Group #16817913 — subgroups

Subgroups — #16817913

Order 3712.

218 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14271216	C₂	1	1	yes	maximal, characteristic
2	C₄×#90777	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#6597	C₂	1	4	yes	maximal, complemented
4	C₂₉×#533	C₂	1	1	yes	maximal, characteristic
5	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂₉×#125	—	2	1	no	—
8	C₂₉×C₂²⋊Q₈	—	2	1	no	—
9	C₂₉×C₂²⋊Q₈	C₄	1	2	yes	—
10	C₄×Q₈×C₂₉	—	2	1	no	—
11	C₄×Q₈×C₂₉	C₂²	1	2	yes	—
12	C₂₉×#117	—	2	1	no	—
13	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
14	C₂×#90777	—	2	1	no	—
15	C₂²⋊#6597	—	2	1	no	—
16	C₄⋊#6597	C₂²	1	2	yes	—
17	#11745151	—	2	1	no	—
18	C₄×#6597	—	2	1	no	—
19	C₄×#6597	C₂²	1	2	yes	—
20	C₂×Q₈×C₂₉	—	2	1	no	—
21	C₂×Q₈×C₂₉	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
25	C₄⋊C₄×C₂₉	—	2	1	no	—
26	C₄⋊C₄×C₂₉	—	2	1	no	—
27	C₄⋊C₄×C₂₉	—	2	1	no	—
28	C₄⋊C₄×C₂₉	—	2	2	no	—
29	C₄⋊C₄×C₂₉	—	2	2	no	—
30	C₄⋊C₄×C₂₉	C₂×C₄	1	2	yes	—
31	C₂²⋊C₄×C₂₉	—	2	2	no	—
32	C₄²×C₂₉	—	2	2	no	—
33	C₄²×C₂₉	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₂₉	D₄	1	1	yes	characteristic
35	C₄²×C₂₉	D₄	1	1	yes	characteristic
36	C₄²×C₂₉	C₂³	1	1	yes	characteristic
37	#90777	—	2	4	no	—
38	C₂×#6597	—	2	2	no	—
39	C₂×#6597	—	2	2	no	—
40	C₂³×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₂₉	—	2	4	no	—
42	Q₈×C₂₉	—	4	2	no	—
43	C₂×C₄×C₂₉	—	2	2	no	—
44	C₂×C₄×C₂₉	—	2	2	no	—
45	C₂×C₄×C₂₉	—	2	2	no	—
46	C₂×C₄×C₂₉	—	2	2	no	—
47	C₂×C₄×C₂₉	—	2	2	no	—
48	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
54	#6597	—	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	—	4	2	no	—
65	C₄×C₂₉	—	4	2	no	—
66	C₄×C₂₉	#104	1	2	yes	—
67	C₄×C₂₉	#131	1	2	yes	—
68	C₄×C₂₉	#137	1	2	yes	—
69	#533	D₂₉	1	1	yes	characteristic
70	#431	—	29	1	no	—
71	C₄×M₁₆	—	29	1	no	—
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	—	58	1	no	—
78	C₄⋊Q₈	Dic₂₉	1	1	yes	characteristic
79	C₄⋊Q₈	Dic₂₉	1	1	yes	characteristic
80	#125	—	2	1	no	—
81	C₂²⋊Q₈	—	2	1	no	—
82	C₂²⋊Q₈	Dic₂₉	1	2	yes	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	C₂×D₂₉	1	2	yes	—
85	#117	—	2	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	—	58	1	no	—
90	C₄×C₈	—	29	2	no	—
91	C₄×C₈	—	58	1	no	—
92	C₂₉	#1485	1	1	yes	characteristic, complemented, Sylow
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	C₂×Dic₂₉	1	2	yes	—
95	C₂²×C₄	C₂×Dic₂₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
98	M₁₆	—	58	4	no	—
99	C₂×C₈	—	58	2	no	—
100	C₂×C₈	—	58	2	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂×Dic₂₉	1	2	yes	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂×Dic₂₉	1	1	yes	characteristic
110	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
111	C₄²	C₂²⋊D₂₉	1	1	yes	characteristic
112	C₄²	C₂²×D₂₉	1	1	yes	characteristic
113	C₂³	C₂²⋊Dic₂₉	1	1	yes	characteristic
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
122	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊Dic₂₉	1	1	yes	characteristic
124	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
127	C₈	—	58	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×C₂²⋊Dic₂₉	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	#11745185	1	2	yes	—
139	C₄	#11745304	1	2	yes	—
140	C₄	#11745310	1	2	yes	—
141	C₂	—	4	1	no	—
142	C₂	#14271130	1	1	yes	characteristic
143	C₂	#14271172	1	1	yes	characteristic
144	C₂	C₂×#11745185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271216

C₂

yes

maximal, characteristic

C₄×#90777

C₂

yes

maximal, characteristic

Q₈⋊#6597

C₂

yes

maximal, complemented

C₂₉×#533

C₂

yes

maximal, characteristic

C₂₉×C₄⋊Q₈

C₄

yes

characteristic

C₂₉×C₄⋊Q₈

C₄

yes

characteristic

C₂₉×#125

—

C₂₉×C₂²⋊Q₈

—

C₂₉×C₂²⋊Q₈

C₄

yes

—

C₄×Q₈×C₂₉

—

C₄×Q₈×C₂₉

C₂²

yes

—

C₂₉×#117

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×#90777

—

C₂²⋊#6597

—

C₄⋊#6597

C₂²

yes

—

#11745151

—

C₄×#6597

—

C₄×#6597

C₂²

yes

—

C₂×Q₈×C₂₉

—

C₂×Q₈×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₂²⋊C₄×C₂₉

—

C₄²×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

D₄

yes

characteristic

C₄²×C₂₉

C₂³

yes

—

—

—

C₂³×C₂₉

C₂²⋊C₄

yes

characteristic

Q₈×C₂₉

—

Q₈×C₂₉

—

C₂×C₄×C₂₉

—

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

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#104

yes

—

C₄×C₂₉

#131

yes

—

C₄×C₂₉

#137

yes

—

#533

D₂₉

yes

characteristic

#431

—

C₄×M₁₆

—

Q₈⋊C₈

—

C₂×C₂₉

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×M₁₆

—

C₄⋊Q₈

Dic₂₉

yes

characteristic

C₄⋊Q₈

Dic₂₉

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₂₉

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×D₂₉

yes

—

#117

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂₉

#1485

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₉

yes

—

C₂²×C₄

C₂×Dic₂₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉

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

C₂×Dic₂₉

yes

—

107

C₂²⋊C₄

—

108

C₄²

—

109

C₄²

C₂×Dic₂₉

yes

characteristic

110

C₄²

C₂²⋊D₂₉

yes

characteristic

111

C₄²

C₂²⋊D₂₉

yes

characteristic

112

C₄²

C₂²×D₂₉

yes

characteristic

113

C₂³

C₂²⋊Dic₂₉

yes

characteristic

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

122

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

123

C₂×C₄

C₂²⋊Dic₂₉

yes

characteristic

124

C₂×C₄

C₂²×Dic₂₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

127

C₈

—

128

C₈

—

116

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₂²⋊Dic₂₉

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#11745185

yes

—

139

C₄

#11745304

yes

—

140

C₄

#11745310

yes

—

141

C₂

—

142

C₂

#14271130

yes

characteristic

143

C₂

#14271172

yes

characteristic

144

C₂

C₂×#11745185

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₉
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₄²	C₂²×D₂₉
Fitting	C₂₉×#533	C₂
socle	C₂²×C₂₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₉	C₂²⋊C₄
cosocle	C₄²×C₂₉	C₂³
hypercenter	#533	D₂₉
hypocenter	C₂₉	#1485
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₉

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₄²

C₂²×D₂₉

Fitting

C₂₉×#533

C₂

socle

C₂²×C₂₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₉

C₂²⋊C₄

cosocle

C₄²×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. Q₈⋊#6597	C₂	semidirect
92. C₂₉	#1485	semidirect

H ≅ G/N

type

3. Q₈⋊#6597

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
#14271216	2
C₄×#90777	2
Q₈⋊#6597	2
C₂₉×#533	2
#1485	29

subgroup

index

C₂₉×#533

Subgroups — #16817913

Conjugacy classes of subgroups

Complements

Maximal subgroups