Group #28373616 — subgroups

Subgroups — #28373616

Order 13952.

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

normal subgroups only

This group has 368 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	#20295581	C₂	1	2	yes	maximal
2	#20295431	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295376	C₂	1	2	yes	maximal, complemented
4	#20295264	C₂	1	4	yes	maximal
5	Q₁₆×Dic₁₀₉	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
7	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
9	#16812699	C₂²	1	2	yes	—
10	#16812690	—	2	2	no	—
11	#16812679	—	2	2	no	—
12	#16812675	C₂²	1	4	yes	—
13	Q₈×Dic₁₀₉	C₂²	1	4	yes	—
14	#16812664	—	2	2	no	—
15	#16812657	—	2	2	no	—
16	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
17	#16812633	C₂²	1	4	yes	—
18	#16812623	C₂²	1	8	yes	—
19	C₄×#11724784	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₀₉	C₂²	1	2	yes	—
21	C₂×Q₁₆×C₁₀₉	C₂²	1	4	yes	—
22	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
23	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₁₈	—	2	2	no	—
25	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
27	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×#11724784	C₂³	1	2	yes	—
29	C₈×D₁₀₉	—	2	4	no	—
30	C₄⋊Dic₁₀₉	—	2	2	no	—
31	C₄⋊Dic₁₀₉	—	2	4	no	—
32	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
33	#13182008	—	2	4	no	—
34	#13182006	—	2	4	no	—
35	C₄×Dic₁₀₉	—	2	4	no	—
36	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
37	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
38	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	—
40	Q₁₆×C₁₀₉	—	2	8	no	—
41	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
42	C₄⋊C₄×C₁₀₉	—	2	2	no	—
43	C₄⋊C₄×C₁₀₉	—	2	2	no	—
44	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
45	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₁₀₉	—	2	8	no	—
47	C₂×C₄×C₁₀₉	—	2	4	no	—
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₀₉	—	2	4	no	—
52	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
55	C₄×D₁₀₉	—	2	2	no	—
56	C₄×D₁₀₉	—	2	2	no	—
57	C₄×D₁₀₉	Q₁₆	1	4	yes	complemented
58	Dic₂₁₈	—	4	4	no	—
59	C₈×C₁₀₉	#41	1	4	yes	—
60	#11724784	—	2	2	no	—
61	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₁₀₉	—	2	1	no	—
63	C₂×D₁₀₉	—	2	1	no	—
64	C₂×D₁₀₉	C₂×Q₁₆	1	2	yes	—
65	C₄×C₁₀₉	—	4	4	no	—
66	C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
67	C₄×C₁₀₉	#135	1	2	yes	—
68	C₄×C₁₀₉	C₂×#41	1	2	yes	—
69	Dic₁₀₉	—	2	1	no	—
70	Dic₁₀₉	—	2	1	no	—
71	Dic₁₀₉	—	4	4	no	—
72	Dic₁₀₉	C₂×Q₁₆	1	2	yes	—
73	C₂×C₁₀₉	#532	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂²×Q₁₆	1	1	yes	characteristic
75	C₂×C₁₀₉	C₂×#135	1	1	yes	characteristic
76	D₁₀₉	—	2	2	no	—
77	#3170	—	109	1	no	maximal, Sylow
78	C₁₀₉	#3170	1	1	yes	characteristic, complemented, Sylow
79	#533	—	109	2	no	—
80	#500	—	109	1	no	—
81	#494	D₁₀₉	1	1	yes	characteristic, complemented
82	#487	—	109	2	no	—
83	#467	—	109	4	no	—
84	C₄×Q₁₆	—	109	4	no	—
85	C₂×C₄×C₈	—	109	1	no	—
86	C₂×Q₁₆	C₂×D₁₀₉	1	4	yes	—
87	C₂²×C₈	—	109	2	no	—
88	C₄⋊Q₈	—	109	2	no	—
89	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	—
90	#125	—	218	2	no	—
91	C₂²⋊Q₈	—	109	4	no	—
92	C₂²⋊Q₈	—	218	2	no	—
93	C₄×Q₈	—	109	4	no	—
94	C₄×Q₈	—	218	2	no	—
95	#117	—	218	2	no	—
96	C₂×C₄²	—	109	1	no	—
97	#107	—	109	4	no	—
98	#103	—	109	8	no	—
99	C₄×C₈	—	109	2	no	—
100	C₄×C₈	—	109	1	no	—
101	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
102	C₂×Q₈	—	218	2	no	—
103	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	—
104	C₂²×C₄	—	109	1	no	—
105	C₂²×C₄	—	109	1	no	—
106	C₂²×C₄	—	109	1	no	—
107	Q₁₆	—	2	8	no	—
108	C₂×C₈	—	109	2	no	—
109	C₂×C₈	—	218	4	no	—
110	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	109	4	no	—
114	C₄⋊C₄	—	218	4	no	—
115	C₄⋊C₄	—	218	4	no	—
116	C₄⋊C₄	—	218	2	no	—
117	C₂²⋊C₄	—	218	4	no	—
118	C₄²	—	109	1	no	—
119	C₄²	—	109	1	no	—
120	C₄²	—	109	1	no	—
121	C₄²	—	218	4	no	—
122	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
123	C₂³	—	109	1	no	—
124	Q₈	—	2	8	no	—
125	Q₈	—	436	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	109	1	no	—
128	C₂×C₄	—	109	4	no	—
129	C₂×C₄	—	109	1	no	—
130	C₂×C₄	—	109	1	no	—
131	C₂×C₄	—	218	4	no	—
132	C₂×C₄	—	218	2	no	—
133	C₂×C₄	—	218	2	no	—
134	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
135	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
136	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
137	C₈	—	218	2	no	—
138	C₈	#13182028	1	4	yes	—
139	C₂²	—	109	2	no	—
140	C₂²	—	218	1	no	—
141	C₂²	—	218	1	no	—
142	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	109	2	no	—
145	C₄	—	436	4	no	—
146	C₄	—	218	1	no	—
147	C₄	—	218	1	no	—
148	C₄	Q₁₆×D₁₀₉	1	2	yes	—
149	C₄	#16812722	1	2	yes	—
150	C₄	C₂×#13182028	1	2	yes	—
151	C₂	—	218	2	no	—
152	C₂	#20295574	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₁₀₉	1	1	yes	characteristic
154	C₂	C₂×#16812722	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295581

C₂

yes

maximal

#20295431

C₂

yes

maximal, characteristic, complemented

#20295376

C₂

yes

maximal, complemented

#20295264

C₂

yes

maximal

Q₁₆×Dic₁₀₉

C₂

yes

maximal, complemented

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#494×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₂²

yes

—

#16812699

C₂²

yes

—

#16812690

—

#16812679

—

#16812675

C₂²

yes

—

Q₈×Dic₁₀₉

C₂²

yes

—

#16812664

—

#16812657

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#16812633

C₂²

yes

—

#16812623

C₂²

yes

—

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

—

C₂×Q₁₆×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂³

yes

—

C₈×D₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂³

yes

—

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

Q₁₆

yes

complemented

Dic₂₁₈

—

C₈×C₁₀₉

#41

yes

—

#11724784

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×Q₁₆

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×Q₁₆

yes

—

C₄×C₁₀₉

#135

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×Q₁₆

yes

—

C₂×C₁₀₉

#532

yes

characteristic

C₂×C₁₀₉

C₂²×Q₁₆

yes

characteristic

C₂×C₁₀₉

C₂×#135

yes

characteristic

D₁₀₉

—

#3170

—

109

maximal, Sylow

C₁₀₉

#3170

yes

characteristic, complemented, Sylow

#533

—

109

—

#500

—

109

—

#494

D₁₀₉

yes

characteristic, complemented

#487

—

109

—

#467

—

109

—

C₄×Q₁₆

—

109

—

C₂×C₄×C₈

—

109

—

C₂×Q₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#125

—

218

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

#117

—

218

—

C₂×C₄²

—

109

—

#107

—

109

—

#103

—

109

—

C₄×C₈

—

109

—

100

C₄×C₈

—

109

—

101

C₄×C₈

C₂×D₁₀₉

yes

characteristic

102

C₂×Q₈

—

218

—

103

C₂×Q₈

C₂²×D₁₀₉

yes

—

104

C₂²×C₄

—

109

—

105

C₂²×C₄

—

109

—

106

C₂²×C₄

—

109

—

107

Q₁₆

—

108

C₂×C₈

—

109

—

109

C₂×C₈

—

218

—

110

C₂×C₈

C₂²×D₁₀₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

109

—

114

C₄⋊C₄

—

218

—

115

C₄⋊C₄

—

218

—

116

C₄⋊C₄

—

218

—

117

C₂²⋊C₄

—

218

—

118

C₄²

—

109

—

119

C₄²

—

109

—

120

C₄²

—

109

—

121

C₄²

—

218

—

122

C₄²

C₂²×D₁₀₉

yes

characteristic

123

C₂³

—

109

—

124

Q₈

—

125

Q₈

—

436

—

126

C₂×C₄

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

109

—

130

C₂×C₄

—

109

—

131

C₂×C₄

—

218

—

132

C₂×C₄

—

218

—

133

C₂×C₄

—

218

—

134

C₂×C₄

D₄×D₁₀₉

yes

characteristic

135

C₂×C₄

D₄×D₁₀₉

yes

characteristic

136

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

137

C₈

—

218

—

138

C₈

#13182028

yes

—

139

C₂²

—

109

—

140

C₂²

—

218

—

141

C₂²

—

218

—

142

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

143

C₄

—

144

C₄

—

109

—

145

C₄

—

436

—

146

C₄

—

218

—

147

C₄

—

218

—

148

C₄

Q₁₆×D₁₀₉

yes

—

149

C₄

#16812722

yes

—

150

C₄

C₂×#13182028

yes

—

151

C₂

—

218

—

152

C₂

#20295574

yes

characteristic

153

C₂

C₂×Q₁₆×D₁₀₉

yes

characteristic

154

C₂

C₂×#16812722

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#494×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂²×D₁₀₉	C₂×D₄
hypercenter	#494	D₁₀₉
hypocenter	C₁₀₉	#3170
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#494×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 ◁ #494

N	H ≅ G/N	type
2. #20295431	C₂	semidirect
3. #20295376	C₂	semidirect
5. Q₁₆×Dic₁₀₉	C₂	semidirect
7. #494×C₁₀₉	C₂	semidirect
57. C₄×D₁₀₉	Q₁₆	semidirect
78. C₁₀₉	#3170	semidirect
81. #494	D₁₀₉	semidirect

H ≅ G/N

type

2. #20295431

C₂

semidirect

3. #20295376

C₂

semidirect

5. Q₁₆×Dic₁₀₉

C₂

semidirect

7. #494×C₁₀₉

C₂

semidirect

57. C₄×D₁₀₉

Q₁₆

semidirect

78. C₁₀₉

#3170

semidirect

81. #494

D₁₀₉

semidirect

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

subgroup	index
#20295581	2
#20295431	2
#20295376	2
#20295264	2
Q₁₆×Dic₁₀₉	2
C₄×C₈×D₁₀₉	2
#494×C₁₀₉	2
#3170	109

subgroup

index

Q₁₆×Dic₁₀₉

C₄×C₈×D₁₀₉

#494×C₁₀₉

#3170

109

Subgroups — #28373616

Conjugacy classes of subgroups

Complements

Maximal subgroups