Group #13136452 — subgroups

Subgroups — #13136452

Order 1728.

472 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 147 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
2	#11720185	C₂	1	4	yes	maximal, complemented
3	#11720184	C₂	1	4	yes	maximal, complemented
4	D₄×Dic₂₇	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₅₄	C₂	1	2	yes	maximal, complemented
6	#10589815	—	3	1	no	maximal
7	C₂×D₄×C₂₇	C₂²	1	4	yes	—
8	C₂×D₄×C₂₇	C₂²	1	2	yes	—
9	C₂²×Dic₂₇	C₂²	1	4	yes	—
10	C₂×Dic₅₄	C₂²	1	2	yes	—
11	C₄²×C₂₇	C₂²	1	1	yes	characteristic
12	#88372	C₂²	1	8	yes	—
13	#88372	C₂²	1	4	yes	—
14	C₄⋊Dic₂₇	C₂²	1	2	yes	—
15	Dic₂₇⋊C₄	C₂²	1	4	yes	—
16	C₄×Dic₂₇	C₂²	1	4	yes	—
17	C₉×#127	D₃	1	1	yes	characteristic
18	#63498	—	3	4	no	—
19	#63497	—	3	4	no	—
20	D₄×Dic₉	—	3	4	no	—
21	C₄×Dic₁₈	—	3	2	no	—
22	C₂³×C₂₇	C₂³	1	4	yes	—
23	D₄×C₂₇	—	2	8	no	—
24	D₄×C₂₇	—	4	2	no	—
25	C₂×C₄×C₂₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₂₇	C₂³	1	2	yes	—
27	C₂×Dic₂₇	—	4	4	no	—
28	C₂×Dic₂₇	C₂³	1	4	yes	—
29	C₂×Dic₂₇	C₂³	1	4	yes	—
30	Dic₅₄	—	2	4	no	—
31	#5713	—	9	1	no	—
32	C₂×D₄×C₉	C₂×D₃	1	4	yes	—
33	C₂×D₄×C₉	C₂×D₃	1	2	yes	—
34	C₂²×Dic₉	—	3	4	no	—
35	C₂×Dic₁₈	—	3	2	no	—
36	C₄²×C₉	C₂×D₃	1	1	yes	characteristic
37	#3676	—	3	8	no	—
38	#3676	—	3	4	no	—
39	C₄⋊Dic₉	—	3	2	no	—
40	Dic₉⋊C₄	—	3	4	no	—
41	C₄×Dic₉	—	3	4	no	—
42	C₂²×C₂₇	—	2	8	no	—
43	C₂²×C₂₇	—	2	4	no	—
44	C₂²×C₂₇	C₂⁴	1	1	yes	characteristic
45	C₄×C₂₇	—	2	1	no	—
46	C₄×C₂₇	#41	1	4	yes	—
47	Dic₂₇	—	2	4	no	—
48	Dic₂₇	—	2	4	no	—
49	C₃×#127	D₉	1	1	yes	characteristic
50	#936	—	9	4	no	—
51	#935	—	9	4	no	—
52	D₄×Dic₃	—	9	4	no	—
53	C₄×Dic₆	—	9	2	no	—
54	C₂³×C₉	C₂²×D₃	1	4	yes	—
55	D₄×C₉	—	2	8	no	—
56	D₄×C₉	—	4	2	no	—
57	C₂×C₄×C₉	C₂²×D₃	1	1	yes	characteristic
58	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
59	C₂×Dic₉	—	3	4	no	—
60	C₂×Dic₉	—	3	4	no	—
61	C₂×Dic₉	—	12	4	no	—
62	Dic₁₈	—	6	4	no	—
63	#559	—	27	1	no	Sylow
64	C₂×C₂₇	—	4	4	no	—
65	C₂×C₂₇	C₂×#41	1	2	yes	—
66	C₂×C₂₇	H(5,𝔽₂)	1	1	yes	characteristic
67	C₂×C₃×D₄	C₂×D₉	1	4	yes	—
68	C₂×C₃×D₄	C₂×D₉	1	2	yes	—
69	C₂²×Dic₃	—	9	4	no	—
70	C₂×Dic₆	—	9	2	no	—
71	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
72	#217	—	9	8	no	—
73	#217	—	9	4	no	—
74	C₄⋊Dic₃	—	9	2	no	—
75	Dic₃⋊C₄	—	9	4	no	—
76	C₄×Dic₃	—	9	4	no	—
77	C₂²×C₉	—	2	8	no	—
78	C₂²×C₉	—	2	4	no	—
79	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
80	C₄×C₉	—	2	1	no	—
81	C₄×C₉	#237	1	4	yes	—
82	Dic₉	—	6	4	no	—
83	Dic₉	—	6	4	no	—
84	#127	D₂₇	1	1	yes	characteristic
85	#124	—	27	4	no	—
86	#123	—	27	4	no	—
87	C₄×Q₈	—	27	2	no	—
88	C₄×D₄	—	27	4	no	—
89	C₂₇	#559	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₃	C₂²×D₉	1	4	yes	—
91	C₃×D₄	—	2	8	no	—
92	C₃×D₄	—	4	2	no	—
93	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
94	C₂×C₃×C₄	C₂²×D₉	1	2	yes	—
95	C₂×Dic₃	—	9	4	no	—
96	C₂×Dic₃	—	9	4	no	—
97	C₂×Dic₃	—	36	4	no	—
98	Dic₆	—	18	4	no	—
99	C₂×C₉	—	4	4	no	—
100	C₂×C₉	C₂×#237	1	2	yes	—
101	C₂×C₉	#1004	1	1	yes	characteristic
102	C₂×Q₈	—	27	2	no	—
103	C₂×D₄	C₂×D₂₇	1	4	yes	—
104	C₂×D₄	C₂×D₂₇	1	2	yes	—
105	C₂²×C₄	—	27	4	no	—
106	C₄⋊C₄	—	27	2	no	—
107	C₄⋊C₄	—	27	4	no	—
108	C₂²⋊C₄	—	27	8	no	—
109	C₂²⋊C₄	—	27	4	no	—
110	C₄²	—	27	4	no	—
111	C₄²	C₂×D₂₇	1	1	yes	characteristic
112	C₂²×C₃	—	2	8	no	—
113	C₂²×C₃	—	2	4	no	—
114	C₂²×C₃	C₂³×D₉	1	1	yes	characteristic
115	C₃×C₄	—	2	1	no	—
116	C₃×C₄	#3699	1	4	yes	—
117	Dic₃	—	18	4	no	—
118	Dic₃	—	18	4	no	—
119	C₉	#5713	1	1	yes	characteristic
120	C₂³	C₂²×D₂₇	1	4	yes	—
121	Q₈	—	54	4	no	—
122	D₄	—	2	8	no	—
123	D₄	—	4	2	no	—
124	C₂×C₄	—	27	4	no	—
125	C₂×C₄	—	27	4	no	—
126	C₂×C₄	—	108	4	no	—
127	C₂×C₄	C₂²×D₂₇	1	1	yes	characteristic
128	C₂×C₄	C₂²×D₂₇	1	2	yes	—
129	C₂×C₃	—	4	4	no	—
130	C₂×C₃	C₂×#3699	1	2	yes	—
131	C₂×C₃	#63710	1	1	yes	characteristic
132	C₂²	—	2	8	no	—
133	C₂²	—	2	4	no	—
134	C₂²	C₂³×D₂₇	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	54	4	no	—
137	C₄	—	54	4	no	—
138	C₄	#88401	1	4	yes	—
139	C₃	#10589815	1	1	yes	characteristic
140	C₂	—	4	4	no	—
141	C₂	C₂×#88401	1	2	yes	—
142	C₂	#11720670	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₇×#127

C₂

yes

maximal, characteristic

#11720185

C₂

yes

maximal, complemented

#11720184

C₂

yes

maximal, complemented

D₄×Dic₂₇

C₂

yes

maximal, complemented

C₄×Dic₅₄

C₂

yes

maximal, complemented

#10589815

—

maximal

C₂×D₄×C₂₇

C₂²

yes

—

C₂×D₄×C₂₇

C₂²

yes

—

C₂²×Dic₂₇

C₂²

yes

—

C₂×Dic₅₄

C₂²

yes

—

C₄²×C₂₇

C₂²

yes

characteristic

#88372

C₂²

yes

—

#88372

C₂²

yes

—

C₄⋊Dic₂₇

C₂²

yes

—

Dic₂₇⋊C₄

C₂²

yes

—

C₄×Dic₂₇

C₂²

yes

—

C₉×#127

D₃

yes

characteristic

#63498

—

#63497

—

D₄×Dic₉

—

C₄×Dic₁₈

—

C₂³×C₂₇

C₂³

yes

—

D₄×C₂₇

—

D₄×C₂₇

—

C₂×C₄×C₂₇

C₂³

yes

characteristic

C₂×C₄×C₂₇

C₂³

yes

—

C₂×Dic₂₇

—

C₂×Dic₂₇

C₂³

yes

—

C₂×Dic₂₇

C₂³

yes

—

Dic₅₄

—

#5713

—

C₂×D₄×C₉

C₂×D₃

yes

—

C₂×D₄×C₉

C₂×D₃

yes

—

C₂²×Dic₉

—

C₂×Dic₁₈

—

C₄²×C₉

C₂×D₃

yes

characteristic

#3676

—

#3676

—

C₄⋊Dic₉

—

Dic₉⋊C₄

—

C₄×Dic₉

—

C₂²×C₂₇

—

C₂²×C₂₇

—

C₂²×C₂₇

C₂⁴

yes

characteristic

C₄×C₂₇

—

C₄×C₂₇

#41

yes

—

Dic₂₇

—

Dic₂₇

—

C₃×#127

D₉

yes

characteristic

#936

—

#935

—

D₄×Dic₃

—

C₄×Dic₆

—

C₂³×C₉

C₂²×D₃

yes

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

C₂²×D₃

yes

characteristic

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

#559

—

Sylow

C₂×C₂₇

—

C₂×C₂₇

C₂×#41

yes

—

C₂×C₂₇

H(5,𝔽₂)

yes

characteristic

C₂×C₃×D₄

C₂×D₉

yes

—

C₂×C₃×D₄

C₂×D₉

yes

—

C₂²×Dic₃

—

C₂×Dic₆

—

C₃×C₄²

C₂×D₉

yes

characteristic

#217

—

#217

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂³×D₃

yes

characteristic

C₄×C₉

—

C₄×C₉

#237

yes

—

Dic₉

—

Dic₉

—

#127

D₂₇

yes

characteristic

#124

—

#123

—

C₄×Q₈

—

C₄×D₄

—

C₂₇

#559

yes

characteristic, complemented, Sylow

C₂³×C₃

C₂²×D₉

yes

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

C₂²×D₉

yes

characteristic

C₂×C₃×C₄

C₂²×D₉

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₂×C₉

—

100

C₂×C₉

C₂×#237

yes

—

101

C₂×C₉

#1004

yes

characteristic

102

C₂×Q₈

—

103

C₂×D₄

C₂×D₂₇

yes

—

104

C₂×D₄

C₂×D₂₇

yes

—

105

C₂²×C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₄²

C₂×D₂₇

yes

characteristic

112

C₂²×C₃

—

113

C₂²×C₃

—

114

C₂²×C₃

C₂³×D₉

yes

characteristic

115

C₃×C₄

—

116

C₃×C₄

#3699

yes

—

117

Dic₃

—

118

Dic₃

—

119

C₉

#5713

yes

characteristic

120

C₂³

C₂²×D₂₇

yes

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

108

—

127

C₂×C₄

C₂²×D₂₇

yes

characteristic

128

C₂×C₄

C₂²×D₂₇

yes

—

129

C₂×C₃

—

130

C₂×C₃

C₂×#3699

yes

—

131

C₂×C₃

#63710

yes

characteristic

132

C₂²

—

133

C₂²

—

134

C₂²

C₂³×D₂₇

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#88401

yes

—

139

C₃

#10589815

yes

characteristic

140

C₂

—

141

C₂

C₂×#88401

yes

—

142

C₂

#11720670

yes

characteristic

	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
prime radical	C₃×#127	D₉
hypercenter	#127	D₂₇
hypocenter	C₂₇	#559
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

D₉

D₂₇

C₂₇

G ▷ C₂²×C₂₇ ▷ C₂₇

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
2. #11720185	C₂	semidirect
3. #11720184	C₂	semidirect
4. D₄×Dic₂₇	C₂	semidirect
5. C₄×Dic₅₄	C₂	semidirect
89. C₂₇	#559	semidirect

H ≅ G/N

type

2. #11720185

C₂

semidirect

3. #11720184

C₂

semidirect

4. D₄×Dic₂₇

C₂

semidirect

5. C₄×Dic₅₄

C₂

semidirect

89. C₂₇

#559

semidirect

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

subgroup	index
C₂₇×#127	2
#11720185	2
#11720184	2
D₄×Dic₂₇	2
C₄×Dic₅₄	2
#10589815	3

subgroup

index

C₂₇×#127

#11720185

#11720184

D₄×Dic₂₇

C₄×Dic₅₄

#10589815

Subgroups — #13136452

Conjugacy classes of subgroups

Complements

Maximal subgroups