Group #24999674 — subgroups

Subgroups — #24999674

Order 11008.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19135418	C₂	1	3	yes	maximal
2	#19132476	C₂	1	3	yes	maximal, complemented
3	C₄₃×#2845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13063322	C₂²	1	1	yes	characteristic
5	C₂×#13063245	—	2	6	no	—
6	C₂×#13063236	C₂²	1	3	yes	—
7	C₄₃×#398	—	2	6	no	—
8	C₄₃×#382	C₂²	1	3	yes	—
9	C₂²×D₁₇₂	—	2	3	no	—
10	C₂²×C₄₃⋊C₈	—	2	3	no	—
11	#13063322	—	2	3	no	—
12	#13063322	Q₈	1	2	yes	—
13	#13063245	—	4	12	no	—
14	#13063236	—	2	6	no	—
15	C₂×C₄⋊C₄×C₄₃	—	2	6	no	—
16	C₂×C₄⋊C₄×C₄₃	D₄	1	6	yes	—
17	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
18	#95×C₄₃	—	2	2	no	—
19	#95×C₄₃	—	2	6	no	—
20	C₂³×D₄₃	—	4	1	no	—
21	C₂²×C₄×C₄₃	—	2	6	no	—
22	C₂²×C₄×C₄₃	C₂×D₄	1	3	yes	—
23	C₂×D₁₇₂	—	2	6	no	—
24	C₂×D₁₇₂	—	4	3	no	—
25	C₂×D₁₇₂	—	4	6	no	—
26	C₂×D₁₇₂	—	4	6	no	—
27	C₄⋊C₄×C₄₃	—	4	6	no	—
28	C₄⋊C₄×C₄₃	—	8	6	no	—
29	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₄₃	#41	1	3	yes	—
31	C₂×C₄₃⋊C₈	—	2	6	no	—
32	C₂×C₄₃⋊C₈	—	4	3	no	—
33	C₂×C₄₃⋊C₈	—	4	3	no	—
34	C₂³×C₄₃	#120	1	1	yes	characteristic
35	C₂²×D₄₃	—	4	2	no	—
36	C₂²×D₄₃	—	8	3	no	—
37	C₂²×D₄₃	—	8	3	no	—
38	C₂×C₄×C₄₃	—	2	3	no	—
39	C₂×C₄×C₄₃	—	2	3	no	—
40	C₂×C₄×C₄₃	—	4	6	no	—
41	C₂×C₄×C₄₃	—	4	6	no	—
42	C₂×C₄×C₄₃	—	4	6	no	—
43	C₂×C₄×C₄₃	C₂²⋊Q₈	1	3	yes	—
44	C₂×C₄×C₄₃	#124	1	3	yes	—
45	D₁₇₂	—	4	12	no	—
46	D₁₇₂	—	8	6	no	—
47	C₄₃⋊C₈	—	4	6	no	—
48	#10360	—	43	1	no	maximal, Sylow
49	C₂²×C₄₃	#393	1	1	yes	characteristic
50	C₂²×C₄₃	#453	1	3	yes	—
51	C₂²×C₄₃	#454	1	3	yes	—
52	C₂×D₄₃	—	8	6	no	—
53	C₂×D₄₃	—	16	3	no	—
54	C₂×D₄₃	—	16	1	no	—
55	C₄×C₄₃	—	2	3	no	—
56	C₄×C₄₃	—	2	3	no	—
57	C₄×C₄₃	—	8	6	no	—
58	#2845	D₄₃	1	1	yes	characteristic, complemented
59	#1961	—	43	3	no	—
60	#1295	—	43	3	no	—
61	C₂×C₄₃	#1656	1	3	yes	—
62	C₂×C₄₃	#1658	1	1	yes	characteristic
63	C₂×C₄₃	#2027	1	3	yes	—
64	D₄₃	—	16	2	no	—
65	C₂×#127	—	43	1	no	—
66	C₂×#102	—	86	6	no	—
67	C₂×#97	—	43	3	no	—
68	#398	—	2	6	no	—
69	#382	C₂×D₄₃	1	3	yes	—
70	C₄₃	#10360	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	86	3	no	—
72	C₂²×C₈	—	86	3	no	—
73	#127	—	43	2	no	—
74	#127	—	86	3	no	—
75	C₂×C₄⋊C₄	—	2	6	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₄₃	1	6	yes	—
77	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
78	#102	—	172	12	no	—
79	#97	—	86	6	no	—
80	#95	—	2	2	no	—
81	#95	—	2	6	no	—
82	C₂⁴	—	172	1	no	—
83	C₂×D₄	—	86	6	no	—
84	C₂×D₄	—	172	6	no	—
85	C₂×D₄	—	172	6	no	—
86	C₂×D₄	—	172	3	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
89	C₂×C₈	—	86	6	no	—
90	C₂×C₈	—	172	3	no	—
91	C₂×C₈	—	172	3	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	Q₈×D₄₃	1	1	yes	characteristic
95	C₄²	#10621902	1	3	yes	—
96	C₂³	—	344	3	no	—
97	C₂³	—	344	3	no	—
98	C₂³	—	172	2	no	—
99	C₂³	#13063294	1	1	yes	characteristic
100	D₄	—	344	6	no	—
101	D₄	—	172	12	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	#13063301	1	3	yes	—
108	C₂×C₄	#13063314	1	3	yes	—
109	C₈	—	172	6	no	—
110	C₂²	—	344	6	no	—
111	C₂²	—	688	3	no	—
112	C₂²	—	688	1	no	—
113	C₂²	#14582810	1	1	yes	characteristic
114	C₂²	#14583067	1	3	yes	—
115	C₂²	#14583072	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	688	2	no	—
120	C₂	#19133888	1	3	yes	—
121	C₂	#19133897	1	1	yes	characteristic
122	C₂	#19135748	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135418

C₂

yes

maximal

#19132476

C₂

yes

maximal, complemented

C₄₃×#2845

C₂

yes

maximal, characteristic, complemented

C₂×#13063322

C₂²

yes

characteristic

C₂×#13063245

—

C₂×#13063236

C₂²

yes

—

C₄₃×#398

—

C₄₃×#382

C₂²

yes

—

C₂²×D₁₇₂

—

C₂²×C₄₃⋊C₈

—

#13063322

—

#13063322

Q₈

yes

—

#13063245

—

#13063236

—

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#95×C₄₃

—

#95×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

#124

yes

—

D₁₇₂

—

D₁₇₂

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#454

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

#2845

D₄₃

yes

characteristic, complemented

#1961

—

#1295

—

C₂×C₄₃

#1656

yes

—

C₂×C₄₃

#1658

yes

characteristic

C₂×C₄₃

#2027

yes

—

D₄₃

—

C₂×#127

—

C₂×#102

—

C₂×#97

—

#398

—

#382

C₂×D₄₃

yes

—

C₄₃

#10360

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₄₃

yes

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

#102

—

172

—

#97

—

#95

—

#95

—

C₂⁴

—

172

—

C₂×D₄

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

—

172

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₄₃

yes

characteristic

C₄²

#10621902

yes

—

C₂³

—

344

—

C₂³

—

344

—

C₂³

—

172

—

C₂³

#13063294

yes

characteristic

100

D₄

—

344

—

101

D₄

—

172

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#13063301

yes

—

108

C₂×C₄

#13063314

yes

—

109

C₈

—

172

—

110

C₂²

—

344

—

111

C₂²

—

688

—

112

C₂²

—

688

—

113

C₂²

#14582810

yes

characteristic

114

C₂²

#14583067

yes

—

115

C₂²

#14583072

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

688

—

120

C₂

#19133888

yes

—

121

C₂

#19133897

yes

characteristic

122

C₂

#19135748

yes

—

	subgroup	G/H
center	C₂³	#13063294
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#2845	C₂
socle	C₂³×C₄₃	#120
prime radical	C₂×#13063322	C₂²
hypercenter	#2845	D₄₃
hypocenter	C₄₃	#10360
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#13063294

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#2845

C₂

C₂³×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
2. #19132476	C₂	semidirect
3. C₄₃×#2845	C₂	semidirect
58. #2845	D₄₃	semidirect
70. C₄₃	#10360	semidirect

H ≅ G/N

type

2. #19132476

C₂

semidirect

3. C₄₃×#2845

C₂

semidirect

58. #2845

D₄₃

semidirect

70. C₄₃

#10360

semidirect

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

subgroup	index
#19135418	2
#19132476	2
C₄₃×#2845	2
#10360	43

subgroup

index

#19135418

#19132476

C₄₃×#2845

#10360

Subgroups — #24999674

Conjugacy classes of subgroups

Complements

Maximal subgroups