Group #26120774 — subgroups

Subgroups — #26120774

Order 12032.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20243557	C₂	1	3	yes	maximal
2	#20241269	C₂	1	3	yes	maximal
3	C₄₇×#2844	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093957	—	2	3	no	—
5	C₂×#13093955	—	2	3	no	—
6	C₂×#13093935	C₂²	1	3	yes	—
7	#15688993	C₂²	1	1	yes	characteristic
8	C₄₇×#400	—	2	3	no	—
9	C₄₇×#395	—	2	3	no	—
10	C₄₇×#382	C₂²	1	3	yes	—
11	C₂²×C₄₇⋊C₈	—	2	3	no	—
12	C₂×C₄⋊Dic₄₇	—	2	3	no	—
13	#13093957	—	4	6	no	—
14	#13093955	—	4	6	no	—
15	#13093935	—	2	3	no	—
16	#13093935	Q₈	1	6	yes	complemented
17	C₂²×Q₈×C₄₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₄₇	—	2	3	no	—
19	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
20	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
21	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
22	#95×C₄₇	—	2	3	no	—
23	#95×C₄₇	—	2	1	no	—
24	#95×C₄₇	—	2	6	no	—
25	C₂×Q₈×C₄₇	—	8	3	no	—
26	C₂²×C₄×C₄₇	—	2	3	no	—
27	C₂²×C₄×C₄₇	—	2	3	no	—
28	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
29	C₂²×Dic₄₇	—	4	1	no	—
30	C₄⋊C₄×C₄₇	—	4	3	no	—
31	C₄⋊C₄×C₄₇	—	4	3	no	—
32	C₄⋊C₄×C₄₇	—	8	3	no	—
33	C₄²×C₄₇	C₂×Q₈	1	3	yes	—
34	C₄²×C₄₇	#41	1	1	yes	characteristic
35	C₄⋊Dic₄₇	—	4	3	no	—
36	C₄⋊Dic₄₇	—	4	3	no	—
37	C₂×C₄₇⋊C₈	—	2	6	no	—
38	C₂×C₄₇⋊C₈	—	2	6	no	—
39	C₂×C₄₇⋊C₈	—	2	6	no	—
40	C₂³×C₄₇	#120	1	1	yes	characteristic
41	Q₈×C₄₇	—	8	4	no	—
42	C₂×C₄×C₄₇	—	2	3	no	—
43	C₂×C₄×C₄₇	—	2	3	no	—
44	C₂×C₄×C₄₇	—	4	3	no	—
45	C₂×C₄×C₄₇	—	4	3	no	—
46	C₂×C₄×C₄₇	—	4	6	no	—
47	C₂×C₄×C₄₇	—	4	6	no	—
48	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
49	C₂×C₄×C₄₇	C₄⋊Q₈	1	3	yes	—
50	C₂×Dic₄₇	—	8	3	no	—
51	C₄₇⋊C₈	—	2	12	no	—
52	#10583	—	47	1	no	maximal, Sylow
53	C₂²×C₄₇	#395	1	1	yes	characteristic
54	C₂²×C₄₇	#453	1	3	yes	—
55	C₂²×C₄₇	#456	1	3	yes	—
56	C₄×C₄₇	—	2	3	no	—
57	C₄×C₄₇	—	2	3	no	—
58	C₄×C₄₇	—	8	3	no	—
59	C₄×C₄₇	—	8	3	no	—
60	Dic₄₇	—	16	1	no	—
61	#2844	D₄₇	1	1	yes	characteristic
62	#1847	—	47	3	no	—
63	#1295	—	47	3	no	—
64	C₂×C₄₇	#1655	1	1	yes	characteristic
65	C₂×C₄₇	#1660	1	3	yes	—
66	C₂×C₄₇	#2060	1	3	yes	—
67	C₂×#107	—	94	3	no	—
68	C₂×#106	—	94	3	no	—
69	C₂×#97	—	47	3	no	—
70	#400	—	2	3	no	—
71	#395	—	2	3	no	—
72	#384	—	47	1	no	—
73	#382	C₂×D₄₇	1	3	yes	—
74	C₄₇	#10583	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×C₈	—	94	3	no	—
77	C₂×C₄⋊C₄	—	2	3	no	—
78	C₂×C₄⋊C₄	—	94	3	no	—
79	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
80	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
81	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
82	#107	—	188	6	no	—
83	#106	—	188	6	no	—
84	#97	—	47	6	no	—
85	#97	—	94	3	no	—
86	#95	—	2	3	no	—
87	#95	—	2	1	no	—
88	#95	—	2	6	no	—
89	C₂×Q₈	—	8	3	no	—
90	C₂²×C₄	—	2	3	no	—
91	C₂²×C₄	—	2	3	no	—
92	C₂²×C₄	—	188	1	no	—
93	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
94	C₂×C₈	—	94	6	no	—
95	C₂×C₈	—	94	6	no	—
96	C₂×C₈	—	94	6	no	—
97	C₄⋊C₄	—	4	3	no	—
98	C₄⋊C₄	—	4	3	no	—
99	C₄⋊C₄	—	8	3	no	—
100	C₄⋊C₄	—	188	3	no	—
101	C₄⋊C₄	—	188	3	no	—
102	C₄²	Q₈×D₄₇	1	3	yes	—
103	C₄²	#10625354	1	1	yes	characteristic
104	C₂³	#13093993	1	1	yes	characteristic
105	Q₈	—	8	4	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	4	3	no	—
109	C₂×C₄	—	4	3	no	—
110	C₂×C₄	—	4	6	no	—
111	C₂×C₄	—	4	6	no	—
112	C₂×C₄	—	376	3	no	—
113	C₂×C₄	#13094000	1	3	yes	—
114	C₂×C₄	#13094022	1	3	yes	—
115	C₈	—	94	12	no	—
116	C₂²	#15689042	1	1	yes	characteristic
117	C₂²	#15689291	1	3	yes	—
118	C₂²	#15689306	1	3	yes	—
119	C₄	—	2	3	no	—
120	C₄	—	2	3	no	—
121	C₄	—	8	3	no	—
122	C₄	—	8	3	no	—
123	C₄	—	752	1	no	—
124	C₂	#20242677	1	1	yes	characteristic
125	C₂	#20242699	1	3	yes	—
126	C₂	#20244730	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243557

C₂

yes

maximal

#20241269

C₂

yes

maximal

C₄₇×#2844

C₂

yes

maximal, characteristic

C₂×#13093957

—

C₂×#13093955

—

C₂×#13093935

C₂²

yes

—

#15688993

C₂²

yes

characteristic

C₄₇×#400

—

C₄₇×#395

—

C₄₇×#382

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₂×C₄⋊Dic₄₇

—

—

—

—

Q₈

yes

complemented

C₂²×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#95×C₄₇

—

#95×C₄₇

—

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

#41

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#456

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2844

D₄₇

yes

characteristic

#1847

—

#1295

—

C₂×C₄₇

#1655

yes

characteristic

C₂×C₄₇

#1660

yes

—

C₂×C₄₇

#2060

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

—

—

—

—

C₂×D₄₇

yes

—

C₄₇

#10583

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#107

—

188

—

#106

—

188

—

—

—

—

—

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

188

—

101

C₄⋊C₄

—

188

—

102

C₄²

Q₈×D₄₇

yes

—

103

C₄²

#10625354

yes

characteristic

104

C₂³

#13093993

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

376

—

113

C₂×C₄

#13094000

yes

—

114

C₂×C₄

#13094022

yes

—

115

C₈

—

116

C₂²

#15689042

yes

characteristic

117

C₂²

#15689291

yes

—

118

C₂²

#15689306

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

752

—

124

C₂

#20242677

yes

characteristic

125

C₂

#20242699

yes

—

126

C₂

#20244730

yes

—

	subgroup	G/H
center	C₂³	#13093993
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2844	C₂
socle, prime radical	C₂³×C₄₇	#120
hypercenter	#2844	D₄₇
hypocenter	C₄₇	#10583
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#13093993

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#2844

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
16. #13093935	Q₈	semidirect
74. C₄₇	#10583	semidirect

H ≅ G/N

type

16. #13093935

Q₈

semidirect

74. C₄₇

#10583

semidirect

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

subgroup	index
#20243557	2
#20241269	2
C₄₇×#2844	2
#10583	47

subgroup

index

#20243557

#20241269

C₄₇×#2844

#10583

Subgroups — #26120774

Conjugacy classes of subgroups

Complements

Maximal subgroups