Group #26136790 — subgroups

Subgroups — #26136790

Order 12032.

178 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 79 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×#15688956	C₂	1	1	yes	maximal, characteristic
2	#368×D₄₇	C₂	1	2	yes	maximal
3	#20241592	C₂	1	1	yes	maximal, characteristic, complemented
4	#20241591	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1425	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₄₇	C₂²	1	1	yes	characteristic
7	#108×D₄₇	—	2	2	no	—
8	#15688956	C₄	1	4	yes	—
9	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic
10	#15688953	C₂²	1	2	yes	—
11	#15688952	—	2	2	no	—
12	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#368	C₂²	1	2	yes	—
14	M₁₆×D₄₇	—	4	2	no	—
15	C₂×C₈×D₄₇	D₄	1	1	yes	characteristic
16	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
17	C₁₆×D₄₇	C₂×C₄	1	2	yes	—
18	C₁₆×D₄₇	Q₈	1	2	yes	—
19	#13093960	—	2	2	no	—
20	#13093959	—	4	2	no	—
21	C₄₇⋊C₃₂	C₂×C₄	1	2	yes	—
22	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
23	#108×C₄₇	—	2	2	no	—
24	C₃₂×C₄₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×D₄₇	D₈	1	1	yes	characteristic
26	M₁₆×C₄₇	—	4	2	no	—
27	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
28	#10625329	—	4	2	no	—
29	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
30	#10625324	—	8	2	no	—
31	C₈×D₄₇	—	8	2	no	—
32	C₈×D₄₇	C₄⋊C₄	1	2	yes	—
33	C₈×D₄₇	Q₁₆	1	2	yes	—
34	C₁₆×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₁₆×C₄₇	C₂×Q₈	1	1	yes	characteristic
36	C₄₇⋊C₁₆	C₄⋊C₄	1	2	yes	—
37	C₂²×D₄₇	Q₃₂	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
39	C₂×Dic₄₇	Q₃₂	1	1	yes	characteristic
40	C₄×D₄₇	#107	1	2	yes	—
41	C₄×D₄₇	D₁₆	1	2	yes	—
42	C₈×C₄₇	—	8	2	no	—
43	C₈×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
45	C₄₇⋊C₈	—	8	2	no	—
46	C₄₇⋊C₈	#107	1	2	yes	—
47	#13636	—	47	1	no	maximal, Sylow
48	C₂²×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
49	C₂×D₄₇	—	4	1	no	—
50	C₂×D₄₇	#366	1	2	yes	—
51	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
52	C₄×C₄₇	C₂×D₁₆	1	1	yes	characteristic
53	Dic₄₇	#366	1	2	yes	—
54	C₂×M₆₄	—	47	1	no	—
55	C₂×#368	—	47	2	no	—
56	#1425	—	47	1	no	—
57	#1425	—	47	2	no	—
58	#1425	D₄₇	1	1	yes	characteristic, complemented
59	C₂×C₄₇	—	2	1	no	—
60	C₂×C₄₇	C₂×#366	1	1	yes	characteristic
61	D₄₇	—	2	2	no	—
62	C₂²×C₁₆	—	47	1	no	—
63	C₂×#108	—	94	2	no	—
64	M₆₄	—	47	4	no	—
65	C₂×C₃₂	—	47	1	no	—
66	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
67	#368	—	47	2	no	—
68	#368	—	94	2	no	—
69	#368	C₂×D₄₇	1	2	yes	—
70	C₄₇	#13636	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	188	2	no	—
72	C₂²×C₈	—	47	1	no	—
73	C₂×C₁₆	—	47	1	no	—
74	C₂×C₁₆	—	47	2	no	—
75	C₂×C₁₆	—	47	2	no	—
76	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
77	#108	—	2	2	no	—
78	#108	—	94	2	no	—
79	#108	—	188	2	no	—
80	C₃₂	—	47	2	no	—
81	C₃₂	C₄×D₄₇	1	2	yes	—
82	C₂²×C₄	—	47	1	no	—
83	M₁₆	—	4	2	no	—
84	M₁₆	—	376	2	no	—
85	M₁₆	—	188	2	no	—
86	C₂×C₈	—	47	1	no	—
87	C₂×C₈	—	47	2	no	—
88	C₂×C₈	—	47	2	no	—
89	C₂×C₈	—	376	2	no	—
90	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
91	C₁₆	—	47	2	no	—
92	C₁₆	C₂×C₄×D₄₇	1	1	yes	characteristic
93	C₁₆	Q₈×D₄₇	1	1	yes	characteristic
94	C₂³	—	47	1	no	—
95	C₂×C₄	—	47	1	no	—
96	C₂×C₄	—	47	2	no	—
97	C₂×C₄	—	47	2	no	—
98	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
99	C₈	—	8	2	no	—
100	C₈	—	47	2	no	—
101	C₈	—	376	2	no	—
102	C₈	C₄⋊C₄×D₄₇	1	1	yes	characteristic
103	C₈	Q₁₆×D₄₇	1	1	yes	characteristic
104	C₂²	—	47	2	no	—
105	C₂²	—	188	1	no	—
106	C₂²	Q₃₂×D₄₇	1	1	yes	characteristic
107	C₄	—	47	2	no	—
108	C₄	#107×D₄₇	1	1	yes	characteristic
109	C₄	D₁₆×D₄₇	1	1	yes	characteristic
110	C₂	—	2	1	no	—
111	C₂	—	94	2	no	—
112	C₂	#366×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688956

C₂

yes

maximal, characteristic

#368×D₄₇

C₂

yes

maximal

#20241592

C₂

yes

maximal, characteristic, complemented

#20241591

C₂

yes

maximal, complemented

C₄₇×#1425

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₄₇

C₂²

yes

characteristic

#108×D₄₇

—

#15688956

C₄

yes

—

C₂×C₄₇⋊C₃₂

C₂²

yes

characteristic

#15688953

C₂²

yes

—

#15688952

—

C₂×C₃₂×C₄₇

C₂²

yes

characteristic

C₄₇×#368

C₂²

yes

—

M₁₆×D₄₇

—

C₂×C₈×D₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

D₄

yes

characteristic

C₁₆×D₄₇

C₂×C₄

yes

—

C₁₆×D₄₇

Q₈

yes

—

#13093960

—

#13093959

—

C₄₇⋊C₃₂

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

C₃₂×C₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

D₈

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

D₈

yes

characteristic

#10625324

—

C₈×D₄₇

—

C₈×D₄₇

C₄⋊C₄

yes

—

C₈×D₄₇

Q₁₆

yes

—

C₁₆×C₄₇

C₂²×C₄

yes

characteristic

C₁₆×C₄₇

C₂×Q₈

yes

characteristic

C₄₇⋊C₁₆

C₄⋊C₄

yes

—

C₂²×D₄₇

Q₃₂

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

Q₃₂

yes

characteristic

C₄×D₄₇

#107

yes

—

C₄×D₄₇

D₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₈×C₄₇

C₂×Q₁₆

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

#107

yes

—

#13636

—

maximal, Sylow

C₂²×C₄₇

C₂×Q₃₂

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

#366

yes

—

C₄×C₄₇

C₂×#107

yes

characteristic

C₄×C₄₇

C₂×D₁₆

yes

characteristic

Dic₄₇

#366

yes

—

C₂×M₆₄

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×C₄₇

—

C₂×C₄₇

C₂×#366

yes

characteristic

D₄₇

—

C₂²×C₁₆

—

C₂×#108

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

—

C₄₇

#13636

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

188

—

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₄₇

yes

—

—

—

188

—

C₃₂

—

C₃₂

C₄×D₄₇

yes

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

376

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

376

—

C₂×C₈

D₄×D₄₇

yes

characteristic

C₁₆

—

C₁₆

C₂×C₄×D₄₇

yes

characteristic

C₁₆

Q₈×D₄₇

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×D₄₇

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

376

—

102

C₈

C₄⋊C₄×D₄₇

yes

characteristic

103

C₈

Q₁₆×D₄₇

yes

characteristic

104

C₂²

—

105

C₂²

—

188

—

106

C₂²

Q₃₂×D₄₇

yes

characteristic

107

C₄

—

108

C₄

#107×D₄₇

yes

characteristic

109

C₄

D₁₆×D₄₇

yes

characteristic

110

C₂

—

111

C₂

—

112

C₂

#366×D₄₇

yes

characteristic

	subgroup	G/H
center	C₄	D₁₆×D₄₇
commutator	C₁₆×C₄₇	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#1425	C₂
socle	C₂×C₄₇	C₂×#366
prime radical	C₂²×D₄₇	Q₃₂
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	#1425	D₄₇
hypocenter	C₄₇	#13636
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

subgroup

G/H

center

C₄

D₁₆×D₄₇

commutator

C₁₆×C₄₇

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting

C₄₇×#1425

C₂

socle

C₂×C₄₇

C₂×#366

prime radical

C₂²×D₄₇

Q₃₂

cosocle

C₂×C₁₆×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1425

N	H ≅ G/N	type
3. #20241592	C₂	semidirect
4. #20241591	C₂	semidirect
5. C₄₇×#1425	C₂	semidirect
58. #1425	D₄₇	semidirect
70. C₄₇	#13636	semidirect

H ≅ G/N

type

3. #20241592

C₂

semidirect

4. #20241591

C₂

semidirect

5. C₄₇×#1425

C₂

semidirect

58. #1425

D₄₇

semidirect

70. C₄₇

#13636

semidirect

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

subgroup	index
C₂×#15688956	2
#368×D₄₇	2
#20241592	2
#20241591	2
C₄₇×#1425	2
#13636	47

subgroup

index

#368×D₄₇

C₄₇×#1425

Subgroups — #26136790

Conjugacy classes of subgroups

Complements

Maximal subgroups