Finite Group Databasebeta

← #13120701

Subgroups — #13120701

Order 1664.

500 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 131 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×#88141C211yesmaximal, characteristic, complemented
3C13×#394C211yesmaximal, characteristic
4#11718883C211yesmaximal, characteristic
5#11718640C211yesmaximal, characteristic, complemented
6#11718612C212yesmaximal, complemented
7#11718610C212yesmaximal, complemented
8#11718608C212yesmaximal, complemented
9#11718606C212yesmaximal, complemented
10#11718317C211yesmaximal, characteristic, complemented
11#11718312C211yesmaximal, characteristic, complemented
12C22×D4×C13C2211yescharacteristic
13C22×Dic26C2211yescharacteristic
14C22×Dic26C2211yescharacteristic
15C2×C13×C4⋊C4C2211yescharacteristic
16C2×C13×C22⋊C4C2212yes—
17C2×C13×C22⋊C4C2212yes—
18C2×C22⋊Dic13C2212yes—
19C2×C22⋊Dic13C2212yes—
20C2×C22⋊Dic13C2212yes—
21C2×C22⋊Dic13C2212yes—
22#88190—24no—
23C2×C4⋊Dic13C2212yes—
24C2×#6101C2211yescharacteristic
25C2×#6101C2211yescharacteristic
26C2×#6101C2212yes—
27C2×C4×Dic13C2211yescharacteristic
28C2×C4×Dic13C2211yescharacteristic
29C2×C4×Dic13C2211yescharacteristic
30#88141—24no—
31C13×#95C2211yescharacteristic
32#88071C2212yes—
33#88071C2212yes—
34#88071C2211yescharacteristic
35#88071C2212yes—
36#88071C2212yes—
37C24×C13C2312yes—
38C2×D4×C13—22no—
39C2×D4×C13—42no—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C22×C4×C13C2311yescharacteristic
44C22×Dic13C2312yes—
45C22×Dic13C2312yes—
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2311yescharacteristic
48C22×Dic13C2311yescharacteristic
49C22×Dic13C2311yescharacteristic
50C2×Dic26—22no—
51C2×Dic26—22no—
52C2×Dic26—22no—
53C2×Dic26—22no—
54C2×Dic26—42no—
55C13×C4⋊C4—22no—
56C13×C22⋊C4—24no—
57C13×C22⋊C4—42no—
58C22⋊Dic13—24no—
59C22⋊Dic13—24no—
60C22⋊Dic13—24no—
61C22⋊Dic13—42no—
62C4⋊Dic13—42no—
63#6101—22no—
64#6101—22no—
65#6101—42no—
66C4×Dic13—22no—
67C4×Dic13—22no—
68C4×Dic13D414yes—
69#2632—131nomaximal, Sylow
70C23×C13—22no—
71C23×C13—22no—
72C23×C13—22no—
73C23×C13—22no—
74C23×C13—22no—
75C23×C13—22no—
76C23×C13—22no—
77C23×C13C2411yescharacteristic
78D4×C13—44no—
79C2×C4×C13—21no—
80C2×C4×C13—21no—
81C2×C4×C13—22no—
82C2×C4×C13—21no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13—22no—
87C2×C4×C13C2×D412yes—
88C2×C4×C13#4112yes—
89C2×Dic13—21no—
90C2×Dic13—21no—
91C2×Dic13—21no—
92C2×Dic13—21no—
93C2×Dic13—22no—
94C2×Dic13—21no—
95C2×Dic13—22no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13—22no—
99C2×Dic13—22no—
100C2×Dic13—22no—
101C2×Dic13—22no—
102C2×Dic13C2×D412yes—
103C2×Dic13C2×D412yes—
104C2×Dic13#4112yes—
105Dic26—28no—
106Dic26—44no—
107C2×#124—131no—
108C2×C4×Q8—131no—
109#399—132no—
110#395—131no—
111#395—131no—
112#394—131no—
113#394D1311yescharacteristic
114#393—132no—
115#388—132no—
116#388—132no—
117#383—131no—
118C22×C13—42no—
119C22×C13—42no—
120C22×C13—42no—
121C22×C13—42no—
122C22×C13—42no—
123C22×C13—42no—
124C22×C13—42no—
125C22×C13C22×D411yescharacteristic
126C22×C13C2×#4111yescharacteristic
127C22×C13C2×#4111yescharacteristic
128C22×C13H(5,𝔽2)12yes—
129C22×C132−1+412yes—
130C4×C13—22no—
131C4×C13—22no—
132C4×C13—41no—
133C4×C13—42no—
134Dic13—22no—
135Dic13—22no—
136Dic13—22no—
137Dic13—42no—
138Dic13—41no—
139Dic13—42no—
140C22×Q8—131no—
141C22×Q8—131no—
142C22×D4C2×D1311yescharacteristic
143#124—264no—
144C4×Q8—264no—
145C2×C4⋊C4—131no—
146C2×C4⋊C4—131no—
147C2×C4⋊C4—132no—
148C2×C4⋊C4—132no—
149C2×C4⋊C4C2×D1311yescharacteristic
150C2×C22⋊C4—132no—
151C2×C22⋊C4—132no—
152C2×C22⋊C4—132no—
153C2×C22⋊C4—132no—
154C2×C22⋊C4C2×D1312yes—
155C2×C22⋊C4C2×D1312yes—
156C2×C42—131no—
157C2×C42—131no—
158C2×C42—131no—
159#95—132no—
160#95—132no—
161#95—131no—
162#95—132no—
163#95—132no—
164#95C2×D1311yescharacteristic
165C2×C13—82no—
166C2×C13#52511yescharacteristic
167C2×C13#53511yescharacteristic
168C2×C13#53611yescharacteristic
169C2×C13#54212yes—
170C2×C13#54212yes—
171C24C22×D1312yes—
172C2×Q8—262no—
173C2×Q8—262no—
174C2×Q8—262no—
175C2×Q8—262no—
176C2×Q8—522no—
177C2×D4—22no—
178C2×D4—42no—
179C22×C4—131no—
180C22×C4—131no—
181C22×C4—131no—
182C22×C4—132no—
183C22×C4—132no—
184C22×C4—131no—
185C22×C4C22×D1312yes—
186C22×C4C22×D1311yescharacteristic
187C22×C4C22×D1311yescharacteristic
188C22×C4C22×D1311yescharacteristic
189C4⋊C4—22no—
190C4⋊C4—262no—
191C4⋊C4—262no—
192C4⋊C4—522no—
193C4⋊C4—522no—
194C22⋊C4—24no—
195C22⋊C4—42no—
196C22⋊C4—264no—
197C22⋊C4—264no—
198C22⋊C4—264no—
199C22⋊C4—522no—
200C42—134no—
201C42—262no—
202C42—262no—
203C13#263211yescharacteristic, complemented, Sylow
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—22no—
211C23C23×D1311yescharacteristic
212Q8—268no—
213Q8—524no—
214D4—44no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—132no—
224C2×C4—132no—
225C2×C4—132no—
226C2×C4—261no—
227C2×C4—261no—
228C2×C4—261no—
229C2×C4—261no—
230C2×C4—262no—
231C2×C4—261no—
232C2×C4—262no—
233C2×C4—262no—
234C2×C4—262no—
235C2×C4—262no—
236C2×C4—262no—
237C2×C4—262no—
238C2×C4—262no—
239C2×C4D4×D1312yes—
240C2×C4#612912yes—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22C2×D4×D1311yescharacteristic
249C22C2×#612911yescharacteristic
250C22#8825112yes—
251C22#41×D1311yescharacteristic
252C22#8825712yes—
253C4—22no—
254C4—22no—
255C4—41no—
256C4—42no—
257C4—262no—
258C4—262no—
259C4—262no—
260C4—521no—
261C4—522no—
262C4—522no—
263C2—82no—
264C2#1171929711yescharacteristic
265C2#1171930812yes—
266C2#1171931111yescharacteristic
267C2#1171936712yes—
268C2#1171938011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#394C2
prime radicalC22×D4×C13C22
hypercenter#394D13
hypocenterC13#2632
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#88141C2semidirect
5. #11718640C2semidirect
6. #11718612C2semidirect
7. #11718610C2semidirect
8. #11718608C2semidirect
9. #11718606C2semidirect
10. #11718317C2semidirect
11. #11718312C2semidirect
203. C13#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#881902
C2×#881412
C13×#3942
#117188832
#117186402
#117186122
#117186102
#117186082
#117186062
#117183172
#117183122
#263213