Group #16925486 — subgroups

Subgroups — #16925486

Order 4352.

390 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 179 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	#14527532	C₂	1	2	yes	maximal
2	#14527076	C₂	1	2	yes	maximal
3	#14526617	C₂	1	2	yes	maximal
4	C₄²×C₈×C₁₇	C₂	1	1	yes	maximal, characteristic
5	#11762862	C₂²	1	4	yes	—
6	#11762562	C₄	1	4	yes	complemented
7	#11762560	C₄	1	4	yes	complemented
8	#11762531	—	2	4	no	—
9	C₂×C₄×C₈×C₁₇	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₁₇	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
13	C₄³×C₁₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₇	C₂×C₄	1	2	yes	—
16	C₂²×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₁₇	—	2	2	no	—
18	C₂×C₄²×C₁₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₁₇	—	2	4	no	—
21	C₄×C₈×C₁₇	—	2	4	no	—
22	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₇	D₄	1	2	yes	—
25	C₄×C₈×C₁₇	Q₈	1	2	yes	—
26	C₂²⋊#3622	—	2	4	no	—
27	C₄⋊#3622	C₂×C₄	1	8	yes	—
28	C₄×#3622	—	2	4	no	—
29	C₂²×C₄×C₁₇	—	2	2	no	—
30	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₇	#41	1	2	yes	—
32	C₂×C₈×C₁₇	—	2	2	no	—
33	C₂×C₈×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	—	2	2	no	—
35	C₂×C₈×C₁₇	C₄²	1	2	yes	—
36	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₁₇	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₁₇	C₄⋊C₄	1	4	yes	—
39	C₂×C₈×C₁₇	M₁₆	1	4	yes	—
40	C₄²×C₁₇	—	2	8	no	—
41	C₄²×C₁₇	—	2	2	no	—
42	C₄²×C₁₇	M₁₆	1	4	yes	—
43	C₄²×C₁₇	C₂²×C₄	1	2	yes	—
44	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₁₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×#3622	—	2	8	no	—
47	#8267	—	17	1	no	maximal, Sylow
48	C₂³×C₁₇	#117	1	1	yes	characteristic
49	C₂×C₄×C₁₇	—	2	8	no	—
50	C₂×C₄×C₁₇	—	2	4	no	—
51	C₂×C₄×C₁₇	—	2	4	no	—
52	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
53	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₁₇	C₄×D₄	1	2	yes	—
55	C₂×C₄×C₁₇	C₄×Q₈	1	2	yes	—
56	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×M₁₆	1	2	yes	—
58	C₈×C₁₇	—	2	2	no	—
59	C₈×C₁₇	—	2	4	no	—
60	C₈×C₁₇	#97	1	2	yes	—
61	C₈×C₁₇	#97	1	2	yes	—
62	#3622	—	4	8	no	—
63	C₄²×C₈	D₁₇	1	1	yes	characteristic
64	#1564	—	17	2	no	—
65	#1452	—	17	2	no	—
66	#1276	—	17	2	no	—
67	C₂²×C₁₇	—	2	2	no	—
68	C₂²×C₁₇	C₄×C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₇	#432	1	2	yes	—
70	C₄×C₁₇	—	2	8	no	—
71	C₄×C₁₇	—	2	2	no	—
72	C₄×C₁₇	C₂×#97	1	1	yes	characteristic
73	C₄×C₁₇	C₄×M₁₆	1	1	yes	characteristic
74	C₄×C₁₇	#423	1	2	yes	—
75	C₄×C₁₇	#443	1	4	yes	—
76	#431	—	17	4	no	—
77	C₂×C₄×C₈	Dic₁₇	1	2	yes	—
78	C₂×C₄×C₈	Dic₁₇	1	2	yes	—
79	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
81	C₄³	C₂×D₁₇	1	1	yes	characteristic
82	#335	—	17	4	no	—
83	#334	—	17	4	no	—
84	#322	—	34	4	no	—
85	C₂×C₁₇	—	2	2	no	—
86	C₂×C₁₇	#1767	1	1	yes	characteristic
87	C₂×C₁₇	#1772	1	2	yes	—
88	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
89	C₂²×C₈	C₂×Dic₁₇	1	2	yes	—
90	C₂²×C₈	C₂×Dic₁₇	1	1	yes	characteristic
91	C₂×C₄²	—	2	2	no	—
92	C₂×C₄²	C₂×Dic₁₇	1	2	yes	—
93	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
94	C₄⋊C₈	—	17	8	no	—
95	C₂²⋊C₈	—	34	4	no	—
96	C₄×C₈	—	2	4	no	—
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	34	4	no	—
99	C₄×C₈	Dic₃₄	1	2	yes	—
100	C₄×C₈	C₄×D₁₇	1	2	yes	—
101	C₄×C₈	C₄×D₁₇	1	2	yes	—
102	C₄×C₈	D₆₈	1	2	yes	—
103	C₁₇	#8267	1	1	yes	characteristic, complemented, Sylow
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	#63252	1	2	yes	—
106	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	34	8	no	—
111	C₂×C₈	#63223	1	4	yes	—
112	C₂×C₈	C₄×Dic₁₇	1	2	yes	—
113	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
114	C₂×C₈	C₄×Dic₁₇	1	1	yes	characteristic
115	C₂×C₈	C₄⋊Dic₁₇	1	4	yes	—
116	C₄²	—	2	8	no	—
117	C₄²	—	2	2	no	—
118	C₄²	#63223	1	4	yes	—
119	C₄²	C₂×Dic₃₄	1	1	yes	characteristic
120	C₄²	C₂×C₄×D₁₇	1	2	yes	—
121	C₄²	C₂×D₆₈	1	1	yes	characteristic
122	C₂³	#10587612	1	1	yes	characteristic
123	C₂×C₄	—	2	8	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	C₄×Dic₃₄	1	2	yes	—
127	C₂×C₄	C₄×D₆₈	1	2	yes	—
128	C₂×C₄	C₂×#63223	1	2	yes	—
129	C₂×C₄	C₂×#63223	1	2	yes	—
130	C₂×C₄	C₂×C₄×Dic₁₇	1	1	yes	characteristic
131	C₂×C₄	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	2	4	no	—
134	C₈	—	68	8	no	—
135	C₈	#10587474	1	2	yes	—
136	C₈	#10587474	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	C₄×C₄⋊Dic₁₇	1	1	yes	characteristic
139	C₂²	#11762866	1	2	yes	—
140	C₄	—	2	8	no	—
141	C₄	—	2	2	no	—
142	C₄	C₂×#10587474	1	1	yes	characteristic
143	C₄	C₄×#63223	1	1	yes	characteristic
144	C₄	#11762833	1	2	yes	—
145	C₄	#11762927	1	4	yes	—
146	C₂	—	2	2	no	—
147	C₂	#14528637	1	1	yes	characteristic
148	C₂	#14528663	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14527532

C₂

yes

maximal

#14527076

C₂

yes

maximal

#14526617

C₂

yes

maximal

C₄²×C₈×C₁₇

C₂

yes

maximal, characteristic

#11762862

C₂²

yes

—

#11762562

C₄

yes

complemented

#11762560

C₄

yes

complemented

#11762531

—

C₂×C₄×C₈×C₁₇

C₄

yes

—

C₂×C₄×C₈×C₁₇

C₄

yes

—

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₇

C₂²

yes

characteristic

C₄³×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₇

C₂×C₄

yes

—

C₂²×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₄×C₈×C₁₇

D₄

yes

—

C₄×C₈×C₁₇

Q₈

yes

—

C₂²⋊#3622

—

C₄⋊#3622

C₂×C₄

yes

—

C₄×#3622

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

#41

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

C₄²

yes

—

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄²

yes

characteristic

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₂×C₈×C₁₇

M₁₆

yes

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

M₁₆

yes

—

C₄²×C₁₇

C₂²×C₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

characteristic

C₄²×C₁₇

C₂×Q₈

yes

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄×D₄

yes

—

C₂×C₄×C₁₇

C₄×Q₈

yes

—

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

C₂×C₄×C₁₇

C₂×M₁₆

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

#97

yes

—

C₈×C₁₇

#97

yes

—

#3622

—

C₄²×C₈

D₁₇

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

C₄×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

#432

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×#97

yes

characteristic

C₄×C₁₇

C₄×M₁₆

yes

characteristic

C₄×C₁₇

#423

yes

—

C₄×C₁₇

#443

yes

—

#431

—

C₂×C₄×C₈

Dic₁₇

yes

—

C₂×C₄×C₈

Dic₁₇

yes

—

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄×C₈

C₂×D₁₇

yes

characteristic

C₄³

C₂×D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#1767

yes

characteristic

C₂×C₁₇

#1772

yes

—

C₂²×C₈

C₂×Dic₁₇

yes

characteristic

C₂²×C₈

C₂×Dic₁₇

yes

—

C₂²×C₈

C₂×Dic₁₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×Dic₁₇

yes

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₃₄

yes

—

100

C₄×C₈

C₄×D₁₇

yes

—

101

C₄×C₈

C₄×D₁₇

yes

—

102

C₄×C₈

D₆₈

yes

—

103

C₁₇

#8267

yes

characteristic, complemented, Sylow

104

C₂²×C₄

—

105

C₂²×C₄

#63252

yes

—

106

C₂²×C₄

C₂²×Dic₁₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#63223

yes

—

112

C₂×C₈

C₄×Dic₁₇

yes

—

113

C₂×C₈

C₄×Dic₁₇

yes

characteristic

114

C₂×C₈

C₄×Dic₁₇

yes

characteristic

115

C₂×C₈

C₄⋊Dic₁₇

yes

—

116

C₄²

—

117

C₄²

—

118

C₄²

#63223

yes

—

119

C₄²

C₂×Dic₃₄

yes

characteristic

120

C₄²

C₂×C₄×D₁₇

yes

—

121

C₄²

C₂×D₆₈

yes

characteristic

122

C₂³

#10587612

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₄×Dic₃₄

yes

—

127

C₂×C₄

C₄×D₆₈

yes

—

128

C₂×C₄

C₂×#63223

yes

—

129

C₂×C₄

C₂×#63223

yes

—

130

C₂×C₄

C₂×C₄×Dic₁₇

yes

characteristic

131

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

#10587474

yes

—

136

C₈

#10587474

yes

—

137

C₂²

—

138

C₂²

C₄×C₄⋊Dic₁₇

yes

characteristic

139

C₂²

#11762866

yes

—

140

C₄

—

141

C₄

—

142

C₄

C₂×#10587474

yes

characteristic

143

C₄

C₄×#63223

yes

characteristic

144

C₄

#11762833

yes

—

145

C₄

#11762927

yes

—

146

C₂

—

147

C₂

#14528637

yes

characteristic

148

C₂

#14528663

yes

—

	subgroup	G/H
center	C₄²	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₄²×C₈×C₁₇	C₂
socle	C₂²×C₁₇	C₄×C₄⋊C₄
prime radical	C₂³×C₁₇	#117
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	C₄²×C₈	D₁₇
hypocenter	C₁₇	#8267
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈

subgroup

G/H

center

C₄²

C₂×D₆₈

commutator

C₂×C₄×C₁₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₄²×C₈×C₁₇

C₂

socle

C₂²×C₁₇

C₄×C₄⋊C₄

prime radical

C₂³×C₁₇

#117

cosocle

C₂×C₄²×C₁₇

C₂³

hypercenter

C₄²×C₈

D₁₇

hypocenter

C₁₇

#8267

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₄²×C₈

N	H ≅ G/N	type
6. #11762562	C₄	semidirect
7. #11762560	C₄	semidirect
103. C₁₇	#8267	semidirect

H ≅ G/N

type

6. #11762562

C₄

semidirect

7. #11762560

C₄

semidirect

103. C₁₇

#8267

semidirect

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

subgroup	index
#14527532	2
#14527076	2
#14526617	2
C₄²×C₈×C₁₇	2
#8267	17

subgroup

index

C₄²×C₈×C₁₇

Subgroups — #16925486

Conjugacy classes of subgroups

Complements

Maximal subgroups