Group #26103459 — subgroups

Subgroups — #26103459

Order 12032.

208 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 107 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×#1848	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093964	—	2	2	no	—
4	#15688944	—	2	2	no	—
5	#15688944	C₄	1	4	yes	—
6	C₂×#108×C₄₇	—	2	2	no	—
7	C₄₇×#423	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	#13093964	—	2	4	no	—
10	#13093964	—	4	2	no	—
11	C₂×C₄₇⋊C₁₆	—	4	2	no	—
12	C₂×M₁₆×C₄₇	—	2	2	no	—
13	C₂²×C₈×C₄₇	D₄	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
15	#108×C₄₇	—	4	2	no	—
16	#108×C₄₇	—	4	2	no	—
17	C₄⋊C₈×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	C₂²⋊C₄	1	1	yes	characteristic
23	M₁₆×C₄₇	—	4	4	no	—
24	C₂×C₈×C₄₇	—	4	2	no	—
25	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
26	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₄₇	D₈	1	2	yes	—
28	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
29	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
30	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
31	C₄²×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	C₄²	1	1	yes	characteristic
33	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
34	C₄₇⋊C₁₆	—	4	4	no	—
35	C₂³×C₄₇	#101	1	1	yes	characteristic
36	C₂×C₄×C₄₇	—	2	1	no	—
37	C₂×C₄×C₄₇	—	2	1	no	—
38	C₂×C₄×C₄₇	—	4	2	no	—
39	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
40	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
41	C₂×C₄×C₄₇	#102	1	2	yes	—
42	C₂×C₄×C₄₇	#103	1	2	yes	—
43	C₈×C₄₇	—	8	2	no	—
44	C₈×C₄₇	#102	1	2	yes	—
45	C₈×C₄₇	#103	1	2	yes	—
46	C₈×C₄₇	#106	1	2	yes	—
47	C₈×C₄₇	#107	1	2	yes	—
48	#7272	—	47	1	no	maximal, Sylow
49	C₂²×C₄₇	—	4	1	no	—
50	C₂²×C₄₇	#332	1	2	yes	—
51	C₂²×C₄₇	#343	1	1	yes	characteristic
52	C₄×C₄₇	—	2	2	no	—
53	C₄×C₄₇	—	4	1	no	—
54	C₄×C₄₇	#331	1	1	yes	characteristic
55	C₄×C₄₇	#333	1	1	yes	characteristic
56	C₄×C₄₇	#340	1	2	yes	—
57	C₂×#364	—	47	2	no	—
58	#1848	D₄₇	1	1	yes	characteristic
59	C₂×C₄₇	—	2	1	no	—
60	C₂×C₄₇	—	2	2	no	—
61	C₂×C₄₇	#1296	1	1	yes	characteristic
62	C₂×M₃₂	—	94	2	no	—
63	C₂×#108	—	2	2	no	—
64	#423	Dic₄₇	1	2	yes	—
65	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
66	#364	—	47	4	no	—
67	#364	—	94	2	no	—
68	C₄₇	#7272	1	1	yes	characteristic, complemented, Sylow
69	C₂×M₁₆	—	2	2	no	—
70	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
71	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
72	M₃₂	—	94	4	no	—
73	M₃₂	—	188	2	no	—
74	C₂×C₁₆	—	188	2	no	—
75	#108	—	4	2	no	—
76	#108	—	4	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₄×C₈	Dic₉₄	1	1	yes	characteristic
79	C₄×C₈	C₄×D₄₇	1	2	yes	—
80	C₄×C₈	D₁₈₈	1	1	yes	characteristic
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
83	M₁₆	—	4	4	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
86	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
87	C₂×C₈	#10625334	1	2	yes	—
88	C₂×C₈	#10625335	1	2	yes	—
89	C₂×C₈	#10625336	1	2	yes	—
90	C₂×C₈	#10625337	1	2	yes	—
91	C₄²	—	2	1	no	—
92	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
93	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
94	C₁₆	—	188	4	no	—
95	C₂³	#13093943	1	1	yes	characteristic
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
100	C₂×C₄	#13093941	1	1	yes	characteristic
101	C₂×C₄	#13093945	1	2	yes	—
102	C₂×C₄	#13093948	1	2	yes	—
103	C₈	—	8	2	no	—
104	C₈	#13093944	1	2	yes	—
105	C₈	#13093947	1	2	yes	—
106	C₈	#13093955	1	2	yes	—
107	C₈	#13093957	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	#15688869	1	2	yes	—
110	C₂²	#15688892	1	1	yes	characteristic
111	C₄	—	2	2	no	—
112	C₄	—	4	1	no	—
113	C₄	#15688867	1	1	yes	characteristic
114	C₄	#15688872	1	1	yes	characteristic
115	C₄	#15688885	1	2	yes	—
116	C₂	—	2	1	no	—
117	C₂	—	2	2	no	—
118	C₂	#20241271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688944

C₂

yes

maximal

C₄₇×#1848

C₂

yes

maximal, characteristic

C₂×#13093964

—

#15688944

—

#15688944

C₄

yes

—

C₂×#108×C₄₇

—

C₄₇×#423

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

#108×C₄₇

—

#108×C₄₇

—

C₄⋊C₈×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

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

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

—

C₄²×C₄₇

—

C₄²×C₄₇

C₄²

yes

characteristic

C₄²×C₄₇

C₄⋊C₄

yes

characteristic

C₄₇⋊C₁₆

—

C₂³×C₄₇

#101

yes

characteristic

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

—

C₈×C₄₇

#102

yes

—

C₈×C₄₇

#103

yes

—

C₈×C₄₇

#106

yes

—

C₈×C₄₇

#107

yes

—

#7272

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#332

yes

—

C₂²×C₄₇

#343

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×#364

—

#1848

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1296

yes

characteristic

C₂×M₃₂

—

C₂×#108

—

#423

Dic₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#364

—

#364

—

C₄₇

#7272

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

M₃₂

—

M₃₂

—

188

—

C₂×C₁₆

—

188

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

Dic₉₄

yes

characteristic

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

D₁₈₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

M₁₆

—

C₂×C₈

—

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

—

C₄²

—

C₄²

C₄×Dic₄₇

yes

characteristic

C₄²

C₄⋊Dic₄₇

yes

characteristic

C₁₆

—

188

—

C₂³

#13093943

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄₇⋊#95

yes

characteristic

100

C₂×C₄

#13093941

yes

characteristic

101

C₂×C₄

#13093945

yes

—

102

C₂×C₄

#13093948

yes

—

103

C₈

—

104

C₈

#13093944

yes

—

105

C₈

#13093947

yes

—

106

C₈

#13093955

yes

—

107

C₈

#13093957

yes

—

108

C₂²

—

109

C₂²

#15688869

yes

—

110

C₂²

#15688892

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

#15688867

yes

characteristic

114

C₄

#15688872

yes

characteristic

115

C₄

#15688885

yes

—

116

C₂

—

117

C₂

—

118

C₂

#20241271

yes

characteristic

	subgroup	G/H
center	C₄	#15688872
commutator	C₄²×C₄₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1848	C₂
socle	C₂×C₄₇	#1296
prime radical	C₂³×C₄₇	#101
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1848	D₄₇
hypocenter	C₄₇	#7272
LCS	G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1848

subgroup

G/H

center

C₄

#15688872

commutator

C₄²×C₄₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₄₇×#1848

C₂

C₂×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

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

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1848

N	H ≅ G/N	type
68. C₄₇	#7272	semidirect

H ≅ G/N

type

68. C₄₇

#7272

semidirect

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

subgroup	index
C₂×#15688944	2
C₄₇×#1848	2
#7272	47

subgroup

index

C₂×#15688944

C₄₇×#1848

#7272

Subgroups — #26103459

Conjugacy classes of subgroups

Complements

Maximal subgroups