Group #16818860 — subgroups

Subgroups — #16818860

Order 3712.

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

normal subgroups only

This group has 156 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	#14271542	C₂	1	1	yes	maximal, characteristic
2	#14271448	C₂	1	1	yes	maximal, characteristic
3	Q₈×#6597	C₂	1	1	yes	maximal, characteristic
4	#14270915	C₂	1	1	yes	maximal, characteristic
5	#14270909	C₂	1	1	yes	maximal, characteristic
6	#14270908	C₂	1	1	yes	maximal, characteristic
7	C₂₉×#475	C₂	1	1	yes	maximal, characteristic
8	C₂₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
10	C₂×#90785	—	2	1	no	—
11	C₂×#90785	—	2	1	no	—
12	#11745231	C₂²	1	1	yes	characteristic
13	C₂₉×#106	—	2	1	no	—
14	C₂₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂₉×#103	—	2	1	no	—
16	#11745183	—	2	1	no	—
17	#11745156	—	2	1	no	—
18	C₄⋊#6597	—	2	1	no	—
19	C₄⋊#6597	C₂²	1	1	yes	characteristic
20	C₄⋊#6597	C₂²	1	1	yes	characteristic
21	C₄×#6597	—	2	1	no	—
22	C₄×#6597	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₂₉	—	2	1	no	—
24	C₂×Q₈×C₂₉	D₄	1	1	yes	characteristic
25	C₂×Dic₅₈	—	2	1	no	—
26	C₂×C₈×C₂₉	—	2	1	no	—
27	C₄⋊C₄×C₂₉	—	2	1	no	—
28	C₄⋊C₄×C₂₉	—	2	1	no	—
29	C₄⋊C₄×C₂₉	—	2	1	no	—
30	C₄⋊C₄×C₂₉	D₄	1	1	yes	characteristic
31	C₄²×C₂₉	—	2	1	no	—
32	C₄²×C₂₉	C₂³	1	1	yes	characteristic
33	#90785	—	4	2	no	—
34	#90785	—	4	2	no	—
35	C₄⋊Dic₂₉	—	2	1	no	—
36	C₄⋊Dic₂₉	—	2	1	no	—
37	C₂×#6597	—	2	1	no	—
38	C₂×#6597	—	2	1	no	—
39	C₂×#6597	D₄	1	1	yes	characteristic
40	C₂×#6597	D₄	1	1	yes	characteristic
41	Q₈×C₂₉	—	2	1	no	—
42	Q₈×C₂₉	—	4	1	no	—
43	Q₈×C₂₉	—	4	1	no	—
44	Q₈×C₂₉	Q₁₆	1	2	yes	complemented
45	C₂×C₄×C₂₉	—	2	1	no	—
46	C₂×C₄×C₂₉	—	2	1	no	—
47	C₂×C₄×C₂₉	—	4	1	no	—
48	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
51	C₂×Dic₂₉	—	4	1	no	—
52	Dic₅₈	—	4	1	no	—
53	Dic₅₈	—	4	1	no	—
54	C₈×C₂₉	—	4	1	no	—
55	#6597	—	2	1	no	—
56	#6597	—	4	1	no	—
57	#6597	—	4	1	no	—
58	#6597	SD₁₆	1	2	yes	—
59	#1670	—	29	1	no	maximal, Sylow
60	C₂²×C₂₉	#121	1	1	yes	characteristic
61	C₄×C₂₉	—	2	2	no	—
62	C₄×C₂₉	—	2	1	no	—
63	C₄×C₂₉	—	4	1	no	—
64	C₄×C₂₉	—	8	1	no	—
65	C₄×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
67	C₄×C₂₉	#135	1	1	yes	characteristic
68	C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	Dic₂₉	—	8	1	no	—
70	#494	—	29	1	no	—
71	#475	—	29	1	no	—
72	#475	D₂₉	1	1	yes	characteristic
73	C₈×Q₈	—	29	1	no	—
74	#335	—	29	1	no	—
75	#332	—	29	1	no	—
76	#332	—	29	1	no	—
77	C₂×C₂₉	#460	1	1	yes	characteristic
78	C₂×C₂₉	#467	1	1	yes	characteristic
79	C₂×C₂₉	#473	1	1	yes	characteristic
80	C₂×Q₁₆	—	58	1	no	—
81	C₂×Q₁₆	—	58	1	no	—
82	C₄⋊Q₈	—	29	1	no	—
83	C₄⋊Q₈	C₂×D₂₉	1	1	yes	characteristic
84	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
85	#106	—	2	1	no	—
86	#106	—	58	1	no	—
87	C₄⋊C₈	—	29	1	no	—
88	C₄⋊C₈	—	29	1	no	—
89	C₄⋊C₈	—	58	1	no	—
90	C₄⋊C₈	C₂×D₂₉	1	1	yes	characteristic
91	#103	—	2	1	no	—
92	#103	—	58	1	no	—
93	C₄×C₈	—	29	1	no	—
94	C₄×C₈	—	58	1	no	—
95	C₂₉	#1670	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	58	1	no	—
98	C₂×Q₈	C₂²⋊D₂₉	1	1	yes	characteristic
99	Q₁₆	—	116	2	no	—
100	Q₁₆	—	116	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	29	1	no	—
103	C₂×C₈	—	29	1	no	—
104	C₂×C₈	—	58	1	no	—
105	C₂×C₈	—	58	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	58	1	no	—
110	C₄⋊C₄	—	58	1	no	—
111	C₄⋊C₄	C₂²⋊D₂₉	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₂₉	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	116	1	no	—
118	Q₈	—	116	1	no	—
119	Q₈	#90785	1	2	yes	complemented
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	116	1	no	—
124	C₂×C₄	D₄×D₂₉	1	1	yes	characteristic
125	C₂×C₄	#90807	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	29	2	no	—
129	C₈	—	58	1	no	—
130	C₈	—	116	1	no	—
131	C₈	—	116	1	no	—
132	C₂²	#11745301	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	232	1	no	—
138	C₄	SD₁₆×D₂₉	1	1	yes	characteristic
139	C₄	#11745276	1	1	yes	characteristic
140	C₄	C₂×#90785	1	1	yes	characteristic
141	C₄	#11745312	1	1	yes	characteristic
142	C₂	#14271361	1	1	yes	characteristic
143	C₂	#14271402	1	1	yes	characteristic
144	C₂	#14271436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271542

C₂

yes

maximal, characteristic

#14271448

C₂

yes

maximal, characteristic

Q₈×#6597

C₂

yes

maximal, characteristic

#14270915

C₂

yes

maximal, characteristic

#14270909

C₂

yes

maximal, characteristic

#14270908

C₂

yes

maximal, characteristic

C₂₉×#475

C₂

yes

maximal, characteristic

C₂₉×C₄⋊Q₈

C₂²

yes

characteristic

C₄×Q₈×C₂₉

C₂²

yes

—

—

C₂²

yes

characteristic

C₂₉×#106

—

C₂₉×C₄⋊C₈

C₂²

yes

characteristic

C₂₉×#103

—

—

—

—

C₂²

yes

characteristic

C₄⋊#6597

C₂²

yes

characteristic

C₄×#6597

—

C₄×#6597

C₂²

yes

characteristic

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

D₄

yes

characteristic

C₂×Dic₅₈

—

C₂×C₈×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

—

C₄²×C₂₉

C₂³

yes

characteristic

#90785

—

#90785

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₂×#6597

—

C₂×#6597

—

C₂×#6597

D₄

yes

characteristic

C₂×#6597

D₄

yes

characteristic

Q₈×C₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

—

Q₈×C₂₉

Q₁₆

yes

complemented

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

#41

yes

characteristic

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

—

—

—

SD₁₆

yes

—

—

C₂²×C₂₉

yes

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂×SD₁₆

yes

characteristic

C₄×C₂₉

C₂×Q₁₆

yes

C₄×C₂₉

yes

C₄×C₂₉

yes

Dic₂₉

—

#494

—

#475

—

#475

D₂₉

yes

characteristic

C₈×Q₈

—

#335

—

#332

—

#332

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₉

yes

characteristic

C₄×Q₈

C₂×D₂₉

yes

characteristic

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₉

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₂₉

#1670

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₉

yes

characteristic

Q₁₆

—

116

—

100

Q₁₆

—

116

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²⋊D₂₉

yes

characteristic

112

C₄²

—

113

C₄²

C₂²×D₂₉

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

116

—

118

Q₈

—

116

—

119

Q₈

#90785

yes

complemented

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

116

—

124

C₂×C₄

D₄×D₂₉

yes

characteristic

125

C₂×C₄

#90807

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

116

—

131

C₈

—

116

—

132

C₂²

#11745301

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

232

—

138

C₄

SD₁₆×D₂₉

yes

characteristic

139

C₄

#11745276

yes

characteristic

140

C₄

C₂×#90785

yes

characteristic

141

C₄

#11745312

yes

characteristic

142

C₂

#14271361

yes

characteristic

143

C₂

#14271402

yes

characteristic

144

C₂

#14271436

yes

characteristic

	subgroup	G/H
center	C₂²	#11745301
commutator, cosocle	C₄²×C₂₉	C₂³
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×#475	C₂
socle, prime radical	C₂²×C₂₉	#121
hypercenter	#475	D₂₉
hypocenter	C₂₉	#1670
LCS	G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#11745301

commutator, cosocle

C₄²×C₂₉

C₂³

Frattini

C₄²

C₂²×D₂₉

Fitting

C₂₉×#475

C₂

C₂²×C₂₉

D₂₉

C₂₉

G ▷ C₄²×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
44. Q₈×C₂₉	Q₁₆	semidirect
95. C₂₉	#1670	semidirect
119. Q₈	#90785	semidirect

H ≅ G/N

type

44. Q₈×C₂₉

Q₁₆

semidirect

95. C₂₉

#1670

semidirect

119. Q₈

#90785

semidirect

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

subgroup	index
#14271542	2
#14271448	2
Q₈×#6597	2
#14270915	2
#14270909	2
#14270908	2
C₂₉×#475	2
#1670	29

subgroup

index

C₂₉×#475

Subgroups — #16818860

Conjugacy classes of subgroups

Complements

Maximal subgroups