Group #26103106 — subgroups

Subgroups — #26103106

Order 12032.

306 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 207 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×C₄×C₄₇⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₁₆	C₄	1	4	yes	—
4	C₄×C₄₇⋊C₁₆	—	2	2	no	—
5	C₄×C₄₇⋊C₁₆	—	2	2	no	—
6	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₈²×C₄₇	—	2	2	no	—
9	C₂×C₄₇⋊C₁₆	—	2	4	no	—
10	C₂×C₄₇⋊C₁₆	—	2	4	no	—
11	C₂×C₄₇⋊C₁₆	C₈	1	8	yes	complemented
12	C₂²×C₈×C₄₇	C₈	1	4	yes	—
13	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₄₇	—	2	2	no	—
16	C₄×C₈×C₄₇	—	2	2	no	—
17	C₄×C₈×C₄₇	D₄	1	2	yes	—
18	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
19	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂×C₈	1	2	yes	—
22	C₂×C₈×C₄₇	—	2	4	no	—
23	C₂×C₈×C₄₇	—	2	4	no	—
24	C₂×C₈×C₄₇	C₁₆	1	8	yes	complemented
25	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
26	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
28	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
29	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
31	C₄²×C₄₇	M₁₆	1	2	yes	—
32	C₄₇⋊C₁₆	—	2	8	no	—
33	C₂³×C₄₇	C₄×C₈	1	1	yes	characteristic
34	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
36	C₂×C₄×C₄₇	#97	1	2	yes	—
37	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
39	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
40	C₂×C₄×C₄₇	C₂×C₁₆	1	4	yes	—
41	C₂×C₄×C₄₇	M₃₂	1	4	yes	—
42	C₈×C₄₇	—	2	8	no	—
43	C₈×C₄₇	C₂²⋊C₈	1	4	yes	—
44	C₈×C₄₇	C₄⋊C₈	1	4	yes	—
45	#7136	—	47	1	no	maximal, Sylow
46	C₂²×C₄₇	C₈²	1	1	yes	characteristic
47	C₂²×C₄₇	#322	1	1	yes	characteristic
48	C₂²×C₄₇	#336	1	1	yes	characteristic
49	C₂²×C₄₇	C₄×C₁₆	1	2	yes	—
50	C₂²×C₄₇	#346	1	2	yes	—
51	C₄×C₄₇	#322	1	2	yes	—
52	C₄×C₄₇	#336	1	2	yes	—
53	C₄×C₄₇	C₂²⋊C₁₆	1	4	yes	—
54	C₄×C₄₇	C₄⋊C₁₆	1	4	yes	—
55	C₂×C₄×C₁₆	—	47	2	no	—
56	C₂×C₈²	D₄₇	1	1	yes	characteristic
57	C₂×C₄₇	#1273	1	1	yes	characteristic
58	C₂×C₄₇	C₈×C₁₆	1	1	yes	characteristic
59	C₂×C₄₇	#1311	1	1	yes	characteristic
60	C₂×C₄₇	#1312	1	2	yes	—
61	C₂×C₄₇	#1374	1	2	yes	—
62	C₂²×C₁₆	—	47	4	no	—
63	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
64	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
65	C₄×C₁₆	—	94	2	no	—
66	C₄×C₁₆	—	94	2	no	—
67	C₈²	—	2	2	no	—
68	C₄₇	#7136	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	C₂×C₁₆	—	47	8	no	—
73	C₂×C₁₆	—	94	4	no	—
74	C₂×C₁₆	—	94	4	no	—
75	C₄×C₈	—	2	2	no	—
76	C₄×C₈	—	2	2	no	—
77	C₄×C₈	Dic₉₄	1	1	yes	characteristic
78	C₄×C₈	D₁₈₈	1	1	yes	characteristic
79	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
80	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
81	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₄₇⋊C₁₆	1	8	yes	complemented
85	C₂×C₈	C₈×D₄₇	1	4	yes	—
86	C₂×C₈	#10625324	1	4	yes	—
87	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
88	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
89	C₄²	#10625329	1	2	yes	—
90	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
91	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
92	C₁₆	—	94	8	no	—
93	C₂³	C₄×C₄₇⋊C₈	1	1	yes	characteristic
94	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
95	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
96	C₂×C₄	#13093934	1	2	yes	—
97	C₂×C₄	#13093935	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
99	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
100	C₂×C₄	C₂×C₄₇⋊C₁₆	1	4	yes	—
101	C₂×C₄	#13093964	1	4	yes	—
102	C₈	—	2	8	no	—
103	C₈	(C₂×D₄₇)⋊C₈	1	4	yes	—
104	C₈	Dic₄₇⋊C₈	1	4	yes	—
105	C₂²	C₈×C₄₇⋊C₈	1	1	yes	characteristic
106	C₂²	#15688844	1	1	yes	characteristic
107	C₂²	#15688878	1	1	yes	characteristic
108	C₂²	C₄×C₄₇⋊C₁₆	1	2	yes	—
109	C₂²	#15688899	1	2	yes	—
110	C₄	#15688845	1	2	yes	—
111	C₄	#15688877	1	2	yes	—
112	C₄	C₂²⋊(C₄₇⋊C₁₆)	1	4	yes	—
113	C₄	C₄⋊(C₄₇⋊C₁₆)	1	4	yes	—
114	C₂	C₄₇⋊#1273	1	1	yes	characteristic
115	C₂	C₈×C₄₇⋊C₁₆	1	1	yes	characteristic
116	C₂	#20241305	1	1	yes	characteristic
117	C₂	#20241306	1	2	yes	—
118	C₂	#20241466	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₄₇⋊C₁₆

C₂

yes

maximal

C₂×C₈²×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₄

yes

—

C₄×C₄₇⋊C₁₆

—

C₄×C₄₇⋊C₁₆

—

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₈²×C₄₇

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₈

yes

complemented

C₂²×C₈×C₄₇

C₈

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₁₆

yes

complemented

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄²×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

M₁₆

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

C₄×C₈

yes

—

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₁₆

yes

—

C₂×C₄×C₄₇

M₃₂

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊C₈

yes

—

C₈×C₄₇

C₄⋊C₈

yes

—

#7136

—

maximal, Sylow

C₂²×C₄₇

C₈²

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

C₄×C₁₆

yes

—

C₂²×C₄₇

#346

yes

—

C₄×C₄₇

#322

yes

—

C₄×C₄₇

#336

yes

—

C₄×C₄₇

C₂²⋊C₁₆

yes

—

C₄×C₄₇

C₄⋊C₁₆

yes

—

C₂×C₄×C₁₆

—

C₂×C₈²

D₄₇

yes

characteristic

C₂×C₄₇

#1273

yes

characteristic

C₂×C₄₇

C₈×C₁₆

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1374

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₄₇

#7136

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄₇⋊C₈

yes

—

C₂²×C₈

C₄×D₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₉₄

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄₇⋊C₁₆

yes

complemented

C₂×C₈

C₈×D₄₇

yes

—

C₂×C₈

#10625324

yes

—

C₂×C₈

Dic₄₇⋊C₄

yes

—

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

C₄²

#10625329

yes

—

C₄²

C₄⋊Dic₄₇

yes

characteristic

C₄²

C₂²⋊Dic₄₇

yes

characteristic

C₁₆

—

C₂³

C₄×C₄₇⋊C₈

yes

characteristic

C₂×C₄

C₄₇⋊#95

yes

characteristic

C₂×C₄

C₈×Dic₄₇

yes

—

C₂×C₄

#13093934

yes

—

C₂×C₄

#13093935

yes

characteristic

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

100

C₂×C₄

C₂×C₄₇⋊C₁₆

yes

—

101

C₂×C₄

#13093964

yes

—

102

C₈

—

103

C₈

(C₂×D₄₇)⋊C₈

yes

—

104

C₈

Dic₄₇⋊C₈

yes

—

105

C₂²

C₈×C₄₇⋊C₈

yes

characteristic

106

C₂²

#15688844

yes

characteristic

107

C₂²

#15688878

yes

characteristic

108

C₂²

C₄×C₄₇⋊C₁₆

yes

—

109

C₂²

#15688899

yes

—

110

C₄

#15688845

yes

—

111

C₄

#15688877

yes

—

112

C₄

C₂²⋊(C₄₇⋊C₁₆)

yes

—

113

C₄

C₄⋊(C₄₇⋊C₁₆)

yes

—

114

C₂

C₄₇⋊#1273

yes

characteristic

115

C₂

C₈×C₄₇⋊C₁₆

yes

characteristic

116

C₂

#20241305

yes

characteristic

117

C₂

#20241306

yes

—

118

C₂

#20241466

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄×C₈	C₂×D₄₇
commutator	C₂×C₄₇	C₈×C₁₆
Fitting	C₂×C₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#7136
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

subgroup

G/H

center, Frattini

C₂×C₄×C₈

C₂×D₄₇

commutator

C₂×C₄₇

C₈×C₁₆

Fitting

C₂×C₈²×C₄₇

C₂

socle, prime radical

C₂³×C₄₇

C₄×C₈

cosocle

C₂×C₄×C₈×C₄₇

C₂²

hypercenter

C₂×C₈²

D₄₇

C₄₇

G ▷ C₂×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

N	H ≅ G/N	type
11. C₂×C₄₇⋊C₁₆	C₈	semidirect
24. C₂×C₈×C₄₇	C₁₆	semidirect
68. C₄₇	#7136	semidirect
84. C₂×C₈	C₄₇⋊C₁₆	semidirect

H ≅ G/N

type

11. C₂×C₄₇⋊C₁₆

C₈

semidirect

24. C₂×C₈×C₄₇

C₁₆

semidirect

68. C₄₇

#7136

semidirect

84. C₂×C₈

C₄₇⋊C₁₆

semidirect

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

subgroup	index
C₂×C₄×C₄₇⋊C₁₆	2
C₂×C₈²×C₄₇	2
#7136	47

subgroup

index

C₂×C₄×C₄₇⋊C₁₆

C₂×C₈²×C₄₇

#7136

Subgroups — #26103106

Conjugacy classes of subgroups

Complements

Maximal subgroups