Group #26103451 — subgroups

Subgroups — #26103451

Order 12032.

238 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 111 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×#15688944	C₂	1	2	yes	maximal
2	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093964	—	2	2	no	—
4	#15688944	—	2	2	no	—
5	#15688944	C₄	1	4	yes	complemented
6	C₂×#107×C₄₇	—	2	2	no	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#384	C₄	1	2	yes	—
9	#13093964	—	2	4	no	—
10	#13093964	—	4	2	no	—
11	C₂×C₄₇⋊C₁₆	—	4	2	no	—
12	C₂²×C₈×C₄₇	D₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
15	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
16	#107×C₄₇	—	4	2	no	—
17	#107×C₄₇	—	4	2	no	—
18	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
20	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄×C₄₇	—	4	2	no	—
23	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₄₇	D₈	1	2	yes	—
27	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
28	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
29	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
30	C₄⋊C₄×C₄₇	—	4	4	no	—
31	C₄⋊C₄×C₄₇	—	8	4	no	—
32	C₄²×C₄₇	—	2	1	no	—
33	C₄²×C₄₇	C₄²	1	1	yes	characteristic
34	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
35	C₄₇⋊C₁₆	—	4	4	no	—
36	C₂³×C₄₇	#100	1	1	yes	characteristic
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	2	1	no	—
40	C₂×C₄×C₄₇	—	2	1	no	—
41	C₂×C₄×C₄₇	—	8	4	no	—
42	C₂×C₄×C₄₇	—	8	2	no	—
43	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
44	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
45	C₂×C₄×C₄₇	#102	1	2	yes	—
46	C₂×C₄×C₄₇	#103	1	2	yes	—
47	C₈×C₄₇	#102	1	2	yes	—
48	C₈×C₄₇	#103	1	2	yes	—
49	C₈×C₄₇	#106	1	2	yes	—
50	C₈×C₄₇	#107	1	2	yes	—
51	#7269	—	47	1	no	maximal, Sylow
52	C₂²×C₄₇	—	2	1	no	—
53	C₂²×C₄₇	—	2	1	no	—
54	C₂²×C₄₇	#331	1	1	yes	characteristic
55	C₂²×C₄₇	#333	1	1	yes	characteristic
56	C₂²×C₄₇	#343	1	1	yes	characteristic
57	C₄×C₄₇	—	2	2	no	—
58	C₄×C₄₇	—	2	2	no	—
59	C₄×C₄₇	—	16	2	no	—
60	C₄×C₄₇	#332	1	2	yes	—
61	C₄×C₄₇	#340	1	2	yes	—
62	C₂×#364	—	47	2	no	—
63	#1845	D₄₇	1	1	yes	characteristic
64	C₂×C₄₇	—	2	1	no	—
65	C₂×C₄₇	—	2	1	no	—
66	C₂×C₄₇	#1296	1	1	yes	characteristic
67	C₂×C₄₇	#1357	1	1	yes	characteristic
68	C₂×C₄₇	#1358	1	1	yes	characteristic
69	C₂×M₃₂	—	94	2	no	—
70	C₂×#107	—	2	2	no	—
71	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
72	#384	Dic₄₇	1	2	yes	—
73	#364	—	47	4	no	—
74	#364	—	94	2	no	—
75	C₄₇	#7269	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	4	2	no	—
79	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
80	M₃₂	—	94	4	no	—
81	M₃₂	—	188	2	no	—
82	C₂×C₁₆	—	188	2	no	—
83	#107	—	4	2	no	—
84	#107	—	4	2	no	—
85	C₄×C₈	Dic₉₄	1	1	yes	characteristic
86	C₄×C₈	C₄×D₄₇	1	2	yes	—
87	C₄×C₈	D₁₈₈	1	1	yes	characteristic
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	2	no	—
90	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
91	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
92	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
93	C₂×C₈	#10625334	1	2	yes	—
94	C₂×C₈	#10625335	1	2	yes	—
95	C₂×C₈	#10625336	1	2	yes	—
96	C₂×C₈	#10625337	1	2	yes	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄²	—	2	1	no	—
100	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
101	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
102	C₁₆	—	188	4	no	—
103	C₂³	#13093941	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	8	4	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
111	C₂×C₄	#13093943	1	1	yes	characteristic
112	C₂×C₄	#13093945	1	2	yes	—
113	C₂×C₄	#13093948	1	2	yes	—
114	C₈	#13093944	1	2	yes	—
115	C₈	#13093947	1	2	yes	—
116	C₈	#13093955	1	2	yes	—
117	C₈	#13093957	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#15688867	1	1	yes	characteristic
121	C₂²	#15688872	1	1	yes	characteristic
122	C₂²	#15688892	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	16	2	no	—
126	C₄	#15688869	1	2	yes	—
127	C₄	#15688885	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#20241271	1	1	yes	characteristic
131	C₂	#20241425	1	1	yes	characteristic
132	C₂	#20241427	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688944

C₂

yes

maximal

C₄₇×#1845

C₂

yes

maximal, characteristic

C₂×#13093964

—

#15688944

—

#15688944

C₄

yes

complemented

C₂×#107×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₄

yes

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

#107×C₄₇

—

#107×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₄²

yes

characteristic

C₄²×C₄₇

C₄⋊C₄

yes

characteristic

C₄₇⋊C₁₆

—

C₂³×C₄₇

#100

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₈×C₄₇

#102

yes

—

C₈×C₄₇

#103

yes

—

C₈×C₄₇

#106

yes

—

C₈×C₄₇

#107

yes

—

#7269

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#332

yes

—

C₄×C₄₇

#340

yes

—

C₂×#364

—

#1845

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×M₃₂

—

C₂×#107

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#384

Dic₄₇

yes

—

#364

—

#364

—

C₄₇

#7269

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

M₃₂

—

M₃₂

—

188

—

C₂×C₁₆

—

188

—

#107

—

#107

—

C₄×C₈

Dic₉₄

yes

characteristic

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

D₁₈₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₈

Dic₄₇⋊C₄

yes

—

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

C₂×C₈

#10625334

yes

—

C₂×C₈

#10625335

yes

—

C₂×C₈

#10625336

yes

—

C₂×C₈

#10625337

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

C₄×Dic₄₇

yes

characteristic

101

C₄²

C₄⋊Dic₄₇

yes

characteristic

102

C₁₆

—

188

—

103

C₂³

#13093941

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₄₇⋊#95

yes

characteristic

111

C₂×C₄

#13093943

yes

characteristic

112

C₂×C₄

#13093945

yes

—

113

C₂×C₄

#13093948

yes

—

114

C₈

#13093944

yes

—

115

C₈

#13093947

yes

—

116

C₈

#13093955

yes

—

117

C₈

#13093957

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

#15688867

yes

characteristic

121

C₂²

#15688872

yes

characteristic

122

C₂²

#15688892

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#15688869

yes

—

127

C₄

#15688885

yes

—

128

C₂

—

129

C₂

—

130

C₂

#20241271

yes

characteristic

131

C₂

#20241425

yes

characteristic

132

C₂

#20241427

yes

characteristic

	subgroup	G/H
center	C₂²	#15688867
commutator	C₄²×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1845	C₂
socle	C₂²×C₄₇	#331
prime radical	C₂³×C₄₇	#100
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1845	D₄₇
hypocenter	C₄₇	#7269
LCS	G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂²

#15688867

commutator

C₄²×C₄₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₄₇×#1845

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
5. #15688944	C₄	semidirect
75. C₄₇	#7269	semidirect

H ≅ G/N

type

5. #15688944

C₄

semidirect

75. C₄₇

#7269

semidirect

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

subgroup	index
C₂×#15688944	2
C₄₇×#1845	2
#7269	47

subgroup

index

C₂×#15688944

C₄₇×#1845

#7269

Subgroups — #26103451

Conjugacy classes of subgroups

Complements

Maximal subgroups