Finite Group Databasebeta

← #14531950

Subgroups — #14531950

Order 2176.

464 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762716C212yesmaximal, complemented
2#11762708C212yesmaximal, complemented
3#11762702C212yesmaximal
4#11762702C212yesmaximal
5#11762695C212yesmaximal, complemented
6#11762695C212yesmaximal, complemented
7#11762672C211yesmaximal, characteristic
8#11762669C211yesmaximal, characteristic
9C2×C17×C22⋊Q8C211yesmaximal, characteristic
10C22×Q8×C17C2211yescharacteristic
11C23×C4×C17C2211yescharacteristic
12C17×C22⋊Q8C418yescomplemented
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2212yes—
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊Dic17C2211yescharacteristic
17C2×C22⋊Dic17C2211yescharacteristic
18C2×C22⋊Dic17C2212yes—
19C2×C4⋊Dic17C2212yes—
20C2×C4⋊Dic17C2212yes—
21C2×C4⋊Dic17C2212yes—
22C2×C4×Dic17C2212yes—
23C2×C4×Dic17C2212yes—
24#10587502C2212yes—
25#10587502C2212yes—
26#10587502C2211yescharacteristic
27#10587502C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2211yescharacteristic
32C24×C17C2311yescharacteristic
33C2×Q8×C17—42no—
34C2×Q8×C17C2×C414yes—
35C22×C4×C17—42no—
36C22×C4×C17C2×C414yes—
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2312yes—
40C22×C4×C17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2312yes—
45C4⋊C4×C17C2×C414yes—
46C4⋊C4×C17C2×C418yes—
47C22⋊C4×C17C2×C418yes—
48C22⋊Dic17—41no—
49C22⋊Dic17—42no—
50C22⋊Dic17—41no—
51C4⋊Dic17—42no—
52C4⋊Dic17—42no—
53C4⋊Dic17—42no—
54C4×Dic17—42no—
55C4×Dic17—42no—
56C23×C17—21no—
57C23×C17—22no—
58C23×C17—22no—
59C23×C17—21no—
60C23×C17C22×C412yes—
61C23×C17C2411yescharacteristic
62Q8×C17—44no—
63C2×C4×C17—22no—
64C2×C4×C17—22no—
65C2×C4×C17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—44no—
70C2×C4×C17C22×C414yes—
71C2×C4×C17C22×C414yes—
72C2×C4×C17C22×C412yes—
73C2×C4×C17C22×C412yes—
74C2×Dic17—22no—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86#2378—171nomaximal, Sylow
87C22×C17—22no—
88C22×C17—22no—
89C22×C17—22no—
90C22×C17—42no—
91C22×C17—42no—
92C22×C17C23×C411yescharacteristic
93C22×C17H(5,𝔽2)12yes—
94C22×C17H(5,𝔽2)11yescharacteristic
95C22×C172−1+412yes—
96C22×C172−1+411yescharacteristic
97C4×C17—22no—
98C4×C17—22no—
99C4×C17—24no—
100C4×C17—24no—
101Dic17—42no—
102Dic17—42no—
103Dic17—42no—
104Dic17—42no—
105C2×C22⋊Q8D1711yescharacteristic
106#391—172no—
107#389—172no—
108#388—172no—
109#388—172no—
110#387—172no—
111#387—172no—
112#381—171no—
113#380—171no—
114C2×C17—42no—
115C2×C17#51811yescharacteristic
116C2×C17#51911yescharacteristic
117C2×C17#52011yescharacteristic
118C2×C17#56212yes—
119C2×C17#56312yes—
120C22×Q8C2×D1711yescharacteristic
121C23×C4C2×D1711yescharacteristic
122C22⋊Q8Dic1718yescomplemented
123C2×C4⋊C4—172no—
124C2×C4⋊C4—172no—
125C2×C4⋊C4—172no—
126C2×C4⋊C4C2×D1711yescharacteristic
127C2×C4⋊C4C2×D1712yes—
128C2×C22⋊C4—171no—
129C2×C22⋊C4—171no—
130C2×C22⋊C4—172no—
131C2×C22⋊C4C2×D1712yes—
132C2×C42—172no—
133C2×C42—172no—
134#95—172no—
135#95—171no—
136#95—172no—
137#95—171no—
138#95—172no—
139#95—172no—
140#95—172no—
141#95—172no—
142C17#237811yescharacteristic, complemented, Sylow
143C24C22×D1711yescharacteristic
144C2×Q8—42no—
145C2×Q8C2×Dic1714yes—
146C22×C4—42no—
147C22×C4—172no—
148C22×C4—172no—
149C22×C4—172no—
150C22×C4—172no—
151C22×C4C2×Dic1714yes—
152C22×C4C22×D1711yescharacteristic
153C22×C4C22×D1712yes—
154C22×C4C22×D1712yes—
155C22×C4C22×D1711yescharacteristic
156C4⋊C4—682no—
157C4⋊C4—682no—
158C4⋊C4—682no—
159C4⋊C4C2×Dic1714yes—
160C4⋊C4C2×Dic1718yes—
161C22⋊C4—682no—
162C22⋊C4—681no—
163C22⋊C4—681no—
164C22⋊C4C2×Dic1718yes—
165C42—682no—
166C42—682no—
167C23—22no—
168C23—22no—
169C23—21no—
170C23—21no—
171C23C22×Dic1712yes—
172C23C23×D1711yescharacteristic
173Q8—44no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—44no—
181C2×C4—342no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—342no—
187C2×C4—342no—
188C2×C4—342no—
189C2×C4—342no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4C22×Dic1714yes—
194C2×C4C22×Dic1714yes—
195C2×C4C22×Dic1712yes—
196C2×C4C22×Dic1712yes—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—42no—
201C22—42no—
202C22#1058769112yes—
203C22#1058769412yes—
204C22#1058769611yescharacteristic
205C22#1058769711yescharacteristic
206C22C23×Dic1711yescharacteristic
207C4—22no—
208C4—22no—
209C4—24no—
210C4—24no—
211C4—682no—
212C4—682no—
213C4—682no—
214C4—682no—
215C2—42no—
216C2#1176328411yescharacteristic
217C2#1176328711yescharacteristic
218C2#1176329211yescharacteristic
219C2#1176355512yes—
220C2#1176356012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutatorC22×C17C23×C4
FittingC2×C17×C22⋊Q8C2
socle, cosocleC23×C17C24
prime radicalC24×C17C23
hypercenterC2×C22⋊Q8D17
hypocenterC17#2378
LCSG ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11762716C2semidirect
2. #11762708C2semidirect
5. #11762695C2semidirect
6. #11762695C2semidirect
12. C17×C22⋊Q8C4semidirect
122. C22⋊Q8Dic17semidirect
142. C17#2378semidirect

Maximal subgroups

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

subgroupindex
#117627162
#117627082
#117627022
#117626952
#117626722
#117626692
C2×C17×C22⋊Q82
#237817