Group #26131027 — subgroups

Subgroups — #26131027

Order 12032.

246 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20245668	C₂	1	2	yes	maximal
2	#20245661	C₂	1	2	yes	maximal
3	#20243006	C₂	1	1	yes	maximal, characteristic
4	#20241305	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1710	C₂	1	1	yes	maximal, characteristic
6	#15689528	—	2	1	no	—
7	#15689528	C₂²	1	2	yes	—
8	#15689503	—	2	1	no	—
9	#15688950	—	2	4	no	—
10	#15688948	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄₇×#500	C₂²	1	1	yes	characteristic
13	C₄₇×#500	C₂²	1	1	yes	characteristic
14	C₄₇×#487	—	2	1	no	—
15	C₄₇×#487	—	2	1	no	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	4	2	no	—
18	#13094024	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
20	#13093958	—	2	2	no	—
21	#13093958	—	2	2	no	—
22	#13093958	—	4	2	no	—
23	C₄⋊Q₈×C₄₇	—	2	1	no	—
24	C₄⋊Q₈×C₄₇	—	2	1	no	—
25	#107×C₄₇	—	2	2	no	—
26	#107×C₄₇	—	2	2	no	—
27	#103×C₄₇	—	4	2	no	—
28	#103×C₄₇	—	4	2	no	—
29	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₄₇	—	4	1	no	—
33	C₂×Q₈×C₄₇	—	4	1	no	—
34	C₂×Dic₉₄	—	4	2	no	—
35	C₂×C₈×C₄₇	D₈	1	2	yes	—
36	C₂×C₈×C₄₇	D₈	1	2	yes	—
37	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
38	C₄⋊C₄×C₄₇	—	4	1	no	—
39	C₄⋊C₄×C₄₇	—	4	1	no	—
40	C₄⋊C₄×C₄₇	—	4	1	no	—
41	C₄⋊C₄×C₄₇	—	4	1	no	—
42	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₄₇	—	4	2	no	—
44	C₄⋊Dic₄₇	—	4	2	no	—
45	Dic₁₈₈	—	8	4	no	—
46	C₄₇⋊C₁₆	Q₁₆	1	8	yes	complemented
47	Q₈×C₄₇	—	8	2	no	—
48	Q₈×C₄₇	—	8	2	no	—
49	C₂×C₄×C₄₇	—	8	1	no	—
50	C₂×C₄×C₄₇	—	8	1	no	—
51	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
54	C₂×Dic₄₇	—	8	2	no	—
55	Dic₉₄	—	8	4	no	—
56	C₈×C₄₇	—	2	2	no	—
57	C₈×C₄₇	—	2	2	no	—
58	C₈×C₄₇	C₂×Q₁₆	1	4	yes	—
59	#12524	—	47	1	no	maximal, Sylow
60	C₂²×C₄₇	#493	1	1	yes	characteristic
61	C₄×C₄₇	—	16	1	no	—
62	C₄×C₄₇	—	16	1	no	—
63	C₄×C₄₇	#494	1	2	yes	—
64	C₄×C₄₇	#509	1	2	yes	—
65	C₄×C₄₇	#510	1	2	yes	—
66	Dic₄₇	—	16	2	no	—
67	#2254	—	47	2	no	—
68	#2252	—	47	2	no	—
69	#1713	—	47	1	no	—
70	#1710	D₄₇	1	1	yes	characteristic
71	#1311	—	47	1	no	—
72	C₂×C₄₇	#1713	1	1	yes	characteristic
73	C₂×C₄₇	#2250	1	1	yes	characteristic
74	C₂×C₄₇	#2251	1	1	yes	characteristic
75	#500	—	47	2	no	—
76	#500	—	94	1	no	—
77	#500	C₂×D₄₇	1	1	yes	characteristic
78	#500	C₂×D₄₇	1	1	yes	characteristic
79	#494	—	94	1	no	—
80	#487	—	2	1	no	—
81	#487	—	2	1	no	—
82	#367	—	94	4	no	—
83	#366	—	94	4	no	—
84	C₄×C₁₆	—	47	2	no	—
85	C₈²	C₂×D₄₇	1	1	yes	characteristic
86	C₄₇	#12524	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	188	2	no	—
88	C₄⋊Q₈	—	2	1	no	—
89	C₄⋊Q₈	—	2	1	no	—
90	C₄⋊Q₈	—	94	2	no	—
91	C₂×C₁₆	—	47	4	no	—
92	#107	—	2	2	no	—
93	#107	—	2	2	no	—
94	#107	—	94	2	no	—
95	#107	—	94	2	no	—
96	#107	—	188	2	no	—
97	#103	—	4	2	no	—
98	#103	—	4	2	no	—
99	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
100	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
101	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
102	C₂×Q₈	—	4	1	no	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×Q₈	—	188	2	no	—
105	Q₁₆	—	376	4	no	—
106	C₂×C₈	#10625334	1	2	yes	—
107	C₂×C₈	#10625334	1	2	yes	—
108	C₂×C₈	D₄×D₄₇	1	2	yes	—
109	C₄⋊C₄	—	4	1	no	—
110	C₄⋊C₄	—	4	1	no	—
111	C₄⋊C₄	—	4	1	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	188	2	no	—
114	C₄⋊C₄	—	188	2	no	—
115	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
116	C₁₆	—	47	8	no	—
117	Q₈	—	8	2	no	—
118	Q₈	—	8	2	no	—
119	Q₈	—	376	4	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	—	376	2	no	—
123	C₂×C₄	#13094020	1	1	yes	characteristic
124	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
125	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	Q₁₆×D₄₇	1	4	yes	—
129	C₂²	#15689499	1	1	yes	characteristic
130	C₄	—	16	1	no	—
131	C₄	—	16	1	no	—
132	C₄	—	752	2	no	—
133	C₄	#15689501	1	2	yes	—
134	C₄	#15689558	1	2	yes	—
135	C₄	#15689563	1	2	yes	—
136	C₂	#20243004	1	1	yes	characteristic
137	C₂	#20245655	1	1	yes	characteristic
138	C₂	#20245660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245668

C₂

yes

maximal

#20245661

C₂

yes

maximal

#20243006

C₂

yes

maximal, characteristic

#20241305

C₂

yes

maximal, characteristic

C₄₇×#1710

C₂

yes

maximal, characteristic

#15689528

—

#15689528

C₂²

yes

—

#15689503

—

#15688950

—

#15688948

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#500

C₂²

yes

characteristic

C₄₇×#500

C₂²

yes

characteristic

C₄₇×#487

—

C₄₇×#487

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093958

—

#13093958

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

#107×C₄₇

—

#107×C₄₇

—

#103×C₄₇

—

#103×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

C₄₇⋊C₁₆

Q₁₆

yes

complemented

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₁₆

yes

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#494

yes

—

C₄×C₄₇

#509

yes

—

C₄×C₄₇

#510

yes

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

#1311

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

C₂×D₄₇

yes

characteristic

#500

C₂×D₄₇

yes

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12524

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

188

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

—

—

—

—

—

188

—

#103

—

#103

—

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

100

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

101

C₄×C₈

C₂²×D₄₇

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

188

—

105

Q₁₆

—

376

—

106

C₂×C₈

#10625334

yes

—

107

C₂×C₈

#10625334

yes

—

108

C₂×C₈

D₄×D₄₇

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

188

—

114

C₄⋊C₄

—

188

—

115

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

116

C₁₆

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

376

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

376

—

123

C₂×C₄

#13094020

yes

characteristic

124

C₂×C₄

C₂×#10625334

yes

characteristic

125

C₂×C₄

C₂×#10625334

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

Q₁₆×D₄₇

yes

—

129

C₂²

#15689499

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

752

—

133

C₄

#15689501

yes

—

134

C₄

#15689558

yes

—

135

C₄

#15689563

yes

—

136

C₂

#20243004

yes

characteristic

137

C₂

#20245655

yes

characteristic

138

C₂

#20245660

yes

characteristic

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1710	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#1710	D₄₇
hypocenter	C₄₇	#12524
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

subgroup

G/H

center

C₂²

#15689499

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1710

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1710

N	H ≅ G/N	type
46. C₄₇⋊C₁₆	Q₁₆	semidirect
86. C₄₇	#12524	semidirect

H ≅ G/N

type

46. C₄₇⋊C₁₆

Q₁₆

semidirect

86. C₄₇

#12524

semidirect

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

subgroup	index
#20245668	2
#20245661	2
#20243006	2
#20241305	2
C₄₇×#1710	2
#12524	47

subgroup

index

C₄₇×#1710

Subgroups — #26131027

Conjugacy classes of subgroups

Complements

Maximal subgroups