Group #17961664 — subgroups

Subgroups — #17961664

Order 4352.

278 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 104 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	#14545127	C₂	1	2	yes	maximal
2	#14545125	C₂	1	1	yes	maximal, characteristic
3	#14545121	C₂	1	1	yes	maximal, characteristic
4	#14545119	C₂	1	2	yes	maximal
5	#14528538	C₂	1	1	yes	maximal, characteristic
6	C₂×#10587538	C₂²	1	2	yes	—
7	C₂×#10587538	C₂²	1	1	yes	characteristic
8	C₂²×#63216	C₂²	1	1	yes	characteristic
9	#11763732	—	2	4	no	—
10	#11763731	—	2	2	no	—
11	#11763730	—	2	2	no	—
12	C₄⋊#63216	—	2	4	no	—
13	#11762767	C₂²	1	1	yes	characteristic
14	#11762765	C₂²	1	2	yes	—
15	#11762567	C₄	1	1	yes	characteristic
16	#11762565	C₄	1	2	yes	—
17	C₁₇×H(3,ℤ/4)	C₄	1	1	yes	characteristic
18	C₂×M₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₁₇×#117	C₂×C₄	1	2	yes	—
20	#10587612	C₂×C₄	1	1	yes	characteristic
21	C₂×#63223	C₂×C₄	1	2	yes	—
22	C₂²×#3622	C₂³	1	1	yes	characteristic
23	#10587538	—	2	2	no	—
24	#10587538	—	2	2	no	—
25	#10587538	—	2	2	no	—
26	C₂×#63216	—	2	2	no	—
27	C₂×#63216	—	2	2	no	—
28	C₂×#63216	—	2	1	no	—
29	C₂×#63216	—	2	1	no	—
30	#10587486	Q₈	1	2	yes	—
31	C₈×Dic₁₇	D₄	1	2	yes	—
32	#10587474	Q₈	1	4	yes	—
33	C₄×#3622	D₄	1	4	yes	—
34	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₁₇	—	2	2	no	—
36	C₂×C₈×C₁₇	C₄⋊C₄	1	2	yes	—
37	C₄⋊C₄×C₁₇	—	2	2	no	—
38	C₂²⋊C₄×C₁₇	—	2	2	no	—
39	C₄²×C₁₇	C₄⋊C₄	1	4	yes	—
40	C₂²⋊Dic₁₇	—	2	1	no	—
41	C₄⋊Dic₁₇	—	2	1	no	—
42	C₄×Dic₁₇	C₄⋊C₄	1	2	yes	—
43	#63223	—	2	2	no	—
44	#63223	—	2	2	no	—
45	C₂×#3622	C₄⋊C₄	1	4	yes	—
46	C₂×#3622	C₂×D₄	1	2	yes	—
47	C₂×#3622	C₂×D₄	1	1	yes	characteristic
48	C₂×#3622	C₂×Q₈	1	2	yes	—
49	C₂×#3622	C₂×Q₈	1	1	yes	characteristic
50	#63216	—	4	1	no	—
51	#63216	—	4	2	no	—
52	#63216	—	4	1	no	—
53	#12902	—	17	1	no	maximal, Sylow
54	C₂³×C₁₇	#131	1	1	yes	characteristic
55	C₂×C₄×C₁₇	—	2	4	no	—
56	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₇	#131	1	2	yes	—
59	C₂×C₄×C₁₇	#131	1	1	yes	characteristic
60	C₂×Dic₁₇	—	2	2	no	—
61	C₈×C₁₇	—	2	2	no	—
62	#3622	—	2	4	no	—
63	#3622	#127	1	1	yes	characteristic
64	#3622	C₄⋊Q₈	1	2	yes	—
65	#3622	C₄⋊Q₈	1	1	yes	characteristic
66	#2153	—	17	1	no	—
67	#2153	—	17	2	no	—
68	#2151	—	17	2	no	—
69	#2151	—	17	1	no	—
70	#1747	—	17	1	no	—
71	C₂²×C₁₇	—	4	1	no	—
72	C₂²×C₁₇	#424	1	2	yes	—
73	C₂²×C₁₇	#424	1	1	yes	characteristic
74	C₄×C₁₇	—	4	4	no	—
75	C₄×C₁₇	#384	1	1	yes	characteristic
76	C₄×C₁₇	#424	1	1	yes	characteristic
77	C₄×C₁₇	#424	1	2	yes	—
78	Dic₁₇	—	4	2	no	—
79	C₂×M₃₂	—	17	2	no	—
80	C₂×M₃₂	—	17	1	no	—
81	C₂²×C₁₆	—	17	1	no	—
82	#405	—	17	1	no	—
83	#405	—	17	2	no	—
84	#364	—	34	2	no	—
85	#364	—	34	4	no	—
86	C₄⋊C₁₆	—	34	4	no	—
87	C₄⋊C₁₆	—	34	2	no	—
88	#338	—	17	1	no	—
89	H(3,ℤ/4)	—	17	2	no	—
90	H(3,ℤ/4)	#595	1	1	yes	characteristic
91	C₂×C₁₇	—	2	1	no	—
92	C₂×C₁₇	—	2	2	no	—
93	C₂×C₁₇	#1843	1	1	yes	characteristic
94	C₂×M₁₆	—	17	2	no	—
95	C₂×M₁₆	C₂×#595	1	1	yes	characteristic
96	C₂²×C₈	—	17	1	no	—
97	#117	—	17	1	no	—
98	#117	C₂×#595	1	2	yes	—
99	M₃₂	—	34	2	no	—
100	M₃₂	—	34	2	no	—
101	M₃₂	—	34	2	no	—
102	C₂×C₁₆	—	34	2	no	—
103	C₂×C₁₆	—	34	1	no	—
104	C₂×C₁₆	—	34	2	no	—
105	C₂×C₁₆	—	34	1	no	—
106	#97	—	17	2	no	—
107	#97	—	17	4	no	—
108	C₄×C₈	—	17	2	no	—
109	C₄×C₈	—	17	4	no	—
110	C₁₇	#12902	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₄	C₂²×#595	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	34	2	no	—
114	M₁₆	—	34	2	no	—
115	C₂×C₈	—	17	1	no	—
116	C₂×C₈	—	17	2	no	—
117	C₂×C₈	—	17	1	no	—
118	C₂×C₈	—	17	2	no	—
119	C₂×C₈	—	17	4	no	—
120	C₂×C₈	C₄⋊#595	1	2	yes	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	34	1	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	34	1	no	—
125	C₄²	—	17	2	no	—
126	C₄²	C₄⋊#595	1	4	yes	—
127	C₁₆	—	68	1	no	—
128	C₁₆	—	68	2	no	—
129	C₁₆	—	68	1	no	—
130	C₂³	#10587678	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	34	2	no	—
133	C₂×C₄	C₂×C₄⋊#595	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊#595	1	2	yes	—
135	C₂×C₄	#10587678	1	2	yes	—
136	C₂×C₄	#10587680	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	17	1	no	—
139	C₈	—	17	1	no	—
140	C₈	—	17	2	no	—
141	C₈	—	34	4	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#11763815	1	2	yes	—
144	C₂²	#11763816	1	1	yes	characteristic
145	C₄	—	4	4	no	—
146	C₄	—	68	2	no	—
147	C₄	#11763755	1	1	yes	characteristic
148	C₄	#11763815	1	1	yes	characteristic
149	C₄	#11763816	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	—	2	1	no	—
152	C₂	#14544718	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14545127

C₂

yes

maximal

#14545125

C₂

yes

maximal, characteristic

#14545121

C₂

yes

maximal, characteristic

#14545119

C₂

yes

maximal

#14528538

C₂

yes

maximal, characteristic

C₂×#10587538

C₂²

yes

—

C₂×#10587538

C₂²

yes

characteristic

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

Q₈

yes

—

C₄×#3622

D₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

C₄⋊C₄

yes

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

C₄⋊C₄

yes

—

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

C₄⋊C₄

yes

—

#63223

—

#63223

—

C₂×#3622

C₄⋊C₄

yes

—

C₂×#3622

C₂×D₄

yes

—

C₂×#3622

C₂×D₄

yes

characteristic

C₂×#3622

C₂×Q₈

yes

—

C₂×#3622

C₂×Q₈

yes

—

—

—

—

C₂³×C₁₇

yes

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

#131

yes

—

C₂×C₄×C₁₇

#131

yes

characteristic

C₂×Dic₁₇

—

C₈×C₁₇

—

—

yes

C₄⋊Q₈

yes

—

#3622

C₄⋊Q₈

yes

—

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

#424

yes

—

C₂²×C₁₇

#424

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

Dic₁₇

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

—

—

—

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

yes

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#1843

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×#595

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×#595

yes

—

M₃₂

—

100

M₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

#97

—

107

#97

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₁₇

#12902

yes

characteristic, complemented, Sylow

111

C₂²×C₄

C₂²×#595

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₄⋊#595

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

C₄⋊#595

yes

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

130

C₂³

#10587678

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₄⋊#595

yes

characteristic

134

C₂×C₄

C₂×C₄⋊#595

yes

—

135

C₂×C₄

#10587678

yes

—

136

C₂×C₄

#10587680

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₂²

—

143

C₂²

#11763815

yes

—

144

C₂²

#11763816

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

#11763755

yes

characteristic

148

C₄

#11763815

yes

characteristic

149

C₄

#11763816

yes

—

150

C₂

—

151

C₂

—

152

C₂

#14544718

yes

characteristic

	subgroup	G/H
center	C₄	#11763755
commutator	C₂²×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#595
Fitting	C₁₇×H(3,ℤ/4)	C₄
socle	C₂×C₁₇	#1843
prime radical	C₂³×C₁₇	#131
cosocle	C₂²×#3622	C₂³
hypercenter	H(3,ℤ/4)	#595
hypocenter	C₁₇	#12902
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#11763755

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
110. C₁₇	#12902	semidirect

H ≅ G/N

type

110. C₁₇

#12902

semidirect

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

subgroup	index
#14545127	2
#14545125	2
#14545121	2
#14545119	2
#14528538	2
#12902	17

subgroup

index

Subgroups — #17961664

Conjugacy classes of subgroups

Complements

Maximal subgroups