Group #28362803 — subgroups

Subgroups — C₈×C₂²⋊Dic₁₀₉ #28362803

Order 13952.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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×C₂²⋊#11724784	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
3	C₄×C₂²⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294780	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	2	yes	—
7	C₂×C₂²⋊Dic₁₀₉	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
9	C₂²⋊#11724784	C₄	1	4	yes	complemented
10	C₈×Dic₁₀₉	—	2	4	no	—
11	#16812606	C₄	1	2	yes	—
12	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	—	2	4	no	—
14	C₂²×C₈×C₁₀₉	C₄	1	4	yes	complemented
15	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×Dic₁₀₉	C₂×C₄	1	4	yes	—
18	C₂×#11724784	—	2	2	no	—
19	C₂×#11724784	—	2	2	no	—
20	C₂×#11724784	C₂×C₄	1	4	yes	—
21	C₂²⋊Dic₁₀₉	C₈	1	8	yes	direct product factor
22	C₄×Dic₁₀₉	—	2	2	no	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	C₂⁴×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₀₉	—	2	2	no	—
26	C₂²×C₄×C₁₀₉	—	2	2	no	—
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	—	2	8	no	—
33	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₂×C₈×C₁₀₉	D₄	1	4	yes	—
36	C₂³×C₁₀₉	—	2	2	no	—
37	C₂³×C₁₀₉	—	2	2	no	—
38	C₂³×C₁₀₉	C₄²	1	2	yes	—
39	C₂³×C₁₀₉	C₂×C₈	1	4	yes	—
40	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₀₉	#41	1	2	yes	—
50	C₂×Dic₁₀₉	—	2	4	no	—
51	C₂×Dic₁₀₉	—	2	4	no	—
52	C₂×Dic₁₀₉	C₂×C₈	1	8	yes	complemented
53	C₈×C₁₀₉	—	2	2	no	—
54	C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	complemented
55	#11724784	—	2	4	no	—
56	C₂²×C₁₀₉	—	2	2	no	—
57	C₂²×C₁₀₉	—	2	2	no	—
58	C₂²×C₁₀₉	—	2	8	no	—
59	C₂²×C₁₀₉	C₄×C₈	1	4	yes	complemented
60	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
61	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
62	C₂²×C₁₀₉	C₂²×C₈	1	2	yes	—
63	C₂²×C₁₀₉	#131	1	2	yes	—
64	C₄×C₁₀₉	—	2	2	no	—
65	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₄×C₁₀₉	#117	1	1	yes	characteristic
67	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
68	Dic₁₀₉	—	2	8	no	—
69	C₂×C₁₀₉	—	2	4	no	—
70	C₂×C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
72	C₂×C₁₀₉	#405	1	1	yes	characteristic
73	C₂×C₁₀₉	C₈×D₄	1	4	yes	—
74	C₈×C₂²⋊C₄	—	109	1	no	maximal, Sylow
75	C₁₀₉	C₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
76	C₂³×C₈	D₁₀₉	1	1	yes	characteristic
77	C₂×C₂²⋊C₈	—	109	1	no	—
78	C₂×C₄×C₈	—	109	2	no	—
79	C₄×C₂²⋊C₄	—	109	1	no	—
80	#336	—	109	2	no	—
81	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
82	C₂²×C₈	—	2	4	no	—
83	C₂²×C₈	—	109	2	no	—
84	C₂²×C₈	Dic₁₀₉	1	4	yes	complemented
85	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
87	C₂×C₂²⋊C₄	—	109	2	no	—
88	C₂×C₄²	—	109	2	no	—
89	C₂²⋊C₈	—	109	4	no	—
90	C₄×C₈	—	218	4	no	—
91	#95	—	109	2	no	—
92	C₂⁴	C₄×D₁₀₉	1	1	yes	characteristic
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	109	4	no	—
96	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
98	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	—	2	8	no	—
102	C₂×C₈	—	109	4	no	—
103	C₂×C₈	—	218	2	no	—
104	C₂×C₈	—	218	2	no	—
105	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
106	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
107	C₂×C₈	C₂²⋊D₁₀₉	1	4	yes	—
108	C₂²⋊C₄	—	109	8	no	—
109	C₄²	—	218	2	no	—
110	C₄²	—	218	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	2	2	no	—
113	C₂³	C₄×Dic₁₀₉	1	2	yes	—
114	C₂³	C₈×D₁₀₉	1	4	yes	—
115	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	109	8	no	—
121	C₂×C₄	—	218	4	no	—
122	C₂×C₄	—	218	4	no	—
123	C₂×C₄	C₄×Dic₁₀₉	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
125	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
127	C₂×C₄	#13182029	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	218	4	no	—
130	C₈	C₂²⋊Dic₁₀₉	1	4	yes	direct product factor
131	C₂²	—	2	2	no	—
132	C₂²	—	2	8	no	—
133	C₂²	—	2	2	no	—
134	C₂²	C₈×Dic₁₀₉	1	4	yes	complemented
135	C₂²	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
136	C₂²	C₄×C₂²⋊D₁₀₉	1	2	yes	—
137	C₂²	C₂×C₈×D₁₀₉	1	2	yes	—
138	C₂²	#16812705	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	218	8	no	—
141	C₄	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
142	C₄	#16812658	1	1	yes	characteristic
143	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
144	C₂	—	2	4	no	—
145	C₂	C₄×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
146	C₂	C₂×C₈×Dic₁₀₉	1	1	yes	characteristic
147	C₂	#20294981	1	1	yes	characteristic
148	C₂	C₈×C₂²⋊D₁₀₉	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#11724784

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, complemented

C₄×C₂²⋊Dic₁₀₉

C₂

yes

maximal, characteristic

#20294780

C₂

yes

maximal, complemented

C₂³×C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₂²⋊#11724784

C₄

yes

complemented

C₈×Dic₁₀₉

—

#16812606

C₄

yes

—

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₂²⋊Dic₁₀₉

C₈

yes

direct product factor

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂⁴×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₄²

yes

—

C₂³×C₁₀₉

C₂×C₈

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₈

yes

complemented

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂²⋊C₄

yes

complemented

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×C₈

yes

complemented

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

—

C₂²×C₁₀₉

#131

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

#405

yes

characteristic

C₂×C₁₀₉

C₈×D₄

yes

—

C₈×C₂²⋊C₄

—

109

maximal, Sylow

C₁₀₉

C₈×C₂²⋊C₄

yes

characteristic, complemented, Sylow

C₂³×C₈

D₁₀₉

yes

characteristic

C₂×C₂²⋊C₈

—

109

—

C₂×C₄×C₈

—

109

—

C₄×C₂²⋊C₄

—

109

—

#336

—

109

—

C₂³×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

109

—

C₂²×C₈

Dic₁₀₉

yes

complemented

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂²⋊C₈

—

109

—

C₄×C₈

—

218

—

#95

—

109

—

C₂⁴

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

109

—

103

C₂×C₈

—

218

—

104

C₂×C₈

—

218

—

105

C₂×C₈

C₂×Dic₁₀₉

yes

—

106

C₂×C₈

C₂×Dic₁₀₉

yes

—

107

C₂×C₈

C₂²⋊D₁₀₉

yes

—

108

C₂²⋊C₄

—

109

—

109

C₄²

—

218

—

110

C₄²

—

218

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₄×Dic₁₀₉

yes

—

114

C₂³

C₈×D₁₀₉

yes

—

115

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

109

—

121

C₂×C₄

—

218

—

122

C₂×C₄

—

218

—

123

C₂×C₄

C₄×Dic₁₀₉

yes

—

124

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

125

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

127

C₂×C₄

#13182029

yes

—

128

C₈

—

129

C₈

—

218

—

130

C₈

C₂²⋊Dic₁₀₉

yes

direct product factor

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₈×Dic₁₀₉

yes

complemented

135

C₂²

C₂×C₄×Dic₁₀₉

yes

characteristic

136

C₂²

C₄×C₂²⋊D₁₀₉

yes

—

137

C₂²

C₂×C₈×D₁₀₉

yes

—

138

C₂²

#16812705

yes

—

139

C₄

—

140

C₄

—

218

—

141

C₄

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

142

C₄

#16812658

yes

characteristic

143

C₄

C₄×C₂²⋊D₁₀₉

yes

—

144

C₂

—

145

C₂

C₄×C₂²⋊Dic₁₀₉

yes

characteristic

146

C₂

C₂×C₈×Dic₁₀₉

yes

characteristic

147

C₂

#20294981

yes

characteristic

148

C₂

C₈×C₂²⋊D₁₀₉

yes

—

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂³×C₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂²×C₄
prime radical	C₂⁴×C₁₀₉	C₂×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂³×C₈	D₁₀₉
hypocenter	C₁₀₉	C₈×C₂²⋊C₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

subgroup

G/H

center

C₂²×C₈

C₂×D₁₀₉

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂³×C₈×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂²×C₄

prime radical

C₂⁴×C₁₀₉

C₂×C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂³×C₈

D₁₀₉

hypocenter

C₁₀₉

C₈×C₂²⋊C₄

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

N	H ≅ G/N	type
2. C₂×C₈×Dic₁₀₉	C₂	semidirect
4. #20294780	C₂	semidirect
9. C₂²⋊#11724784	C₄	semidirect
14. C₂²×C₈×C₁₀₉	C₄	semidirect
21. C₂²⋊Dic₁₀₉	C₈	direct (N × H)
52. C₂×Dic₁₀₉	C₂×C₈	semidirect
54. C₈×C₁₀₉	C₂²⋊C₄	semidirect
59. C₂²×C₁₀₉	C₄×C₈	semidirect
75. C₁₀₉	C₈×C₂²⋊C₄	semidirect
84. C₂²×C₈	Dic₁₀₉	semidirect
130. C₈	C₂²⋊Dic₁₀₉	direct (N × H)
134. C₂²	C₈×Dic₁₀₉	semidirect

H ≅ G/N

type

2. C₂×C₈×Dic₁₀₉

C₂

semidirect

4. #20294780

C₂

semidirect

9. C₂²⋊#11724784

C₄

semidirect

14. C₂²×C₈×C₁₀₉

C₄

semidirect

21. C₂²⋊Dic₁₀₉

C₈

direct (N × H)

52. C₂×Dic₁₀₉

C₂×C₈

semidirect

54. C₈×C₁₀₉

C₂²⋊C₄

semidirect

59. C₂²×C₁₀₉

C₄×C₈

semidirect

75. C₁₀₉

C₈×C₂²⋊C₄

semidirect

84. C₂²×C₈

Dic₁₀₉

semidirect

130. C₈

C₂²⋊Dic₁₀₉

direct (N × H)

134. C₂²

C₈×Dic₁₀₉

semidirect

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

subgroup	index
C₂×C₂²⋊#11724784	2
C₂×C₈×Dic₁₀₉	2
C₄×C₂²⋊Dic₁₀₉	2
#20294780	2
C₂³×C₈×C₁₀₉	2
C₈×C₂²⋊C₄	109

subgroup

index

C₂×C₂²⋊#11724784

C₂×C₈×Dic₁₀₉

C₄×C₂²⋊Dic₁₀₉

#20294780

C₂³×C₈×C₁₀₉

C₈×C₂²⋊C₄

109

Subgroups — C₈×C₂²⋊Dic₁₀₉ #28362803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×C22⋊Dic109 #28362803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂²⋊Dic₁₀₉ #28362803