Group #16819103 — subgroups

Subgroups — #16819103

Order 3712.

178 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14271515	C₂	1	1	yes	maximal, characteristic, complemented
2	#14271514	C₂	1	1	yes	maximal, characteristic, complemented
3	#14271513	C₂	1	1	yes	maximal, characteristic, complemented
4	#14271509	C₂	1	1	yes	maximal, characteristic
5	#14271509	C₂	1	1	yes	maximal, characteristic
6	C₈×#6597	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₉×#486	C₂	1	1	yes	maximal, characteristic
8	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂₉×#127	C₂²	1	1	yes	characteristic
10	#11745231	C₂²	1	1	yes	characteristic
11	#11745231	C₂²	1	1	yes	characteristic
12	C₂₉×#102	—	2	2	no	—
13	C₄×C₈×C₂₉	C₂²	1	1	yes	characteristic
14	#11745183	—	2	2	no	—
15	#11745180	—	2	2	no	—
16	#11745180	—	2	2	no	—
17	#11745164	—	2	2	no	—
18	#11745158	—	2	2	no	—
19	C₄×#6597	C₂²	1	1	yes	characteristic
20	C₄×#6597	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₂₉	—	2	1	no	—
22	C₂×D₄×C₂₉	—	2	1	no	—
23	C₂×D₄×C₂₉	—	2	1	no	—
24	C₂×D₄×C₂₉	—	2	1	no	—
25	C₂×Dic₅₈	—	2	1	no	—
26	C₂×Dic₅₈	—	2	1	no	—
27	C₂×C₈×C₂₉	D₄	1	2	yes	—
28	C₄⋊C₄×C₂₉	—	2	1	no	—
29	C₄⋊C₄×C₂₉	—	2	1	no	—
30	C₄²×C₂₉	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₂₉	—	2	1	no	—
32	C₄⋊Dic₂₉	—	2	1	no	—
33	C₄⋊Dic₂₉	—	2	1	no	—
34	C₄⋊Dic₂₉	—	2	1	no	—
35	C₂×#6597	D₄	1	2	yes	—
36	C₂×#6597	D₄	1	2	yes	—
37	C₂³×C₂₉	—	4	1	no	—
38	Q₈×C₂₉	—	4	2	no	—
39	D₄×C₂₉	—	4	2	no	—
40	D₄×C₂₉	—	4	2	no	—
41	D₄×C₂₉	—	4	2	no	—
42	C₂×C₄×C₂₉	—	4	1	no	—
43	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₂₉	—	4	1	no	—
47	C₂×Dic₂₉	—	4	1	no	—
48	Dic₅₈	—	4	2	no	—
49	Dic₅₈	—	4	2	no	—
50	C₈×C₂₉	—	2	2	no	—
51	#6597	—	2	2	no	—
52	#6597	—	2	2	no	—
53	#1711	—	29	1	no	maximal, Sylow
54	C₂²×C₂₉	—	8	1	no	—
55	C₂²×C₂₉	—	8	1	no	—
56	C₂²×C₂₉	—	8	1	no	—
57	C₂²×C₂₉	#127	1	1	yes	characteristic
58	C₄×C₂₉	—	8	1	no	—
59	C₄×C₂₉	#135	1	2	yes	—
60	C₄×C₂₉	#135	1	2	yes	—
61	C₄×C₂₉	#135	1	2	yes	—
62	Dic₂₉	—	8	1	no	—
63	Dic₂₉	—	8	1	no	—
64	#487	—	29	1	no	—
65	#487	—	29	1	no	—
66	#487	—	29	1	no	—
67	#486	—	29	1	no	—
68	#486	—	29	1	no	—
69	#486	D₂₉	1	1	yes	characteristic
70	C₈²	—	29	1	no	—
71	C₂×C₂₉	—	16	1	no	—
72	C₂×C₂₉	#495	1	1	yes	characteristic
73	C₂×C₂₉	#495	1	1	yes	characteristic
74	C₂×C₂₉	#495	1	1	yes	characteristic
75	C₄⋊Q₈	—	29	1	no	—
76	C₄⋊Q₈	—	29	1	no	—
77	C₄⋊Q₈	C₂×D₂₉	1	1	yes	characteristic
78	#127	C₂×D₂₉	1	1	yes	characteristic
79	#103	—	58	2	no	—
80	#103	—	58	2	no	—
81	#103	—	58	2	no	—
82	#102	—	2	2	no	—
83	#102	—	58	2	no	—
84	#102	—	58	2	no	—
85	C₄×C₈	—	29	1	no	—
86	C₄×C₈	—	29	1	no	—
87	C₄×C₈	C₂×D₂₉	1	1	yes	characteristic
88	C₂₉	#1711	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	58	1	no	—
91	C₂×Q₈	—	58	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×C₈	—	29	2	no	—
96	C₂×C₈	—	29	2	no	—
97	C₂×C₈	C₂²⋊D₂₉	1	2	yes	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	58	1	no	—
101	C₄⋊C₄	—	58	1	no	—
102	C₄⋊C₄	—	58	1	no	—
103	C₄⋊C₄	—	58	1	no	—
104	C₄²	C₂²×D₂₉	1	1	yes	characteristic
105	C₂³	—	4	1	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	116	2	no	—
108	Q₈	—	116	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	116	1	no	—
114	C₂×C₄	—	116	1	no	—
115	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
116	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	58	2	no	—
120	C₈	—	58	2	no	—
121	C₂²	—	8	1	no	—
122	C₂²	—	8	1	no	—
123	C₂²	—	8	1	no	—
124	C₂²	#11745302	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	232	1	no	—
127	C₄	—	232	1	no	—
128	C₄	#11745274	1	2	yes	—
129	C₄	#11745278	1	2	yes	—
130	C₄	#11745312	1	2	yes	—
131	C₂	—	16	1	no	—
132	C₂	#14271544	1	1	yes	characteristic
133	C₂	#14271545	1	1	yes	characteristic
134	C₂	#14271546	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271515

C₂

yes

maximal, characteristic, complemented

#14271514

C₂

yes

maximal, characteristic, complemented

#14271513

C₂

yes

maximal, characteristic, complemented

#14271509

C₂

yes

maximal, characteristic

#14271509

C₂

yes

maximal, characteristic

C₈×#6597

C₂

yes

maximal, characteristic, complemented

C₂₉×#486

C₂

yes

maximal, characteristic

C₂₉×C₄⋊Q₈

C₂²

yes

characteristic

C₂₉×#127

C₂²

yes

characteristic

#11745231

C₂²

yes

characteristic

#11745231

C₂²

yes

characteristic

C₂₉×#102

—

C₄×C₈×C₂₉

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#6597

C₂²

yes

characteristic

C₂×Q₈×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×D₄×C₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×C₈×C₂₉

D₄

yes

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄²×C₂₉

C₂³

yes

characteristic

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₂×#6597

D₄

yes

—

C₂×#6597

D₄

yes

—

C₂³×C₂₉

—

Q₈×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

#127

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

#135

yes

—

C₄×C₂₉

#135

yes

—

C₄×C₂₉

#135

yes

—

Dic₂₉

—

Dic₂₉

—

—

—

—

—

—

D₂₉

yes

characteristic

C₈²

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₉

yes

characteristic

#127

C₂×D₂₉

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₉

yes

characteristic

C₂₉

#1711

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂²×D₂₉

yes

characteristic

105

C₂³

—

106

Q₈

—

107

Q₈

—

116

—

108

Q₈

—

116

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

116

—

114

C₂×C₄

—

116

—

115

C₂×C₄

D₄×D₂₉

yes

characteristic

116

C₂×C₄

D₄×D₂₉

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#11745302

yes

characteristic

125

C₄

—

126

C₄

—

232

—

127

C₄

—

232

—

128

C₄

#11745274

yes

—

129

C₄

#11745278

yes

—

130

C₄

#11745312

yes

—

131

C₂

—

132

C₂

#14271544

yes

characteristic

133

C₂

#14271545

yes

characteristic

134

C₂

#14271546

yes

characteristic

	subgroup	G/H
center	C₂²	#11745302
commutator, cosocle	C₄²×C₂₉	C₂³
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×#486	C₂
socle	C₂²×C₂₉	#127
prime radical	C₂₉×#127	C₂²
hypercenter	#486	D₂₉
hypocenter	C₂₉	#1711
LCS	G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#11745302

commutator, cosocle

C₄²×C₂₉

C₂³

Frattini

C₄²

C₂²×D₂₉

Fitting

C₂₉×#486

C₂

socle

C₂²×C₂₉

#127

prime radical

C₂₉×#127

C₂²

D₂₉

C₂₉

G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #14271515	C₂	semidirect
2. #14271514	C₂	semidirect
3. #14271513	C₂	semidirect
6. C₈×#6597	C₂	semidirect
88. C₂₉	#1711	semidirect

H ≅ G/N

type

1. #14271515

C₂

semidirect

2. #14271514

C₂

semidirect

3. #14271513

C₂

semidirect

6. C₈×#6597

C₂

semidirect

88. C₂₉

#1711

semidirect

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

subgroup	index
#14271515	2
#14271514	2
#14271513	2
#14271509	2
C₈×#6597	2
C₂₉×#486	2
#1711	29

subgroup

index

C₂₉×#486

Subgroups — #16819103

Conjugacy classes of subgroups

Complements

Maximal subgroups