Group #16830023 — subgroups

Subgroups — #16830023

Order 3712.

340 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 165 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	#14271791	C₂	1	2	yes	maximal
2	#14271572	C₂	1	4	yes	maximal, complemented
3	#14271565	C₂	1	2	yes	maximal, complemented
4	#14271561	C₂	1	2	yes	maximal, complemented
5	#14271205	C₂	1	2	yes	maximal
6	#14271201	C₂	1	1	yes	maximal, characteristic
7	C₂×#11745165	C₂	1	1	yes	maximal, characteristic
8	C₂₉×#405	C₂	1	1	yes	maximal, characteristic
9	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
11	C₂₉×#117	C₂²	1	1	yes	characteristic
12	#11745288	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
14	#11745252	—	2	2	no	—
15	#11745250	—	2	2	no	—
16	#11745240	—	2	4	no	—
17	#11745232	C₂²	1	4	yes	—
18	#11745231	C₂²	1	4	yes	—
19	C₂₉×#97	C₂²	1	2	yes	—
20	C₄×C₈×C₂₉	C₂²	1	2	yes	—
21	#11745166	C₂²	1	4	yes	—
22	#11745166	C₂²	1	2	yes	—
23	#11745165	C₂²	1	1	yes	characteristic
24	#11745165	C₂²	1	2	yes	—
25	#11745165	C₂²	1	4	yes	—
26	#11745165	C₂²	1	1	yes	characteristic
27	#11745165	C₂²	1	2	yes	—
28	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
29	C₂²×Dic₂₉	—	2	2	no	—
30	C₂×Dic₅₈	—	2	4	no	—
31	M₁₆×C₂₉	Q₈	1	4	yes	complemented
32	C₂×C₈×C₂₉	Q₈	1	4	yes	complemented
33	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₂₉	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₂₉	C₂³	1	2	yes	—
36	C₄⋊C₄×C₂₉	D₄	1	2	yes	—
37	C₂²⋊C₄×C₂₉	D₄	1	2	yes	—
38	C₄²×C₂₉	C₂³	1	2	yes	—
39	C₂²⋊Dic₂₉	—	2	2	no	—
40	C₄⋊Dic₂₉	—	2	2	no	—
41	C₄⋊Dic₂₉	—	2	4	no	—
42	C₄⋊Dic₂₉	C₂³	1	4	yes	—
43	C₄⋊Dic₂₉	C₂³	1	2	yes	—
44	C₄⋊Dic₂₉	C₂³	1	2	yes	—
45	#90779	—	2	4	no	—
46	#90779	—	2	4	no	—
47	C₄×Dic₂₉	—	2	2	no	—
48	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₂₉	—	2	2	no	—
55	C₂×Dic₂₉	—	2	2	no	—
56	C₂×Dic₂₉	—	2	4	no	—
57	C₂×Dic₂₉	—	8	2	no	—
58	Dic₅₈	—	8	4	no	—
59	C₈×C₂₉	C₂×Q₈	1	2	yes	—
60	C₈×C₂₉	C₂×Q₈	1	2	yes	—
61	C₈×C₂₉	C₂×Q₈	1	4	yes	—
62	#3163	—	29	1	no	maximal, Sylow
63	C₂²×C₂₉	—	2	2	no	—
64	C₂²×C₂₉	C₄⋊Q₈	1	2	yes	—
65	C₂²×C₂₉	C₂²×D₄	1	1	yes	characteristic
66	C₄×C₂₉	—	2	4	no	—
67	C₄×C₂₉	C₄⋊Q₈	1	2	yes	—
68	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₂₉	C₂²×Q₈	1	1	yes	characteristic
70	Dic₂₉	—	4	2	no	—
71	Dic₂₉	—	4	4	no	—
72	Dic₂₉	—	4	2	no	—
73	#544	—	29	2	no	—
74	#501	—	29	4	no	—
75	#499	—	29	2	no	—
76	#498	—	29	2	no	—
77	#428	—	29	2	no	—
78	#427	—	29	1	no	—
79	C₂×#106	—	29	1	no	—
80	#405	D₂₉	1	1	yes	characteristic
81	C₂×C₂₉	—	2	2	no	—
82	C₂×C₂₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₂₉	#577	1	2	yes	—
84	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
86	C₄⋊Q₈	—	29	4	no	—
87	C₄⋊Q₈	—	58	2	no	—
88	#125	—	29	4	no	—
89	#125	—	58	2	no	—
90	C₂²⋊Q₈	—	58	4	no	—
91	#117	—	29	2	no	—
92	#117	C₂×D₂₉	1	1	yes	characteristic
93	C₂×C₄⋊C₄	—	29	2	no	—
94	#107	—	29	2	no	—
95	#107	—	29	4	no	—
96	#106	—	29	2	no	—
97	#106	—	29	2	no	—
98	#106	—	29	1	no	—
99	#106	—	29	1	no	—
100	#106	—	29	4	no	—
101	#97	C₂×D₂₉	1	2	yes	—
102	C₄×C₈	C₂×D₂₉	1	2	yes	—
103	C₂₉	#3163	1	1	yes	characteristic, complemented, Sylow
104	C₂×Q₈	—	58	4	no	—
105	C₂²×C₄	—	58	2	no	—
106	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
107	M₁₆	Dic₅₈	1	4	yes	complemented
108	C₂×C₈	Dic₅₈	1	4	yes	complemented
109	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₂₉	1	2	yes	—
111	C₂×C₈	C₂²×D₂₉	1	1	yes	characteristic
112	C₄⋊C₄	—	29	4	no	—
113	C₄⋊C₄	—	29	2	no	—
114	C₄⋊C₄	—	29	2	no	—
115	C₄⋊C₄	—	58	2	no	—
116	C₄⋊C₄	—	58	4	no	—
117	C₄⋊C₄	—	58	4	no	—
118	C₄⋊C₄	—	58	4	no	—
119	C₄⋊C₄	D₁₁₆	1	2	yes	—
120	C₂²⋊C₄	—	58	2	no	—
121	C₂²⋊C₄	D₁₁₆	1	2	yes	—
122	C₄²	—	58	2	no	—
123	C₄²	C₂²×D₂₉	1	2	yes	—
124	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
125	Q₈	—	232	4	no	—
126	C₂×C₄	—	58	2	no	—
127	C₂×C₄	—	58	2	no	—
128	C₂×C₄	—	58	4	no	—
129	C₂×C₄	—	232	2	no	—
130	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
131	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
132	C₂×C₄	C₂×D₁₁₆	1	4	yes	—
133	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
135	C₈	C₂×Dic₅₈	1	2	yes	—
136	C₈	C₂×Dic₅₈	1	2	yes	—
137	C₈	C₂×Dic₅₈	1	4	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#11745231	1	2	yes	—
140	C₂²	C₂²×D₁₁₆	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	116	4	no	—
143	C₄	—	116	2	no	—
144	C₄	—	116	2	no	—
145	C₄	#11745231	1	2	yes	—
146	C₄	C₂²×Dic₅₈	1	1	yes	characteristic
147	C₄	C₂²×Dic₅₈	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#11745231	1	1	yes	characteristic
150	C₂	#14271975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14271791

C₂

yes

maximal

#14271572

C₂

yes

maximal, complemented

#14271565

C₂

yes

maximal, complemented

#14271561

C₂

yes

maximal, complemented

#14271205

C₂

yes

maximal

#14271201

C₂

yes

maximal, characteristic

C₂×#11745165

C₂

yes

maximal, characteristic

C₂₉×#405

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₉

C₂²

yes

characteristic

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂₉×#117

C₂²

yes

characteristic

#11745288

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

—

—

—

C₂²

yes

—

#11745231

C₂²

yes

—

C₂₉×#97

C₂²

yes

—

C₄×C₈×C₂₉

C₂²

yes

—

#11745166

C₂²

yes

—

#11745166

C₂²

yes

—

#11745165

C₂²

yes

characteristic

#11745165

C₂²

yes

—

#11745165

C₂²

yes

—

#11745165

C₂²

yes

characteristic

#11745165

C₂²

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

M₁₆×C₂₉

Q₈

yes

complemented

C₂×C₈×C₂₉

Q₈

yes

complemented

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₂×C₈×C₂₉

C₂³

yes

characteristic

C₂×C₈×C₂₉

C₂³

yes

—

C₄⋊C₄×C₂₉

D₄

yes

—

C₂²⋊C₄×C₂₉

D₄

yes

—

C₄²×C₂₉

C₂³

yes

—

C₂²⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂³

yes

—

C₄⋊Dic₂₉

C₂³

yes

—

C₄⋊Dic₂₉

C₂³

yes

—

#90779

—

#90779

—

C₄×Dic₂₉

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

C₂⁴

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

C₈×C₂₉

C₂×Q₈

yes

—

C₈×C₂₉

C₂×Q₈

yes

—

C₈×C₂₉

C₂×Q₈

yes

—

#3163

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

C₄⋊Q₈

yes

—

C₂²×C₂₉

C₂²×D₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

C₄⋊Q₈

yes

—

C₄×C₂₉

C₂²×Q₈

yes

characteristic

C₄×C₂₉

C₂²×Q₈

yes

characteristic

Dic₂₉

—

Dic₂₉

—

Dic₂₉

—

—

—

—

—

—

—

—

D₂₉

yes

characteristic

C₂×C₂₉

—

C₂×C₂₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₉

#577

yes

—

C₂×M₁₆

C₂×D₂₉

yes

characteristic

C₂²×C₈

C₂×D₂₉

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₂₉

yes

characteristic

C₂×C₄⋊C₄

—

—

—

—

—

—

—

100

#106

—

101

#97

C₂×D₂₉

yes

—

102

C₄×C₈

C₂×D₂₉

yes

—

103

C₂₉

#3163

yes

characteristic, complemented, Sylow

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₂₉

yes

characteristic

107

M₁₆

Dic₅₈

yes

complemented

108

C₂×C₈

Dic₅₈

yes

complemented

109

C₂×C₈

C₂²×D₂₉

yes

characteristic

110

C₂×C₈

C₂²×D₂₉

yes

—

111

C₂×C₈

C₂²×D₂₉

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

D₁₁₆

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

D₁₁₆

yes

—

122

C₄²

—

123

C₄²

C₂²×D₂₉

yes

—

124

C₂³

C₂×D₁₁₆

yes

characteristic

125

Q₈

—

232

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

232

—

130

C₂×C₄

C₂×Dic₅₈

yes

—

131

C₂×C₄

C₂×Dic₅₈

yes

—

132

C₂×C₄

C₂×D₁₁₆

yes

—

133

C₂×C₄

C₂×D₁₁₆

yes

characteristic

134

C₂×C₄

C₂³×D₂₉

yes

characteristic

135

C₈

C₂×Dic₅₈

yes

—

136

C₈

C₂×Dic₅₈

yes

—

137

C₈

C₂×Dic₅₈

yes

—

138

C₂²

—

139

C₂²

#11745231

yes

—

140

C₂²

C₂²×D₁₁₆

yes

characteristic

141

C₄

—

142

C₄

—

116

—

143

C₄

—

116

—

144

C₄

—

116

—

145

C₄

#11745231

yes

—

146

C₄

C₂²×Dic₅₈

yes

characteristic

147

C₄

C₂²×Dic₅₈

yes

characteristic

148

C₂

—

149

C₂

C₂×#11745231

yes

characteristic

150

C₂

#14271975

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₁₆
commutator, cosocle	C₂×C₄×C₂₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉
Fitting	C₂₉×#405	C₂
socle	C₂²×C₂₉	C₂²×D₄
prime radical	C₂³×C₂₉	C₂×D₄
hypercenter	#405	D₂₉
hypocenter	C₂₉	#3163
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₁₁₆

commutator, cosocle

C₂×C₄×C₂₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉

Fitting

C₂₉×#405

C₂

socle

C₂²×C₂₉

C₂²×D₄

prime radical

C₂³×C₂₉

C₂×D₄

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #14271572	C₂	semidirect
3. #14271565	C₂	semidirect
4. #14271561	C₂	semidirect
31. M₁₆×C₂₉	Q₈	semidirect
32. C₂×C₈×C₂₉	Q₈	semidirect
103. C₂₉	#3163	semidirect
107. M₁₆	Dic₅₈	semidirect
108. C₂×C₈	Dic₅₈	semidirect

H ≅ G/N

type

2. #14271572

C₂

semidirect

3. #14271565

C₂

semidirect

4. #14271561

C₂

semidirect

31. M₁₆×C₂₉

Q₈

semidirect

32. C₂×C₈×C₂₉

Q₈

semidirect

103. C₂₉

#3163

semidirect

107. M₁₆

Dic₅₈

semidirect

108. C₂×C₈

Dic₅₈

semidirect

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

subgroup	index
#14271791	2
#14271572	2
#14271565	2
#14271561	2
#14271205	2
#14271201	2
C₂×#11745165	2
C₂₉×#405	2
#3163	29

subgroup

index

C₂₉×#405

Subgroups — #16830023

Conjugacy classes of subgroups

Complements

Maximal subgroups