Group #13179626 — subgroups

Subgroups — #13179626

Order 1728.

2884 conjugacy classes of subgroups; only the 891 normal subgroups are shown.

normal subgroups only

This group has 2884 conjugacy classes of subgroups — too many to draw. Only its 891 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×C₃³×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	#11723967	C₂	1	2	yes	maximal
3	#11723965	C₂	1	4	yes	maximal
4	#11723931	C₂	1	4	yes	maximal, complemented
5	#11723929	C₂	1	4	yes	maximal, complemented
6	#10595971	—	3	13	no	maximal
7	C₂²×C₃³×C₄	C₂²	1	2	yes	—
8	C₂²×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×#6308	C₂²	1	4	yes	—
10	C₃³×C₄⋊C₄	C₂²	1	4	yes	—
11	#88864	C₂²	1	4	yes	—
12	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
13	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	4	yes	—
14	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂²	1	2	yes	—
15	#88857	C₂²	1	8	yes	—
16	C₄×C₃⋊(C₃⋊Dic₃)	C₂²	1	4	yes	—
17	C₂×C₃²×C₄⋊C₄	D₃	1	13	yes	—
18	#64135	—	3	26	no	—
19	#64133	—	3	52	no	—
20	#64099	—	3	52	no	—
21	#64097	—	3	52	no	—
22	C₂³×C₃³	C₂³	1	1	yes	characteristic
23	C₂×C₃³×C₄	—	4	2	no	—
24	C₂×C₃³×C₄	Q₈	1	4	yes	complemented
25	C₂×C₃³×C₄	C₂³	1	4	yes	—
26	C₂×C₃³×C₄	C₂³	1	2	yes	—
27	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
28	C₂×C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
29	#6308	—	4	4	no	—
30	#5500	—	9	13	no	—
31	C₂²×C₃²×C₄	C₂×D₃	1	26	yes	—
32	C₂²×C₃²×C₄	C₂×D₃	1	13	yes	—
33	C₂×#634	—	3	52	no	—
34	C₃²×C₄⋊C₄	C₂×D₃	1	52	yes	—
35	#3757	—	3	52	no	—
36	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
37	C₄⋊(C₃⋊Dic₃)	—	3	52	no	—
38	C₄⋊(C₃⋊Dic₃)	—	3	26	no	—
39	#3750	—	3	104	no	—
40	C₄×C₃⋊Dic₃	—	3	52	no	—
41	C₂²×C₃³	—	2	1	no	—
42	C₂²×C₃³	—	2	1	no	—
43	C₂²×C₃³	C₂×Q₈	1	2	yes	—
44	C₂²×C₃³	C₂⁴	1	1	yes	characteristic
45	C₃³×C₄	—	2	4	no	—
46	C₃³×C₄	C₂×Q₈	1	4	yes	—
47	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
48	C₃⋊(C₃⋊Dic₃)	—	2	4	no	—
49	C₂×C₃×C₄⋊C₄	C₃⋊D₃	1	13	yes	—
50	#926	—	9	26	no	—
51	#924	—	9	52	no	—
52	#890	—	9	52	no	—
53	#888	—	9	52	no	—
54	C₂³×C₃²	C₂²×D₃	1	13	yes	—
55	C₂×C₃²×C₄	—	4	26	no	—
56	C₂×C₃²×C₄	Dic₆	1	52	yes	complemented
57	C₂×C₃²×C₄	C₂²×D₃	1	52	yes	—
58	C₂×C₃²×C₄	C₂²×D₃	1	26	yes	—
59	C₂×C₃⋊Dic₃	—	3	52	no	—
60	C₂×C₃⋊Dic₃	—	3	52	no	—
61	#634	—	12	52	no	—
62	#544	—	27	1	no	Sylow
63	C₂×C₃³	—	2	2	no	—
64	C₂×C₃³	C₂²×Q₈	1	1	yes	characteristic
65	C₂×C₃³	H(5,𝔽₂)	1	1	yes	characteristic
66	C₂×C₃³	2₋¹⁺⁴	1	1	yes	characteristic
67	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
68	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
69	C₂×Dic₆	—	9	52	no	—
70	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	52	yes	—
71	#217	—	9	52	no	—
72	C₄⋊Dic₃	—	9	26	no	—
73	C₄⋊Dic₃	—	9	52	no	—
74	C₄⋊Dic₃	—	9	26	no	—
75	Dic₃⋊C₄	—	9	104	no	—
76	C₄×Dic₃	—	9	52	no	—
77	C₂²×C₃²	—	2	13	no	—
78	C₂²×C₃²	—	2	13	no	—
79	C₂²×C₃²	C₂×Dic₆	1	26	yes	—
80	C₂²×C₃²	C₂³×D₃	1	13	yes	—
81	C₃²×C₄	—	2	52	no	—
82	C₃²×C₄	C₂×Dic₆	1	52	yes	—
83	C₃⋊Dic₃	—	6	52	no	—
84	C₃⋊Dic₃	—	6	52	no	—
85	C₄⋊Q₈	—	27	4	no	—
86	#125	—	27	4	no	—
87	C₂²⋊Q₈	—	27	4	no	—
88	#117	—	27	2	no	—
89	C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
90	C₃³	#544	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₃	C₂²×C₃⋊D₃	1	13	yes	—
92	C₂×C₃×C₄	—	4	26	no	—
93	C₂×C₃×C₄	#634	1	52	yes	complemented
94	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	52	yes	—
95	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	26	yes	—
96	C₂×Dic₃	—	9	52	no	—
97	C₂×Dic₃	—	9	52	no	—
98	Dic₆	—	36	52	no	—
99	C₂×C₃²	—	2	26	no	—
100	C₂×C₃²	C₂²×Dic₆	1	13	yes	—
101	C₂×C₃²	#1004	1	13	yes	—
102	C₂×C₃²	#1007	1	13	yes	—
103	C₂×Q₈	—	27	4	no	—
104	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
105	C₂²×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
106	C₄⋊C₄	—	27	2	no	—
107	C₄⋊C₄	—	27	2	no	—
108	C₄⋊C₄	—	27	4	no	—
109	C₄⋊C₄	—	27	8	no	—
110	C₄⋊C₄	C₂×C₃⋊(C₃⋊D₃)	1	4	yes	—
111	C₂²⋊C₄	—	27	4	no	—
112	C₄²	—	27	4	no	—
113	C₂²×C₃	—	2	13	no	—
114	C₂²×C₃	—	2	13	no	—
115	C₂²×C₃	C₂×#634	1	26	yes	—
116	C₂²×C₃	C₂³×C₃⋊D₃	1	13	yes	—
117	C₃×C₄	—	2	52	no	—
118	C₃×C₄	C₂×#634	1	52	yes	—
119	Dic₃	—	18	52	no	—
120	Dic₃	—	18	52	no	—
121	C₃²	#5500	1	13	yes	complemented
122	C₂³	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
123	Q₈	—	108	4	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	27	4	no	—
126	C₂×C₄	—	27	4	no	—
127	C₂×C₄	#6308	1	4	yes	complemented
128	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	4	yes	—
129	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	2	yes	—
130	C₂×C₃	—	2	26	no	—
131	C₂×C₃	C₂²×#634	1	13	yes	—
132	C₂×C₃	#64361	1	13	yes	—
133	C₂×C₃	#64364	1	13	yes	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂×#6308	1	2	yes	—
137	C₂²	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	54	4	no	—
140	C₄	—	54	4	no	—
141	C₄	C₂×#6308	1	4	yes	—
142	C₃	#10595971	1	13	yes	complemented
143	C₂	—	2	2	no	—
144	C₂	C₂²×#6308	1	1	yes	characteristic
145	C₂	#11724679	1	1	yes	characteristic
146	C₂	#11724682	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃³×C₄⋊C₄

C₂

yes

maximal, characteristic

#11723967

C₂

yes

maximal

#11723965

C₂

yes

maximal

#11723931

C₂

yes

maximal, complemented

#11723929

C₂

yes

maximal, complemented

#10595971

—

maximal

C₂²×C₃³×C₄

C₂²

yes

—

C₂²×C₃³×C₄

C₂²

yes

characteristic

C₂×#6308

C₂²

yes

—

C₃³×C₄⋊C₄

C₂²

yes

—

#88864

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

C₄⋊(C₃⋊(C₃⋊Dic₃))

C₂²

yes

—

#88857

C₂²

yes

—

C₄×C₃⋊(C₃⋊Dic₃)

C₂²

yes

—

C₂×C₃²×C₄⋊C₄

D₃

yes

—

—

—

—

—

C₂³×C₃³

C₂³

yes

characteristic

C₂×C₃³×C₄

—

C₂×C₃³×C₄

Q₈

yes

complemented

C₂×C₃³×C₄

C₂³

yes

—

C₂×C₃³×C₄

C₂³

yes

—

C₂×C₃⋊(C₃⋊Dic₃)

C₂³

yes

—

C₂×C₃⋊(C₃⋊Dic₃)

C₂³

yes

—

#6308

—

#5500

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂×#634

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

#3757

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

—

#3750

—

104

—

C₄×C₃⋊Dic₃

—

C₂²×C₃³

—

C₂²×C₃³

—

C₂²×C₃³

C₂×Q₈

yes

—

C₂²×C₃³

C₂⁴

yes

characteristic

C₃³×C₄

—

C₃³×C₄

C₂×Q₈

yes

—

C₃⋊(C₃⋊Dic₃)

—

C₃⋊(C₃⋊Dic₃)

—

C₂×C₃×C₄⋊C₄

C₃⋊D₃

yes

—

—

—

—

—

C₂³×C₃²

C₂²×D₃

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

Dic₆

yes

complemented

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#544

—

Sylow

C₂×C₃³

—

C₂×C₃³

C₂²×Q₈

yes

characteristic

C₂×C₃³

H(5,𝔽₂)

yes

characteristic

C₂×C₃³

2₋¹⁺⁴

yes

characteristic

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂²×C₃×C₄

C₂×C₃⋊D₃

yes

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

104

—

C₄×Dic₃

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₂×Dic₆

yes

—

C₂²×C₃²

C₂³×D₃

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

—

C₂×C₄⋊C₄

C₃⋊(C₃⋊D₃)

yes

characteristic

C₃³

#544

yes

characteristic, complemented, Sylow

C₂³×C₃

C₂²×C₃⋊D₃

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#634

yes

complemented

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₂×C₃²

—

100

C₂×C₃²

C₂²×Dic₆

yes

—

101

C₂×C₃²

#1004

yes

—

102

C₂×C₃²

#1007

yes

—

103

C₂×Q₈

—

104

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

105

C₂²×C₄

C₂×C₃⋊(C₃⋊D₃)

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂×C₃⋊(C₃⋊D₃)

yes

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₂²×C₃

—

114

C₂²×C₃

—

115

C₂²×C₃

C₂×#634

yes

—

116

C₂²×C₃

C₂³×C₃⋊D₃

yes

—

117

C₃×C₄

—

118

C₃×C₄

C₂×#634

yes

—

119

Dic₃

—

120

Dic₃

—

121

C₃²

#5500

yes

complemented

122

C₂³

C₂²×C₃⋊(C₃⋊D₃)

yes

characteristic

123

Q₈

—

108

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

#6308

yes

complemented

128

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

—

129

C₂×C₄

C₂²×C₃⋊(C₃⋊D₃)

yes

—

130

C₂×C₃

—

131

C₂×C₃

C₂²×#634

yes

—

132

C₂×C₃

#64361

yes

—

133

C₂×C₃

#64364

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂×#6308

yes

—

137

C₂²

C₂³×C₃⋊(C₃⋊D₃)

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₂×#6308

yes

—

142

C₃

#10595971

yes

complemented

143

C₂

—

144

C₂

C₂²×#6308

yes

characteristic

145

C₂

#11724679

yes

characteristic

146

C₂

#11724682

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂⁴
Fitting	C₂×C₃³×C₄⋊C₄	C₂
prime radical	C₂³×C₃³	C₂³
hypercenter	C₂×C₄⋊C₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#544
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×C₃⋊(C₃⋊D₃)

commutator, socle, cosocle

C₂²×C₃³

C₂⁴

Fitting

C₂×C₃³×C₄⋊C₄

C₂

prime radical

C₂³×C₃³

C₂³

hypercenter

C₂×C₄⋊C₄

C₃⋊(C₃⋊D₃)

C₃³

G ▷ C₂²×C₃³ ▷ C₃³

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
4. #11723931	C₂	semidirect
5. #11723929	C₂	semidirect
24. C₂×C₃³×C₄	Q₈	semidirect
56. C₂×C₃²×C₄	Dic₆	semidirect
90. C₃³	#544	semidirect
93. C₂×C₃×C₄	#634	semidirect
121. C₃²	#5500	semidirect
127. C₂×C₄	#6308	semidirect
142. C₃	#10595971	semidirect

H ≅ G/N

type

4. #11723931

C₂

semidirect

5. #11723929

C₂

semidirect

24. C₂×C₃³×C₄

Q₈

semidirect

56. C₂×C₃²×C₄

Dic₆

semidirect

90. C₃³

#544

semidirect

93. C₂×C₃×C₄

#634

semidirect

121. C₃²

#5500

semidirect

127. C₂×C₄

#6308

semidirect

142. C₃

#10595971

semidirect

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

subgroup	index
C₂×C₃³×C₄⋊C₄	2
#11723967	2
#11723965	2
#11723931	2
#11723929	2
#10595971	3

subgroup

index

C₂×C₃³×C₄⋊C₄

Subgroups — #13179626

Conjugacy classes of subgroups

Complements

Maximal subgroups