Group #13136254 — subgroups

Subgroups — #13136254

Order 1728.

532 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 211 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×Dic₅₄	C₂	1	1	yes	maximal, characteristic
2	C₂₇×#117	C₂	1	1	yes	maximal, characteristic
3	#11720175	C₂	1	1	yes	maximal, characteristic
4	#11720136	C₂	1	4	yes	maximal, complemented
5	#11720126	C₂	1	4	yes	maximal
6	C₄×Dic₅₄	C₂	1	4	yes	maximal, complemented
7	#10589618	—	3	1	no	maximal
8	C₂²×C₄×C₂₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₇	C₂²	1	2	yes	—
10	C₂×Dic₅₄	C₄	1	8	yes	complemented
11	C₂×Dic₅₄	C₂²	1	2	yes	—
12	C₂×Dic₅₄	C₂²	1	2	yes	—
13	C₄⋊C₄×C₂₇	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₂₇	C₂²	1	2	yes	—
15	C₄²×C₂₇	C₂²	1	2	yes	—
16	#88372	C₂²	1	2	yes	—
17	C₄⋊Dic₂₇	C₂²	1	2	yes	—
18	Dic₂₇⋊C₄	C₂²	1	8	yes	—
19	C₄×Dic₂₇	C₂²	1	8	yes	—
20	C₄×Dic₂₇	C₂²	1	2	yes	—
21	C₂²×Dic₁₈	—	3	1	no	—
22	C₉×#117	D₃	1	1	yes	characteristic
23	#63488	—	3	1	no	—
24	#63449	—	3	4	no	—
25	#63439	—	3	4	no	—
26	C₄×Dic₁₈	—	3	4	no	—
27	C₂³×C₂₇	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₂₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×C₂₇	C₂³	1	4	yes	—
30	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₂₇	C₂×C₄	1	8	yes	—
33	C₂×Dic₂₇	C₂³	1	4	yes	—
34	C₂×Dic₂₇	C₂³	1	4	yes	—
35	Dic₅₄	C₂×C₄	1	8	yes	—
36	Dic₅₄	C₂×C₄	1	8	yes	—
37	#5516	—	9	1	no	—
38	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
39	C₂²×Dic₉	—	3	2	no	—
40	C₂×Dic₁₈	—	3	8	no	—
41	C₂×Dic₁₈	—	3	2	no	—
42	C₂×Dic₁₈	—	3	2	no	—
43	C₉×C₄⋊C₄	C₂×D₃	1	2	yes	—
44	C₉×C₂²⋊C₄	C₂×D₃	1	2	yes	—
45	C₄²×C₉	C₂×D₃	1	2	yes	—
46	#3676	—	3	2	no	—
47	C₄⋊Dic₉	—	3	2	no	—
48	Dic₉⋊C₄	—	3	8	no	—
49	C₄×Dic₉	—	3	2	no	—
50	C₄×Dic₉	—	3	8	no	—
51	C₂²×C₂₇	—	2	2	no	—
52	C₂²×C₂₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
54	C₄×C₂₇	—	2	4	no	—
55	C₄×C₂₇	C₂²×C₄	1	2	yes	—
56	C₄×C₂₇	C₂²×C₄	1	2	yes	—
57	Dic₂₇	—	2	4	no	—
58	Dic₂₇	C₂²×C₄	1	8	yes	—
59	C₂²×Dic₆	—	9	1	no	—
60	C₃×#117	D₉	1	1	yes	characteristic
61	#926	—	9	1	no	—
62	#887	—	9	4	no	—
63	#877	—	9	4	no	—
64	C₄×Dic₆	—	9	4	no	—
65	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
66	C₂×C₄×C₉	C₄×D₃	1	4	yes	—
67	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
68	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
69	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
70	C₂×Dic₉	—	3	8	no	—
71	C₂×Dic₉	—	3	4	no	—
72	C₂×Dic₉	—	3	4	no	—
73	Dic₁₈	—	3	8	no	—
74	Dic₁₈	—	3	8	no	—
75	#519	—	27	1	no	Sylow
76	C₂×C₂₇	—	2	2	no	—
77	C₂×C₂₇	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₂₇	2₋¹⁺⁴	1	2	yes	—
79	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
80	C₂²×Dic₃	—	9	2	no	—
81	C₂×Dic₆	—	9	8	no	—
82	C₂×Dic₆	—	9	2	no	—
83	C₂×Dic₆	—	9	2	no	—
84	C₃×C₄⋊C₄	C₂×D₉	1	2	yes	—
85	C₃×C₂²⋊C₄	C₂×D₉	1	2	yes	—
86	C₃×C₄²	C₂×D₉	1	2	yes	—
87	#217	—	9	2	no	—
88	C₄⋊Dic₃	—	9	2	no	—
89	Dic₃⋊C₄	—	9	8	no	—
90	C₄×Dic₃	—	9	8	no	—
91	C₄×Dic₃	—	9	2	no	—
92	C₂²×C₉	—	2	2	no	—
93	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
94	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
95	C₄×C₉	—	2	4	no	—
96	C₄×C₉	C₂×C₄×D₃	1	2	yes	—
97	C₄×C₉	C₂×C₄×D₃	1	2	yes	—
98	Dic₉	—	3	8	no	—
99	Dic₉	—	6	4	no	—
100	C₂²×Q₈	—	27	1	no	—
101	C₄×Q₈	—	27	4	no	—
102	C₄×Q₈	—	27	4	no	—
103	#117	—	27	1	no	—
104	#117	—	27	4	no	—
105	#117	D₂₇	1	1	yes	characteristic
106	C₂₇	#519	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
108	C₂×C₃×C₄	C₄×D₉	1	4	yes	—
109	C₂×C₃×C₄	C₂²×D₉	1	4	yes	—
110	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
111	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
112	C₂×Dic₃	—	9	8	no	—
113	C₂×Dic₃	—	9	4	no	—
114	C₂×Dic₃	—	9	4	no	—
115	Dic₆	—	9	8	no	—
116	Dic₆	—	9	8	no	—
117	C₂×C₉	—	2	2	no	—
118	C₂×C₉	C₂²×C₄×D₃	1	1	yes	characteristic
119	C₂×C₉	#1010	1	2	yes	—
120	C₂×Q₈	—	27	8	no	—
121	C₂×Q₈	—	27	2	no	—
122	C₂×Q₈	—	27	2	no	—
123	C₂²×C₄	—	27	2	no	—
124	C₂²×C₄	C₂×D₂₇	1	1	yes	characteristic
125	C₄⋊C₄	—	27	8	no	—
126	C₄⋊C₄	—	27	2	no	—
127	C₄⋊C₄	C₂×D₂₇	1	2	yes	—
128	C₂²⋊C₄	—	27	2	no	—
129	C₂²⋊C₄	C₂×D₂₇	1	2	yes	—
130	C₄²	—	27	8	no	—
131	C₄²	—	27	2	no	—
132	C₄²	C₂×D₂₇	1	2	yes	—
133	C₂²×C₃	—	2	2	no	—
134	C₂²×C₃	C₂×C₄×D₉	1	2	yes	—
135	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
136	C₃×C₄	—	2	4	no	—
137	C₃×C₄	C₂×C₄×D₉	1	2	yes	—
138	C₃×C₄	C₂×C₄×D₉	1	2	yes	—
139	Dic₃	—	9	8	no	—
140	Dic₃	—	18	4	no	—
141	C₉	#5516	1	1	yes	characteristic
142	C₂³	C₂²×D₂₇	1	1	yes	characteristic
143	Q₈	—	27	8	no	—
144	Q₈	—	27	8	no	—
145	C₂×C₄	—	27	4	no	—
146	C₂×C₄	—	27	8	no	—
147	C₂×C₄	—	27	4	no	—
148	C₂×C₄	C₄×D₂₇	1	4	yes	—
149	C₂×C₄	C₂²×D₂₇	1	4	yes	—
150	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
151	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
152	C₂×C₃	—	2	2	no	—
153	C₂×C₃	C₂²×C₄×D₉	1	1	yes	characteristic
154	C₂×C₃	#63716	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	C₂×C₄×D₂₇	1	2	yes	—
157	C₂²	C₂³×D₂₇	1	1	yes	characteristic
158	C₄	—	2	4	no	—
159	C₄	—	27	8	no	—
160	C₄	—	54	4	no	—
161	C₄	C₂×C₄×D₂₇	1	2	yes	—
162	C₄	C₂×C₄×D₂₇	1	2	yes	—
163	C₃	#10589618	1	1	yes	characteristic
164	C₂	—	2	2	no	—
165	C₂	C₂²×C₄×D₂₇	1	1	yes	characteristic
166	C₂	#11720676	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₄

C₂

yes

maximal, characteristic

C₂₇×#117

C₂

yes

maximal, characteristic

#11720175

C₂

yes

maximal, characteristic

#11720136

C₂

yes

maximal, complemented

#11720126

C₂

yes

maximal

C₄×Dic₅₄

C₂

yes

maximal, complemented

#10589618

—

maximal

C₂²×C₄×C₂₇

C₂²

yes

characteristic

C₂²×Dic₂₇

C₂²

yes

—

C₂×Dic₅₄

C₄

yes

complemented

C₂×Dic₅₄

C₂²

yes

—

C₂×Dic₅₄

C₂²

yes

—

C₄⋊C₄×C₂₇

C₂²

yes

—

C₂²⋊C₄×C₂₇

C₂²

yes

—

C₄²×C₂₇

C₂²

yes

—

#88372

C₂²

yes

—

C₄⋊Dic₂₇

C₂²

yes

—

Dic₂₇⋊C₄

C₂²

yes

—

C₄×Dic₂₇

C₂²

yes

—

C₄×Dic₂₇

C₂²

yes

—

C₂²×Dic₁₈

—

C₉×#117

D₃

yes

—

—

—

C₄×Dic₁₈

—

C₂³×C₂₇

C₂³

yes

characteristic

C₂×C₄×C₂₇

C₂×C₄

yes

—

C₂×C₄×C₂₇

C₂³

yes

—

C₂×C₄×C₂₇

C₂³

yes

characteristic

C₂×C₄×C₂₇

C₂³

yes

characteristic

C₂×Dic₂₇

C₂×C₄

yes

—

C₂×Dic₂₇

C₂³

yes

—

C₂×Dic₂₇

C₂³

yes

—

Dic₅₄

C₂×C₄

yes

—

Dic₅₄

C₂×C₄

yes

—

#5516

—

C₂²×C₄×C₉

C₂×D₃

yes

characteristic

C₂²×Dic₉

—

C₂×Dic₁₈

—

C₂×Dic₁₈

—

C₂×Dic₁₈

—

C₉×C₄⋊C₄

C₂×D₃

yes

—

C₉×C₂²⋊C₄

C₂×D₃

yes

—

C₄²×C₉

C₂×D₃

yes

—

#3676

—

C₄⋊Dic₉

—

Dic₉⋊C₄

—

C₄×Dic₉

—

C₄×Dic₉

—

C₂²×C₂₇

—

C₂²×C₂₇

C₂²×C₄

yes

—

C₂²×C₂₇

C₂⁴

yes

characteristic

C₄×C₂₇

—

C₄×C₂₇

C₂²×C₄

yes

—

C₄×C₂₇

C₂²×C₄

yes

—

Dic₂₇

—

Dic₂₇

C₂²×C₄

yes

—

C₂²×Dic₆

—

C₃×#117

D₉

yes

—

—

—

C₄×Dic₆

—

C₂³×C₉

C₂²×D₃

yes

characteristic

C₂×C₄×C₉

C₄×D₃

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

#519

—

Sylow

C₂×C₂₇

—

C₂×C₂₇

C₂³×C₄

yes

characteristic

C₂×C₂₇

2₋¹⁺⁴

yes

—

C₂²×C₃×C₄

C₂×D₉

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×D₉

yes

—

C₃×C₂²⋊C₄

C₂×D₉

yes

—

C₃×C₄²

C₂×D₉

yes

—

#217

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×C₄×D₃

yes

—

C₂²×C₉

C₂³×D₃

yes

characteristic

C₄×C₉

—

C₄×C₉

C₂×C₄×D₃

yes

—

C₄×C₉

C₂×C₄×D₃

yes

—

Dic₉

—

Dic₉

—

100

C₂²×Q₈

—

101

C₄×Q₈

—

102

C₄×Q₈

—

103

#117

—

104

#117

—

105

#117

D₂₇

yes

characteristic

106

C₂₇

#519

yes

characteristic, complemented, Sylow

107

C₂³×C₃

C₂²×D₉

yes

characteristic

108

C₂×C₃×C₄

C₄×D₉

yes

—

109

C₂×C₃×C₄

C₂²×D₉

yes

—

110

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

111

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

112

C₂×Dic₃

—

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

Dic₆

—

116

Dic₆

—

117

C₂×C₉

—

118

C₂×C₉

C₂²×C₄×D₃

yes

characteristic

119

C₂×C₉

#1010

yes

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

C₂×D₂₇

yes

characteristic

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

C₂×D₂₇

yes

—

128

C₂²⋊C₄

—

129

C₂²⋊C₄

C₂×D₂₇

yes

—

130

C₄²

—

131

C₄²

—

132

C₄²

C₂×D₂₇

yes

—

133

C₂²×C₃

—

134

C₂²×C₃

C₂×C₄×D₉

yes

—

135

C₂²×C₃

C₂³×D₉

yes

characteristic

136

C₃×C₄

—

137

C₃×C₄

C₂×C₄×D₉

yes

—

138

C₃×C₄

C₂×C₄×D₉

yes

—

139

Dic₃

—

140

Dic₃

—

141

C₉

#5516

yes

characteristic

142

C₂³

C₂²×D₂₇

yes

characteristic

143

Q₈

—

144

Q₈

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₄×D₂₇

yes

—

149

C₂×C₄

C₂²×D₂₇

yes

—

150

C₂×C₄

C₂²×D₂₇

yes

characteristic

151

C₂×C₄

C₂²×D₂₇

yes

characteristic

152

C₂×C₃

—

153

C₂×C₃

C₂²×C₄×D₉

yes

characteristic

154

C₂×C₃

#63716

yes

—

155

C₂²

—

156

C₂²

C₂×C₄×D₂₇

yes

—

157

C₂²

C₂³×D₂₇

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₂×C₄×D₂₇

yes

—

162

C₄

C₂×C₄×D₂₇

yes

—

163

C₃

#10589618

yes

characteristic

164

C₂

—

165

C₂

C₂²×C₄×D₂₇

yes

characteristic

166

C₂

#11720676

yes

—

	subgroup	G/H
center	C₂²	C₂³×D₂₇
commutator	C₂×C₂₇	C₂³×C₄
Frattini	C₂²×C₉	C₂³×D₃
Fitting	C₂₇×#117	C₂
socle	C₂²×C₃	C₂³×D₉
prime radical	C₂³×C₃	C₂²×D₉
cosocle	C₂²×C₂₇	C₂⁴
hypercenter	#117	D₂₇
hypocenter	C₂₇	#519
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center

C₂²

C₂³×D₂₇

commutator

C₂×C₂₇

C₂³×C₄

Frattini

C₂²×C₉

C₂³×D₃

Fitting

C₂₇×#117

C₂

socle

C₂²×C₃

C₂³×D₉

prime radical

C₂³×C₃

C₂²×D₉

cosocle

C₂²×C₂₇

C₂⁴

D₂₇

C₂₇

G ▷ C₂×C₂₇ ▷ C₂₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
4. #11720136	C₂	semidirect
6. C₄×Dic₅₄	C₂	semidirect
10. C₂×Dic₅₄	C₄	semidirect
106. C₂₇	#519	semidirect

H ≅ G/N

type

4. #11720136

C₂

semidirect

6. C₄×Dic₅₄

C₂

semidirect

10. C₂×Dic₅₄

C₄

semidirect

106. C₂₇

#519

semidirect

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

subgroup	index
C₂²×Dic₅₄	2
C₂₇×#117	2
#11720175	2
#11720136	2
#11720126	2
C₄×Dic₅₄	2
#10589618	3

subgroup

index

C₂²×Dic₅₄

C₂₇×#117

C₄×Dic₅₄

Subgroups — #13136254

Conjugacy classes of subgroups

Complements

Maximal subgroups