Group #13136458 — subgroups

Subgroups — #13136458

Order 1728.

504 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₂₇×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	#11720188	C₂	1	6	yes	maximal, complemented
3	#11720186	C₂	1	3	yes	maximal, complemented
4	#11720184	C₂	1	3	yes	maximal, complemented
5	#11720125	C₂	1	2	yes	maximal, complemented
6	#10589822	—	3	1	no	maximal
7	C₂³×D₂₇	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₂₇	C₂²	1	3	yes	—
9	C₂×D₄×C₂₇	C₂²	1	3	yes	complemented
10	C₂×#6171	C₂²	1	6	yes	—
11	C₂²×Dic₂₇	C₂²	1	3	yes	complemented
12	C₄²×C₂₇	C₂²	1	1	yes	characteristic, complemented
13	#88372	C₂²	1	3	yes	—
14	#88372	C₂²	1	3	yes	complemented
15	(C₂×D₂₇)⋊C₄	C₂²	1	6	yes	—
16	Dic₂₇⋊C₄	C₂²	1	6	yes	complemented
17	C₉×#127	D₃	1	1	yes	characteristic
18	#63501	—	3	6	no	—
19	#63499	—	3	3	no	—
20	#63497	—	3	3	no	—
21	#63438	—	3	2	no	—
22	C₂³×C₂₇	C₂³	1	1	yes	characteristic
23	C₂³×C₂₇	C₂³	1	3	yes	—
24	C₂²×D₂₇	—	4	3	no	—
25	C₂²×D₂₇	C₂³	1	2	yes	—
26	D₄×C₂₇	—	4	3	no	—
27	D₄×C₂₇	—	4	3	no	—
28	C₂×C₄×C₂₇	C₂³	1	3	yes	—
29	#6171	—	4	6	no	—
30	C₂×Dic₂₇	—	4	3	no	—
31	C₂×Dic₂₇	C₂³	1	6	yes	—
32	#5720	—	9	1	no	—
33	C₂³×D₉	—	3	1	no	—
34	C₂×D₄×C₉	C₂×D₃	1	3	yes	—
35	C₂×D₄×C₉	C₂×D₃	1	3	yes	—
36	C₂×#611	—	3	6	no	—
37	C₂²×Dic₉	—	3	3	no	—
38	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
39	#3676	—	3	3	no	—
40	#3676	—	3	3	no	—
41	(C₂×D₉)⋊C₄	—	3	6	no	—
42	Dic₉⋊C₄	—	3	6	no	—
43	C₂²×C₂₇	—	2	3	no	—
44	C₂²×C₂₇	—	2	6	no	—
45	C₂²×C₂₇	—	2	3	no	—
46	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
47	C₂×D₂₇	—	2	6	no	—
48	C₂×D₂₇	—	4	4	no	—
49	C₄×C₂₇	—	2	3	no	—
50	Dic₂₇	—	2	6	no	—
51	C₃×#127	D₉	1	1	yes	characteristic
52	#939	—	9	6	no	—
53	#937	—	9	3	no	—
54	#935	—	9	3	no	—
55	#876	—	9	2	no	—
56	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
57	C₂³×C₉	C₂²×D₃	1	3	yes	—
58	C₂²×D₉	—	3	2	no	—
59	C₂²×D₉	—	12	3	no	—
60	D₄×C₉	—	4	3	no	—
61	D₄×C₉	—	4	3	no	—
62	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
63	#611	—	12	6	no	—
64	C₂×Dic₉	—	3	6	no	—
65	C₂×Dic₉	—	12	3	no	—
66	#560	—	27	1	no	Sylow
67	C₂×C₂₇	—	4	1	no	—
68	C₂×C₂₇	—	4	3	no	—
69	C₂×C₂₇	H(5,𝔽₂)	1	3	yes	—
70	D₂₇	—	4	2	no	—
71	C₂³×D₃	—	9	1	no	—
72	C₂×C₃×D₄	C₂×D₉	1	3	yes	—
73	C₂×C₃×D₄	C₂×D₉	1	3	yes	—
74	C₂×#67	—	9	6	no	—
75	C₂²×Dic₃	—	9	3	no	—
76	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
77	#217	—	9	3	no	—
78	#217	—	9	3	no	—
79	(C₂×D₃)⋊C₄	—	9	6	no	—
80	Dic₃⋊C₄	—	9	6	no	—
81	C₂²×C₉	—	2	3	no	—
82	C₂²×C₉	—	2	6	no	—
83	C₂²×C₉	—	2	3	no	—
84	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
85	C₂×D₉	—	6	6	no	—
86	C₂×D₉	—	12	4	no	—
87	C₄×C₉	—	2	3	no	—
88	Dic₉	—	6	6	no	—
89	#127	D₂₇	1	1	yes	characteristic, complemented
90	#126	—	27	2	no	—
91	#123	—	27	3	no	—
92	#121	—	27	6	no	—
93	#120	—	27	3	no	—
94	C₂₇	#560	1	1	yes	characteristic, complemented, Sylow
95	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
96	C₂³×C₃	C₂²×D₉	1	3	yes	—
97	C₂²×D₃	—	9	2	no	—
98	C₂²×D₃	—	36	3	no	—
99	C₃×D₄	—	4	3	no	—
100	C₃×D₄	—	4	3	no	—
101	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
102	#67	—	36	6	no	—
103	C₂×Dic₃	—	9	6	no	—
104	C₂×Dic₃	—	36	3	no	—
105	C₂×C₉	—	4	1	no	—
106	C₂×C₉	—	4	3	no	—
107	C₂×C₉	#1004	1	3	yes	—
108	D₉	—	12	2	no	—
109	C₂⁴	—	27	1	no	—
110	C₂×D₄	—	27	6	no	—
111	C₂×D₄	C₂×D₂₇	1	3	yes	—
112	C₂×D₄	C₂×D₂₇	1	3	yes	complemented
113	C₂²×C₄	—	27	3	no	—
114	C₄⋊C₄	—	27	6	no	—
115	C₂²⋊C₄	—	27	3	no	—
116	C₂²⋊C₄	—	27	3	no	—
117	C₂²⋊C₄	—	27	6	no	—
118	C₄²	C₂×D₂₇	1	1	yes	characteristic, complemented
119	C₂²×C₃	—	2	3	no	—
120	C₂²×C₃	—	2	6	no	—
121	C₂²×C₃	—	2	3	no	—
122	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
123	C₂×D₃	—	18	6	no	—
124	C₂×D₃	—	36	4	no	—
125	C₃×C₄	—	2	3	no	—
126	Dic₃	—	18	6	no	—
127	C₉	#5720	1	1	yes	characteristic
128	C₂³	—	27	2	no	—
129	C₂³	—	108	3	no	—
130	C₂³	C₂²×D₂₇	1	1	yes	characteristic
131	C₂³	C₂²×D₂₇	1	3	yes	—
132	D₄	—	4	3	no	—
133	D₄	—	4	3	no	—
134	D₄	—	108	6	no	—
135	C₂×C₄	—	27	6	no	—
136	C₂×C₄	—	108	3	no	—
137	C₂×C₄	C₂²×D₂₇	1	3	yes	—
138	C₂×C₃	—	4	1	no	—
139	C₂×C₃	—	4	3	no	—
140	C₂×C₃	#63710	1	3	yes	—
141	D₃	—	36	2	no	—
142	C₂²	—	2	3	no	—
143	C₂²	—	2	6	no	—
144	C₂²	—	2	3	no	—
145	C₂²	—	54	6	no	—
146	C₂²	—	108	4	no	—
147	C₂²	C₂³×D₂₇	1	1	yes	characteristic
148	C₄	—	2	3	no	—
149	C₄	—	54	6	no	—
150	C₃	#10589822	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	—	4	3	no	—
153	C₂	—	108	2	no	—
154	C₂	#11720670	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₇×#127

C₂

yes

maximal, characteristic, complemented

#11720188

C₂

yes

maximal, complemented

#11720186

C₂

yes

maximal, complemented

#11720184

C₂

yes

maximal, complemented

#11720125

C₂

yes

maximal, complemented

#10589822

—

maximal

C₂³×D₂₇

C₂²

yes

characteristic

C₂×D₄×C₂₇

C₂²

yes

—

C₂×D₄×C₂₇

C₂²

yes

complemented

C₂×#6171

C₂²

yes

—

C₂²×Dic₂₇

C₂²

yes

complemented

C₄²×C₂₇

C₂²

yes

characteristic, complemented

#88372

C₂²

yes

—

#88372

C₂²

yes

complemented

(C₂×D₂₇)⋊C₄

C₂²

yes

—

Dic₂₇⋊C₄

C₂²

yes

complemented

C₉×#127

D₃

yes

—

—

—

—

C₂³×C₂₇

C₂³

yes

characteristic

C₂³×C₂₇

C₂³

yes

—

C₂²×D₂₇

—

C₂²×D₂₇

C₂³

yes

—

D₄×C₂₇

—

D₄×C₂₇

—

C₂×C₄×C₂₇

C₂³

yes

—

#6171

—

C₂×Dic₂₇

—

C₂×Dic₂₇

C₂³

yes

—

#5720

—

C₂³×D₉

—

C₂×D₄×C₉

C₂×D₃

yes

—

C₂×D₄×C₉

C₂×D₃

yes

—

C₂×#611

—

C₂²×Dic₉

—

C₄²×C₉

C₂×D₃

yes

characteristic

#3676

—

#3676

—

(C₂×D₉)⋊C₄

—

Dic₉⋊C₄

—

C₂²×C₂₇

—

C₂²×C₂₇

—

C₂²×C₂₇

—

C₂²×C₂₇

C₂⁴

yes

characteristic

C₂×D₂₇

—

C₂×D₂₇

—

C₄×C₂₇

—

Dic₂₇

—

C₃×#127

D₉

yes

—

—

—

—

C₂³×C₉

C₂²×D₃

yes

characteristic

C₂³×C₉

C₂²×D₃

yes

—

C₂²×D₉

—

C₂²×D₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

C₂²×D₃

yes

—

#611

—

C₂×Dic₉

—

C₂×Dic₉

—

#560

—

Sylow

C₂×C₂₇

—

C₂×C₂₇

—

C₂×C₂₇

H(5,𝔽₂)

yes

—

D₂₇

—

C₂³×D₃

—

C₂×C₃×D₄

C₂×D₉

yes

—

C₂×C₃×D₄

C₂×D₉

yes

—

C₂×#67

—

C₂²×Dic₃

—

C₃×C₄²

C₂×D₉

yes

characteristic

#217

—

#217

—

(C₂×D₃)⋊C₄

—

Dic₃⋊C₄

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂³×D₃

yes

characteristic

C₂×D₉

—

C₂×D₉

—

C₄×C₉

—

Dic₉

—

#127

D₂₇

yes

characteristic, complemented

—

—

—

—

C₂₇

#560

yes

characteristic, complemented, Sylow

C₂³×C₃

C₂²×D₉

yes

characteristic

C₂³×C₃

C₂²×D₉

yes

—

C₂²×D₃

—

C₂²×D₃

—

C₃×D₄

—

100

C₃×D₄

—

101

C₂×C₃×C₄

C₂²×D₉

yes

—

102

#67

—

103

C₂×Dic₃

—

104

C₂×Dic₃

—

105

C₂×C₉

—

106

C₂×C₉

—

107

C₂×C₉

#1004

yes

—

108

D₉

—

109

C₂⁴

—

110

C₂×D₄

—

111

C₂×D₄

C₂×D₂₇

yes

—

112

C₂×D₄

C₂×D₂₇

yes

complemented

113

C₂²×C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

C₂×D₂₇

yes

characteristic, complemented

119

C₂²×C₃

—

120

C₂²×C₃

—

121

C₂²×C₃

—

122

C₂²×C₃

C₂³×D₉

yes

characteristic

123

C₂×D₃

—

124

C₂×D₃

—

125

C₃×C₄

—

126

Dic₃

—

127

C₉

#5720

yes

characteristic

128

C₂³

—

129

C₂³

—

108

—

130

C₂³

C₂²×D₂₇

yes

characteristic

131

C₂³

C₂²×D₂₇

yes

—

132

D₄

—

133

D₄

—

134

D₄

—

108

—

135

C₂×C₄

—

136

C₂×C₄

—

108

—

137

C₂×C₄

C₂²×D₂₇

yes

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

#63710

yes

—

141

D₃

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

108

—

147

C₂²

C₂³×D₂₇

yes

characteristic

148

C₄

—

149

C₄

—

150

C₃

#10589822

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

108

—

154

C₂

#11720670

yes

—

	subgroup	G/H
center	C₂²	C₂³×D₂₇
commutator, cosocle	C₂²×C₂₇	C₂⁴
Frattini	C₂²×C₉	C₂³×D₃
Fitting	C₂₇×#127	C₂
socle	C₂²×C₃	C₂³×D₉
hypercenter	#127	D₂₇
hypocenter	C₂₇	#560
LCS	G ▷ C₂²×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center

C₂²

C₂³×D₂₇

commutator, cosocle

C₂²×C₂₇

C₂⁴

Frattini

C₂²×C₉

C₂³×D₃

Fitting

C₂₇×#127

C₂

socle

C₂²×C₃

C₂³×D₉

D₂₇

C₂₇

G ▷ C₂²×C₂₇ ▷ C₂₇

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. C₂₇×#127	C₂	semidirect
2. #11720188	C₂	semidirect
3. #11720186	C₂	semidirect
4. #11720184	C₂	semidirect
5. #11720125	C₂	semidirect
9. C₂×D₄×C₂₇	C₂²	semidirect
11. C₂²×Dic₂₇	C₂²	semidirect
12. C₄²×C₂₇	C₂²	semidirect
14. #88372	C₂²	semidirect
16. Dic₂₇⋊C₄	C₂²	semidirect
89. #127	D₂₇	semidirect
94. C₂₇	#560	semidirect
112. C₂×D₄	C₂×D₂₇	semidirect
118. C₄²	C₂×D₂₇	semidirect

H ≅ G/N

type

1. C₂₇×#127

C₂

semidirect

2. #11720188

C₂

semidirect

3. #11720186

C₂

semidirect

4. #11720184

C₂

semidirect

5. #11720125

C₂

semidirect

9. C₂×D₄×C₂₇

C₂²

semidirect

11. C₂²×Dic₂₇

C₂²

semidirect

12. C₄²×C₂₇

C₂²

semidirect

14. #88372

C₂²

semidirect

16. Dic₂₇⋊C₄

C₂²

semidirect

89. #127

D₂₇

semidirect

94. C₂₇

#560

semidirect

112. C₂×D₄

C₂×D₂₇

semidirect

118. C₄²

C₂×D₂₇

semidirect

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

subgroup	index
C₂₇×#127	2
#11720188	2
#11720186	2
#11720184	2
#11720125	2
#10589822	3

subgroup

index

C₂₇×#127

Subgroups — #13136458

Conjugacy classes of subgroups

Complements

Maximal subgroups