Group #44543708 — subgroups

Subgroups — #44543708

Order 31936.

192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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	#30630349	C₂	1	1	yes	maximal, characteristic, complemented
2	#30630344	C₂	1	2	yes	maximal
3	#30630344	C₂	1	2	yes	maximal
4	#30630342	C₂	1	1	yes	maximal, characteristic, complemented
5	#30630342	C₂	1	2	yes	maximal, complemented
6	#30630341	C₂	1	2	yes	maximal, complemented
7	#30630330	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₄₉₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#30630316	C₂	1	1	yes	maximal, characteristic, complemented
10	#30630310	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₄₉₉	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₉₉₈	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₄₉₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₄₉₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₄₉₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₄₉₉	C₂²	1	2	yes	—
17	C₄⋊Dic₄₉₉	C₂²	1	2	yes	—
18	Dic₄₉₉⋊C₄	C₂²	1	2	yes	—
19	Dic₄₉₉⋊C₄	C₂²	1	2	yes	—
20	Dic₄₉₉⋊C₄	C₂²	1	2	yes	—
21	Dic₄₉₉⋊C₄	C₂²	1	2	yes	—
22	Dic₄₉₉⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊Dic₄₉₉	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₄₉₉	C₂²	1	1	yes	characteristic
25	C₂²⋊Dic₄₉₉	C₂²	1	2	yes	—
26	C₄×Dic₄₉₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₄₉₉	C₂²	1	1	yes	characteristic
28	C₄×Dic₄₉₉	C₂²	1	2	yes	—
29	C₄×Dic₄₉₉	C₂²	1	2	yes	—
30	C₂×Q₈×C₄₉₉	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₄₉₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₄₉₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₄₉₉	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₄₉₉	C₂²	1	2	yes	—
35	Dic₉₉₈	—	2	2	no	—
36	C₂×Dic₄₉₉	C₂³	1	1	yes	characteristic
37	C₂×Dic₄₉₉	C₂³	1	1	yes	characteristic
38	C₂×Dic₄₉₉	C₂³	1	1	yes	characteristic
39	C₂×Dic₄₉₉	C₂³	1	2	yes	—
40	C₂×Dic₄₉₉	C₂³	1	1	yes	characteristic
41	C₂×Dic₄₉₉	C₂³	1	2	yes	—
42	C₂³×C₄₉₉	C₂³	1	1	yes	characteristic
43	Q₈×C₄₉₉	—	2	2	no	—
44	C₂×C₄×C₄₉₉	—	2	2	no	—
45	C₂×C₄×C₄₉₉	C₂³	1	2	yes	—
46	C₂×C₄×C₄₉₉	C₂³	1	2	yes	—
47	C₂×C₄×C₄₉₉	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₄₉₉	C₂³	1	1	yes	characteristic
49	C₂²×C₄₉₉	—	2	1	no	—
50	C₂²×C₄₉₉	—	2	1	no	—
51	C₂²×C₄₉₉	—	2	1	no	—
52	C₂²×C₄₉₉	C₂⁴	1	1	yes	characteristic
53	C₄×C₄₉₉	—	2	1	no	—
54	C₄×C₄₉₉	—	2	2	no	—
55	C₄×C₄₉₉	—	2	2	no	—
56	C₄×C₄₉₉	#41	1	2	yes	—
57	Dic₄₉₉	—	2	2	no	—
58	Dic₄₉₉	—	2	1	no	—
59	Dic₄₉₉	—	2	1	no	—
60	Dic₄₉₉	—	2	1	no	—
61	Dic₄₉₉	—	2	1	no	—
62	Dic₄₉₉	—	2	2	no	—
63	C₂×C₄₉₉	—	4	1	no	—
64	C₂×C₄₉₉	C₂×#41	1	1	yes	characteristic
65	C₂×C₄₉₉	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂×C₄₉₉	2₋¹⁺⁴	1	1	yes	characteristic
67	C₄₉₉	#541	1	1	yes	characteristic, complemented, Sylow
68	#541	—	499	1	no	maximal, Sylow
69	C₄⋊Q₈	—	499	1	no	—
70	#126	—	499	2	no	—
71	#126	—	499	2	no	—
72	#125	—	499	2	no	—
73	#125	—	499	1	no	—
74	#125	—	499	2	no	—
75	C₂²⋊Q₈	—	499	1	no	—
76	C₂²⋊Q₈	D₄₉₉	1	1	yes	characteristic
77	C₄×Q₈	—	499	1	no	—
78	C₄×Q₈	—	499	1	no	—
79	#117	—	499	1	no	—
80	C₂×Q₈	—	499	1	no	—
81	C₂×Q₈	C₂×D₄₉₉	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄₉₉	1	1	yes	characteristic
83	C₄⋊C₄	—	499	2	no	—
84	C₄⋊C₄	—	499	2	no	—
85	C₄⋊C₄	—	499	2	no	—
86	C₄⋊C₄	—	499	2	no	—
87	C₄⋊C₄	—	499	2	no	—
88	C₄⋊C₄	—	499	1	no	—
89	C₄⋊C₄	—	499	1	no	—
90	C₄⋊C₄	—	499	1	no	—
91	C₄⋊C₄	—	499	2	no	—
92	C₄⋊C₄	—	499	2	no	—
93	C₄⋊C₄	C₂×D₄₉₉	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₄₉₉	1	2	yes	—
95	C₂²⋊C₄	—	499	1	no	—
96	C₂²⋊C₄	—	499	1	no	—
97	C₂²⋊C₄	—	499	2	no	—
98	C₂²⋊C₄	C₂×D₄₉₉	1	2	yes	—
99	C₄²	—	499	1	no	—
100	C₄²	—	499	1	no	—
101	C₄²	—	499	2	no	—
102	C₄²	—	499	2	no	—
103	C₂³	C₂²×D₄₉₉	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	Q₈	—	998	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	499	1	no	—
108	C₂×C₄	—	499	1	no	—
109	C₂×C₄	—	499	1	no	—
110	C₂×C₄	—	499	1	no	—
111	C₂×C₄	—	499	2	no	—
112	C₂×C₄	—	499	2	no	—
113	C₂×C₄	C₂²×D₄₉₉	1	2	yes	—
114	C₂×C₄	C₂²×D₄₉₉	1	2	yes	—
115	C₂×C₄	C₂²×D₄₉₉	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₄₉₉	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂³×D₄₉₉	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	998	2	no	—
125	C₄	—	998	1	no	—
126	C₄	—	998	1	no	—
127	C₄	—	998	1	no	—
128	C₄	—	998	1	no	—
129	C₄	—	998	2	no	—
130	C₄	#22547810	1	2	yes	—
131	C₂	—	4	1	no	—
132	C₂	C₂×#22547810	1	1	yes	characteristic
133	C₂	#30630398	1	1	yes	characteristic
134	C₂	#30630399	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30630349

C₂

yes

maximal, characteristic, complemented

#30630344

C₂

yes

maximal

#30630344

C₂

yes

maximal

#30630342

C₂

yes

maximal, characteristic, complemented

#30630342

C₂

yes

maximal, complemented

#30630341

C₂

yes

maximal, complemented

#30630330

C₂

yes

maximal, characteristic

Q₈×Dic₄₉₉

C₂

yes

maximal, characteristic, complemented

#30630316

C₂

yes

maximal, characteristic, complemented

#30630310

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₄₉₉

C₂

yes

maximal, characteristic

C₂×Dic₉₉₈

C₂²

yes

characteristic

C₄⋊Dic₄₉₉

C₂²

yes

characteristic

C₄⋊Dic₄₉₉

C₂²

yes

characteristic

C₄⋊Dic₄₉₉

C₂²

yes

characteristic

C₄⋊Dic₄₉₉

C₂²

yes

—

C₄⋊Dic₄₉₉

C₂²

yes

—

Dic₄₉₉⋊C₄

C₂²

yes

—

Dic₄₉₉⋊C₄

C₂²

yes

—

Dic₄₉₉⋊C₄

C₂²

yes

—

Dic₄₉₉⋊C₄

C₂²

yes

—

Dic₄₉₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₄₉₉

C₂²

yes

characteristic

C₂²⋊Dic₄₉₉

C₂²

yes

characteristic

C₂²⋊Dic₄₉₉

C₂²

yes

—

C₄×Dic₄₉₉

C₂²

yes

characteristic

C₄×Dic₄₉₉

C₂²

yes

characteristic

C₄×Dic₄₉₉

C₂²

yes

—

C₄×Dic₄₉₉

C₂²

yes

—

C₂×Q₈×C₄₉₉

C₂²

yes

characteristic

C₂²×C₄×C₄₉₉

C₂²

yes

characteristic

C₄⋊C₄×C₄₉₉

C₂²

yes

characteristic

C₄⋊C₄×C₄₉₉

C₂²

yes

—

C₂²⋊C₄×C₄₉₉

C₂²

yes

—

Dic₉₉₈

—

C₂×Dic₄₉₉

C₂³

yes

characteristic

C₂×Dic₄₉₉

C₂³

yes

characteristic

C₂×Dic₄₉₉

C₂³

yes

characteristic

C₂×Dic₄₉₉

C₂³

yes

—

C₂×Dic₄₉₉

C₂³

yes

characteristic

C₂×Dic₄₉₉

C₂³

yes

—

C₂³×C₄₉₉

C₂³

yes

characteristic

Q₈×C₄₉₉

—

C₂×C₄×C₄₉₉

—

C₂×C₄×C₄₉₉

C₂³

yes

—

C₂×C₄×C₄₉₉

C₂³

yes

—

C₂×C₄×C₄₉₉

C₂³

yes

characteristic

C₂×C₄×C₄₉₉

C₂³

yes

characteristic

C₂²×C₄₉₉

—

C₂²×C₄₉₉

—

C₂²×C₄₉₉

—

C₂²×C₄₉₉

C₂⁴

yes

characteristic

C₄×C₄₉₉

—

C₄×C₄₉₉

—

C₄×C₄₉₉

—

C₄×C₄₉₉

#41

yes

—

Dic₄₉₉

—

Dic₄₉₉

—

Dic₄₉₉

—

Dic₄₉₉

—

Dic₄₉₉

—

Dic₄₉₉

—

C₂×C₄₉₉

—

C₂×C₄₉₉

C₂×#41

yes

characteristic

C₂×C₄₉₉

2₋¹⁺⁴

yes

characteristic

C₂×C₄₉₉

2₋¹⁺⁴

yes

characteristic

C₄₉₉

#541

yes

characteristic, complemented, Sylow

#541

—

499

maximal, Sylow

C₄⋊Q₈

—

499

—

#126

—

499

—

#126

—

499

—

#125

—

499

—

#125

—

499

—

#125

—

499

—

C₂²⋊Q₈

—

499

—

C₂²⋊Q₈

D₄₉₉

yes

characteristic

C₄×Q₈

—

499

—

C₄×Q₈

—

499

—

#117

—

499

—

C₂×Q₈

—

499

—

C₂×Q₈

C₂×D₄₉₉

yes

characteristic

C₂²×C₄

C₂×D₄₉₉

yes

characteristic

C₄⋊C₄

—

499

—

C₄⋊C₄

—

499

—

C₄⋊C₄

—

499

—

C₄⋊C₄

—

499

—

C₄⋊C₄

—

499

—

C₄⋊C₄

—

499

—

C₄⋊C₄

—

499

—

C₄⋊C₄

—

499

—

C₄⋊C₄

—

499

—

C₄⋊C₄

—

499

—

C₄⋊C₄

C₂×D₄₉₉

yes

characteristic

C₄⋊C₄

C₂×D₄₉₉

yes

—

C₂²⋊C₄

—

499

—

C₂²⋊C₄

—

499

—

C₂²⋊C₄

—

499

—

C₂²⋊C₄

C₂×D₄₉₉

yes

—

C₄²

—

499

—

100

C₄²

—

499

—

101

C₄²

—

499

—

102

C₄²

—

499

—

103

C₂³

C₂²×D₄₉₉

yes

characteristic

104

Q₈

—

105

Q₈

—

998

—

106

C₂×C₄

—

107

C₂×C₄

—

499

—

108

C₂×C₄

—

499

—

109

C₂×C₄

—

499

—

110

C₂×C₄

—

499

—

111

C₂×C₄

—

499

—

112

C₂×C₄

—

499

—

113

C₂×C₄

C₂²×D₄₉₉

yes

—

114

C₂×C₄

C₂²×D₄₉₉

yes

—

115

C₂×C₄

C₂²×D₄₉₉

yes

characteristic

116

C₂×C₄

C₂²×D₄₉₉

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₄₉₉

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

998

—

125

C₄

—

998

—

126

C₄

—

998

—

127

C₄

—

998

—

128

C₄

—

998

—

129

C₄

—

998

—

130

C₄

#22547810

yes

—

131

C₂

—

132

C₂

C₂×#22547810

yes

characteristic

133

C₂

#30630398

yes

characteristic

134

C₂

#30630399

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₉₉
commutator, socle, cosocle	C₂²×C₄₉₉	C₂⁴
Fitting	C₂²⋊Q₈×C₄₉₉	C₂
prime radical	C₂³×C₄₉₉	C₂³
hypercenter	C₂²⋊Q₈	D₄₉₉
hypocenter	C₄₉₉	#541
LCS	G ▷ C₂²×C₄₉₉ ▷ C₄₉₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₉₉

commutator, socle, cosocle

C₂²×C₄₉₉

C₂⁴

Fitting

C₂²⋊Q₈×C₄₉₉

C₂

prime radical

C₂³×C₄₉₉

C₂³

hypercenter

C₂²⋊Q₈

D₄₉₉

C₄₉₉

G ▷ C₂²×C₄₉₉ ▷ C₄₉₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #30630349	C₂	semidirect
4. #30630342	C₂	semidirect
5. #30630342	C₂	semidirect
6. #30630341	C₂	semidirect
8. Q₈×Dic₄₉₉	C₂	semidirect
9. #30630316	C₂	semidirect
67. C₄₉₉	#541	semidirect

H ≅ G/N

type

1. #30630349

C₂

semidirect

4. #30630342

C₂

semidirect

5. #30630342

C₂

semidirect

6. #30630341

C₂

semidirect

8. Q₈×Dic₄₉₉

C₂

semidirect

9. #30630316

C₂

semidirect

67. C₄₉₉

#541

semidirect

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

subgroup	index
#30630349	2
#30630344	2
#30630342	2
#30630341	2
#30630330	2
Q₈×Dic₄₉₉	2
#30630316	2
#30630310	2
C₂²⋊Q₈×C₄₉₉	2
#541	499

subgroup

index

Q₈×Dic₄₉₉

C₂²⋊Q₈×C₄₉₉

499

Subgroups — #44543708

Conjugacy classes of subgroups

Complements

Maximal subgroups