Group #28371609 — subgroups

Subgroups — C₂×#20295061 #28371609

Order 13952.

512 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 229 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	C₂²×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812657	C₂	1	2	yes	maximal
3	#20295061	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16812632	C₂	1	2	yes	maximal, complemented
5	C₂×#16812631	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₁₀₉	C₄	1	8	yes	complemented
10	C₂×C₈×D₁₀₉	C₂²	1	4	yes	—
11	#16812657	—	2	4	no	—
12	#16812657	C₂²	1	8	yes	—
13	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
14	C₂×#13182006	—	2	2	no	—
15	C₂×C₄×Dic₁₀₉	—	2	2	no	—
16	#16812632	C₂²	1	8	yes	complemented
17	#16812631	C₂²	1	4	yes	complemented
18	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
20	#106×C₁₀₉	C₂²	1	4	yes	complemented
21	C₂³×D₁₀₉	D₄	1	1	yes	characteristic
22	C₂²×Dic₁₀₉	—	2	2	no	—
23	C₂²×Dic₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₄×D₁₀₉	C₂×C₄	1	4	yes	—
25	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
26	C₂×C₄×D₁₀₉	Q₈	1	4	yes	—
27	C₂×C₄×D₁₀₉	C₂³	1	2	yes	—
28	C₂×#11724784	C₂×C₄	1	4	yes	—
29	C₂×#11724784	C₂³	1	2	yes	—
30	C₈×D₁₀₉	C₂×C₄	1	16	yes	complemented
31	C₄⋊Dic₁₀₉	—	2	2	no	—
32	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
33	#13182006	—	2	8	no	—
34	C₄×Dic₁₀₉	—	2	8	no	—
35	C₂²×C₄×C₁₀₉	—	2	2	no	—
36	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
38	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
39	C₄⋊C₄×C₁₀₉	—	2	2	no	—
40	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
41	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₁₀₉	—	2	4	no	—
43	C₂²×D₁₀₉	C₄⋊C₄	1	4	yes	—
44	C₂²×D₁₀₉	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	—	2	4	no	—
46	C₂×C₄×C₁₀₉	—	4	4	no	—
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₀₉	—	2	4	no	—
52	C₂×Dic₁₀₉	—	4	4	no	—
53	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
54	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
55	C₄×D₁₀₉	C₄⋊C₄	1	8	yes	—
56	C₄×D₁₀₉	C₂²×C₄	1	4	yes	—
57	C₄×D₁₀₉	C₂×Q₈	1	4	yes	—
58	C₈×C₁₀₉	C₂²×C₄	1	4	yes	—
59	#11724784	C₂²×C₄	1	4	yes	—
60	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
61	C₂²×C₁₀₉	#135	1	4	yes	—
62	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
63	C₂×D₁₀₉	—	2	4	no	—
64	C₂×D₁₀₉	—	2	8	no	—
65	C₂×D₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
66	C₄×C₁₀₉	—	4	4	no	—
67	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
68	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
70	Dic₁₀₉	—	4	4	no	—
71	Dic₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
72	C₂×C₁₀₉	#427	1	4	yes	complemented
73	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂×#135	1	2	yes	—
75	D₁₀₉	—	2	4	no	—
76	C₂×#427	—	109	1	no	maximal, Sylow
77	C₁₀₉	C₂×#427	1	1	yes	characteristic, complemented, Sylow
78	C₂³×C₈	—	109	1	no	—
79	C₂×#117	—	109	2	no	—
80	#427	—	109	8	no	—
81	C₂×#107	—	109	2	no	—
82	C₂×#106	—	109	1	no	—
83	C₂×#106	D₁₀₉	1	1	yes	characteristic, complemented
84	C₂³×C₄	—	109	1	no	—
85	C₂²×C₈	—	109	1	no	—
86	C₂²×C₈	—	109	8	no	—
87	C₂²×C₈	—	109	4	no	—
88	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
89	#117	—	109	8	no	—
90	#117	—	218	4	no	—
91	C₂×C₄⋊C₄	—	109	2	no	—
92	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
93	C₂×C₂²⋊C₄	—	218	2	no	—
94	C₂×C₄²	—	218	2	no	—
95	#107	—	109	8	no	—
96	#106	—	109	4	no	—
97	#106	C₂×D₁₀₉	1	4	yes	complemented
98	C₂⁴	—	109	1	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	109	1	no	—
101	C₂²×C₄	—	109	4	no	—
102	C₂²×C₄	—	109	2	no	—
103	C₂²×C₄	—	109	4	no	—
104	C₂²×C₄	—	109	2	no	—
105	C₂²×C₄	—	218	2	no	—
106	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
107	C₂×C₈	—	109	4	no	—
108	C₂×C₈	—	109	2	no	—
109	C₂×C₈	—	109	16	no	—
110	C₂×C₈	C₄×D₁₀₉	1	4	yes	—
111	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	109	4	no	—
114	C₄⋊C₄	—	218	2	no	—
115	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
116	C₂²⋊C₄	—	218	8	no	—
117	C₄²	—	218	8	no	—
118	C₂³	—	109	4	no	—
119	C₂³	—	109	2	no	—
120	C₂³	—	218	4	no	—
121	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	109	4	no	—
125	C₂×C₄	—	109	2	no	—
126	C₂×C₄	—	109	8	no	—
127	C₂×C₄	—	109	4	no	—
128	C₂×C₄	—	109	4	no	—
129	C₂×C₄	—	436	4	no	—
130	C₂×C₄	—	218	4	no	—
131	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
132	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
134	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
135	C₈	—	109	4	no	—
136	C₈	C₂×C₄×D₁₀₉	1	4	yes	—
137	C₂²	—	109	4	no	—
138	C₂²	—	218	4	no	—
139	C₂²	—	218	8	no	—
140	C₂²	C₄⋊C₄×D₁₀₉	1	2	yes	—
141	C₂²	#16812720	1	4	yes	—
142	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
143	C₄	—	4	4	no	—
144	C₄	—	109	4	no	—
145	C₄	—	436	4	no	—
146	C₄	C₄⋊C₄×D₁₀₉	1	2	yes	—
147	C₄	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
148	C₄	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
149	C₂	—	218	4	no	—
150	C₂	#20295061	1	4	yes	direct product factor
151	C₂	C₂×C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
152	C₂	C₂×#16812720	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#16812657

C₂

yes

maximal

#20295061

C₂

yes

maximal, direct product factor

C₂×#16812632

C₂

yes

maximal, complemented

C₂×#16812631

C₂

yes

maximal, characteristic, complemented

C₂×#106×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₀₉

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₄

yes

complemented

C₂×C₈×D₁₀₉

C₂²

yes

—

#16812657

—

#16812657

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182006

—

C₂×C₄×Dic₁₀₉

—

#16812632

C₂²

yes

complemented

#16812631

C₂²

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

C₂²

yes

complemented

C₂³×D₁₀₉

D₄

yes

characteristic

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

D₄

yes

—

C₂×C₄×D₁₀₉

Q₈

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂³

yes

—

C₈×D₁₀₉

C₂×C₄

yes

complemented

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182006

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₄⋊C₄

yes

—

C₂²×D₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₄×D₁₀₉

C₄⋊C₄

yes

—

C₄×D₁₀₉

C₂²×C₄

yes

—

C₄×D₁₀₉

C₂×Q₈

yes

—

C₈×C₁₀₉

C₂²×C₄

yes

—

#11724784

C₂²×C₄

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂×C₄⋊C₄

yes

—

C₂×C₁₀₉

#427

yes

complemented

C₂×C₁₀₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×#135

yes

—

D₁₀₉

—

C₂×#427

—

109

maximal, Sylow

C₁₀₉

C₂×#427

yes

characteristic, complemented, Sylow

C₂³×C₈

—

109

—

C₂×#117

—

109

—

#427

—

109

—

C₂×#107

—

109

—

C₂×#106

—

109

—

C₂×#106

D₁₀₉

yes

characteristic, complemented

C₂³×C₄

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

#117

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₂²⋊C₄

—

218

—

C₂×C₄²

—

218

—

#107

—

109

—

#106

—

109

—

#106

C₂×D₁₀₉

yes

complemented

C₂⁴

—

109

—

C₂²×C₄

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

—

109

—

102

C₂²×C₄

—

109

—

103

C₂²×C₄

—

109

—

104

C₂²×C₄

—

109

—

105

C₂²×C₄

—

218

—

106

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

107

C₂×C₈

—

109

—

108

C₂×C₈

—

109

—

109

C₂×C₈

—

109

—

110

C₂×C₈

C₄×D₁₀₉

yes

—

111

C₂×C₈

C₂²×D₁₀₉

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

109

—

114

C₄⋊C₄

—

218

—

115

C₄⋊C₄

C₂²×D₁₀₉

yes

—

116

C₂²⋊C₄

—

218

—

117

C₄²

—

218

—

118

C₂³

—

109

—

119

C₂³

—

109

—

120

C₂³

—

218

—

121

C₂³

D₄×D₁₀₉

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

109

—

127

C₂×C₄

—

109

—

128

C₂×C₄

—

109

—

129

C₂×C₄

—

436

—

130

C₂×C₄

—

218

—

131

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

132

C₂×C₄

D₄×D₁₀₉

yes

characteristic

133

C₂×C₄

Q₈×D₁₀₉

yes

—

134

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

135

C₈

—

109

—

136

C₈

C₂×C₄×D₁₀₉

yes

—

137

C₂²

—

109

—

138

C₂²

—

218

—

139

C₂²

—

218

—

140

C₂²

C₄⋊C₄×D₁₀₉

yes

—

141

C₂²

#16812720

yes

—

142

C₂²

C₂×D₄×D₁₀₉

yes

characteristic

143

C₄

—

144

C₄

—

109

—

145

C₄

—

436

—

146

C₄

C₄⋊C₄×D₁₀₉

yes

—

147

C₄

C₂²×C₄×D₁₀₉

yes

characteristic

148

C₄

C₂×Q₈×D₁₀₉

yes

characteristic

149

C₂

—

218

—

150

C₂

#20295061

yes

direct product factor

151

C₂

C₂×C₄⋊C₄×D₁₀₉

yes

characteristic

152

C₂

C₂×#16812720

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#106×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂×D₄
prime radical	C₂³×D₁₀₉	D₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×#106	D₁₀₉
hypocenter	C₁₀₉	C₂×#427
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#106×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂×D₄

prime radical

C₂³×D₁₀₉

D₄

cosocle

C₂×C₄×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #20295061	C₂	direct (N × H)
4. C₂×#16812632	C₂	semidirect
5. C₂×#16812631	C₂	semidirect
6. C₂×#106×C₁₀₉	C₂	semidirect
9. C₂×C₈×D₁₀₉	C₄	semidirect
16. #16812632	C₂²	semidirect
17. #16812631	C₂²	semidirect
20. #106×C₁₀₉	C₂²	semidirect
30. C₈×D₁₀₉	C₂×C₄	semidirect
72. C₂×C₁₀₉	#427	semidirect
77. C₁₀₉	C₂×#427	semidirect
83. C₂×#106	D₁₀₉	semidirect
97. #106	C₂×D₁₀₉	semidirect
150. C₂	#20295061	direct (N × H)

H ≅ G/N

type

3. #20295061

C₂

direct (N × H)

4. C₂×#16812632

C₂

semidirect

5. C₂×#16812631

C₂

semidirect

6. C₂×#106×C₁₀₉

C₂

semidirect

9. C₂×C₈×D₁₀₉

C₄

semidirect

16. #16812632

C₂²

semidirect

17. #16812631

C₂²

semidirect

20. #106×C₁₀₉

C₂²

semidirect

30. C₈×D₁₀₉

C₂×C₄

semidirect

72. C₂×C₁₀₉

#427

semidirect

77. C₁₀₉

C₂×#427

semidirect

83. C₂×#106

D₁₀₉

semidirect

97. #106

C₂×D₁₀₉

semidirect

150. C₂

#20295061

direct (N × H)

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

subgroup	index
C₂²×C₈×D₁₀₉	2
C₂×#16812657	2
#20295061	2
C₂×#16812632	2
C₂×#16812631	2
C₂×#106×C₁₀₉	2
C₂×#427	109

subgroup

index

C₂²×C₈×D₁₀₉

C₂×#106×C₁₀₉

109

Subgroups — C₂×#20295061 #28371609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295061 #28371609

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295061 #28371609