Group #28373545 — subgroups

Subgroups — #28373545

Order 13952.

356 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 141 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	#20295582	C₂	1	2	yes	maximal
2	#20295430	C₂	1	2	yes	maximal, complemented
3	#20295428	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295425	C₂	1	2	yes	maximal, complemented
5	#20295423	C₂	1	2	yes	maximal, complemented
6	#20295063	C₂	1	2	yes	maximal
7	#20295061	C₂	1	2	yes	maximal
8	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
9	#499×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
12	#16812698	C₂²	1	4	yes	—
13	#16812691	C₂²	1	2	yes	—
14	#16812677	—	2	4	no	—
15	#16812664	—	2	4	no	—
16	#16812657	C₂²	1	4	yes	—
17	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
18	#16812633	C₂²	1	2	yes	—
19	#16812632	C₂²	1	4	yes	—
20	#16812631	C₂²	1	2	yes	—
21	#16812630	C₂²	1	4	yes	—
22	C₄×#11724784	C₂²	1	1	yes	characteristic
23	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	#125×C₁₀₉	C₂²	1	2	yes	—
26	#107×C₁₀₉	C₂²	1	2	yes	—
27	#106×C₁₀₉	C₂²	1	2	yes	—
28	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
29	C₂×Dic₂₁₈	—	2	4	no	—
30	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
31	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
32	C₂×#11724784	C₂³	1	1	yes	characteristic
33	C₂×#11724784	C₂³	1	1	yes	characteristic
34	C₈×D₁₀₉	Q₈	1	4	yes	complemented
35	C₈×D₁₀₉	Q₈	1	4	yes	complemented
36	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
37	#13182008	—	2	4	no	—
38	#13182008	—	2	4	no	—
39	#13182006	—	2	4	no	—
40	C₄×Dic₁₀₉	—	2	4	no	—
41	C₄×Dic₁₀₉	D₄	1	2	yes	—
42	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₁₀₉	—	2	4	no	—
46	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
47	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
48	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	—	2	4	no	—
50	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₁₀₉	—	2	4	no	—
53	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
55	C₄×D₁₀₉	—	2	4	no	—
56	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
57	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
58	Dic₂₁₈	—	4	8	no	—
59	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
60	C₈×C₁₀₉	C₂×Q₈	1	2	yes	—
61	#11724784	C₂×Q₈	1	2	yes	—
62	#11724784	C₂×Q₈	1	2	yes	—
63	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₁₀₉	—	2	2	no	—
65	C₂×D₁₀₉	C₄⋊Q₈	1	2	yes	—
66	C₄×C₁₀₉	—	2	2	no	—
67	C₄×C₁₀₉	—	4	4	no	—
68	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
70	Dic₁₀₉	—	4	4	no	—
71	Dic₁₀₉	C₄⋊Q₈	1	2	yes	—
72	Dic₁₀₉	#135	1	4	yes	—
73	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂×#135	1	2	yes	—
75	D₁₀₉	—	2	2	no	—
76	#3160	—	109	1	no	maximal, Sylow
77	C₁₀₉	#3160	1	1	yes	characteristic, complemented, Sylow
78	#533	—	109	2	no	—
79	#500	—	109	2	no	—
80	#499	—	109	2	no	—
81	#499	—	109	1	no	—
82	#499	D₁₀₉	1	1	yes	characteristic, complemented
83	#498	—	109	2	no	—
84	#427	—	109	2	no	—
85	#427	—	109	2	no	—
86	C₂×C₄×C₈	—	109	1	no	—
87	C₂²×C₈	—	109	1	no	—
88	C₂²×C₈	—	109	1	no	—
89	C₄⋊Q₈	—	109	4	no	—
90	#125	—	109	2	no	—
91	#125	C₂×D₁₀₉	1	2	yes	—
92	C₂²⋊Q₈	—	218	4	no	—
93	C₄×Q₈	—	218	4	no	—
94	#117	—	109	4	no	—
95	C₂×C₄²	—	109	1	no	—
96	#107	—	109	2	no	—
97	#107	—	109	4	no	—
98	#107	C₂×D₁₀₉	1	2	yes	—
99	#106	—	109	2	no	—
100	#106	—	109	4	no	—
101	#106	C₂×D₁₀₉	1	2	yes	—
102	C₄×C₈	—	109	1	no	—
103	C₄×C₈	—	109	1	no	—
104	C₄×C₈	—	109	1	no	—
105	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
106	C₂×Q₈	—	218	4	no	—
107	C₂²×C₄	—	109	1	no	—
108	C₂²×C₄	—	109	2	no	—
109	C₂×C₈	—	109	1	no	—
110	C₂×C₈	—	109	1	no	—
111	C₂×C₈	—	109	4	no	—
112	C₂×C₈	—	109	4	no	—
113	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	109	4	no	—
117	C₄⋊C₄	—	218	4	no	—
118	C₄⋊C₄	—	218	4	no	—
119	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
120	C₂²⋊C₄	—	218	4	no	—
121	C₄²	—	109	2	no	—
122	C₄²	—	109	1	no	—
123	C₄²	—	218	4	no	—
124	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
125	C₂³	—	109	1	no	—
126	Q₈	—	436	8	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	109	1	no	—
129	C₂×C₄	—	109	2	no	—
130	C₂×C₄	—	109	2	no	—
131	C₂×C₄	—	109	2	no	—
132	C₂×C₄	—	218	4	no	—
133	C₂×C₄	—	218	4	no	—
134	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
135	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
136	C₈	—	109	2	no	—
137	C₈	—	109	2	no	—
138	C₈	Q₈×D₁₀₉	1	2	yes	—
139	C₈	Q₈×D₁₀₉	1	2	yes	—
140	C₂²	—	109	2	no	—
141	C₂²	—	218	2	no	—
142	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	109	4	no	—
146	C₄	—	109	2	no	—
147	C₄	—	436	4	no	—
148	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
150	C₂	—	218	2	no	—
151	C₂	C₄⋊Q₈×D₁₀₉	1	1	yes	characteristic
152	C₂	#135×D₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295582

C₂

yes

maximal

#20295430

C₂

yes

maximal, complemented

#20295428

C₂

yes

maximal, characteristic, complemented

#20295425

C₂

yes

maximal, complemented

#20295423

C₂

yes

maximal, complemented

#20295063

C₂

yes

maximal

#20295061

C₂

yes

maximal

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#499×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812698

C₂²

yes

—

#16812691

C₂²

yes

—

#16812677

—

#16812664

—

#16812657

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812633

C₂²

yes

—

#16812632

C₂²

yes

—

#16812631

C₂²

yes

—

#16812630

C₂²

yes

—

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

#125×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×C₄×D₁₀₉

D₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

Q₈

yes

complemented

C₈×D₁₀₉

Q₈

yes

complemented

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×Q₈

yes

—

C₄×D₁₀₉

C₂×Q₈

yes

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×Q₈

yes

—

C₈×C₁₀₉

C₂×Q₈

yes

—

#11724784

C₂×Q₈

yes

—

#11724784

C₂×Q₈

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₄⋊Q₈

yes

—

Dic₁₀₉

#135

yes

—

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#135

yes

—

D₁₀₉

—

#3160

—

109

maximal, Sylow

C₁₀₉

#3160

yes

characteristic, complemented, Sylow

#533

—

109

—

#500

—

109

—

#499

—

109

—

#499

—

109

—

#499

D₁₀₉

yes

characteristic, complemented

#498

—

109

—

#427

—

109

—

#427

—

109

—

C₂×C₄×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

#125

—

109

—

#125

C₂×D₁₀₉

yes

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

218

—

#117

—

109

—

C₂×C₄²

—

109

—

#107

—

109

—

#107

—

109

—

#107

C₂×D₁₀₉

yes

—

#106

—

109

—

100

#106

—

109

—

101

#106

C₂×D₁₀₉

yes

—

102

C₄×C₈

—

109

—

103

C₄×C₈

—

109

—

104

C₄×C₈

—

109

—

105

C₄×C₈

C₂×D₁₀₉

yes

characteristic

106

C₂×Q₈

—

218

—

107

C₂²×C₄

—

109

—

108

C₂²×C₄

—

109

—

109

C₂×C₈

—

109

—

110

C₂×C₈

—

109

—

111

C₂×C₈

—

109

—

112

C₂×C₈

—

109

—

113

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

114

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

109

—

117

C₄⋊C₄

—

218

—

118

C₄⋊C₄

—

218

—

119

C₄⋊C₄

C₂²×D₁₀₉

yes

—

120

C₂²⋊C₄

—

218

—

121

C₄²

—

109

—

122

C₄²

—

109

—

123

C₄²

—

218

—

124

C₄²

C₂²×D₁₀₉

yes

characteristic

125

C₂³

—

109

—

126

Q₈

—

436

—

127

C₂×C₄

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

109

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

109

—

132

C₂×C₄

—

218

—

133

C₂×C₄

—

218

—

134

C₂×C₄

D₄×D₁₀₉

yes

—

135

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

136

C₈

—

109

—

137

C₈

—

109

—

138

C₈

Q₈×D₁₀₉

yes

—

139

C₈

Q₈×D₁₀₉

yes

—

140

C₂²

—

109

—

141

C₂²

—

218

—

142

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

109

—

146

C₄

—

109

—

147

C₄

—

436

—

148

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

149

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

150

C₂

—

218

—

151

C₂

C₄⋊Q₈×D₁₀₉

yes

characteristic

152

C₂

#135×D₁₀₉

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#499×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
hypercenter	#499	D₁₀₉
hypocenter	C₁₀₉	#3160
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#499×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂²×D₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #20295430	C₂	semidirect
3. #20295428	C₂	semidirect
4. #20295425	C₂	semidirect
5. #20295423	C₂	semidirect
9. #499×C₁₀₉	C₂	semidirect
34. C₈×D₁₀₉	Q₈	semidirect
35. C₈×D₁₀₉	Q₈	semidirect
77. C₁₀₉	#3160	semidirect
82. #499	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295430

C₂

semidirect

3. #20295428

C₂

semidirect

4. #20295425

C₂

semidirect

5. #20295423

C₂

semidirect

9. #499×C₁₀₉

C₂

semidirect

34. C₈×D₁₀₉

Q₈

semidirect

35. C₈×D₁₀₉

Q₈

semidirect

77. C₁₀₉

#3160

semidirect

82. #499

D₁₀₉

semidirect

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

subgroup	index
#20295582	2
#20295430	2
#20295428	2
#20295425	2
#20295423	2
#20295063	2
#20295061	2
C₄×C₈×D₁₀₉	2
#499×C₁₀₉	2
#3160	109

subgroup

index

C₄×C₈×D₁₀₉

#499×C₁₀₉

#3160

109

Subgroups — #28373545

Conjugacy classes of subgroups

Complements

Maximal subgroups