Group #50306634 — subgroups

Subgroups — #50306634

Order 38336.

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	#34062707	C₂	1	1	yes	maximal, characteristic, complemented
2	#34062703	C₂	1	1	yes	maximal, characteristic, complemented
3	#34062699	C₂	1	2	yes	maximal, complemented
4	#34062697	C₂	1	2	yes	maximal, complemented
5	#34062692	C₂	1	2	yes	maximal, complemented
6	#34062691	C₂	1	2	yes	maximal, complemented
7	#34062688	C₂	1	2	yes	maximal, complemented
8	#34062687	C₂	1	2	yes	maximal, complemented
9	#124×C₅₉₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₁₉₈	C₂²	1	1	yes	characteristic
11	C₂×D₂₃₉₆	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₅₉₉	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₅₉₉	C₂²	1	2	yes	—
14	C₂²×Dic₅₉₉	C₂²	1	2	yes	—
15	C₂×C₄×D₅₉₉	C₂²	1	2	yes	—
16	C₄⋊Dic₅₉₉	C₂²	1	2	yes	—
17	C₄⋊Dic₅₉₉	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₅₉₉	C₂²	1	1	yes	characteristic
19	Dic₅₉₉⋊C₄	C₂²	1	2	yes	—
20	Dic₅₉₉⋊C₄	C₂²	1	2	yes	—
21	Dic₅₉₉⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₅₉₉	C₂²	1	2	yes	—
23	(C₂×D₅₉₉)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₅₉₉)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₅₉₉)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×C₅₉₉	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₅₉₉	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₅₉₉	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₅₉₉	C₂²	1	2	yes	—
30	C₄²×C₅₉₉	C₂²	1	1	yes	characteristic
31	C₂²×D₅₉₉	C₂³	1	2	yes	—
32	Dic₁₁₉₈	—	4	1	no	—
33	D₂₃₉₆	—	4	1	no	—
34	C₂²⋊D₅₉₉	—	4	2	no	—
35	C₂²⋊D₅₉₉	—	4	2	no	—
36	C₂×Dic₅₉₉	—	4	2	no	—
37	C₂×Dic₅₉₉	C₂³	1	2	yes	—
38	C₂×Dic₅₉₉	C₂³	1	2	yes	—
39	C₂×Dic₅₉₉	C₂³	1	2	yes	—
40	C₄×D₅₉₉	—	4	2	no	—
41	C₂³×C₅₉₉	C₂³	1	2	yes	—
42	Q₈×C₅₉₉	—	4	1	no	—
43	D₄×C₅₉₉	—	4	1	no	—
44	C₂×C₄×C₅₉₉	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₅₉₉	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₅₉₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₅₉₉	C₂³	1	2	yes	—
48	C₂²×C₅₉₉	—	2	2	no	—
49	C₂²×C₅₉₉	—	2	2	no	—
50	C₂²×C₅₉₉	—	2	2	no	—
51	C₂²×C₅₉₉	C₂⁴	1	1	yes	characteristic
52	C₂×D₅₉₉	—	2	2	no	—
53	C₂×D₅₉₉	—	2	2	no	—
54	C₂×D₅₉₉	—	2	2	no	—
55	C₄×C₅₉₉	—	2	1	no	—
56	C₄×C₅₉₉	—	2	2	no	—
57	C₄×C₅₉₉	—	2	1	no	—
58	C₄×C₅₉₉	—	2	1	no	—
59	Dic₅₉₉	—	2	2	no	—
60	Dic₅₉₉	—	2	2	no	—
61	Dic₅₉₉	—	2	2	no	—
62	C₂×C₅₉₉	—	4	2	no	—
63	C₂×C₅₉₉	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₅₉₉	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₅₉₉	2₋¹⁺⁴	1	1	yes	characteristic
66	D₅₉₉	—	4	2	no	—
67	C₅₉₉	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	599	1	no	maximal, Sylow
69	#125	—	599	1	no	—
70	#124	—	599	1	no	—
71	#124	D₅₉₉	1	1	yes	characteristic, complemented
72	#123	—	599	2	no	—
73	#123	—	599	2	no	—
74	C₂²⋊Q₈	—	599	2	no	—
75	C₂²⋊Q₈	—	599	2	no	—
76	#121	—	599	2	no	—
77	#121	—	599	2	no	—
78	C₂×Q₈	—	599	1	no	—
79	C₂×Q₈	C₂×D₅₉₉	1	1	yes	characteristic
80	C₂×D₄	—	599	2	no	—
81	C₂×D₄	—	599	2	no	—
82	C₂×D₄	—	599	1	no	—
83	C₂×D₄	C₂×D₅₉₉	1	1	yes	characteristic
84	C₂²×C₄	—	599	2	no	—
85	C₂²×C₄	—	599	2	no	—
86	C₄⋊C₄	—	599	2	no	—
87	C₄⋊C₄	—	599	2	no	—
88	C₄⋊C₄	—	599	2	no	—
89	C₄⋊C₄	—	599	2	no	—
90	C₄⋊C₄	—	599	1	no	—
91	C₄⋊C₄	—	599	1	no	—
92	C₂²⋊C₄	—	599	2	no	—
93	C₂²⋊C₄	—	599	2	no	—
94	C₂²⋊C₄	—	599	2	no	—
95	C₂²⋊C₄	—	599	2	no	—
96	C₂²⋊C₄	C₂×D₅₉₉	1	2	yes	—
97	C₂²⋊C₄	C₂×D₅₉₉	1	2	yes	—
98	C₄²	C₂×D₅₉₉	1	1	yes	characteristic
99	C₂³	—	599	2	no	—
100	C₂³	C₂²×D₅₉₉	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	2396	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	2396	1	no	—
105	D₄	—	2396	2	no	—
106	D₄	—	2396	2	no	—
107	C₂×C₄	—	599	2	no	—
108	C₂×C₄	—	599	2	no	—
109	C₂×C₄	—	599	2	no	—
110	C₂×C₄	—	2396	2	no	—
111	C₂×C₄	—	2396	2	no	—
112	C₂×C₄	C₂²×D₅₉₉	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₅₉₉	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₅₉₉	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₅₉₉	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	1198	2	no	—
120	C₂²	—	1198	2	no	—
121	C₂²	—	1198	2	no	—
122	C₂²	C₂³×D₅₉₉	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	—	1198	2	no	—
128	C₄	—	1198	2	no	—
129	C₄	—	1198	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	2396	2	no	—
132	C₂	#34062760	1	1	yes	characteristic
133	C₂	#34062761	1	1	yes	characteristic
134	C₂	#34062762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34062707

C₂

yes

maximal, characteristic, complemented

#34062703

C₂

yes

maximal, characteristic, complemented

#34062699

C₂

yes

maximal, complemented

#34062697

C₂

yes

maximal, complemented

#34062692

C₂

yes

maximal, complemented

#34062691

C₂

yes

maximal, complemented

#34062688

C₂

yes

maximal, complemented

#34062687

C₂

yes

maximal, complemented

#124×C₅₉₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₁₉₈

C₂²

yes

characteristic

C₂×D₂₃₉₆

C₂²

yes

characteristic

C₂×C₂²⋊D₅₉₉

C₂²

yes

—

C₂×C₂²⋊D₅₉₉

C₂²

yes

—

C₂²×Dic₅₉₉

C₂²

yes

—

C₂×C₄×D₅₉₉

C₂²

yes

—

C₄⋊Dic₅₉₉

C₂²

yes

—

C₄⋊Dic₅₉₉

C₂²

yes

characteristic

C₄⋊Dic₅₉₉

C₂²

yes

characteristic

Dic₅₉₉⋊C₄

C₂²

yes

—

Dic₅₉₉⋊C₄

C₂²

yes

—

Dic₅₉₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₅₉₉

C₂²

yes

—

(C₂×D₅₉₉)⋊C₄

C₂²

yes

—

(C₂×D₅₉₉)⋊C₄

C₂²

yes

—

(C₂×D₅₉₉)⋊C₄

C₂²

yes

—

C₂×Q₈×C₅₉₉

C₂²

yes

characteristic

C₂×D₄×C₅₉₉

C₂²

yes

characteristic

C₂²⋊C₄×C₅₉₉

C₂²

yes

—

C₂²⋊C₄×C₅₉₉

C₂²

yes

—

C₄²×C₅₉₉

C₂²

yes

characteristic

C₂²×D₅₉₉

C₂³

yes

—

Dic₁₁₉₈

—

D₂₃₉₆

—

C₂²⋊D₅₉₉

—

C₂²⋊D₅₉₉

—

C₂×Dic₅₉₉

—

C₂×Dic₅₉₉

C₂³

yes

—

C₂×Dic₅₉₉

C₂³

yes

—

C₂×Dic₅₉₉

C₂³

yes

—

C₄×D₅₉₉

—

C₂³×C₅₉₉

C₂³

yes

—

Q₈×C₅₉₉

—

D₄×C₅₉₉

—

C₂×C₄×C₅₉₉

C₂³

yes

characteristic

C₂×C₄×C₅₉₉

C₂³

yes

characteristic

C₂×C₄×C₅₉₉

C₂³

yes

characteristic

C₂×C₄×C₅₉₉

C₂³

yes

—

C₂²×C₅₉₉

—

C₂²×C₅₉₉

—

C₂²×C₅₉₉

—

C₂²×C₅₉₉

C₂⁴

yes

characteristic

C₂×D₅₉₉

—

C₂×D₅₉₉

—

C₂×D₅₉₉

—

C₄×C₅₉₉

—

C₄×C₅₉₉

—

C₄×C₅₉₉

—

C₄×C₅₉₉

—

Dic₅₉₉

—

Dic₅₉₉

—

Dic₅₉₉

—

C₂×C₅₉₉

—

C₂×C₅₉₉

yes

C₂×C₅₉₉

yes

C₂×C₅₉₉

2₋¹⁺⁴

yes

characteristic

D₅₉₉

—

C₅₉₉

#562

yes

characteristic, complemented, Sylow

#562

—

599

maximal, Sylow

#125

—

599

—

#124

—

599

—

#124

D₅₉₉

yes

characteristic, complemented

#123

—

599

—

#123

—

599

—

C₂²⋊Q₈

—

599

—

C₂²⋊Q₈

—

599

—

#121

—

599

—

#121

—

599

—

C₂×Q₈

—

599

—

C₂×Q₈

C₂×D₅₉₉

yes

characteristic

C₂×D₄

—

599

—

C₂×D₄

—

599

—

C₂×D₄

—

599

—

C₂×D₄

C₂×D₅₉₉

yes

characteristic

C₂²×C₄

—

599

—

C₂²×C₄

—

599

—

C₄⋊C₄

—

599

—

C₄⋊C₄

—

599

—

C₄⋊C₄

—

599

—

C₄⋊C₄

—

599

—

C₄⋊C₄

—

599

—

C₄⋊C₄

—

599

—

C₂²⋊C₄

—

599

—

C₂²⋊C₄

—

599

—

C₂²⋊C₄

—

599

—

C₂²⋊C₄

—

599

—

C₂²⋊C₄

C₂×D₅₉₉

yes

—

C₂²⋊C₄

C₂×D₅₉₉

yes

—

C₄²

C₂×D₅₉₉

yes

characteristic

C₂³

—

599

—

100

C₂³

C₂²×D₅₉₉

yes

—

101

Q₈

—

102

Q₈

—

2396

—

103

D₄

—

104

D₄

—

2396

—

105

D₄

—

2396

—

106

D₄

—

2396

—

107

C₂×C₄

—

599

—

108

C₂×C₄

—

599

—

109

C₂×C₄

—

599

—

110

C₂×C₄

—

2396

—

111

C₂×C₄

—

2396

—

112

C₂×C₄

C₂²×D₅₉₉

yes

characteristic

113

C₂×C₄

C₂²×D₅₉₉

yes

characteristic

114

C₂×C₄

C₂²×D₅₉₉

yes

characteristic

115

C₂×C₄

C₂²×D₅₉₉

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

1198

—

120

C₂²

—

1198

—

121

C₂²

—

1198

—

122

C₂²

C₂³×D₅₉₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

1198

—

128

C₄

—

1198

—

129

C₄

—

1198

—

130

C₂

—

131

C₂

—

2396

—

132

C₂

#34062760

yes

characteristic

133

C₂

#34062761

yes

characteristic

134

C₂

#34062762

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₉₉
commutator, socle, cosocle	C₂²×C₅₉₉	C₂⁴
Fitting	#124×C₅₉₉	C₂
hypercenter	#124	D₅₉₉
hypocenter	C₅₉₉	#562
LCS	G ▷ C₂²×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₉₉

commutator, socle, cosocle

C₂²×C₅₉₉

C₂⁴

Fitting

#124×C₅₉₉

C₂

D₅₉₉

C₅₉₉

G ▷ C₂²×C₅₉₉ ▷ C₅₉₉

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #34062707	C₂	semidirect
2. #34062703	C₂	semidirect
3. #34062699	C₂	semidirect
4. #34062697	C₂	semidirect
5. #34062692	C₂	semidirect
6. #34062691	C₂	semidirect
7. #34062688	C₂	semidirect
8. #34062687	C₂	semidirect
9. #124×C₅₉₉	C₂	semidirect
67. C₅₉₉	#562	semidirect
71. #124	D₅₉₉	semidirect

H ≅ G/N

type

1. #34062707

C₂

semidirect

2. #34062703

C₂

semidirect

3. #34062699

C₂

semidirect

4. #34062697

C₂

semidirect

5. #34062692

C₂

semidirect

6. #34062691

C₂

semidirect

7. #34062688

C₂

semidirect

8. #34062687

C₂

semidirect

9. #124×C₅₉₉

C₂

semidirect

67. C₅₉₉

#562

semidirect

71. #124

D₅₉₉

semidirect

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

subgroup	index
#34062707	2
#34062703	2
#34062699	2
#34062697	2
#34062692	2
#34062691	2
#34062688	2
#34062687	2
#124×C₅₉₉	2
#562	599

subgroup

index

#124×C₅₉₉

599

Subgroups — #50306634

Conjugacy classes of subgroups

Complements

Maximal subgroups