Group #17961629 — subgroups

Subgroups — #17961629

Order 4352.

234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 84 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	#14545125	C₂	1	1	yes	maximal, characteristic
2	#14545121	C₂	1	1	yes	maximal, characteristic
3	#14543838	C₂	1	2	yes	maximal
4	#14543834	C₂	1	2	yes	maximal
5	#14528538	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	1	yes	characteristic
7	C₂×#10587538	C₂²	1	2	yes	—
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	#11763731	—	2	2	no	—
10	#11763730	—	2	2	no	—
11	#11762767	C₂²	1	1	yes	characteristic
12	#11762765	C₂²	1	2	yes	—
13	#11762567	C₄	1	1	yes	characteristic
14	#11762565	C₄	1	2	yes	—
15	C₁₇×H(3,ℤ/4)	C₄	1	1	yes	characteristic
16	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₁₇×#117	C₂×C₄	1	2	yes	—
18	#10587612	C₂×C₄	1	1	yes	characteristic
19	C₂×#63223	C₂×C₄	1	2	yes	—
20	C₂²×#3622	C₂³	1	1	yes	characteristic
21	#10587538	—	2	2	no	—
22	#10587538	—	2	2	no	—
23	#10587538	—	2	2	no	—
24	C₂×#63216	—	2	1	no	—
25	C₂×#63216	—	2	1	no	—
26	C₂×#63216	—	2	2	no	—
27	C₂×#63216	—	2	2	no	—
28	#10587486	Q₈	1	2	yes	—
29	C₈×Dic₁₇	D₄	1	2	yes	—
30	#10587474	—	2	2	no	—
31	C₄×#3622	—	2	2	no	—
32	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₁₇	—	2	2	no	—
34	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
35	C₄⋊C₄×C₁₇	—	2	2	no	—
36	C₂²⋊C₄×C₁₇	—	2	2	no	—
37	C₄²×C₁₇	—	2	2	no	—
38	C₂²⋊Dic₁₇	—	2	1	no	—
39	C₄⋊Dic₁₇	—	2	1	no	—
40	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
41	#63223	—	2	2	no	—
42	#63223	—	2	2	no	—
43	C₂×#3622	—	2	2	no	—
44	C₂×#3622	C₂×D₄	1	1	yes	characteristic
45	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
46	C₂×#3622	#41	1	2	yes	—
47	C₂×#3622	#41	1	2	yes	—
48	#63216	—	4	1	no	—
49	#63216	—	4	1	no	—
50	#63216	—	4	2	no	—
51	#12885	—	17	1	no	maximal, Sylow
52	C₂³×C₁₇	#131	1	1	yes	characteristic
53	C₂×C₄×C₁₇	—	4	2	no	—
54	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₇	#117	1	2	yes	—
56	C₂×C₄×C₁₇	#131	1	2	yes	—
57	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
58	C₂×Dic₁₇	—	2	2	no	—
59	C₈×C₁₇	—	2	2	no	—
60	#3622	—	4	2	no	—
61	#3622	#124	1	1	yes	characteristic
62	#3622	#124	1	1	yes	characteristic
63	#3622	#125	1	2	yes	—
64	#2153	—	17	1	no	—
65	#2151	—	17	1	no	—
66	#1747	—	17	1	no	—
67	#1379	—	17	2	no	—
68	#1378	—	17	2	no	—
69	C₂²×C₁₇	—	4	1	no	—
70	C₂²×C₁₇	#424	1	1	yes	characteristic
71	C₂²×C₁₇	#433	1	2	yes	—
72	C₄×C₁₇	—	8	2	no	—
73	C₄×C₁₇	#382	1	1	yes	characteristic
74	C₄×C₁₇	#424	1	1	yes	characteristic
75	C₄×C₁₇	#433	1	2	yes	—
76	Dic₁₇	—	4	2	no	—
77	C₂×M₃₂	—	17	1	no	—
78	C₂×M₃₂	—	17	2	no	—
79	C₂²×C₁₆	—	17	1	no	—
80	#405	—	17	1	no	—
81	#405	—	17	2	no	—
82	#364	—	34	2	no	—
83	C₄⋊C₁₆	—	34	2	no	—
84	#338	—	17	1	no	—
85	H(3,ℤ/4)	—	17	2	no	—
86	H(3,ℤ/4)	#595	1	1	yes	characteristic
87	C₂×C₁₇	—	2	1	no	—
88	C₂×C₁₇	—	2	2	no	—
89	C₂×C₁₇	#1832	1	1	yes	characteristic
90	C₂×M₁₆	—	17	2	no	—
91	C₂×M₁₆	C₂×#595	1	1	yes	characteristic
92	C₂²×C₈	—	17	1	no	—
93	#117	—	17	1	no	—
94	#117	C₂×#595	1	2	yes	—
95	M₃₂	—	34	2	no	—
96	M₃₂	—	34	2	no	—
97	M₃₂	—	34	2	no	—
98	C₂×C₁₆	—	34	1	no	—
99	C₂×C₁₆	—	34	1	no	—
100	C₂×C₁₆	—	34	2	no	—
101	C₂×C₁₆	—	34	2	no	—
102	#97	—	17	2	no	—
103	#97	—	34	2	no	—
104	C₄×C₈	—	17	2	no	—
105	C₄×C₈	—	34	2	no	—
106	C₁₇	#12885	1	1	yes	characteristic, complemented, Sylow
107	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	34	2	no	—
110	M₁₆	—	34	2	no	—
111	C₂×C₈	—	17	1	no	—
112	C₂×C₈	—	17	1	no	—
113	C₂×C₈	—	17	2	no	—
114	C₂×C₈	—	17	2	no	—
115	C₂×C₈	—	34	2	no	—
116	C₂×C₈	C₄⋊#595	1	2	yes	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	34	1	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	34	1	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	17	2	no	—
123	C₁₆	—	68	1	no	—
124	C₁₆	—	68	1	no	—
125	C₁₆	—	68	2	no	—
126	C₂³	#10587678	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	34	2	no	—
129	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
130	C₂×C₄	#10587677	1	2	yes	—
131	C₂×C₄	#10587678	1	2	yes	—
132	C₂×C₄	#10587680	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	17	1	no	—
135	C₈	—	17	1	no	—
136	C₈	—	17	2	no	—
137	C₈	—	68	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	#11763816	1	1	yes	characteristic
140	C₂²	#11763833	1	2	yes	—
141	C₄	—	8	2	no	—
142	C₄	—	68	2	no	—
143	C₄	#11763753	1	1	yes	characteristic
144	C₄	#11763815	1	1	yes	characteristic
145	C₄	#11763834	1	2	yes	—
146	C₂	—	2	2	no	—
147	C₂	—	2	1	no	—
148	C₂	#14544702	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545125

C₂

yes

maximal, characteristic

#14545121

C₂

yes

maximal, characteristic

#14543838

C₂

yes

maximal

#14543834

C₂

yes

maximal

#14528538

C₂

yes

maximal, characteristic

C₂×#10587538

C₂²

yes

characteristic

C₂×#10587538

C₂²

yes

—

C₂²×#63216

C₂²

yes

—

—

C₂²

yes

characteristic

#11762765

C₂²

yes

—

#11762567

C₄

yes

characteristic

#11762565

C₄

yes

—

C₁₇×H(3,ℤ/4)

C₄

yes

characteristic

C₂×M₁₆×C₁₇

C₂×C₄

yes

characteristic

C₁₇×#117

C₂×C₄

yes

—

#10587612

C₂×C₄

yes

characteristic

C₂×#63223

C₂×C₄

yes

—

C₂²×#3622

C₂³

yes

—

—

—

—

—

—

—

Q₈

yes

—

C₈×Dic₁₇

D₄

yes

—

#10587474

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

—

—

—

—

C₂×D₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

characteristic

C₂×#3622

#41

yes

—

C₂×#3622

#41

yes

—

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#117

yes

—

C₂×C₄×C₁₇

#131

yes

—

C₂×C₄×C₁₇

#131

yes

characteristic

C₂×Dic₁₇

—

C₈×C₁₇

—

—

yes

yes

yes

—

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#424

yes

characteristic

C₂²×C₁₇

#433

yes

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#405

—

#364

—

C₄⋊C₁₆

—

—

—

yes

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1832

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×#595

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×#595

yes

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

#97

—

103

#97

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₁₇

#12885

yes

characteristic, complemented, Sylow

107

C₂²×C₄

C₂²×#595

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄⋊#595

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

#10587678

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

130

C₂×C₄

#10587677

yes

—

131

C₂×C₄

#10587678

yes

—

132

C₂×C₄

#10587680

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₂²

—

139

C₂²

#11763816

yes

characteristic

140

C₂²

#11763833

yes

—

141

C₄

—

142

C₄

—

143

C₄

#11763753

yes

characteristic

144

C₄

#11763815

yes

characteristic

145

C₄

#11763834

yes

—

146

C₂

—

147

C₂

—

148

C₂

#14544702

yes

characteristic

	subgroup	G/H
center	C₄	#11763753
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×H(3,ℤ/4)	C₄
socle	C₂×C₁₇	#1832
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×#3622	C₂³
hypercenter	H(3,ℤ/4)	#595
hypocenter	C₁₇	#12885
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#11763753

commutator

C₂²×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#595

Fitting

C₁₇×H(3,ℤ/4)

C₄

C₂×C₁₇

C₂³×C₁₇

C₂²×#3622

C₂³

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
106. C₁₇	#12885	semidirect

H ≅ G/N

type

106. C₁₇

#12885

semidirect

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

subgroup	index
#14545125	2
#14545121	2
#14543838	2
#14543834	2
#14528538	2
#12885	17

subgroup

index

Subgroups — #17961629

Conjugacy classes of subgroups

Complements

Maximal subgroups