Group #26103102 — subgroups

Subgroups — #26103102

Order 12032.

230 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20241221	C₂	1	2	yes	maximal
2	C₄₇×#1994	C₂	1	1	yes	maximal, characteristic
3	#15689176	—	2	2	no	—
4	C₄₇×#533	C₄	1	1	yes	characteristic
5	C₄₇×#423	C₄	1	1	yes	characteristic
6	C₄³×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×#339	—	2	2	no	—
8	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
9	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
10	C₄×C₄₇⋊C₈	—	4	2	no	—
11	C₂×M₁₆×C₄₇	—	2	1	no	—
12	C₄⋊Q₈×C₄₇	—	2	1	no	—
13	#125×C₄₇	C₈	1	2	yes	complemented
14	C₂²⋊Q₈×C₄₇	—	4	1	no	—
15	C₄×Q₈×C₄₇	—	4	1	no	—
16	#117×C₄₇	—	2	1	no	—
17	C₂×C₄²×C₄₇	—	4	1	no	—
18	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄²×C₄₇	D₄	1	2	yes	—
20	C₄⋊C₈×C₄₇	—	4	1	no	—
21	C₂×Q₈×C₄₇	—	4	1	no	—
22	C₂²×C₄×C₄₇	—	2	1	no	—
23	C₂²×C₄×C₄₇	—	2	2	no	—
24	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	M₁₆×C₄₇	—	4	2	no	—
26	C₂×C₈×C₄₇	—	4	1	no	—
27	C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₄⋊C₄×C₄₇	—	4	1	no	—
29	C₄⋊C₄×C₄₇	—	4	2	no	—
30	C₂²⋊C₄×C₄₇	—	4	1	no	—
31	C₄²×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	—	2	2	no	—
33	C₄²×C₄₇	—	4	4	no	—
34	C₄²×C₄₇	—	4	1	no	—
35	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
37	C₄²×C₄₇	SD₁₆	1	2	yes	—
38	C₄²×C₄₇	Q₁₆	1	2	yes	—
39	C₂×C₄₇⋊C₈	—	4	4	no	—
40	C₂³×C₄₇	#99	1	1	yes	characteristic
41	Q₈×C₄₇	—	8	2	no	—
42	C₂×C₄×C₄₇	—	2	1	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	4	4	no	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#103	1	2	yes	—
52	C₂×C₄×C₄₇	#104	1	2	yes	—
53	C₈×C₄₇	—	8	1	no	—
54	C₄₇⋊C₈	—	8	4	no	—
55	#7134	—	47	1	no	maximal, Sylow
56	C₂²×C₄₇	—	4	1	no	—
57	C₂²×C₄₇	#323	1	1	yes	characteristic
58	C₂²×C₄₇	#328	1	2	yes	—
59	C₄×C₄₇	—	2	4	no	—
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	—	4	1	no	—
62	C₄×C₄₇	—	4	2	no	—
63	C₄×C₄₇	—	8	2	no	—
64	C₄×C₄₇	Q₈⋊C₈	1	2	yes	—
65	C₄×C₄₇	#329	1	1	yes	characteristic
66	C₄×C₄₇	#333	1	1	yes	characteristic
67	#1994	D₄₇	1	1	yes	characteristic
68	#1276	—	47	2	no	—
69	C₂×C₄₇	—	2	1	no	—
70	C₂×C₄₇	—	2	2	no	—
71	C₂×C₄₇	#1272	1	1	yes	characteristic
72	#533	Dic₄₇	1	1	yes	characteristic
73	#431	—	94	2	no	—
74	#423	Dic₄₇	1	1	yes	characteristic
75	C₄³	C₂×D₄₇	1	1	yes	characteristic
76	#339	—	2	2	no	—
77	C₄₇	#7134	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₁₆	—	2	1	no	—
79	C₄⋊Q₈	—	2	1	no	—
80	#125	C₄₇⋊C₈	1	2	yes	complemented
81	C₂²⋊Q₈	—	4	1	no	—
82	C₄×Q₈	—	4	1	no	—
83	#117	—	2	1	no	—
84	C₂×C₄²	—	4	1	no	—
85	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
86	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
87	C₄⋊C₈	—	4	1	no	—
88	C₄⋊C₈	—	94	4	no	—
89	C₂²⋊C₈	—	188	2	no	—
90	C₄×C₈	—	188	2	no	—
91	C₂×Q₈	—	4	1	no	—
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
95	M₁₆	—	4	2	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	188	4	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₂²⋊C₄	—	4	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	4	4	no	—
105	C₄²	—	4	1	no	—
106	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
107	C₄²	#10625329	1	1	yes	characteristic
108	C₄²	#10625336	1	2	yes	—
109	C₄²	#10625337	1	2	yes	—
110	C₂³	#13093939	1	1	yes	characteristic
111	Q₈	—	8	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
120	C₂×C₄	#13093941	1	1	yes	characteristic
121	C₂×C₄	#13093948	1	2	yes	—
122	C₂×C₄	#13093951	1	2	yes	—
123	C₈	—	8	1	no	—
124	C₈	—	376	4	no	—
125	C₂²	—	4	1	no	—
126	C₂²	#15688847	1	1	yes	characteristic
127	C₂²	#15688860	1	2	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	1	no	—
131	C₄	—	4	2	no	—
132	C₄	—	8	2	no	—
133	C₄	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
134	C₄	#15688863	1	1	yes	characteristic
135	C₄	#15688872	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	2	2	no	—
138	C₂	#20241214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20241221

C₂

yes

maximal

C₄₇×#1994

C₂

yes

maximal, characteristic

#15689176

—

C₄₇×#533

C₄

yes

characteristic

C₄₇×#423

C₄

yes

characteristic

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#339

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

—

#125×C₄₇

C₈

yes

complemented

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₄⋊C₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

M₁₆

yes

characteristic

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#99

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7134

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#323

yes

characteristic

C₂²×C₄₇

#328

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

Q₈⋊C₈

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

characteristic

#1276

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1272

yes

characteristic

#533

Dic₄₇

yes

characteristic

#431

—

#423

Dic₄₇

yes

characteristic

C₄³

C₂×D₄₇

yes

characteristic

#339

—

C₄₇

#7134

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

C₄₇⋊C₈

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

C₂²⋊D₄₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

188

—

C₄×C₈

—

188

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

107

C₄²

#10625329

yes

characteristic

108

C₄²

#10625336

yes

—

109

C₄²

#10625337

yes

—

110

C₂³

#13093939

yes

characteristic

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

120

C₂×C₄

#13093941

yes

characteristic

121

C₂×C₄

#13093948

yes

—

122

C₂×C₄

#13093951

yes

—

123

C₈

—

124

C₈

—

376

—

125

C₂²

—

126

C₂²

#15688847

yes

characteristic

127

C₂²

#15688860

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

Q₈⋊(C₄₇⋊C₈)

yes

—

134

C₄

#15688863

yes

characteristic

135

C₄

#15688872

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#20241214

yes

characteristic

	subgroup	G/H
center	C₄	#15688872
commutator	C₄²×C₄₇	C₂×C₈
Frattini	C₄³	C₂×D₄₇
Fitting	C₄₇×#1994	C₂
socle	C₂×C₄₇	#1272
prime radical	C₂³×C₄₇	#99
cosocle	C₄³×C₄₇	C₂²
hypercenter	#1994	D₄₇
hypocenter	C₄₇	#7134
LCS	G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄³ ◁ #1994

subgroup

G/H

center

C₄

#15688872

commutator

C₄²×C₄₇

C₂×C₈

Frattini

C₄³

C₂×D₄₇

Fitting

C₄₇×#1994

C₂

C₂×C₄₇

C₂³×C₄₇

C₄³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄³ ◁ #1994

N	H ≅ G/N	type
13. #125×C₄₇	C₈	semidirect
77. C₄₇	#7134	semidirect
80. #125	C₄₇⋊C₈	semidirect

H ≅ G/N

type

13. #125×C₄₇

C₈

semidirect

77. C₄₇

#7134

semidirect

80. #125

C₄₇⋊C₈

semidirect

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

subgroup	index
#20241221	2
C₄₇×#1994	2
#7134	47

subgroup

index

#20241221

C₄₇×#1994

#7134

Subgroups — #26103102

Conjugacy classes of subgroups

Complements

Maximal subgroups