Group #16818223 — subgroups

Subgroups — #16818223

Order 3712.

202 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 73 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	#14271720	C₂	1	1	yes	maximal, characteristic
2	#14271187	C₂	1	1	yes	maximal, characteristic
3	#14270911	C₂	1	2	yes	maximal, complemented
4	#14270908	C₂	1	2	yes	maximal, complemented
5	C₂₉×#432	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
7	#11745288	—	2	1	no	—
8	#11745286	—	2	1	no	—
9	#11745286	C₄	1	2	yes	—
10	C₂×#90777	—	2	1	no	—
11	#11745232	—	2	1	no	—
12	#11745231	C₂²	1	1	yes	characteristic
13	#11745231	C₂²	1	1	yes	characteristic
14	C₄×Dic₅₈	—	2	1	no	—
15	C₄×Dic₅₈	C₄	1	2	yes	—
16	C₂₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂₉×C₂²⋊C₈	—	2	1	no	—
18	C₂₉×#97	—	2	1	no	—
19	C₄⋊#6597	—	2	1	no	—
20	C₄⋊#6597	C₂²	1	2	yes	—
21	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
24	C₂×Dic₅₈	—	2	1	no	—
25	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₂₉	—	2	2	no	—
27	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₂₉	D₄	1	1	yes	characteristic
29	C₄²×C₂₉	D₄	1	1	yes	characteristic
30	C₄²×C₂₉	C₂³	1	1	yes	characteristic
31	C₂²⋊Dic₂₉	—	2	2	no	—
32	C₄⋊Dic₂₉	—	2	1	no	—
33	C₄⋊Dic₂₉	—	2	1	no	—
34	C₄⋊Dic₂₉	—	2	1	no	—
35	C₄⋊Dic₂₉	C₂×C₄	1	2	yes	—
36	#90779	—	2	2	no	—
37	#90779	—	2	2	no	—
38	C₄×Dic₂₉	—	2	2	no	—
39	#90777	—	4	2	no	—
40	C₂×#6597	—	2	2	no	—
41	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	—	2	1	no	—
43	C₂×C₄×C₂₉	—	2	2	no	—
44	C₂×C₄×C₂₉	—	2	1	no	—
45	C₂×C₄×C₂₉	—	4	1	no	—
46	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₂₉	—	2	2	no	—
53	C₂×Dic₂₉	—	2	2	no	—
54	Dic₅₈	—	4	1	no	—
55	Dic₅₈	—	4	1	no	—
56	Dic₅₈	—	4	2	no	—
57	C₈×C₂₉	—	4	2	no	—
58	#6597	—	4	2	no	—
59	#1559	—	29	1	no	maximal, Sylow
60	C₂²×C₂₉	—	2	1	no	—
61	C₂²×C₂₉	—	2	1	no	—
62	C₂²×C₂₉	—	2	1	no	—
63	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₂₉	—	2	1	no	—
65	C₄×C₂₉	—	2	1	no	—
66	C₄×C₂₉	—	2	1	no	—
67	C₄×C₂₉	—	2	1	no	—
68	C₄×C₂₉	AGL(1,ℤ/8)	1	1	yes	characteristic
69	C₄×C₂₉	#137	1	2	yes	—
70	C₄×C₂₉	#137	1	1	yes	characteristic
71	Dic₂₉	—	4	2	no	—
72	Dic₂₉	—	4	2	no	—
73	#533	—	29	1	no	—
74	#432	D₂₉	1	1	yes	characteristic
75	#423	—	29	1	no	—
76	#333	—	29	2	no	—
77	#332	—	29	2	no	—
78	C₂×C₂₉	—	4	1	no	—
79	C₂×C₂₉	#413	1	1	yes	characteristic
80	C₂×C₂₉	#417	1	1	yes	characteristic
81	C₂×C₂₉	#419	1	1	yes	characteristic
82	C₂×M₁₆	—	58	1	no	—
83	C₄⋊Q₈	—	29	1	no	—
84	C₄⋊Q₈	—	29	1	no	—
85	#125	—	58	1	no	—
86	C₂²⋊Q₈	—	29	2	no	—
87	C₂²⋊Q₈	—	58	1	no	—
88	C₄×Q₈	—	29	2	no	—
89	C₄×Q₈	—	58	1	no	—
90	#117	—	58	1	no	—
91	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
92	C₄⋊C₈	—	29	2	no	—
93	C₄⋊C₈	—	58	1	no	—
94	C₄⋊C₈	C₂×D₂₉	1	2	yes	—
95	C₂²⋊C₈	—	2	1	no	—
96	#97	—	2	1	no	—
97	C₂₉	#1559	1	1	yes	characteristic, complemented, Sylow
98	C₂×Q₈	—	29	2	no	—
99	C₂×Q₈	—	58	1	no	—
100	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
101	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
102	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
103	M₁₆	—	116	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	58	2	no	—
106	C₄⋊C₄	—	29	2	no	—
107	C₄⋊C₄	—	58	2	no	—
108	C₄⋊C₄	—	58	2	no	—
109	C₄⋊C₄	—	58	1	no	—
110	C₄⋊C₄	—	58	1	no	—
111	C₄⋊C₄	—	58	1	no	—
112	C₂²⋊C₄	—	58	2	no	—
113	C₄²	—	58	2	no	—
114	C₄²	C₄×D₂₉	1	1	yes	characteristic
115	C₄²	D₁₁₆	1	1	yes	characteristic
116	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
117	C₄²	C₂²×D₂₉	1	1	yes	characteristic
118	C₂³	#90781	1	1	yes	characteristic
119	Q₈	—	116	2	no	—
120	Q₈	—	116	1	no	—
121	Q₈	—	116	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	58	2	no	—
127	C₂×C₄	—	58	2	no	—
128	C₂×C₄	#90781	1	1	yes	characteristic
129	C₂×C₄	#90781	1	1	yes	characteristic
130	C₂×C₄	#90781	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
132	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	116	2	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	C₂×#90781	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	116	2	no	—
145	C₄	—	116	2	no	—
146	C₄	#11745270	1	1	yes	characteristic
147	C₄	#11745271	1	1	yes	characteristic
148	C₄	#11745313	1	2	yes	—
149	C₂	—	4	1	no	—
150	C₂	#14271145	1	1	yes	characteristic
151	C₂	#14271165	1	1	yes	characteristic
152	C₂	#14271171	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271720

C₂

yes

maximal, characteristic

#14271187

C₂

yes

maximal, characteristic

#14270911

C₂

yes

maximal, complemented

#14270908

C₂

yes

maximal, complemented

C₂₉×#432

C₂

yes

maximal, characteristic

C₂×C₄²×C₂₉

C₂²

yes

—

—

C₄

yes

—

C₂×#90777

—

#11745232

—

#11745231

C₂²

yes

characteristic

#11745231

C₂²

yes

characteristic

C₄×Dic₅₈

—

C₄×Dic₅₈

C₄

yes

—

C₂₉×C₄⋊C₈

C₂²

yes

—

C₂₉×C₂²⋊C₈

—

C₂₉×#97

—

C₄⋊#6597

—

C₄⋊#6597

C₂²

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×C₈×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

C₂³

yes

characteristic

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

#90779

—

#90779

—

C₄×Dic₂₉

—

#90777

—

C₂×#6597

—

C₂³×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

#6597

—

#1559

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

AGL(1,ℤ/8)

yes

characteristic

C₄×C₂₉

#137

yes

—

C₄×C₂₉

#137

yes

characteristic

Dic₂₉

—

Dic₂₉

—

#533

—

#432

D₂₉

yes

—

—

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₉

yes

—

C₂²⋊C₈

—

#97

—

C₂₉

#1559

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

100

C₂²×C₄

C₄×D₂₉

yes

characteristic

101

C₂²×C₄

D₁₁₆

yes

characteristic

102

C₂²×C₄

C₂²⋊D₂₉

yes

characteristic

103

M₁₆

—

116

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₄×D₂₉

yes

characteristic

115

C₄²

D₁₁₆

yes

characteristic

116

C₄²

C₂²⋊D₂₉

yes

characteristic

117

C₄²

C₂²×D₂₉

yes

characteristic

118

C₂³

#90781

yes

characteristic

119

Q₈

—

116

—

120

Q₈

—

116

—

121

Q₈

—

116

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#90781

yes

characteristic

129

C₂×C₄

#90781

yes

characteristic

130

C₂×C₄

#90781

yes

characteristic

131

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

132

C₂×C₄

C₂×D₁₁₆

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

134

C₈

—

135

C₈

—

116

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

C₂×#90781

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

116

—

145

C₄

—

116

—

146

C₄

#11745270

yes

characteristic

147

C₄

#11745271

yes

characteristic

148

C₄

#11745313

yes

—

149

C₂

—

150

C₂

#14271145

yes

characteristic

151

C₂

#14271165

yes

characteristic

152

C₂

#14271171

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#90781
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×#432	C₂
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₉	C₂²⋊C₄
cosocle	C₄²×C₂₉	C₂³
hypercenter	#432	D₂₉
hypocenter	C₂₉	#1559
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×#90781

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₄²

C₂²×D₂₉

Fitting

C₂₉×#432

C₂

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₉

C₂²⋊C₄

cosocle

C₄²×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂²×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #14270911	C₂	semidirect
4. #14270908	C₂	semidirect
97. C₂₉	#1559	semidirect

H ≅ G/N

type

3. #14270911

C₂

semidirect

4. #14270908

C₂

semidirect

97. C₂₉

#1559

semidirect

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

subgroup	index
#14271720	2
#14271187	2
#14270911	2
#14270908	2
C₂₉×#432	2
#1559	29

subgroup

index

C₂₉×#432

Subgroups — #16818223

Conjugacy classes of subgroups

Complements

Maximal subgroups