Finite Group Databasebeta

← #13118166

Subgroups — #13118166

Order 1664.

864 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 159 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
1C22×#6129C211yesmaximal, characteristic
2C2×C4⋊C4×D13C211yesmaximal, characteristic, complemented
3C2×#88137C212yesmaximal, complemented
4C13×#394C211yesmaximal, characteristic, complemented
5#11718883C211yesmaximal, characteristic, complemented
6#11718640C211yesmaximal, characteristic, complemented
7#11718617C212yesmaximal, complemented
8#11718611C212yesmaximal, complemented
9#11718607C212yesmaximal, complemented
10#11718325C211yesmaximal, characteristic, complemented
11#11718317C211yesmaximal, characteristic, complemented
12C22×D4×C13C2211yescharacteristic
13C22×C22⋊D13C2212yes—
14C23×Dic13C2212yes—
15C2×#6129—28no—
16C2×#6129—24no—
17C22×C4×D13C2211yescharacteristic
18C22×C4×D13C2211yescharacteristic
19C22×C4×D13C2211yescharacteristic
20C22×Dic26C2211yescharacteristic
21C2×C13×C4⋊C4C2211yescharacteristic
22C2×C13×C22⋊C4C2212yes—
23C2×C13×C22⋊C4C2212yes—
24C2×C22⋊Dic13C2212yes—
25C2×C22⋊Dic13C2212yes—
26C2×#6103C2212yes—
27C2×C4⋊Dic13C2212yes—
28C2×C4⋊Dic13C2211yescharacteristic
29C2×#6101C2212yes—
30C2×#6101C2211yescharacteristic
31C2×#6101C2211yescharacteristic
32C4⋊C4×D13—24no—
33#88137—28no—
34C13×#95C2211yescharacteristic
35#88071C2212yes—
36#88071C2212yes—
37#88071C2211yescharacteristic
38#88071C2212yes—
39C24×C13C2312yes—
40C23×D13C2311yescharacteristic
41C2×D4×C13—24no—
42C2×D4×C13—22no—
43C22×C4×C13C2312yes—
44C22×C4×C13C2311yescharacteristic
45C22×C4×C13C2311yescharacteristic
46C22×C4×C13C2311yescharacteristic
47C2×C22⋊D13—24no—
48C2×C22⋊D13—24no—
49C2×C22⋊D13D418yes—
50C22×Dic13—24no—
51C22×Dic13—24no—
52C22×Dic13—24no—
53C22×Dic13C2312yes—
54C22×Dic13C2311yescharacteristic
55C22×Dic13C2312yes—
56C22×Dic13C2311yescharacteristic
57C22×Dic13C2311yescharacteristic
58#6129—232no—
59C2×C4×D13—22no—
60C2×C4×D13—24no—
61C2×C4×D13—22no—
62C2×C4×D13—22no—
63C2×C4×D13—24no—
64C2×C4×D13—22no—
65C2×C4×D13D414yes—
66C2×Dic26—22no—
67C2×Dic26—24no—
68C13×C4⋊C4—22no—
69C13×C22⋊C4—24no—
70C13×C22⋊C4—24no—
71C22⋊Dic13—24no—
72C22⋊Dic13—24no—
73#6103—24no—
74C4⋊Dic13—24no—
75C4⋊Dic13—22no—
76#6101—22no—
77#6101—24no—
78#6101—22no—
79#2406—131nomaximal, Sylow
80C23×C13—22no—
81C23×C13—22no—
82C23×C13—22no—
83C23×C13—22no—
84C23×C13—22no—
85C23×C13—22no—
86C23×C13C2×D414yes—
87C23×C13C2411yescharacteristic
88C22×D13—21no—
89C22×D13—21no—
90C22×D13—22no—
91C22×D13C2×D414yes—
92C22×D13C2×D412yes—
93D4×C13—28no—
94C2×C4×C13—21no—
95C2×C4×C13—22no—
96C2×C4×C13—21no—
97C2×C4×C13—22no—
98C2×C4×C13—21no—
99C2×C4×C13—22no—
100C2×C4×C13—22no—
101C2×C4×C13—22no—
102C2×C4×C13C2×D412yes—
103C2×C4×C13#4112yescomplemented
104C22⋊D13—216no—
105C2×Dic13—216no—
106C2×Dic13—21no—
107C2×Dic13—22no—
108C2×Dic13—22no—
109C2×Dic13—22no—
110C2×Dic13—21no—
111C2×Dic13—21no—
112C2×Dic13—22no—
113C2×Dic13—22no—
114C2×Dic13—22no—
115C2×Dic13—22no—
116C2×Dic13C2×D414yes—
117C2×Dic13C2×D412yes—
118C2×Dic13#4112yescomplemented
119C4×D13—28no—
120C4×D13—28no—
121C4×D13—44no—
122Dic26—28no—
123C22×#41—131no—
124C2×#123—132no—
125C22×C4⋊C4—131no—
126#395—131no—
127#395—131no—
128#394—131no—
129#394D1311yescharacteristic, complemented
130#386—132no—
131#385—132no—
132#385—132no—
133#381—131no—
134C22×C13—24no—
135C22×C13—24no—
136C22×C13—24no—
137C22×C13—42no—
138C22×C13—42no—
139C22×C13—42no—
140C22×C13—42no—
141C22×C13C22×D411yescharacteristic
142C22×C13C22×D412yes—
143C22×C13C2×#4111yescharacteristic
144C22×C13C2×#4111yescharacteristic
145C22×C132−1+412yes—
146C2×D13—22no—
147C2×D13—22no—
148C2×D13—24no—
149C2×D13—24no—
150C2×D13#12014yes—
151C4×C13—22no—
152C4×C13—22no—
153C4×C13—41no—
154C4×C13—42no—
155Dic13—24no—
156Dic13—22no—
157Dic13—22no—
158Dic13—41no—
159Dic13—42no—
160C2×#41—264no—
161C2×#41—268no—
162C22×Q8—131no—
163C22×D4—132no—
164C22×D4C2×D1311yescharacteristic
165C23×C4—131no—
166C23×C4—132no—
167C23×C4—131no—
168C23×C4—131no—
169#123—268no—
170C2×C4⋊C4—131no—
171C2×C4⋊C4—131no—
172C2×C4⋊C4—131no—
173C2×C4⋊C4—132no—
174C2×C4⋊C4—132no—
175C2×C4⋊C4—264no—
176C2×C4⋊C4C2×D1311yescharacteristic
177C2×C22⋊C4—132no—
178C2×C22⋊C4—132no—
179C2×C22⋊C4—132no—
180C2×C22⋊C4C2×D1312yes—
181C2×C22⋊C4C2×D1312yes—
182#95—132no—
183#95—131no—
184#95—132no—
185#95—132no—
186#95C2×D1311yescharacteristic
187C2×C13—44no—
188C2×C13C2×#12011yescharacteristic
189C2×C13#52511yescharacteristic
190C2×C13#53611yescharacteristic
191C2×C13#54712yes—
192C2×C13#54712yes—
193D13—24no—
194C24—131no—
195C24C22×D1312yes—
196#41—2632no—
197C2×Q8—264no—
198C2×Q8—262no—
199C2×D4—24no—
200C2×D4—22no—
201C2×D4—138no—
202C2×D4—264no—
203C2×D4—264no—
204C22×C4—131no—
205C22×C4—131no—
206C22×C4—132no—
207C22×C4—134no—
208C22×C4—132no—
209C22×C4—131no—
210C22×C4—262no—
211C22×C4—262no—
212C22×C4—264no—
213C22×C4—262no—
214C22×C4—264no—
215C22×C4—264no—
216C22×C4—264no—
217C22×C4—264no—
218C22×C4—262no—
219C22×C4C22×D1312yes—
220C22×C4C22×D1311yescharacteristic
221C22×C4C22×D1311yescharacteristic
222C22×C4C22×D1311yescharacteristic
223C4⋊C4—22no—
224C4⋊C4—262no—
225C4⋊C4—262no—
226C4⋊C4—264no—
227C4⋊C4—264no—
228C4⋊C4—262no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—264no—
232C22⋊C4—264no—
233C22⋊C4—264no—
234C13#240611yescharacteristic, complemented, Sylow
235C23—22no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—134no—
242C23—132no—
243C23—261no—
244C23—261no—
245C23—262no—
246C23D4×D1314yes—
247C23C23×D1311yescharacteristic
248Q8—268no—
249D4—28no—
250D4—2616no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—132no—
260C2×C4—134no—
261C2×C4—132no—
262C2×C4—268no—
263C2×C4—268no—
264C2×C4—2616no—
265C2×C4—261no—
266C2×C4—262no—
267C2×C4—262no—
268C2×C4—262no—
269C2×C4—261no—
270C2×C4—262no—
271C2×C4—262no—
272C2×C4—261no—
273C2×C4—262no—
274C2×C4—262no—
275C2×C4—524no—
276C2×C4D4×D1312yes—
277C2×C4#612912yescomplemented
278C22—24no—
279C22—24no—
280C22—24no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—134no—
286C22—262no—
287C22—264no—
288C22—264no—
289C22—262no—
290C22C2×D4×D1311yescharacteristic
291C22C2×D4×D1312yes—
292C22C2×#612911yescharacteristic
293C22#41×D1311yescharacteristic
294C22#8825712yes—
295C4—22no—
296C4—22no—
297C4—41no—
298C4—42no—
299C4—262no—
300C4—264no—
301C4—262no—
302C4—521no—
303C4—522no—
304C2—44no—
305C2—264no—
306C2D13×#12011yescharacteristic
307C2#1171931111yescharacteristic
308C2#1171931612yes—
309C2#1171936012yes—
310C2#1171938011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#394C2
prime radicalC22×#6129C2
hypercenter#394D13
hypocenterC13#2406
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D13C2semidirect
3. C2×#88137C2semidirect
4. C13×#394C2semidirect
5. #11718883C2semidirect
6. #11718640C2semidirect
7. #11718617C2semidirect
8. #11718611C2semidirect
9. #11718607C2semidirect
10. #11718325C2semidirect
11. #11718317C2semidirect
103. C2×C4×C13#41semidirect
118. C2×Dic13#41semidirect
129. #394D13semidirect
234. C13#2406semidirect
277. C2×C4#6129semidirect

Maximal subgroups

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

subgroupindex
C22×#61292
C2×C4⋊C4×D132
C2×#881372
C13×#3942
#117188832
#117186402
#117186172
#117186112
#117186072
#117183252
#117183172
#240613