Group #26125271 — subgroups

Subgroups — #26125271

Order 12032.

322 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 219 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×#13093964	C₂	1	2	yes	maximal
2	#20241306	C₂	1	4	yes	maximal, complemented
3	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	C₄	1	4	yes	—
5	#15688899	C₄	1	4	yes	—
6	C₄×C₄₇⋊C₁₆	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
9	C₈²×C₄₇	C₄	1	1	yes	characteristic
10	C₈²×C₄₇	C₄	1	1	yes	characteristic
11	C₈²×C₄₇	C₂²	1	2	yes	—
12	#13093964	—	2	8	no	—
13	C₂×C₄₇⋊C₁₆	C₂×C₄	1	8	yes	—
14	C₂²×C₈×C₄₇	C₈	1	4	yes	—
15	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
17	C₄×C₈×C₄₇	C₈	1	4	yes	—
18	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
25	C₂×C₈×C₄₇	—	2	8	no	—
26	C₂×C₈×C₄₇	C₄²	1	2	yes	—
27	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
28	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
29	C₂×C₈×C₄₇	M₁₆	1	8	yes	—
30	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
31	C₄²×C₄₇	C₄²	1	1	yes	characteristic
32	C₄²×C₄₇	C₂×C₈	1	2	yes	—
33	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₄₇⋊C₁₆	—	2	8	no	—
35	C₂³×C₄₇	C₄×C₈	1	1	yes	characteristic
36	C₂×C₄×C₄₇	—	2	4	no	—
37	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
38	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₄₇	#97	1	4	yes	—
40	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
42	C₈×C₄₇	—	2	4	no	—
43	C₈×C₄₇	#97	1	4	yes	—
44	C₈×C₄₇	M₃₂	1	8	yes	complemented
45	C₈×C₄₇	C₂×M₁₆	1	4	yes	—
46	#11402	—	47	1	no	maximal, Sylow
47	C₂²×C₄₇	—	2	2	no	—
48	C₂²×C₄₇	#322	1	2	yes	—
49	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
50	C₄×C₄₇	—	2	2	no	—
51	C₄×C₄₇	#322	1	2	yes	—
52	C₄×C₄₇	C₂×#97	1	2	yes	—
53	C₄×C₄₇	C₂×M₃₂	1	4	yes	—
54	C₄×M₃₂	—	47	2	no	—
55	C₂×C₈²	D₄₇	1	1	yes	characteristic
56	#1312	—	47	4	no	—
57	C₂×C₄₇	—	2	2	no	—
58	C₂×C₄₇	C₂×#322	1	1	yes	characteristic
59	C₂×C₄₇	C₄×M₃₂	1	2	yes	—
60	C₂×M₃₂	—	47	4	no	—
61	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
62	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
63	#346	—	47	4	no	—
64	C₄×C₁₆	—	47	4	no	—
65	C₈²	Dic₄₇	1	1	yes	characteristic
66	C₈²	Dic₄₇	1	1	yes	characteristic
67	C₈²	C₂×D₄₇	1	2	yes	—
68	C₄₇	#11402	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	C₄₇⋊C₈	1	4	yes	—
70	C₂²×C₈	C₄×D₄₇	1	2	yes	—
71	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
72	M₃₂	—	94	8	no	—
73	C₂×C₁₆	—	47	8	no	—
74	C₄×C₈	C₄₇⋊C₈	1	4	yes	—
75	C₄×C₈	C₄×D₄₇	1	2	yes	—
76	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
77	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
78	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
79	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
80	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
81	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	#10625324	1	8	yes	—
84	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
85	C₂×C₈	C₂×C₄₇⋊C₈	1	4	yes	—
86	C₂×C₈	C₄×Dic₄₇	1	2	yes	—
87	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
88	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
89	C₄²	C₄×Dic₄₇	1	1	yes	characteristic
90	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
91	C₁₆	—	94	8	no	—
92	C₂³	C₄×C₄₇⋊C₈	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×C₄₇⋊C₈	1	2	yes	—
95	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
96	C₂×C₄	#13093934	1	4	yes	—
97	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₄₇⋊C₈	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₈	#13093934	1	4	yes	—
101	C₈	#13093964	1	8	yes	complemented
102	C₈	C₂×#10625324	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	#15688844	1	2	yes	—
105	C₂²	C₂×C₄×C₄₇⋊C₈	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	#15688844	1	2	yes	—
108	C₄	C₂×#13093934	1	2	yes	—
109	C₄	C₂×#13093964	1	4	yes	—
110	C₂	—	2	2	no	—
111	C₂	C₂×#15688844	1	1	yes	characteristic
112	C₂	C₄×#13093964	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13093964

C₂

yes

maximal

#20241306

C₂

yes

maximal, complemented

C₂×C₈²×C₄₇

C₂

yes

maximal, characteristic

C₂×#13093964

C₄

yes

—

#15688899

C₄

yes

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₈²×C₄₇

C₄

yes

characteristic

C₈²×C₄₇

C₄

yes

characteristic

C₈²×C₄₇

C₂²

yes

—

#13093964

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₈

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₈

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₄²×C₄₇

C₄²

yes

characteristic

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×C₈

yes

—

C₂×C₄×C₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₈

yes

—

C₈×C₄₇

—

C₈×C₄₇

#97

yes

—

C₈×C₄₇

M₃₂

yes

complemented

C₈×C₄₇

C₂×M₁₆

yes

—

#11402

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#322

yes

—

C₂²×C₄₇

C₂×C₄×C₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#322

yes

—

C₄×C₄₇

C₂×#97

yes

—

C₄×C₄₇

C₂×M₃₂

yes

—

C₄×M₃₂

—

C₂×C₈²

D₄₇

yes

characteristic

#1312

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#322

yes

characteristic

C₂×C₄₇

C₄×M₃₂

yes

—

C₂×M₃₂

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#346

—

C₄×C₁₆

—

C₈²

Dic₄₇

yes

characteristic

C₈²

Dic₄₇

yes

characteristic

C₈²

C₂×D₄₇

yes

—

C₄₇

#11402

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄₇⋊C₈

yes

—

C₂²×C₈

C₄×D₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄×C₈

C₄₇⋊C₈

yes

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

#10625324

yes

—

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

C₄×Dic₄₇

yes

—

C₂×C₈

C₂×C₄×D₄₇

yes

—

C₄²

C₂×C₄₇⋊C₈

yes

—

C₄²

C₄×Dic₄₇

yes

characteristic

C₄²

C₂²×Dic₄₇

yes

characteristic

C₁₆

—

C₂³

C₄×C₄₇⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×C₄₇⋊C₈

yes

—

C₂×C₄

C₄×C₄₇⋊C₈

yes

characteristic

C₂×C₄

#13093934

yes

—

C₂×C₄

C₂×C₄×Dic₄₇

yes

characteristic

C₂×C₄

C₂²×C₄₇⋊C₈

yes

—

C₈

—

100

C₈

#13093934

yes

—

101

C₈

#13093964

yes

complemented

102

C₈

C₂×#10625324

yes

—

103

C₂²

—

104

C₂²

#15688844

yes

—

105

C₂²

C₂×C₄×C₄₇⋊C₈

yes

characteristic

106

C₄

—

107

C₄

#15688844

yes

—

108

C₄

C₂×#13093934

yes

—

109

C₄

C₂×#13093964

yes

—

110

C₂

—

111

C₂

C₂×#15688844

yes

characteristic

112

C₂

C₄×#13093964

yes

—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₂×C₈²×C₄₇	C₂
prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₄×C₈×C₄₇	C₂³
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#11402
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₈²

subgroup

G/H

center, Frattini

C₄×C₈

C₂²×D₄₇

commutator, socle

C₂²×C₄₇

C₂×C₄×C₈

Fitting

C₂×C₈²×C₄₇

C₂

prime radical

C₂³×C₄₇

C₄×C₈

cosocle

C₄×C₈×C₄₇

C₂³

hypercenter

C₂×C₈²

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₄×C₈ ◁ C₂×C₈²

N	H ≅ G/N	type
2. #20241306	C₂	semidirect
44. C₈×C₄₇	M₃₂	semidirect
68. C₄₇	#11402	semidirect
101. C₈	#13093964	semidirect

H ≅ G/N

type

2. #20241306

C₂

semidirect

44. C₈×C₄₇

M₃₂

semidirect

68. C₄₇

#11402

semidirect

101. C₈

#13093964

semidirect

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

subgroup	index
C₄×#13093964	2
#20241306	2
C₂×C₈²×C₄₇	2
#11402	47

subgroup

index

C₄×#13093964

#20241306

C₂×C₈²×C₄₇

#11402

Subgroups — #26125271

Conjugacy classes of subgroups

Complements

Maximal subgroups