Group #44529794 — subgroups

Subgroups — #44529794

Order 31168.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#29510207	C₂	1	1	yes	maximal, characteristic, complemented
2	#29510203	C₂	1	1	yes	maximal, characteristic, complemented
3	#29510199	C₂	1	2	yes	maximal, complemented
4	#29510197	C₂	1	2	yes	maximal, complemented
5	#29510192	C₂	1	2	yes	maximal, complemented
6	#29510191	C₂	1	2	yes	maximal, complemented
7	#29510188	C₂	1	2	yes	maximal, complemented
8	#29510187	C₂	1	2	yes	maximal, complemented
9	#124×C₄₈₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₉₇₄	C₂²	1	1	yes	characteristic
11	C₂×D₁₉₄₈	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₄₈₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₄₈₇	C₂²	1	2	yes	—
14	C₂²×Dic₄₈₇	C₂²	1	2	yes	—
15	C₂×C₄×D₄₈₇	C₂²	1	2	yes	—
16	C₄⋊Dic₄₈₇	C₂²	1	2	yes	—
17	C₄⋊Dic₄₈₇	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₄₈₇	C₂²	1	1	yes	characteristic
19	Dic₄₈₇⋊C₄	C₂²	1	2	yes	—
20	Dic₄₈₇⋊C₄	C₂²	1	2	yes	—
21	Dic₄₈₇⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₄₈₇	C₂²	1	2	yes	—
23	(C₂×D₄₈₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₄₈₇)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₄₈₇)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×C₄₈₇	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₄₈₇	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₄₈₇	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₄₈₇	C₂²	1	2	yes	—
30	C₄²×C₄₈₇	C₂²	1	1	yes	characteristic
31	C₂²×D₄₈₇	C₂³	1	2	yes	—
32	Dic₉₇₄	—	4	1	no	—
33	D₁₉₄₈	—	4	1	no	—
34	C₂²⋊D₄₈₇	—	4	2	no	—
35	C₂²⋊D₄₈₇	—	4	2	no	—
36	C₂×Dic₄₈₇	—	4	2	no	—
37	C₂×Dic₄₈₇	C₂³	1	2	yes	—
38	C₂×Dic₄₈₇	C₂³	1	2	yes	—
39	C₂×Dic₄₈₇	C₂³	1	2	yes	—
40	C₄×D₄₈₇	—	4	2	no	—
41	C₂³×C₄₈₇	C₂³	1	2	yes	—
42	Q₈×C₄₈₇	—	4	1	no	—
43	D₄×C₄₈₇	—	4	1	no	—
44	C₂×C₄×C₄₈₇	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₄₈₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₄₈₇	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₄₈₇	C₂³	1	2	yes	—
48	C₂²×C₄₈₇	—	2	2	no	—
49	C₂²×C₄₈₇	—	2	2	no	—
50	C₂²×C₄₈₇	—	2	2	no	—
51	C₂²×C₄₈₇	C₂⁴	1	1	yes	characteristic
52	C₂×D₄₈₇	—	2	2	no	—
53	C₂×D₄₈₇	—	2	2	no	—
54	C₂×D₄₈₇	—	2	2	no	—
55	C₄×C₄₈₇	—	2	1	no	—
56	C₄×C₄₈₇	—	2	2	no	—
57	C₄×C₄₈₇	—	2	1	no	—
58	C₄×C₄₈₇	—	2	1	no	—
59	Dic₄₈₇	—	2	2	no	—
60	Dic₄₈₇	—	2	2	no	—
61	Dic₄₈₇	—	2	2	no	—
62	C₂×C₄₈₇	—	4	2	no	—
63	C₂×C₄₈₇	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₄₈₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₄₈₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₄₈₇	—	4	2	no	—
67	C₄₈₇	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	487	1	no	maximal, Sylow
69	#125	—	487	1	no	—
70	#124	—	487	1	no	—
71	#124	D₄₈₇	1	1	yes	characteristic, complemented
72	#123	—	487	2	no	—
73	#123	—	487	2	no	—
74	C₂²⋊Q₈	—	487	2	no	—
75	C₂²⋊Q₈	—	487	2	no	—
76	#121	—	487	2	no	—
77	#121	—	487	2	no	—
78	C₂×Q₈	—	487	1	no	—
79	C₂×Q₈	C₂×D₄₈₇	1	1	yes	characteristic
80	C₂×D₄	—	487	2	no	—
81	C₂×D₄	—	487	2	no	—
82	C₂×D₄	—	487	1	no	—
83	C₂×D₄	C₂×D₄₈₇	1	1	yes	characteristic
84	C₂²×C₄	—	487	2	no	—
85	C₂²×C₄	—	487	2	no	—
86	C₄⋊C₄	—	487	2	no	—
87	C₄⋊C₄	—	487	2	no	—
88	C₄⋊C₄	—	487	2	no	—
89	C₄⋊C₄	—	487	2	no	—
90	C₄⋊C₄	—	487	1	no	—
91	C₄⋊C₄	—	487	1	no	—
92	C₂²⋊C₄	—	487	2	no	—
93	C₂²⋊C₄	—	487	2	no	—
94	C₂²⋊C₄	—	487	2	no	—
95	C₂²⋊C₄	—	487	2	no	—
96	C₂²⋊C₄	C₂×D₄₈₇	1	2	yes	—
97	C₂²⋊C₄	C₂×D₄₈₇	1	2	yes	—
98	C₄²	C₂×D₄₈₇	1	1	yes	characteristic
99	C₂³	—	487	2	no	—
100	C₂³	C₂²×D₄₈₇	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	1948	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	1948	1	no	—
105	D₄	—	1948	2	no	—
106	D₄	—	1948	2	no	—
107	C₂×C₄	—	1948	2	no	—
108	C₂×C₄	—	1948	2	no	—
109	C₂×C₄	—	487	2	no	—
110	C₂×C₄	—	487	2	no	—
111	C₂×C₄	—	487	2	no	—
112	C₂×C₄	C₂²×D₄₈₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₄₈₇	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₄₈₇	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₄₈₇	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	974	2	no	—
120	C₂²	—	974	2	no	—
121	C₂²	—	974	2	no	—
122	C₂²	C₂³×D₄₈₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	974	2	no	—
128	C₄	—	974	2	no	—
129	C₄	—	974	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	1948	2	no	—
132	C₂	#29510260	1	1	yes	characteristic
133	C₂	#29510261	1	1	yes	characteristic
134	C₂	#29510262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29510207

C₂

yes

maximal, characteristic, complemented

#29510203

C₂

yes

maximal, characteristic, complemented

#29510199

C₂

yes

maximal, complemented

#29510197

C₂

yes

maximal, complemented

#29510192

C₂

yes

maximal, complemented

#29510191

C₂

yes

maximal, complemented

#29510188

C₂

yes

maximal, complemented

#29510187

C₂

yes

maximal, complemented

#124×C₄₈₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₉₇₄

C₂²

yes

characteristic

C₂×D₁₉₄₈

C₂²

yes

characteristic

C₂×C₂²⋊D₄₈₇

C₂²

yes

—

C₂×C₂²⋊D₄₈₇

C₂²

yes

—

C₂²×Dic₄₈₇

C₂²

yes

—

C₂×C₄×D₄₈₇

C₂²

yes

—

C₄⋊Dic₄₈₇

C₂²

yes

—

C₄⋊Dic₄₈₇

C₂²

yes

characteristic

C₄⋊Dic₄₈₇

C₂²

yes

characteristic

Dic₄₈₇⋊C₄

C₂²

yes

—

Dic₄₈₇⋊C₄

C₂²

yes

—

Dic₄₈₇⋊C₄

C₂²

yes

—

C₂²⋊Dic₄₈₇

C₂²

yes

—

(C₂×D₄₈₇)⋊C₄

C₂²

yes

—

(C₂×D₄₈₇)⋊C₄

C₂²

yes

—

(C₂×D₄₈₇)⋊C₄

C₂²

yes

—

C₂×Q₈×C₄₈₇

C₂²

yes

characteristic

C₂×D₄×C₄₈₇

C₂²

yes

characteristic

C₂²⋊C₄×C₄₈₇

C₂²

yes

—

C₂²⋊C₄×C₄₈₇

C₂²

yes

—

C₄²×C₄₈₇

C₂²

yes

characteristic

C₂²×D₄₈₇

C₂³

yes

—

Dic₉₇₄

—

D₁₉₄₈

—

C₂²⋊D₄₈₇

—

C₂²⋊D₄₈₇

—

C₂×Dic₄₈₇

—

C₂×Dic₄₈₇

C₂³

yes

—

C₂×Dic₄₈₇

C₂³

yes

—

C₂×Dic₄₈₇

C₂³

yes

—

C₄×D₄₈₇

—

C₂³×C₄₈₇

C₂³

yes

—

Q₈×C₄₈₇

—

D₄×C₄₈₇

—

C₂×C₄×C₄₈₇

C₂³

yes

characteristic

C₂×C₄×C₄₈₇

C₂³

yes

characteristic

C₂×C₄×C₄₈₇

C₂³

yes

characteristic

C₂×C₄×C₄₈₇

C₂³

yes

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

C₂⁴

yes

characteristic

C₂×D₄₈₇

—

C₂×D₄₈₇

—

C₂×D₄₈₇

—

C₄×C₄₈₇

—

C₄×C₄₈₇

—

C₄×C₄₈₇

—

C₄×C₄₈₇

—

Dic₄₈₇

—

Dic₄₈₇

—

Dic₄₈₇

—

C₂×C₄₈₇

—

C₂×C₄₈₇

yes

C₂×C₄₈₇

yes

C₂×C₄₈₇

2₋¹⁺⁴

yes

characteristic

D₄₈₇

—

C₄₈₇

#562

yes

characteristic, complemented, Sylow

#562

—

487

maximal, Sylow

#125

—

487

—

#124

—

487

—

#124

D₄₈₇

yes

characteristic, complemented

#123

—

487

—

#123

—

487

—

C₂²⋊Q₈

—

487

—

C₂²⋊Q₈

—

487

—

#121

—

487

—

#121

—

487

—

C₂×Q₈

—

487

—

C₂×Q₈

C₂×D₄₈₇

yes

characteristic

C₂×D₄

—

487

—

C₂×D₄

—

487

—

C₂×D₄

—

487

—

C₂×D₄

C₂×D₄₈₇

yes

characteristic

C₂²×C₄

—

487

—

C₂²×C₄

—

487

—

C₄⋊C₄

—

487

—

C₄⋊C₄

—

487

—

C₄⋊C₄

—

487

—

C₄⋊C₄

—

487

—

C₄⋊C₄

—

487

—

C₄⋊C₄

—

487

—

C₂²⋊C₄

—

487

—

C₂²⋊C₄

—

487

—

C₂²⋊C₄

—

487

—

C₂²⋊C₄

—

487

—

C₂²⋊C₄

C₂×D₄₈₇

yes

—

C₂²⋊C₄

C₂×D₄₈₇

yes

—

C₄²

C₂×D₄₈₇

yes

characteristic

C₂³

—

487

—

100

C₂³

C₂²×D₄₈₇

yes

—

101

Q₈

—

102

Q₈

—

1948

—

103

D₄

—

104

D₄

—

1948

—

105

D₄

—

1948

—

106

D₄

—

1948

—

107

C₂×C₄

—

1948

—

108

C₂×C₄

—

1948

—

109

C₂×C₄

—

487

—

110

C₂×C₄

—

487

—

111

C₂×C₄

—

487

—

112

C₂×C₄

C₂²×D₄₈₇

yes

characteristic

113

C₂×C₄

C₂²×D₄₈₇

yes

characteristic

114

C₂×C₄

C₂²×D₄₈₇

yes

characteristic

115

C₂×C₄

C₂²×D₄₈₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

974

—

120

C₂²

—

974

—

121

C₂²

—

974

—

122

C₂²

C₂³×D₄₈₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

974

—

128

C₄

—

974

—

129

C₄

—

974

—

130

C₂

—

131

C₂

—

1948

—

132

C₂

#29510260

yes

characteristic

133

C₂

#29510261

yes

characteristic

134

C₂

#29510262

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₈₇
commutator, socle, cosocle	C₂²×C₄₈₇	C₂⁴
Fitting	#124×C₄₈₇	C₂
hypercenter	#124	D₄₈₇
hypocenter	C₄₈₇	#562
LCS	G ▷ C₂²×C₄₈₇ ▷ C₄₈₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₈₇

commutator, socle, cosocle

C₂²×C₄₈₇

C₂⁴

Fitting

#124×C₄₈₇

C₂

D₄₈₇

C₄₈₇

G ▷ C₂²×C₄₈₇ ▷ C₄₈₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #29510207	C₂	semidirect
2. #29510203	C₂	semidirect
3. #29510199	C₂	semidirect
4. #29510197	C₂	semidirect
5. #29510192	C₂	semidirect
6. #29510191	C₂	semidirect
7. #29510188	C₂	semidirect
8. #29510187	C₂	semidirect
9. #124×C₄₈₇	C₂	semidirect
67. C₄₈₇	#562	semidirect
71. #124	D₄₈₇	semidirect

H ≅ G/N

type

1. #29510207

C₂

semidirect

2. #29510203

C₂

semidirect

3. #29510199

C₂

semidirect

4. #29510197

C₂

semidirect

5. #29510192

C₂

semidirect

6. #29510191

C₂

semidirect

7. #29510188

C₂

semidirect

8. #29510187

C₂

semidirect

9. #124×C₄₈₇

C₂

semidirect

67. C₄₈₇

#562

semidirect

71. #124

D₄₈₇

semidirect

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

subgroup	index
#29510207	2
#29510203	2
#29510199	2
#29510197	2
#29510192	2
#29510191	2
#29510188	2
#29510187	2
#124×C₄₈₇	2
#562	487

subgroup

index

#124×C₄₈₇

487

Subgroups — #44529794

Conjugacy classes of subgroups

Complements

Maximal subgroups