Group #26105621 — subgroups

Subgroups — #26105621

Order 12032.

414 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20244926	C₂	1	3	yes	maximal
2	#20241214	C₂	1	3	yes	maximal
3	C₄₇×#2811	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093957	—	2	3	no	—
5	C₂×#13093948	—	2	3	no	—
6	#15688993	C₂²	1	1	yes	characteristic
7	#15688878	C₂²	1	3	yes	—
8	C₄₇×#395	—	2	3	no	—
9	C₄₇×#391	—	2	3	no	—
10	C₄₇×#391	C₂²	1	3	yes	—
11	C₂²×C₄₇⋊C₈	—	2	3	no	—
12	C₂×C₄⋊Dic₄₇	—	2	3	no	—
13	#13093957	—	4	6	no	—
14	#13093948	—	4	6	no	—
15	C₂²×Q₈×C₄₇	—	2	1	no	—
16	C₂²×Q₈×C₄₇	D₄	1	3	yes	—
17	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
18	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
19	C₂×C₄²×C₄₇	—	2	1	no	—
20	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
21	#95×C₄₇	—	2	3	no	—
22	#95×C₄₇	—	2	6	no	—
23	C₂×Q₈×C₄₇	—	2	3	no	—
24	C₂×Q₈×C₄₇	—	4	3	no	—
25	C₂×Q₈×C₄₇	—	4	3	no	—
26	C₂×Q₈×C₄₇	—	8	3	no	—
27	C₂×Q₈×C₄₇	Q₁₆	1	6	yes	—
28	C₂²×C₄×C₄₇	—	2	3	no	—
29	C₂²×C₄×C₄₇	—	2	3	no	—
30	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
31	C₂²×Dic₄₇	—	4	1	no	—
32	C₄⋊C₄×C₄₇	—	4	3	no	—
33	C₄⋊C₄×C₄₇	—	8	3	no	—
34	C₄²×C₄₇	—	4	1	no	—
35	C₄²×C₄₇	—	8	1	no	—
36	C₄⋊Dic₄₇	—	4	3	no	—
37	C₄⋊Dic₄₇	—	4	3	no	—
38	C₂×C₄₇⋊C₈	—	2	6	no	—
39	C₂×C₄₇⋊C₈	—	4	3	no	—
40	C₂×C₄₇⋊C₈	—	4	3	no	—
41	C₂³×C₄₇	#120	1	1	yes	characteristic
42	Q₈×C₄₇	—	4	6	no	—
43	Q₈×C₄₇	—	8	3	no	—
44	Q₈×C₄₇	—	8	4	no	—
45	C₂×C₄×C₄₇	—	2	3	no	—
46	C₂×C₄×C₄₇	—	2	3	no	—
47	C₂×C₄×C₄₇	—	2	6	no	—
48	C₂×C₄×C₄₇	—	4	3	no	—
49	C₂×C₄×C₄₇	—	4	3	no	—
50	C₂×C₄×C₄₇	—	4	3	no	—
51	C₂×C₄×C₄₇	—	4	6	no	—
52	C₂×C₄×C₄₇	C₂×Q₁₆	1	3	yes	—
53	C₂×C₄×C₄₇	#137	1	3	yes	—
54	C₂×Dic₄₇	—	8	3	no	—
55	C₄₇⋊C₈	—	4	6	no	—
56	#7837	—	47	1	no	maximal, Sylow
57	C₂²×C₄₇	#451	1	3	yes	—
58	C₂²×C₄₇	#453	1	3	yes	—
59	C₂²×C₄₇	#458	1	1	yes	characteristic
60	C₄×C₄₇	—	2	3	no	—
61	C₄×C₄₇	—	2	3	no	—
62	C₄×C₄₇	—	4	6	no	—
63	C₄×C₄₇	—	8	3	no	—
64	Dic₄₇	—	16	1	no	—
65	#2811	D₄₇	1	1	yes	characteristic
66	#2095	—	47	3	no	—
67	#1272	—	47	3	no	—
68	C₂×C₄₇	#1614	1	3	yes	—
69	C₂×C₄₇	#1633	1	3	yes	—
70	C₂×C₄₇	#1655	1	1	yes	characteristic
71	C₂×#107	—	94	3	no	—
72	C₂×#103	—	94	3	no	—
73	#395	—	2	3	no	—
74	#391	—	2	3	no	—
75	#391	C₂×D₄₇	1	3	yes	—
76	#384	—	47	1	no	—
77	#336	—	47	3	no	—
78	C₄₇	#7837	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₈	—	2	1	no	—
80	C₂²×Q₈	C₂²⋊D₄₇	1	3	yes	—
81	C₂²×C₈	—	94	3	no	—
82	C₂×C₄⋊C₄	—	2	3	no	—
83	C₂×C₄⋊C₄	—	94	3	no	—
84	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
87	#107	—	188	6	no	—
88	#103	—	188	6	no	—
89	#95	—	2	3	no	—
90	#95	—	2	6	no	—
91	C₂×Q₈	—	2	3	no	—
92	C₂×Q₈	—	4	3	no	—
93	C₂×Q₈	—	4	3	no	—
94	C₂×Q₈	—	8	3	no	—
95	C₂×Q₈	#10625337	1	6	yes	—
96	C₂²×C₄	—	2	3	no	—
97	C₂²×C₄	—	2	3	no	—
98	C₂²×C₄	—	188	1	no	—
99	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
100	C₂×C₈	—	94	6	no	—
101	C₂×C₈	—	188	3	no	—
102	C₂×C₈	—	188	3	no	—
103	C₄⋊C₄	—	4	3	no	—
104	C₄⋊C₄	—	8	3	no	—
105	C₄⋊C₄	—	188	3	no	—
106	C₄⋊C₄	—	188	3	no	—
107	C₄²	—	4	1	no	—
108	C₄²	—	8	1	no	—
109	C₂³	#13093993	1	1	yes	characteristic
110	Q₈	—	4	6	no	—
111	Q₈	—	8	3	no	—
112	Q₈	—	8	4	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	4	3	no	—
117	C₂×C₄	—	4	3	no	—
118	C₂×C₄	—	4	3	no	—
119	C₂×C₄	—	4	6	no	—
120	C₂×C₄	—	376	3	no	—
121	C₂×C₄	C₂×#10625337	1	3	yes	—
122	C₂×C₄	#13094055	1	3	yes	—
123	C₈	—	188	6	no	—
124	C₂²	#15689279	1	3	yes	—
125	C₂²	#15689291	1	3	yes	—
126	C₂²	#15689313	1	1	yes	characteristic
127	C₄	—	2	3	no	—
128	C₄	—	2	3	no	—
129	C₄	—	4	6	no	—
130	C₄	—	8	3	no	—
131	C₄	—	752	1	no	—
132	C₂	#20242413	1	3	yes	—
133	C₂	#20242528	1	3	yes	—
134	C₂	#20242677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20244926

C₂

yes

maximal

#20241214

C₂

yes

maximal

C₄₇×#2811

C₂

yes

maximal, characteristic

C₂×#13093957

—

C₂×#13093948

—

#15688993

C₂²

yes

characteristic

#15688878

C₂²

yes

—

C₄₇×#395

—

C₄₇×#391

—

C₄₇×#391

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₂×C₄⋊Dic₄₇

—

#13093957

—

#13093948

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#95×C₄₇

—

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

Q₁₆

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

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

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

#7837

—

maximal, Sylow

C₂²×C₄₇

#451

yes

—

C₂²×C₄₇

#453

yes

—

C₂²×C₄₇

#458

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2811

D₄₇

yes

characteristic

#2095

—

#1272

—

C₂×C₄₇

#1614

yes

—

C₂×C₄₇

#1633

yes

—

C₂×C₄₇

yes

—

—

—

—

C₂×D₄₇

yes

—

#384

—

#336

—

C₄₇

#7837

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂²⋊D₄₇

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#107

—

188

—

#103

—

188

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#10625337

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

100

C₂×C₈

—

101

C₂×C₈

—

188

—

102

C₂×C₈

—

188

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

188

—

106

C₄⋊C₄

—

188

—

107

C₄²

—

108

C₄²

—

109

C₂³

#13093993

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

376

—

121

C₂×C₄

C₂×#10625337

yes

—

122

C₂×C₄

#13094055

yes

—

123

C₈

—

188

—

124

C₂²

#15689279

yes

—

125

C₂²

#15689291

yes

—

126

C₂²

#15689313

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

752

—

132

C₂

#20242413

yes

—

133

C₂

#20242528

yes

—

134

C₂

#20242677

yes

characteristic

	subgroup	G/H
center	C₂³	#13093993
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2811	C₂
socle, prime radical	C₂³×C₄₇	#120
hypercenter	#2811	D₄₇
hypocenter	C₄₇	#7837
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2811

subgroup

G/H

center

C₂³

#13093993

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#2811

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂³×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2811

N	H ≅ G/N	type
78. C₄₇	#7837	semidirect

H ≅ G/N

type

78. C₄₇

#7837

semidirect

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

subgroup	index
#20244926	2
#20241214	2
C₄₇×#2811	2
#7837	47

subgroup

index

#20244926

#20241214

C₄₇×#2811

#7837

Subgroups — #26105621

Conjugacy classes of subgroups

Complements

Maximal subgroups