Group #20294948 — subgroups

Subgroups — #20294948

Order 6976.

186 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 63 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×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13182008	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13182006	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182006	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182006	C₂	1	1	yes	maximal, characteristic
6	#16812606	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₀₉	C₂²	1	1	yes	characteristic
9	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₀₉	—	2	2	no	—
13	#13182008	—	2	2	no	—
14	#13182006	—	2	2	no	—
15	#13182006	—	2	2	no	—
16	#13182006	—	2	2	no	—
17	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₉	—	2	2	no	—
21	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂²×D₁₀₉	—	2	1	no	—
23	C₂²×D₁₀₉	—	2	1	no	—
24	C₂²×D₁₀₉	—	2	1	no	—
25	C₂²×D₁₀₉	—	2	1	no	—
26	C₂²×D₁₀₉	—	2	1	no	—
27	C₂²×D₁₀₉	—	2	1	no	—
28	C₂²×D₁₀₉	Q₈	1	2	yes	—
29	C₂×C₄×C₁₀₉	—	2	1	no	—
30	C₂×C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄×C₁₀₉	—	2	1	no	—
34	C₂×C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄×C₁₀₉	—	2	1	no	—
36	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
38	C₂×Dic₁₀₉	—	2	1	no	—
39	C₂×Dic₁₀₉	—	2	1	no	—
40	C₂×Dic₁₀₉	—	2	1	no	—
41	C₂×Dic₁₀₉	—	2	1	no	—
42	C₂×Dic₁₀₉	—	2	1	no	—
43	C₂×Dic₁₀₉	—	2	1	no	—
44	C₂×Dic₁₀₉	—	2	1	no	—
45	C₂×Dic₁₀₉	—	2	1	no	—
46	C₂×Dic₁₀₉	D₄	1	2	yes	—
47	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₉	#41	1	1	yes	characteristic
52	C₂²×C₁₀₉	#41	1	1	yes	characteristic
53	C₂²×C₁₀₉	#41	1	1	yes	characteristic
54	C₂×D₁₀₉	—	2	2	no	—
55	C₂×D₁₀₉	—	2	2	no	—
56	C₂×D₁₀₉	—	2	2	no	—
57	C₂×D₁₀₉	—	4	1	no	—
58	C₂×D₁₀₉	—	4	1	no	—
59	C₂×D₁₀₉	—	4	1	no	—
60	C₂×D₁₀₉	—	4	1	no	—
61	C₄×C₁₀₉	—	2	2	no	—
62	C₄×C₁₀₉	—	2	2	no	—
63	C₄×C₁₀₉	—	4	1	no	—
64	Dic₁₀₉	—	2	2	no	—
65	Dic₁₀₉	—	4	1	no	—
66	Dic₁₀₉	—	4	1	no	—
67	C₂×C₁₀₉	#121	1	1	yes	characteristic
68	C₂×C₁₀₉	#121	1	1	yes	characteristic
69	C₂×C₁₀₉	#121	1	1	yes	characteristic
70	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₁₀₉	#126	1	1	yes	characteristic
74	D₁₀₉	—	4	2	no	—
75	C₁₀₉	#396	1	1	yes	characteristic, complemented, Sylow
76	#396	—	109	1	no	maximal, Sylow
77	C₂×C₄⋊C₄	—	109	1	no	—
78	C₂×C₄⋊C₄	—	109	1	no	—
79	C₂×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic, complemented
80	C₂×C₂²⋊C₄	—	109	1	no	—
81	C₂×C₂²⋊C₄	—	109	1	no	—
82	C₂×C₂²⋊C₄	—	109	1	no	—
83	#95	—	109	1	no	—
84	C₂⁴	—	109	1	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	109	1	no	—
87	C₂²×C₄	—	109	1	no	—
88	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
90	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	218	2	no	—
93	C₄⋊C₄	—	218	2	no	—
94	C₂²⋊C₄	—	218	2	no	—
95	C₂²⋊C₄	—	218	2	no	—
96	C₂²⋊C₄	—	218	2	no	—
97	C₂³	—	109	2	no	—
98	C₂³	—	218	1	no	—
99	C₂³	—	218	1	no	—
100	C₂³	—	218	1	no	—
101	C₂³	—	218	1	no	—
102	C₂³	—	218	1	no	—
103	C₂³	—	218	1	no	—
104	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	109	2	no	—
113	C₂×C₄	—	218	1	no	—
114	C₂×C₄	—	218	1	no	—
115	C₂×C₄	—	218	1	no	—
116	C₂×C₄	—	218	1	no	—
117	C₂×C₄	—	218	1	no	—
118	C₂×C₄	—	218	1	no	—
119	C₂×C₄	—	218	1	no	—
120	C₂×C₄	—	218	1	no	—
121	C₂×C₄	D₄₃₆	1	2	yes	—
122	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
123	C₂²	—	436	1	no	—
124	C₂²	—	436	1	no	—
125	C₂²	—	436	1	no	—
126	C₂²	—	436	1	no	—
127	C₂²	—	218	2	no	—
128	C₂²	—	218	2	no	—
129	C₂²	—	218	2	no	—
130	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
131	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
132	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
133	C₂²	Q₈×D₁₀₉	1	1	yes	characteristic
134	C₂²	#13182028	1	1	yes	characteristic
135	C₂²	#13182029	1	1	yes	characteristic
136	C₂²	#13182030	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	—	4	1	no	—
140	C₄	—	436	1	no	—
141	C₄	—	436	1	no	—
142	C₄	—	218	2	no	—
143	C₂	—	436	2	no	—
144	C₂	#16812671	1	1	yes	characteristic
145	C₂	#16812673	1	1	yes	characteristic
146	C₂	#16812674	1	1	yes	characteristic
147	C₂	#16812675	1	1	yes	characteristic
148	C₂	#16812677	1	1	yes	characteristic
149	C₂	#16812679	1	1	yes	characteristic
150	C₂	#16812694	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182008

C₂

yes

maximal, characteristic, complemented

C₂×#13182006

C₂

yes

maximal, characteristic

C₂×#13182006

C₂

yes

maximal, characteristic

C₂×#13182006

C₂

yes

maximal, characteristic

#16812606

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

—

—

—

—

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

Q₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×Q₈

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂×C₁₀₉

#126

yes

characteristic

D₁₀₉

—

C₁₀₉

#396

yes

characteristic, complemented, Sylow

#396

—

109

maximal, Sylow

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

D₁₀₉

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

#95

—

109

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

100

C₂³

—

218

—

101

C₂³

—

218

—

102

C₂³

—

218

—

103

C₂³

—

218

—

104

C₂³

C₂²×D₁₀₉

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

109

—

113

C₂×C₄

—

218

—

114

C₂×C₄

—

218

—

115

C₂×C₄

—

218

—

116

C₂×C₄

—

218

—

117

C₂×C₄

—

218

—

118

C₂×C₄

—

218

—

119

C₂×C₄

—

218

—

120

C₂×C₄

—

218

—

121

C₂×C₄

D₄₃₆

yes

—

122

C₂×C₄

C₂²⋊D₁₀₉

yes

—

123

C₂²

—

436

—

124

C₂²

—

436

—

125

C₂²

—

436

—

126

C₂²

—

436

—

127

C₂²

—

218

—

128

C₂²

—

218

—

129

C₂²

—

218

—

130

C₂²

D₄×D₁₀₉

yes

characteristic

131

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

132

C₂²

C₂×D₄₃₆

yes

characteristic

133

C₂²

Q₈×D₁₀₉

yes

characteristic

134

C₂²

#13182028

yes

characteristic

135

C₂²

#13182029

yes

characteristic

136

C₂²

#13182030

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

436

—

141

C₄

—

436

—

142

C₄

—

218

—

143

C₂

—

436

—

144

C₂

#16812671

yes

characteristic

145

C₂

#16812673

yes

characteristic

146

C₂

#16812674

yes

characteristic

147

C₂

#16812675

yes

characteristic

148

C₂

#16812677

yes

characteristic

149

C₂

#16812679

yes

characteristic

150

C₂

#16812694

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₀₉
commutator, socle	C₂³×C₁₀₉	C₂³
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₂
hypercenter	C₂×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#396
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₀₉

commutator, socle

C₂³×C₁₀₉

C₂³

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₂

hypercenter

C₂×C₄⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
2. C₂×#13182008	C₂	semidirect
6. #16812606	C₂	semidirect
7. C₂×C₄⋊C₄×C₁₀₉	C₂	semidirect
75. C₁₀₉	#396	semidirect
79. C₂×C₄⋊C₄	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

2. C₂×#13182008

C₂

semidirect

6. #16812606

C₂

semidirect

7. C₂×C₄⋊C₄×C₁₀₉

C₂

semidirect

75. C₁₀₉

#396

semidirect

79. C₂×C₄⋊C₄

D₁₀₉

semidirect

Subgroups — #20294948

Conjugacy classes of subgroups

Complements