Finite Group Databasebeta

← #13121674

Subgroups — #13121674

Order 1664.

512 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#88190C211yesmaximal, characteristic
2C2×#88189C211yesmaximal, characteristic
3C13×#386C211yesmaximal, characteristic
4#11718873C212yesmaximal, complemented
5#11718614C212yesmaximal, complemented
6#11718614C212yesmaximal, complemented
7#11718611C212yesmaximal, complemented
8#11718317C212yesmaximal, complemented
9#11718314C212yesmaximal, complemented
10C22×D4×C13C2211yescharacteristic
11C23×C4×C13C2211yescharacteristic
12C23×Dic13C2211yescharacteristic
13C22×Dic26C2211yescharacteristic
14C2×C13×C22⋊C4C2212yes—
15C2×C13×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic13C2211yescharacteristic
17C2×C22⋊Dic13C2212yes—
18C2×C22⋊Dic13C2212yes—
19C2×C22⋊Dic13C2212yes—
20#88190—24no—
21#88189—24no—
22C2×C4⋊Dic13C2212yes—
23C2×C4⋊Dic13C2212yes—
24C2×C4⋊Dic13C2212yes—
25C2×#6101C2212yes—
26C2×C4×Dic13C2211yescharacteristic
27C13×#95C2212yes—
28#88071C2212yes—
29#88071C2212yes—
30#88071C2212yes—
31#88071C2212yes—
32#88071C2212yes—
33C24×C13C2311yescharacteristic
34C24×C13C2311yescharacteristic
35C2×D4×C13—41no—
36C2×D4×C13—41no—
37C2×D4×C13D414yes—
38C22×C4×C13—24no—
39C22×C4×C13—41no—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2312yes—
43C22×Dic13—22no—
44C22×Dic13—41no—
45C22×Dic13—41no—
46C22×Dic13C2312yes—
47C22×Dic13C2312yes—
48C22×Dic13C2312yes—
49C22×Dic13C2312yes—
50C2×Dic26—22no—
51C2×Dic26—41no—
52C2×Dic26—41no—
53C13×C22⋊C4—24no—
54C13×C22⋊C4—41no—
55C22⋊Dic13—22no—
56C22⋊Dic13—24no—
57C22⋊Dic13—24no—
58C22⋊Dic13—24no—
59C4⋊Dic13—24no—
60C4⋊Dic13—42no—
61C4⋊Dic13—42no—
62#6101—24no—
63C4×Dic13—22no—
64#2717—131nomaximal, Sylow
65C23×C13—22no—
66C23×C13—22no—
67C23×C13—21no—
68C23×C13—21no—
69C23×C13—21no—
70C23×C13—21no—
71C23×C13—22no—
72C23×C13—22no—
73C23×C13C2×D412yes—
74C23×C13C2×D412yes—
75C23×C13C2411yescharacteristic
76D4×C13—44no—
77C2×C4×C13—22no—
78C2×C4×C13—22no—
79C2×C4×C13—21no—
80C2×C4×C13—21no—
81C2×C4×C13—22no—
82C2×C4×C13—22no—
83C2×C4×C13—22no—
84C2×C4×C13—44no—
85C2×C4×C13C2×D412yes—
86C2×C4×C13#4114yes—
87C2×Dic13—22no—
88C2×Dic13—22no—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—44no—
100Dic26—44no—
101C2×#124—131no—
102C2×#123—131no—
103#397—132no—
104#397—132no—
105#395—132no—
106#387—132no—
107#386D1311yescharacteristic
108#385—132no—
109#380—132no—
110C22×C13—22no—
111C22×C13—22no—
112C22×C13—24no—
113C22×C13—24no—
114C22×C13—42no—
115C22×C13—42no—
116C22×C13—41no—
117C22×C13—41no—
118C22×C13—41no—
119C22×C13—41no—
120C22×C13C22×D411yescharacteristic
121C22×C13C2×#4112yes—
122C22×C13H(5,𝔽2)11yescharacteristic
123C22×C13H(5,𝔽2)11yescharacteristic
124C22×C132−1+412yes—
125C4×C13—22no—
126C4×C13—24no—
127C4×C13—42no—
128Dic13—42no—
129Dic13—42no—
130Dic13—42no—
131Dic13—42no—
132C22×Q8—131no—
133C22×D4C2×D1311yescharacteristic
134C23×C4—131no—
135C23×C4C2×D1311yescharacteristic
136#124—264no—
137#123—264no—
138C2×C4⋊C4—132no—
139C2×C4⋊C4—132no—
140C2×C4⋊C4—132no—
141C2×C4⋊C4—132no—
142C2×C22⋊C4—132no—
143C2×C22⋊C4—131no—
144C2×C22⋊C4—132no—
145C2×C22⋊C4—132no—
146C2×C22⋊C4C2×D1312yes—
147C2×C22⋊C4C2×D1311yescharacteristic
148C2×C42—131no—
149#95—132no—
150#95—132no—
151#95—132no—
152#95—132no—
153#95—132no—
154#95C2×D1312yes—
155C2×C13—42no—
156C2×C13—42no—
157C2×C13#53411yescharacteristic
158C2×C13#54212yes—
159C2×C13#54712yes—
160C2×C13#55111yescharacteristic
161C2×C13#56311yescharacteristic
162C24C22×D1311yescharacteristic
163C24C22×D1311yescharacteristic
164C2×Q8—262no—
165C2×Q8—521no—
166C2×Q8—521no—
167C2×D4—41no—
168C2×D4—41no—
169C2×D4D5214yes—
170C22×C4—24no—
171C22×C4—41no—
172C22×C4—132no—
173C22×C4—132no—
174C22×C4—132no—
175C22×C4—132no—
176C22×C4—262no—
177C22×C4—521no—
178C22×C4—521no—
179C22×C4C22×D1312yes—
180C22×C4C22×D1311yescharacteristic
181C22×C4C22×D1312yes—
182C4⋊C4—264no—
183C4⋊C4—264no—
184C4⋊C4—522no—
185C4⋊C4—522no—
186C22⋊C4—24no—
187C22⋊C4—41no—
188C22⋊C4—262no—
189C22⋊C4—264no—
190C22⋊C4—264no—
191C22⋊C4—264no—
192C42—262no—
193C13#271711yescharacteristic, complemented, Sylow
194C23—22no—
195C23—22no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—22no—
201C23—22no—
202C23C2×D5212yes—
203C23C2×D5212yes—
204C23C23×D1311yescharacteristic
205Q8—524no—
206D4—44no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—44no—
215C2×C4—262no—
216C2×C4—262no—
217C2×C4—262no—
218C2×C4—262no—
219C2×C4—262no—
220C2×C4—262no—
221C2×C4—262no—
222C2×C4—262no—
223C2×C4—262no—
224C2×C4—262no—
225C2×C4—262no—
226C2×C4—262no—
227C2×C4—524no—
228C2×C4C2×D5212yes—
229C2×C4#612914yes—
230C22—22no—
231C22—22no—
232C22—24no—
233C22—24no—
234C22—41no—
235C22—41no—
236C22—42no—
237C22—42no—
238C22—41no—
239C22—41no—
240C22C22×D5211yescharacteristic
241C22C2×#612912yes—
242C22#8825111yescharacteristic
243C22#8825111yescharacteristic
244C22#8825712yes—
245C4—22no—
246C4—24no—
247C4—42no—
248C4—522no—
249C4—522no—
250C4—522no—
251C4—522no—
252C2—42no—
253C2—42no—
254C2#1171919811yescharacteristic
255C2#1171926012yes—
256C2#1171929211yescharacteristic
257C2#1171930812yes—
258C2#1171935611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#386C2
prime radicalC22×D4×C13C22
hypercenter#386D13
hypocenterC13#2717
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
4. #11718873C2semidirect
5. #11718614C2semidirect
6. #11718614C2semidirect
7. #11718611C2semidirect
8. #11718317C2semidirect
9. #11718314C2semidirect
193. C13#2717semidirect

Maximal subgroups

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

subgroupindex
C2×#881902
C2×#881892
C13×#3862
#117188732
#117186142
#117186112
#117183172
#117183142
#271713