Group #16817914 — subgroups

Subgroups — #16817914

Order 3712.

218 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 77 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	#14271216	C₂	1	1	yes	maximal, characteristic
3	#14270895	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂×M₁₆×C₂₉	—	2	1	no	—
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	#11745232	—	2	1	no	—
11	#11745231	C₄	1	1	yes	characteristic
12	#11745231	C₄	1	1	yes	characteristic
13	C₄×Dic₅₈	—	2	1	no	—
14	C₄×Dic₅₈	C₂²	1	2	yes	—
15	C₂₉×#97	—	2	1	no	—
16	C₄×C₈×C₂₉	C₂²	1	2	yes	—
17	C₂²⋊#6597	—	2	1	no	—
18	C₄⋊#6597	C₂²	1	2	yes	—
19	C₄×#6597	—	2	1	no	—
20	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
23	C₂×Dic₅₈	—	2	1	no	—
24	C₂×Dic₅₈	C₂×C₄	1	2	yes	—
25	M₁₆×C₂₉	—	2	4	no	—
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	C₂×#6597	—	2	2	no	—
40	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₉	—	2	2	no	—
42	C₂×C₄×C₂₉	—	2	2	no	—
43	C₂×C₄×C₂₉	—	2	2	no	—
44	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₂₉	—	2	2	no	—
51	C₂×Dic₂₉	—	2	2	no	—
52	Dic₅₈	—	2	4	no	—
53	Dic₅₈	—	4	2	no	—
54	C₈×C₂₉	—	2	4	no	—
55	#6597	—	4	2	no	—
56	#1485	—	29	1	no	maximal, Sylow
57	C₂²×C₂₉	—	2	1	no	—
58	C₂²×C₂₉	—	2	1	no	—
59	C₂²×C₂₉	—	2	1	no	—
60	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₂₉	—	2	1	no	—
62	C₄×C₂₉	—	2	1	no	—
63	C₄×C₂₉	—	2	1	no	—
64	C₄×C₂₉	#104	1	2	yes	—
65	C₄×C₂₉	#131	1	2	yes	—
66	C₄×C₂₉	#137	1	2	yes	—
67	Dic₂₉	—	4	2	no	—
68	Dic₂₉	—	4	2	no	—
69	#533	—	29	1	no	—
70	#431	—	29	1	no	—
71	C₄×M₁₆	D₂₉	1	1	yes	characteristic
72	Q₈⋊C₈	—	29	4	no	—
73	C₂×C₂₉	—	4	1	no	—
74	C₂×C₂₉	#408	1	1	yes	characteristic
75	C₂×C₂₉	#419	1	1	yes	characteristic
76	C₂×C₂₉	C₂×#104	1	1	yes	characteristic
77	C₂×M₁₆	—	2	1	no	—
78	C₄⋊Q₈	—	29	1	no	—
79	C₄⋊Q₈	—	29	1	no	—
80	#125	—	58	1	no	—
81	C₂²⋊Q₈	—	29	2	no	—
82	C₂²⋊Q₈	—	58	1	no	—
83	C₄×Q₈	—	29	2	no	—
84	C₄×Q₈	—	58	1	no	—
85	#117	—	58	1	no	—
86	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
87	C₄⋊C₈	—	29	2	no	—
88	C₂²⋊C₈	—	58	1	no	—
89	#97	—	2	1	no	—
90	C₄×C₈	—	58	1	no	—
91	C₄×C₈	C₂×D₂₉	1	2	yes	—
92	C₂₉	#1485	1	1	yes	characteristic, complemented, Sylow
93	C₂×Q₈	—	29	2	no	—
94	C₂×Q₈	—	58	1	no	—
95	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
96	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
98	M₁₆	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	58	2	no	—
101	C₄⋊C₄	—	29	2	no	—
102	C₄⋊C₄	—	58	1	no	—
103	C₄⋊C₄	—	58	1	no	—
104	C₄⋊C₄	—	58	2	no	—
105	C₄⋊C₄	—	58	1	no	—
106	C₄⋊C₄	—	58	2	no	—
107	C₂²⋊C₄	—	58	2	no	—
108	C₄²	—	58	2	no	—
109	C₄²	C₄×D₂₉	1	1	yes	characteristic
110	C₄²	D₁₁₆	1	1	yes	characteristic
111	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
112	C₄²	C₂²×D₂₉	1	1	yes	characteristic
113	C₂³	#90781	1	1	yes	characteristic
114	Q₈	—	58	4	no	—
115	Q₈	—	116	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	58	2	no	—
120	C₂×C₄	—	58	2	no	—
121	C₂×C₄	#90781	1	1	yes	characteristic
122	C₂×C₄	#90781	1	1	yes	characteristic
123	C₂×C₄	#90781	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	116	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×#90781	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	116	2	no	—
137	C₄	—	116	2	no	—
138	C₄	#11745153	1	2	yes	—
139	C₄	#11745269	1	2	yes	—
140	C₄	#11745271	1	2	yes	—
141	C₂	—	4	1	no	—
142	C₂	#14271131	1	1	yes	characteristic
143	C₂	#14271173	1	1	yes	characteristic
144	C₂	C₂×#11745153	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271720

C₂

yes

maximal, characteristic

#14271216

C₂

yes

maximal, characteristic

#14270895

C₂

yes

maximal, complemented

C₄×M₁₆×C₂₉

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₉

—

C₂×C₄²×C₂₉

C₂²

yes

—

—

C₄

yes

—

#11745232

—

#11745231

C₄

yes

characteristic

#11745231

C₄

yes

characteristic

C₄×Dic₅₈

—

C₄×Dic₅₈

C₂²

yes

—

C₂₉×#97

—

C₄×C₈×C₂₉

C₂²

yes

—

C₂²⋊#6597

—

C₄⋊#6597

C₂²

yes

—

C₄×#6597

—

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

—

M₁₆×C₂₉

—

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

—

C₂×#6597

—

C₂³×C₂₉

C₂²⋊C₄

yes

characteristic

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

—

C₈×C₂₉

—

#6597

—

#1485

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#104

yes

—

C₄×C₂₉

#131

yes

—

C₄×C₂₉

#137

yes

—

Dic₂₉

—

Dic₂₉

—

#533

—

#431

—

C₄×M₁₆

D₂₉

yes

characteristic

Q₈⋊C₈

—

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

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₂₉

yes

—

C₂₉

#1485

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₄×D₂₉

yes

characteristic

C₂²×C₄

D₁₁₆

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉

yes

characteristic

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

C₄×D₂₉

yes

characteristic

110

C₄²

D₁₁₆

yes

characteristic

111

C₄²

C₂²⋊D₂₉

yes

characteristic

112

C₄²

C₂²×D₂₉

yes

characteristic

113

C₂³

#90781

yes

characteristic

114

Q₈

—

115

Q₈

—

116

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#90781

yes

characteristic

122

C₂×C₄

#90781

yes

characteristic

123

C₂×C₄

#90781

yes

characteristic

124

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

125

C₂×C₄

C₂×D₁₁₆

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

127

C₈

—

128

C₈

—

116

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×#90781

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

116

—

137

C₄

—

116

—

138

C₄

#11745153

yes

—

139

C₄

#11745269

yes

—

140

C₄

#11745271

yes

—

141

C₂

—

142

C₂

#14271131

yes

characteristic

143

C₂

#14271173

yes

characteristic

144

C₂

C₂×#11745153

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₁₁₆
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₄²	C₂²×D₂₉
Fitting	C₄×M₁₆×C₂₉	C₂
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₉	C₂²⋊C₄
cosocle	C₄²×C₂₉	C₂³
hypercenter	C₄×M₁₆	D₂₉
hypocenter	C₂₉	#1485
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₁₁₆

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₄²

C₂²×D₂₉

Fitting

C₄×M₁₆×C₂₉

C₂

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₉

C₂²⋊C₄

cosocle

C₄²×C₂₉

C₂³

hypercenter

C₄×M₁₆

D₂₉

hypocenter

C₂₉

#1485

LCS

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #14270895	C₂	semidirect
92. C₂₉	#1485	semidirect

H ≅ G/N

type

3. #14270895

C₂

semidirect

92. C₂₉

#1485

semidirect

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

subgroup	index
#14271720	2
#14271216	2
#14270895	2
C₄×M₁₆×C₂₉	2
#1485	29

subgroup

index

C₄×M₁₆×C₂₉

Subgroups — #16817914

Conjugacy classes of subgroups

Complements

Maximal subgroups