Group #26125256 — subgroups

Subgroups — #26125256

Order 12032.

354 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 199 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	C₄×#13093963	C₂	1	2	yes	maximal
2	C₈²×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#20241305	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×C₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13093963	C₄	1	4	yes	—
7	C₄×C₈×D₄₇	C₄	1	2	yes	—
8	C₄×C₈×D₄₇	C₂²	1	1	yes	characteristic
9	#15688898	C₄	1	4	yes	—
10	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
11	C₈×C₄₇⋊C₈	C₄	1	2	yes	—
12	C₈×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
14	C₈²×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₄₇	C₈	1	4	yes	—
16	C₂×C₈×D₄₇	C₂×C₄	1	2	yes	—
17	C₄²×D₄₇	C₂×C₄	1	1	yes	characteristic
18	#13093963	C₈	1	16	yes	complemented
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
20	C₄×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₈×Dic₄₇	C₈	1	4	yes	—
23	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
24	C₂×C₁₆×C₄₇	C₂×C₄	1	4	yes	—
25	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
26	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₄₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×D₄₇	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
30	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
31	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×Dic₄₇	C₄²	1	1	yes	characteristic
33	C₄×Dic₄₇	C₂×C₈	1	2	yes	—
34	C₂×C₄₇⋊C₈	C₄²	1	2	yes	—
35	C₂×C₄₇⋊C₈	C₂×C₈	1	4	yes	—
36	C₈×D₄₇	—	2	8	no	—
37	C₈×D₄₇	C₂×C₈	1	8	yes	—
38	C₁₆×C₄₇	C₂×C₈	1	8	yes	complemented
39	C₄₇⋊C₁₆	C₂×C₈	1	8	yes	complemented
40	C₂²×D₄₇	C₄×C₈	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
42	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
43	C₂×Dic₄₇	C₄×C₈	1	2	yes	—
44	C₂×Dic₄₇	C₄×C₈	1	1	yes	characteristic
45	C₄×D₄₇	—	2	4	no	—
46	C₄×D₄₇	C₄×C₈	1	4	yes	—
47	C₈×C₄₇	—	2	4	no	—
48	C₈×C₄₇	C₂²×C₈	1	4	yes	—
49	C₄₇⋊C₈	C₄×C₈	1	4	yes	—
50	C₄₇⋊C₈	M₃₂	1	8	yes	complemented
51	C₈×M₃₂	—	47	1	no	maximal, Sylow
52	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
53	C₂×D₄₇	—	2	2	no	—
54	C₂×D₄₇	C₈²	1	2	yes	—
55	C₄×C₄₇	C₂×C₄×C₈	1	2	yes	—
56	C₄×C₄₇	C₂×M₃₂	1	4	yes	—
57	Dic₄₇	—	2	2	no	—
58	Dic₄₇	C₈²	1	2	yes	—
59	C₄×M₃₂	—	47	2	no	—
60	C₂×C₈²	—	47	1	no	—
61	#1311	—	47	1	no	—
62	#1311	D₄₇	1	1	yes	characteristic, complemented
63	C₈×C₁₆	—	47	2	no	—
64	C₂×C₄₇	C₂×C₈²	1	1	yes	characteristic
65	C₂×C₄₇	C₄×M₃₂	1	2	yes	—
66	D₄₇	—	2	2	no	—
67	C₂×M₃₂	—	47	4	no	—
68	C₂×C₄×C₈	—	47	2	no	—
69	C₂×C₄×C₈	—	47	1	no	—
70	#346	—	47	4	no	—
71	C₄×C₁₆	—	47	2	no	—
72	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
73	C₈²	—	47	2	no	—
74	C₈²	—	47	1	no	—
75	C₈²	C₂×D₄₇	1	1	yes	characteristic
76	C₄₇	C₈×M₃₂	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₈	—	47	2	no	—
78	C₂²×C₈	—	47	4	no	—
79	C₂×C₄²	—	47	1	no	—
80	M₃₂	—	47	16	no	—
81	C₂×C₁₆	—	47	4	no	—
82	C₂×C₁₆	C₄×D₄₇	1	4	yes	—
83	C₄×C₈	—	47	4	no	—
84	C₄×C₈	—	47	2	no	—
85	C₄×C₈	—	47	1	no	—
86	C₄×C₈	—	47	2	no	—
87	C₄×C₈	C₄×D₄₇	1	2	yes	—
88	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
89	C₂²×C₄	—	47	1	no	—
90	C₂²×C₄	—	47	2	no	—
91	C₂×C₈	—	47	2	no	—
92	C₂×C₈	—	47	4	no	—
93	C₂×C₈	—	47	8	no	—
94	C₂×C₈	—	94	8	no	—
95	C₂×C₈	C₈×D₄₇	1	4	yes	—
96	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
97	C₄²	—	47	1	no	—
98	C₄²	—	47	2	no	—
99	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
100	C₁₆	—	47	8	no	—
101	C₁₆	C₈×D₄₇	1	8	yes	complemented
102	C₂³	—	47	1	no	—
103	C₂×C₄	—	47	1	no	—
104	C₂×C₄	—	47	4	no	—
105	C₂×C₄	—	47	2	no	—
106	C₂×C₄	—	94	4	no	—
107	C₂×C₄	C₄²×D₄₇	1	1	yes	characteristic
108	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	47	8	no	—
111	C₈	—	47	4	no	—
112	C₈	C₂×C₈×D₄₇	1	4	yes	—
113	C₂²	—	47	2	no	—
114	C₂²	—	94	2	no	—
115	C₂²	C₄×C₈×D₄₇	1	1	yes	characteristic
116	C₄	—	47	2	no	—
117	C₄	—	94	2	no	—
118	C₄	C₄×C₈×D₄₇	1	2	yes	—
119	C₄	M₃₂×D₄₇	1	4	yes	—
120	C₂	—	94	2	no	—
121	C₂	C₈²×D₄₇	1	1	yes	characteristic
122	C₂	#20244981	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13093963

C₂

yes

maximal

C₈²×D₄₇

C₂

yes

maximal, characteristic

#20241305

C₂

yes

maximal, characteristic, complemented

C₁₆×C₄₇⋊C₈

C₂

yes

maximal, complemented

C₄₇×#1311

C₂

yes

maximal, characteristic, complemented

C₂×#13093963

C₄

yes

—

C₄×C₈×D₄₇

C₄

yes

—

C₄×C₈×D₄₇

C₂²

yes

characteristic

#15688898

C₄

yes

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₈×C₄₇⋊C₈

C₄

yes

—

C₈×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×C₈×D₄₇

C₈

yes

—

C₂×C₈×D₄₇

C₂×C₄

yes

—

C₄²×D₄₇

C₂×C₄

yes

characteristic

#13093963

C₈

yes

complemented

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₈×Dic₄₇

C₈

yes

—

C₈×Dic₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄×D₄₇

C₄²

yes

characteristic

C₂×C₄×D₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄×Dic₄₇

C₄²

yes

characteristic

C₄×Dic₄₇

C₂×C₈

yes

—

C₂×C₄₇⋊C₈

C₄²

yes

—

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₈×D₄₇

—

C₈×D₄₇

C₂×C₈

yes

—

C₁₆×C₄₇

C₂×C₈

yes

complemented

C₄₇⋊C₁₆

C₂×C₈

yes

complemented

C₂²×D₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

—

C₂×Dic₄₇

C₄×C₈

yes

—

C₂×Dic₄₇

C₄×C₈

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

C₄×C₈

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₈

yes

—

C₄₇⋊C₈

C₄×C₈

yes

—

C₄₇⋊C₈

M₃₂

yes

complemented

C₈×M₃₂

—

maximal, Sylow

C₂²×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

C₈²

yes

—

C₄×C₄₇

C₂×C₄×C₈

yes

—

C₄×C₄₇

C₂×M₃₂

yes

—

Dic₄₇

—

Dic₄₇

C₈²

yes

—

C₄×M₃₂

—

C₂×C₈²

—

#1311

—

#1311

D₄₇

yes

characteristic, complemented

C₈×C₁₆

—

C₂×C₄₇

C₂×C₈²

yes

characteristic

C₂×C₄₇

C₄×M₃₂

yes

—

D₄₇

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₇

yes

—

C₈²

—

C₈²

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

C₈×M₃₂

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₄₇

yes

—

C₂×C₈

C₂×C₄×D₄₇

yes

—

C₄²

—

C₄²

—

C₄²

C₂×C₄×D₄₇

yes

characteristic

100

C₁₆

—

101

C₁₆

C₈×D₄₇

yes

complemented

102

C₂³

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₄²×D₄₇

yes

characteristic

108

C₂×C₄

C₂×C₈×D₄₇

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

C₂×C₈×D₄₇

yes

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₄×C₈×D₄₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

C₄×C₈×D₄₇

yes

—

119

C₄

M₃₂×D₄₇

yes

—

120

C₂

—

121

C₂

C₈²×D₄₇

yes

characteristic

122

C₂

#20244981

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₇
commutator	C₂×C₄₇	C₂×C₈²
Fitting	C₄₇×#1311	C₂
socle	C₂²×C₄₇	C₂×C₄×C₈
prime radical	C₂²×D₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	#1311	D₄₇
hypocenter	C₄₇	C₈×M₃₂
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ #1311

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₄₇

commutator

C₂×C₄₇

C₂×C₈²

Fitting

C₄₇×#1311

C₂

socle

C₂²×C₄₇

C₂×C₄×C₈

prime radical

C₂²×D₄₇

C₄×C₈

cosocle

C₄×C₈×C₄₇

C₂³

hypercenter

#1311

D₄₇

hypocenter

C₄₇

C₈×M₃₂

LCS

G ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄×C₈ ◁ #1311

N	H ≅ G/N	type
3. #20241305	C₂	semidirect
4. C₁₆×C₄₇⋊C₈	C₂	semidirect
5. C₄₇×#1311	C₂	semidirect
18. #13093963	C₈	semidirect
38. C₁₆×C₄₇	C₂×C₈	semidirect
39. C₄₇⋊C₁₆	C₂×C₈	semidirect
50. C₄₇⋊C₈	M₃₂	semidirect
62. #1311	D₄₇	semidirect
76. C₄₇	C₈×M₃₂	semidirect
101. C₁₆	C₈×D₄₇	semidirect

H ≅ G/N

type

3. #20241305

C₂

semidirect

4. C₁₆×C₄₇⋊C₈

C₂

semidirect

5. C₄₇×#1311

C₂

semidirect

18. #13093963

C₈

semidirect

38. C₁₆×C₄₇

C₂×C₈

semidirect

39. C₄₇⋊C₁₆

C₂×C₈

semidirect

50. C₄₇⋊C₈

M₃₂

semidirect

62. #1311

D₄₇

semidirect

76. C₄₇

C₈×M₃₂

semidirect

101. C₁₆

C₈×D₄₇

semidirect

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

subgroup	index
C₄×#13093963	2
C₈²×D₄₇	2
#20241305	2
C₁₆×C₄₇⋊C₈	2
C₄₇×#1311	2
C₈×M₃₂	47

subgroup

index

C₄×#13093963

C₈²×D₄₇

#20241305

C₁₆×C₄₇⋊C₈

C₄₇×#1311

C₈×M₃₂

Subgroups — #26125256

Conjugacy classes of subgroups

Complements

Maximal subgroups