Group #16927997 — subgroups

Subgroups — #16927997

Order 4352.

322 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 139 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	#14528901	C₂	1	1	yes	maximal, characteristic
2	#14528530	C₂	1	1	yes	maximal, characteristic
3	#14526618	C₂	1	4	yes	maximal
4	C₁₇×#1831	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#3622	C₂²	1	2	yes	—
6	C₂×C₄×#3622	C₂²	1	2	yes	—
7	#11762564	—	2	2	no	—
8	#11762564	C₄	1	2	yes	—
9	#11762564	C₄	1	2	yes	—
10	C₂×C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄³×C₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×#336	—	2	2	no	—
13	C₂²×C₈×C₁₇	—	2	2	no	—
14	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₇	D₄	1	2	yes	—
17	C₂×C₄²×C₁₇	Q₈	1	2	yes	—
18	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
19	C₂²×#3622	—	2	2	no	—
20	C₂²×#3622	C₂×C₄	1	4	yes	—
21	C₁₇×C₄⋊C₈	—	2	4	no	—
22	C₄⋊#3622	—	2	4	no	—
23	C₄×#3622	—	2	2	no	—
24	C₄×#3622	—	2	2	no	—
25	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
26	C₂²×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
27	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₁₇	—	4	2	no	—
31	C₂×C₈×C₁₇	—	4	4	no	—
32	C₄²×C₁₇	—	2	2	no	—
33	C₄²×C₁₇	—	2	2	no	—
34	C₄²×C₁₇	—	2	4	no	—
35	C₄²×C₁₇	—	4	2	no	—
36	C₄²×C₁₇	#41	1	2	yes	—
37	C₄²×C₁₇	#41	1	2	yes	—
38	C₂×#3622	—	2	4	no	—
39	C₂×#3622	—	2	4	no	—
40	C₂×#3622	—	2	4	no	—
41	C₂×#3622	—	4	2	no	—
42	C₂×#3622	—	4	4	no	—
43	#8788	—	17	1	no	maximal, Sylow
44	C₂³×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	—	2	2	no	—
46	C₂×C₄×C₁₇	—	2	2	no	—
47	C₂×C₄×C₁₇	—	2	2	no	—
48	C₂×C₄×C₁₇	—	2	2	no	—
49	C₂×C₄×C₁₇	—	2	2	no	—
50	C₂×C₄×C₁₇	—	2	2	no	—
51	C₂×C₄×C₁₇	—	2	2	no	—
52	C₂×C₄×C₁₇	#117	1	2	yes	—
53	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
54	C₂×C₄×C₁₇	#124	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#125	1	2	yes	—
56	C₂×C₄×C₁₇	#131	1	2	yes	—
57	C₂×C₄×C₁₇	#131	1	2	yes	—
58	C₂×C₄×C₁₇	#135	1	4	yes	—
59	C₈×C₁₇	—	8	2	no	—
60	#3622	—	4	4	no	—
61	#3622	—	8	2	no	—
62	#1831	—	17	1	no	—
63	#1831	D₁₇	1	1	yes	characteristic
64	#1744	—	17	1	no	—
65	#1277	—	17	4	no	—
66	C₂²×C₁₇	#382	1	1	yes	characteristic
67	C₂²×C₁₇	#424	1	1	yes	characteristic
68	C₂²×C₁₇	#424	1	1	yes	characteristic
69	C₂²×C₁₇	#427	1	2	yes	—
70	C₂²×C₁₇	#430	1	2	yes	—
71	C₄×C₁₇	—	2	2	no	—
72	C₄×C₁₇	—	4	2	no	—
73	C₄×C₁₇	—	4	2	no	—
74	C₄×C₁₇	#433	1	4	yes	—
75	C₄×C₁₇	#488	1	2	yes	—
76	C₄×C₁₇	#488	1	2	yes	—
77	C₂×C₄⋊C₈	—	17	2	no	—
78	C₂×C₄⋊C₈	C₂×D₁₇	1	2	yes	—
79	C₂×C₄×C₈	—	17	2	no	—
80	C₄³	C₂×D₁₇	1	1	yes	characteristic
81	#336	—	2	2	no	—
82	#336	—	17	2	no	—
83	#336	—	17	2	no	—
84	#336	—	34	2	no	—
85	C₂×C₁₇	#1572	1	2	yes	—
86	C₂×C₁₇	#1572	1	2	yes	—
87	C₂×C₁₇	#1832	1	1	yes	characteristic
88	C₂×C₁₇	#1835	1	1	yes	characteristic
89	C₂×C₁₇	#1838	1	1	yes	characteristic
90	C₂²×C₈	—	2	2	no	—
91	C₂²×C₈	—	17	4	no	—
92	C₂²×C₈	—	34	2	no	—
93	C₂×C₄²	Dic₃₄	1	2	yes	—
94	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
95	C₂×C₄²	C₄×D₁₇	1	1	yes	characteristic
96	C₂×C₄²	C₂²⋊D₁₇	1	2	yes	—
97	C₂×C₄²	C₂²×D₁₇	1	1	yes	characteristic
98	C₄⋊C₈	—	2	4	no	—
99	C₄⋊C₈	—	34	4	no	—
100	C₄×C₈	—	34	2	no	—
101	C₄×C₈	—	34	2	no	—
102	C₁₇	#8788	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₄	#63225	1	2	yes	—
104	C₂²×C₄	#63225	1	2	yes	—
105	C₂²×C₄	C₂×Dic₃₄	1	1	yes	characteristic
106	C₂²×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
107	C₂²×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	4	4	no	—
110	C₂×C₈	—	34	4	no	—
111	C₂×C₈	—	34	4	no	—
112	C₂×C₈	—	34	4	no	—
113	C₂×C₈	—	68	2	no	—
114	C₂×C₈	—	68	4	no	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	4	2	no	—
119	C₄²	#63254	1	2	yes	—
120	C₄²	#63256	1	2	yes	—
121	C₂³	C₂×#63225	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	#10587576	1	2	yes	—
130	C₂×C₄	#10587578	1	2	yes	—
131	C₂×C₄	#10587587	1	2	yes	—
132	C₂×C₄	#10587593	1	2	yes	—
133	C₂×C₄	#10587622	1	1	yes	characteristic
134	C₂×C₄	#10587633	1	1	yes	characteristic
135	C₂×C₄	#10587636	1	4	yes	—
136	C₈	—	8	2	no	—
137	C₈	—	68	4	no	—
138	C₈	—	136	2	no	—
139	C₂²	#11762674	1	1	yes	characteristic
140	C₂²	#11762835	1	1	yes	characteristic
141	C₂²	#11762837	1	1	yes	characteristic
142	C₂²	#11762845	1	2	yes	—
143	C₂²	#11762857	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	4	2	no	—
146	C₄	—	4	2	no	—
147	C₄	#11762869	1	4	yes	—
148	C₄	#11763163	1	2	yes	—
149	C₄	#11763166	1	2	yes	—
150	C₂	#14527562	1	2	yes	—
151	C₂	#14527565	1	2	yes	—
152	C₂	#14528903	1	1	yes	characteristic
153	C₂	#14528914	1	1	yes	characteristic
154	C₂	#14528924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14528901

C₂

yes

maximal, characteristic

#14528530

C₂

yes

maximal, characteristic

#14526618

C₂

yes

maximal

C₁₇×#1831

C₂

yes

maximal, characteristic

C₂×C₄⋊#3622

C₂²

yes

—

C₂×C₄×#3622

C₂²

yes

—

#11762564

—

#11762564

C₄

yes

—

#11762564

C₄

yes

—

C₂×C₁₇×C₄⋊C₈

C₂²

yes

—

C₄³×C₁₇

C₂²

yes

characteristic

C₁₇×#336

—

C₂²×C₈×C₁₇

—

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

Q₈

yes

—

C₂×C₄²×C₁₇

C₂³

yes

characteristic

C₂²×#3622

—

C₂²×#3622

C₂×C₄

yes

—

C₁₇×C₄⋊C₈

—

C₄⋊#3622

—

C₄×#3622

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

#41

yes

—

C₄²×C₁₇

#41

yes

—

—

—

—

—

—

—

C₂³×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

#117

yes

—

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

#131

yes

—

C₂×C₄×C₁₇

#131

yes

—

C₂×C₄×C₁₇

#135

yes

—

C₈×C₁₇

—

—

—

—

D₁₇

yes

characteristic

#1744

—

#1277

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#430

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#433

yes

—

C₄×C₁₇

#488

yes

—

C₄×C₁₇

#488

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₁₇

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₁₇

yes

—

—

—

—

C₂×C₁₇

#1572

yes

—

C₂×C₁₇

#1572

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

Dic₃₄

yes

—

C₂×C₄²

C₄×D₁₇

yes

characteristic

C₂×C₄²

C₄×D₁₇

yes

characteristic

C₂×C₄²

C₂²⋊D₁₇

yes

—

C₂×C₄²

C₂²×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₁₇

#8788

yes

characteristic, complemented, Sylow

103

C₂²×C₄

#63225

yes

—

104

C₂²×C₄

#63225

yes

—

105

C₂²×C₄

C₂×Dic₃₄

yes

characteristic

106

C₂²×C₄

C₂×C₄×D₁₇

yes

characteristic

107

C₂²×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

#63254

yes

—

120

C₄²

#63256

yes

—

121

C₂³

C₂×#63225

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#10587576

yes

—

130

C₂×C₄

#10587578

yes

—

131

C₂×C₄

#10587587

yes

—

132

C₂×C₄

#10587593

yes

—

133

C₂×C₄

#10587622

yes

characteristic

134

C₂×C₄

#10587633

yes

characteristic

135

C₂×C₄

#10587636

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

136

—

139

C₂²

#11762674

yes

characteristic

140

C₂²

#11762835

yes

characteristic

141

C₂²

#11762837

yes

characteristic

142

C₂²

#11762845

yes

—

143

C₂²

#11762857

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#11762869

yes

—

148

C₄

#11763163

yes

—

149

C₄

#11763166

yes

—

150

C₂

#14527562

yes

—

151

C₂

#14527565

yes

—

152

C₂

#14528903

yes

characteristic

153

C₂

#14528914

yes

characteristic

154

C₂

#14528924

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₇
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₇
Fitting	C₁₇×#1831	C₂
socle, prime radical	C₂³×C₁₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂³
hypercenter	#1831	D₁₇
hypocenter	C₁₇	#8788
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₁₇

commutator

C₂²×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₁₇

Fitting

C₁₇×#1831

C₂

socle, prime radical

C₂³×C₁₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

N	H ≅ G/N	type
102. C₁₇	#8788	semidirect

H ≅ G/N

type

102. C₁₇

#8788

semidirect

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

subgroup	index
#14528901	2
#14528530	2
#14526618	2
C₁₇×#1831	2
#8788	17

subgroup

index

C₁₇×#1831

Subgroups — #16927997

Conjugacy classes of subgroups

Complements

Maximal subgroups