Finite Group Databasebeta

← #13118317

Subgroups — #13118317

Order 1664.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88183C211yesmaximal, characteristic
2C2×#88135C211yesmaximal, characteristic
3C13×#381C211yesmaximal, characteristic, complemented
4C2×#88071C211yesmaximal, characteristic, complemented
5#11718619C211yesmaximal, characteristic
6#11718611C211yesmaximal, characteristic, complemented
7#11718603C211yesmaximal, characteristic, complemented
8#11718336C211yesmaximal, characteristic, complemented
9#11718335C211yesmaximal, characteristic, complemented
10#11718329C211yesmaximal, characteristic, complemented
11#11718325C211yesmaximal, characteristic, complemented
12#11718319C211yesmaximal, characteristic, complemented
13#11718318C211yesmaximal, characteristic, complemented
14#11718315C211yesmaximal, characteristic, complemented
15#11718313C211yesmaximal, characteristic, complemented
16C23×C4×C13C2211yescharacteristic
17C22×C22⋊D13C2211yescharacteristic
18C23×Dic13C2211yescharacteristic
19C23×Dic13C2211yescharacteristic
20C22×C4×D13C2211yescharacteristic
21C2×C13×C22⋊C4C2211yescharacteristic
22C2×C13×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic13C2211yescharacteristic
24#88183—24no—
25C2×#6103C2211yescharacteristic
26C2×#6103C2211yescharacteristic
27C2×#6103C2211yescharacteristic
28C2×#6103C2211yescharacteristic
29C2×#6103C2211yescharacteristic
30C2×C4⋊Dic13C2211yescharacteristic
31C2×C4⋊Dic13C2211yescharacteristic
32C2×#6101C2211yescharacteristic
33C2×#6101C2211yescharacteristic
34C2×#6101C2211yescharacteristic
35C2×#6101C2211yescharacteristic
36C2×#6101C2211yescharacteristic
37C2×C4×Dic13C2211yescharacteristic
38#88135—24no—
39C13×#95C2211yescharacteristic
40C13×#95C2211yescharacteristic
41C13×#95C2211yescharacteristic
42C13×#95C2211yescharacteristic
43#88071C2211yescharacteristic
44#88071C2211yescharacteristic
45#88071C2211yescharacteristic
46#88071C2211yescharacteristic
47#88071C2211yescharacteristic
48#88071C2211yescharacteristic
49#88071C2211yescharacteristic
50#88071C2211yescharacteristic
51#88071C2211yescharacteristic
52#88071C2211yescharacteristic
53C24×C13C2311yescharacteristic
54C23×D13C2311yescharacteristic
55C22×C4×C13—22no—
56C22×C4×C13—22no—
57C22×C4×C13—22no—
58C22×C4×C13C2311yescharacteristic
59C22×C4×C13C2311yescharacteristic
60C22×C4×C13C2311yescharacteristic
61C22×C4×C13C2311yescharacteristic
62C22×C4×C13C2311yescharacteristic
63C22×C4×C13C2311yescharacteristic
64C2×C22⋊D13—22no—
65C2×C22⋊D13—22no—
66C2×C22⋊D13D414yes—
67C22×Dic13—22no—
68C22×Dic13—22no—
69C22×Dic13—22no—
70C22×Dic13—22no—
71C22×Dic13—41no—
72C22×Dic13—41no—
73C22×Dic13C2311yescharacteristic
74C22×Dic13C2311yescharacteristic
75C22×Dic13C2311yescharacteristic
76C22×Dic13C2311yescharacteristic
77C22×Dic13C2311yescharacteristic
78C22×Dic13C2311yescharacteristic
79C22×Dic13C2311yescharacteristic
80C2×C4×D13—22no—
81C2×C4×D13—22no—
82C2×C4×D13—41no—
83C13×C22⋊C4—22no—
84C13×C22⋊C4—22no—
85C22⋊Dic13—22no—
86#6103—22no—
87#6103—22no—
88#6103—22no—
89#6103—22no—
90#6103—41no—
91C4⋊Dic13—22no—
92C4⋊Dic13—41no—
93#6101—22no—
94#6101—22no—
95#6101—22no—
96#6101—22no—
97#6101—41no—
98C4×Dic13—22no—
99#2419—131nomaximal, Sylow
100C23×C13—21no—
101C23×C13—21no—
102C23×C13—21no—
103C23×C13—21no—
104C23×C13C2×D412yes—
105C23×C13#4112yes—
106C23×C13#4112yes—
107C23×C13C2411yescharacteristic
108C22×D13—21no—
109C22×D13—21no—
110C22×D13—21no—
111C22×D13—21no—
112C22×D13—21no—
113C22×D13—21no—
114C22×D13C2×D412yes—
115C2×C4×C13—21no—
116C2×C4×C13—21no—
117C2×C4×C13—21no—
118C2×C4×C13—21no—
119C2×C4×C13—21no—
120C2×C4×C13—21no—
121C2×C4×C13—21no—
122C2×C4×C13—21no—
123C2×C4×C13—21no—
124C2×C4×C13—21no—
125C2×C4×C13—21no—
126C2×C4×C13—21no—
127C2×C4×C13—21no—
128C2×C4×C13—21no—
129C2×C4×C13—21no—
130C2×C4×C13—21no—
131C2×C4×C13—21no—
132C2×C4×C13—21no—
133C2×C4×C13—44no—
134C22⋊D13—28no—
135C2×Dic13—21no—
136C2×Dic13—21no—
137C2×Dic13—21no—
138C2×Dic13—21no—
139C2×Dic13—21no—
140C2×Dic13—21no—
141C2×Dic13—21no—
142C2×Dic13—21no—
143C2×Dic13—21no—
144C2×Dic13—21no—
145C2×Dic13—21no—
146C2×Dic13—21no—
147C2×Dic13—21no—
148C2×Dic13—21no—
149C2×Dic13—21no—
150C2×Dic13—21no—
151C2×Dic13—21no—
152C2×Dic13—21no—
153C2×Dic13—44no—
154C2×Dic13—44no—
155C2×Dic13C2×D412yes—
156C2×Dic13#4112yes—
157C2×Dic13#4112yes—
158C4×D13—44no—
159C2×#123—131no—
160C2×C4×D4—131no—
161#400—131no—
162#398—131no—
163#397—131no—
164#397—131no—
165#387—131no—
166#387—131no—
167#386—131no—
168#385—131no—
169#385—131no—
170#381—131no—
171#381D1311yescharacteristic, complemented
172#380—131no—
173C2×#95—131no—
174C22×C13—22no—
175C22×C13—22no—
176C22×C13—22no—
177C22×C13—22no—
178C22×C13—22no—
179C22×C13—22no—
180C22×C13#12314yes—
181C22×C13C22×D411yescharacteristic
182C22×C13C2×#4111yescharacteristic
183C22×C13C2×#4111yescharacteristic
184C22×C13C2×#4111yescharacteristic
185C22×C13C2×#4111yescharacteristic
186C22×C13H(5,𝔽2)11yescharacteristic
187C22×C132−1+411yescharacteristic
188C2×D13—22no—
189C2×D13—22no—
190C2×D13—22no—
191C2×D13—41no—
192C2×D13—41no—
193C2×D13—41no—
194C2×D13—41no—
195C4×C13—41no—
196C4×C13—41no—
197C4×C13—41no—
198C4×C13—41no—
199C4×C13—41no—
200C4×C13—41no—
201Dic13—22no—
202Dic13—22no—
203Dic13—22no—
204Dic13—41no—
205Dic13—41no—
206Dic13—41no—
207Dic13—41no—
208C22×D4—131no—
209C23×C4—131no—
210C23×C4—131no—
211C23×C4—131no—
212C23×C4C2×D1311yescharacteristic
213#123—264no—
214C4×D4—264no—
215C2×C4⋊C4—131no—
216C2×C4⋊C4—131no—
217C2×C4⋊C4—131no—
218C2×C4⋊C4—131no—
219C2×C4⋊C4—131no—
220C2×C4⋊C4—131no—
221C2×C4⋊C4—131no—
222C2×C22⋊C4—131no—
223C2×C22⋊C4—131no—
224C2×C22⋊C4—131no—
225C2×C22⋊C4—131no—
226C2×C22⋊C4—131no—
227C2×C22⋊C4—131no—
228C2×C22⋊C4C2×D1311yescharacteristic
229C2×C22⋊C4C2×D1311yescharacteristic
230C2×C42—131no—
231#95—131no—
232#95—131no—
233#95—131no—
234#95—131no—
235#95—131no—
236#95—131no—
237#95—131no—
238#95—131no—
239#95—131no—
240#95—131no—
241#95C2×D1311yescharacteristic
242#95C2×D1311yescharacteristic
243#95C2×D1311yescharacteristic
244#95C2×D1311yescharacteristic
245C2×C13—24no—
246C2×C13C2×#12311yescharacteristic
247C2×C13#52911yescharacteristic
248C2×C13#53911yescharacteristic
249C2×C13#54611yescharacteristic
250C2×C13#54711yescharacteristic
251C2×C13#55111yescharacteristic
252C2×C13#55211yescharacteristic
253D13—42no—
254C24—131no—
255C24C22×D1311yescharacteristic
256C2×D4—134no—
257C2×D4—262no—
258C2×D4—262no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—131no—
263C22×C4—131no—
264C22×C4—131no—
265C22×C4—131no—
266C22×C4—131no—
267C22×C4—131no—
268C22×C4—131no—
269C22×C4—262no—
270C22×C4—262no—
271C22×C4—262no—
272C22×C4—262no—
273C22×C4—262no—
274C22×C4—262no—
275C22×C4—521no—
276C22×C4—521no—
277C22×C4—521no—
278C22×C4C22×D1311yescharacteristic
279C22×C4C22×D1311yescharacteristic
280C22×C4C22×D1311yescharacteristic
281C22×C4C22×D1311yescharacteristic
282C22×C4C22×D1311yescharacteristic
283C22×C4C22×D1311yescharacteristic
284C4⋊C4—262no—
285C4⋊C4—262no—
286C4⋊C4—262no—
287C4⋊C4—262no—
288C4⋊C4—262no—
289C4⋊C4—521no—
290C4⋊C4—521no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—262no—
294C22⋊C4—262no—
295C22⋊C4—262no—
296C22⋊C4—262no—
297C22⋊C4—262no—
298C22⋊C4—521no—
299C42—262no—
300C13#241911yescharacteristic, complemented, Sylow
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—132no—
306C23—261no—
307C23—261no—
308C23—261no—
309C23—261no—
310C23—261no—
311C23—261no—
312C23#612712yes—
313C23D4×D1312yes—
314C23#612912yes—
315C23C23×D1311yescharacteristic
316D4—268no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—44no—
336C2×C4—132no—
337C2×C4—132no—
338C2×C4—132no—
339C2×C4—261no—
340C2×C4—261no—
341C2×C4—261no—
342C2×C4—261no—
343C2×C4—261no—
344C2×C4—261no—
345C2×C4—261no—
346C2×C4—261no—
347C2×C4—261no—
348C2×C4—261no—
349C2×C4—261no—
350C2×C4—261no—
351C2×C4—261no—
352C2×C4—261no—
353C2×C4—261no—
354C2×C4—261no—
355C2×C4—261no—
356C2×C4—261no—
357C2×C4—524no—
358C2×C4—524no—
359C2×C4—524no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—262no—
367C22—262no—
368C22—262no—
369C22—521no—
370C22—521no—
371C22—521no—
372C22—521no—
373C22#8813714yes—
374C22C2×#612711yescharacteristic
375C22C2×D4×D1311yescharacteristic
376C22C2×#612911yescharacteristic
377C22#41×D1311yescharacteristic
378C22#41×D1311yescharacteristic
379C22#8825611yescharacteristic
380C22#8825711yescharacteristic
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—262no—
388C4—262no—
389C4—262no—
390C4—521no—
391C4—521no—
392C4—521no—
393C4—521no—
394C2—24no—
395C2—522no—
396C2C2×#8813711yescharacteristic
397C2#1171923211yescharacteristic
398C2#1171923611yescharacteristic
399C2#1171935411yescharacteristic
400C2#1171935911yescharacteristic
401C2#1171936011yescharacteristic
402C2#1171937011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#381C2
prime radicalC22×C22⋊D13C22
hypercenter#381D13
hypocenterC13#2419
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. C13×#381C2semidirect
4. C2×#88071C2semidirect
6. #11718611C2semidirect
7. #11718603C2semidirect
8. #11718336C2semidirect
9. #11718335C2semidirect
10. #11718329C2semidirect
11. #11718325C2semidirect
12. #11718319C2semidirect
13. #11718318C2semidirect
14. #11718315C2semidirect
15. #11718313C2semidirect
171. #381D13semidirect
300. C13#2419semidirect

Maximal subgroups

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

subgroupindex
C2×#881832
C2×#881352
C13×#3812
C2×#880712
#117186192
#117186112
#117186032
#117183362
#117183352
#117183292
#117183252
#117183192
#117183182
#117183152
#117183132
#241913