subgroups — #28373435

Order 13952.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20295576	C₂	1	1	yes	maximal, characteristic
2	#20295396	C₂	1	2	yes	maximal, complemented
3	#20295392	C₂	1	2	yes	maximal, complemented
4	#20295383	C₂	1	4	yes	maximal, complemented
5	#20295025	C₂	1	4	yes	maximal, complemented
6	C₄×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13182029	C₂²	1	2	yes	—
9	C₂×#13182013	C₂²	1	2	yes	—
10	#16812699	C₂²	1	1	yes	characteristic
11	#16812696	C₂²	1	1	yes	characteristic
12	#16812687	C₂²	1	2	yes	—
13	#16812673	—	2	2	no	—
14	C₄×D₄₃₆	—	2	2	no	—
15	#16812622	C₂²	1	8	yes	—
16	#16812619	C₂²	1	8	yes	—
17	#16812610	C₂²	1	2	yes	—
18	C₄×#11724784	C₂²	1	2	yes	—
19	#125×C₁₀₉	—	2	2	no	—
20	#125×C₁₀₉	C₂²	1	2	yes	—
21	#125×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
23	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
25	#13182029	—	2	8	no	—
26	C₂×Dic₂₁₈	C₂³	1	2	yes	—
27	C₂×D₄₃₆	—	2	2	no	—
28	C₂×D₄₃₆	C₂³	1	2	yes	—
29	C₂×C₂²⋊D₁₀₉	—	2	2	no	—
30	C₂×C₄×D₁₀₉	—	2	2	no	—
31	#13182013	—	2	4	no	—
32	C₂×#11724784	C₂³	1	4	yes	—
33	C₄⋊Dic₁₀₉	—	2	2	no	—
34	#13182006	—	2	4	no	—
35	C₂²×C₄×C₁₀₉	—	2	2	no	—
36	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
37	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₁₀₉	—	2	4	no	—
39	C₄⋊C₄×C₁₀₉	—	2	4	no	—
40	C₄⋊C₄×C₁₀₉	—	2	2	no	—
41	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
42	C₄²×C₁₀₉	D₄	1	2	yes	—
43	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₁₀₉	—	2	2	no	—
47	C₂×C₄×C₁₀₉	—	2	4	no	—
48	C₂×C₄×C₁₀₉	—	4	2	no	—
49	C₂×C₄×C₁₀₉	—	4	4	no	—
50	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	#41	1	4	yes	—
54	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₁₀₉	—	4	4	no	—
56	C₂×Dic₁₀₉	—	2	2	no	—
57	D₄₃₆	—	2	4	no	—
58	D₄₃₆	—	4	4	no	—
59	C₄×D₁₀₉	—	4	4	no	—
60	Dic₂₁₈	—	2	4	no	—
61	#11724784	—	2	4	no	—
62	C₂²×C₁₀₉	—	2	2	no	—
63	C₂²×C₁₀₉	#124	1	2	yes	—
64	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₁₀₉	—	4	2	no	—
66	C₂×D₁₀₉	—	4	4	no	—
67	C₄×C₁₀₉	—	2	2	no	—
68	C₄×C₁₀₉	—	2	2	no	—
69	C₄×C₁₀₉	—	4	4	no	—
70	C₄×C₁₀₉	#124	1	2	yes	—
71	C₄×C₁₀₉	C₂×#41	1	2	yes	—
72	Dic₁₀₉	—	4	2	no	—
73	C₂×C₁₀₉	—	2	2	no	—
74	C₂×C₁₀₉	C₂×#124	1	1	yes	characteristic
75	C₂×C₁₀₉	#575	1	2	yes	—
76	D₁₀₉	—	8	2	no	—
77	#3142	—	109	1	no	maximal, Sylow
78	C₁₀₉	#3142	1	1	yes	characteristic, complemented, Sylow
79	#532	—	109	1	no	—
80	C₂×#125	D₁₀₉	1	1	yes	characteristic, complemented
81	#491	—	109	2	no	—
82	#490	—	109	2	no	—
83	#488	—	109	4	no	—
84	#417	—	109	4	no	—
85	C₄×M₁₆	—	109	1	no	—
86	C₂×#41	—	109	2	no	—
87	C₂×M₁₆	—	109	2	no	—
88	C₄⋊Q₈	—	109	1	no	—
89	#127	—	109	1	no	—
90	#125	—	2	2	no	—
91	#125	C₂×D₁₀₉	1	2	yes	—
92	#125	C₂×D₁₀₉	1	2	yes	—
93	#124	—	109	2	no	—
94	#121	—	218	2	no	—
95	C₄×D₄	—	218	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
98	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
99	#103	—	109	8	no	—
100	#102	—	109	8	no	—
101	#97	—	109	2	no	—
102	C₄×C₈	—	109	2	no	—
103	#41	—	218	8	no	—
104	C₂×Q₈	—	109	2	no	—
105	C₂×D₄	—	109	2	no	—
106	C₂×D₄	—	218	2	no	—
107	C₂×D₄	—	218	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	218	2	no	—
110	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
112	M₁₆	—	218	4	no	—
113	C₂×C₈	—	109	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	218	2	no	—
118	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
119	C₂²⋊C₄	—	218	4	no	—
120	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
121	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
122	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
123	C₂³	—	218	2	no	—
124	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
125	Q₈	—	218	4	no	—
126	D₄	—	436	4	no	—
127	D₄	—	436	4	no	—
128	D₄	—	218	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	436	4	no	—
133	C₂×C₄	—	218	2	no	—
134	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
137	C₂×C₄	#13182030	1	4	yes	—
138	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
139	C₈	—	218	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	436	2	no	—
142	C₂²	—	436	4	no	—
143	C₂²	#16812688	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	436	2	no	—
149	C₄	#16812688	1	2	yes	—
150	C₄	C₂×#13182030	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	872	2	no	—
153	C₂	C₂×#16812688	1	1	yes	characteristic
154	C₂	#20295824	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295576

C₂

yes

maximal, characteristic

#20295396

C₂

yes

maximal, complemented

#20295392

C₂

yes

maximal, complemented

#20295383

C₂

yes

maximal, complemented

#20295025

C₂

yes

maximal, complemented

C₄×#13182013

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182029

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

#16812699

C₂²

yes

characteristic

#16812696

C₂²

yes

characteristic

#16812687

C₂²

yes

—

#16812673

—

C₄×D₄₃₆

—

#16812622

C₂²

yes

—

#16812619

C₂²

yes

—

#16812610

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

#125×C₁₀₉

—

#125×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#13182029

—

C₂×Dic₂₁₈

C₂³

yes

—

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂³

yes

—

C₂×C₂²⋊D₁₀₉

—

C₂×C₄×D₁₀₉

—

#13182013

—

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

#13182006

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#124

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×#124

yes

characteristic

C₂×C₁₀₉

#575

yes

—

D₁₀₉

—

#3142

—

109

maximal, Sylow

C₁₀₉

#3142

yes

characteristic, complemented, Sylow

#532

—

109

—

C₂×#125

D₁₀₉

yes

characteristic, complemented

#491

—

109

—

#490

—

109

—

#488

—

109

—

#417

—

109

—

C₄×M₁₆

—

109

—

C₂×#41

—

109

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

—

109

—

#127

—

109

—

#125

—

#125

C₂×D₁₀₉

yes

—

#125

C₂×D₁₀₉

yes

—

#124

—

109

—

#121

—

218

—

C₄×D₄

—

218

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#103

—

109

—

100

#102

—

109

—

101

#97

—

109

—

102

C₄×C₈

—

109

—

103

#41

—

218

—

104

C₂×Q₈

—

109

—

105

C₂×D₄

—

109

—

106

C₂×D₄

—

218

—

107

C₂×D₄

—

218

—

108

C₂²×C₄

—

109

C₂²×C₄

—

218

—

110

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

111

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

112

M₁₆

—

218

—

113

C₂×C₈

—

109

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

218

—

118

C₄⋊C₄

C₂²×D₁₀₉

yes

—

119

C₂²⋊C₄

—

218

—

120

C₄²

C₂²⋊D₁₀₉

yes

—

121

C₄²

C₂²×D₁₀₉

yes

characteristic

122

C₄²

C₂²×D₁₀₉

yes

characteristic

123

C₂³

—

218

—

124

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

125

Q₈

—

218

—

126

D₄

—

436

—

127

D₄

—

436

—

128

D₄

—

218

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

436

—

133

C₂×C₄

—

218

—

134

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

135

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

136

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

137

C₂×C₄

#13182030

yes

—

138

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

139

C₈

—

218

—

140

C₂²

—

141

C₂²

—

436

—

142

C₂²

—

436

—

143

C₂²

#16812688

yes

—

144

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

436

—

149

C₄

#16812688

yes

—

150

C₄

C₂×#13182030

yes

—

151

C₂

—

152

C₂

—

872

—

153

C₂

C₂×#16812688

yes

characteristic

154

C₂

#20295824

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#125×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295576	C₂
hypercenter	C₂×#125	D₁₀₉
hypocenter	C₁₀₉	#3142
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#125×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

C₂

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #20295396	C₂	semidirect
3. #20295392	C₂	semidirect
4. #20295383	C₂	semidirect
5. #20295025	C₂	semidirect
6. C₄×#13182013	C₂	semidirect
7. C₂×#125×C₁₀₉	C₂	semidirect
78. C₁₀₉	#3142	semidirect
80. C₂×#125	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295396

C₂

semidirect

3. #20295392

C₂

semidirect

4. #20295383

C₂

semidirect

5. #20295025

C₂

semidirect

6. C₄×#13182013

C₂

semidirect

7. C₂×#125×C₁₀₉

C₂

semidirect

78. C₁₀₉

#3142

semidirect

80. C₂×#125

D₁₀₉

semidirect

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

subgroup	index
#20295576	2
#20295396	2
#20295392	2
#20295383	2
#20295025	2
C₄×#13182013	2
C₂×#125×C₁₀₉	2
#3142	109

subgroup

index

C₂×#125×C₁₀₉

109

Subgroups — #28373435

Conjugacy classes of subgroups

Complements

Maximal subgroups