Finite Group Databasebeta

← #13121433

Subgroups — #13121433

Order 1664.

506 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88201C211yesmaximal, characteristic
2C2×#88151C211yesmaximal, characteristic, complemented
3C13×#395C211yesmaximal, characteristic, complemented
4#11718891C211yesmaximal, characteristic, complemented
5#11718655C211yesmaximal, characteristic, complemented
6#11718654C211yesmaximal, characteristic, complemented
7#11718652C211yesmaximal, characteristic
8#11718651C211yesmaximal, characteristic, complemented
9#11718642C211yesmaximal, characteristic, complemented
10#11718335C211yesmaximal, characteristic, complemented
11#11718334C211yesmaximal, characteristic, complemented
12#11718330C211yesmaximal, characteristic, complemented
13#11718326C211yesmaximal, characteristic
14#11718325C211yesmaximal, characteristic, complemented
15#11718307C211yesmaximal, characteristic, complemented
16C22×Q8×C13C2211yescharacteristic
17C22×D52C2211yescharacteristic
18C22×C4×D13C2211yescharacteristic
19C22×Dic26C2211yescharacteristic
20C2×C13×C4⋊C4C2211yescharacteristic
21C2×C13×C4⋊C4C2211yescharacteristic
22C2×C13×C4⋊C4C2211yescharacteristic
23#88201—24no—
24C2×#6103C2211yescharacteristic
25C2×#6103C2211yescharacteristic
26C2×#6103C2211yescharacteristic
27C2×#6103C2211yescharacteristic
28C2×#6103C2211yescharacteristic
29C2×#6103C2211yescharacteristic
30C2×#6103C2211yescharacteristic
31C2×#6103C2211yescharacteristic
32C2×#6103C2211yescharacteristic
33C2×#6103C2211yescharacteristic
34C2×#6103C2211yescharacteristic
35C2×C4⋊Dic13C2211yescharacteristic
36C2×C4⋊Dic13C2211yescharacteristic
37C2×#6101C2211yescharacteristic
38C2×C4×Dic13C2211yescharacteristic
39C2×C4×Dic13C2211yescharacteristic
40#88151—24no—
41C13×#95C2211yescharacteristic
42C13×#95C2211yescharacteristic
43C13×#95C2211yescharacteristic
44#88071C2211yescharacteristic
45#88071C2211yescharacteristic
46#88071C2211yescharacteristic
47#88071C2211yescharacteristic
48#88071C2211yescharacteristic
49#88071C2211yescharacteristic
50#88071C2211yescharacteristic
51#88071C2211yescharacteristic
52#88071C2211yescharacteristic
53C23×D13C2311yescharacteristic
54C23×D13C2311yescharacteristic
55C2×Q8×C13—22no—
56C2×Q8×C13—22no—
57C2×Q8×C13—41no—
58C22×C4×C13C2311yescharacteristic
59C22×C4×C13C2311yescharacteristic
60C22×C4×C13C2311yescharacteristic
61C22×C4×C13C2311yescharacteristic
62C22×C4×C13C2311yescharacteristic
63C22×C4×C13C2311yescharacteristic
64C22×C4×C13C2311yescharacteristic
65C22×Dic13C2311yescharacteristic
66C22×Dic13C2311yescharacteristic
67C22×Dic13C2311yescharacteristic
68C22×Dic13C2311yescharacteristic
69C22×Dic13C2311yescharacteristic
70C22×Dic13C2311yescharacteristic
71C2×D52—22no—
72C2×D52—22no—
73C2×D52—41no—
74C2×C4×D13—22no—
75C2×C4×D13—22no—
76C2×C4×D13—41no—
77C2×Dic26—22no—
78C2×Dic26—22no—
79C2×Dic26—41no—
80C13×C4⋊C4—22no—
81C13×C4⋊C4—22no—
82C13×C4⋊C4—41no—
83#6103—22no—
84#6103—22no—
85#6103—22no—
86#6103—22no—
87#6103—22no—
88#6103—41no—
89#6103—41no—
90#6103—41no—
91#6103—41no—
92#6103—41no—
93#6103D414yes—
94C4⋊Dic13—22no—
95C4⋊Dic13—22no—
96#6101—41no—
97C4×Dic13—22no—
98C4×Dic13—41no—
99#2700—131nomaximal, Sylow
100C23×C13C2411yescharacteristic
101C22×D13—21no—
102C22×D13—21no—
103C22×D13—21no—
104C22×D13—21no—
105C22×D13—21no—
106C22×D13—21no—
107C22×D13—21no—
108C22×D13—21no—
109C22×D13—21no—
110C22×D13—21no—
111C22×D13—21no—
112C22×D13—21no—
113C22×D13—21no—
114C22×D13C2×D412yes—
115Q8×C13—44no—
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—21no—
134C2×C4×C13C2×D412yes—
135C2×C4×C13#4112yes—
136C2×C4×C13#4112yes—
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—21no—
154C2×Dic13C2×D412yes—
155D52—44no—
156C4×D13—44no—
157Dic26—44no—
158C2×#124—131no—
159C2×C22⋊Q8—131no—
160#397—131no—
161#397—131no—
162#396—131no—
163#395D1311yescharacteristic, complemented
164#394—131no—
165#393—131no—
166#391—131no—
167#391—131no—
168#388—131no—
169#388—131no—
170#387—131no—
171#386—131no—
172#381—131no—
173C22×C13C22×D411yescharacteristic
174C22×C13C2×#4111yescharacteristic
175C22×C13C2×#4111yescharacteristic
176C22×C13H(5,𝔽2)11yescharacteristic
177C22×C13H(5,𝔽2)11yescharacteristic
178C22×C132−1+411yescharacteristic
179C22×C132−1+411yescharacteristic
180C2×D13—22no—
181C2×D13—22no—
182C2×D13—22no—
183C2×D13—41no—
184C2×D13—41no—
185C2×D13—41no—
186C2×D13—41no—
187C2×D13—41no—
188C2×D13—41no—
189C2×D13—41no—
190C2×D13—41no—
191C2×D13—41no—
192C2×D13—41no—
193C2×D13—41no—
194C4×C13—22no—
195C4×C13—22no—
196C4×C13—22no—
197C4×C13—41no—
198C4×C13—41no—
199C4×C13—41no—
200C4×C13—41no—
201Dic13—22no—
202Dic13—41no—
203Dic13—41no—
204Dic13—41no—
205Dic13—41no—
206Dic13—41no—
207C22×Q8—131no—
208C22×Q8C2×D1311yescharacteristic
209C22×D4—131no—
210C23×C4—131no—
211#124—264no—
212C22⋊Q8—264no—
213C2×C4⋊C4—131no—
214C2×C4⋊C4—131no—
215C2×C4⋊C4—131no—
216C2×C4⋊C4C2×D1311yescharacteristic
217C2×C4⋊C4C2×D1311yescharacteristic
218C2×C4⋊C4C2×D1311yescharacteristic
219C2×C22⋊C4—131no—
220C2×C22⋊C4—131no—
221C2×C22⋊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⋊C4—131no—
229C2×C22⋊C4—131no—
230C2×C42—131no—
231C2×C42—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
244C2×C13#53611yescharacteristic
245C2×C13#54211yescharacteristic
246C2×C13#54211yescharacteristic
247C2×C13#54611yescharacteristic
248C2×C13#54611yescharacteristic
249C2×C13#55611yescharacteristic
250C2×C13#56211yescharacteristic
251D13—42no—
252D13—81no—
253C24—131no—
254C24—131no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—262no—
259C2×Q8—262no—
260C2×Q8—521no—
261C2×D4—262no—
262C2×D4—262no—
263C2×D4—521no—
264C22×C4—131no—
265C22×C4—131no—
266C22×C4—131no—
267C22×C4—131no—
268C22×C4—131no—
269C22×C4—131no—
270C22×C4—262no—
271C22×C4—262no—
272C22×C4—521no—
273C22×C4C22×D1311yescharacteristic
274C22×C4C22×D1311yescharacteristic
275C22×C4C22×D1311yescharacteristic
276C22×C4C22×D1311yescharacteristic
277C22×C4C22×D1311yescharacteristic
278C22×C4C22×D1311yescharacteristic
279C22×C4C22×D1311yescharacteristic
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—41no—
283C4⋊C4—262no—
284C4⋊C4—262no—
285C4⋊C4—521no—
286C22⋊C4—134no—
287C22⋊C4—262no—
288C22⋊C4—262no—
289C22⋊C4—262no—
290C22⋊C4—262no—
291C22⋊C4—262no—
292C22⋊C4—521no—
293C22⋊C4—521no—
294C22⋊C4—521no—
295C22⋊C4—521no—
296C22⋊C4—521no—
297C42—262no—
298C42—521no—
299C13#270011yescharacteristic, complemented, Sylow
300C23—132no—
301C23—261no—
302C23—261no—
303C23—261no—
304C23—261no—
305C23—261no—
306C23—261no—
307C23—261no—
308C23—261no—
309C23—261no—
310C23—261no—
311C23—261no—
312C23—261no—
313C23—261no—
314C23C23×D1311yescharacteristic
315Q8—44no—
316Q8—524no—
317D4—524no—
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—21no—
336C2×C4—132no—
337C2×C4—261no—
338C2×C4—261no—
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—524no—
355C2×C4D4×D1312yes—
356C2×C4#612912yes—
357C2×C4#613112yes—
358C22—262no—
359C22—262no—
360C22—262no—
361C22—521no—
362C22—521no—
363C22—521no—
364C22—521no—
365C22—521no—
366C22—521no—
367C22—521no—
368C22—521no—
369C22—521no—
370C22—521no—
371C22—521no—
372C22C2×D4×D1311yescharacteristic
373C22C2×#612911yescharacteristic
374C22#8825111yescharacteristic
375C22C2×#613111yescharacteristic
376C22#8825411yescharacteristic
377C22#8825411yescharacteristic
378C22#8825611yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—262no—
387C4—521no—
388C4—521no—
389C4—521no—
390C4—521no—
391C4—521no—
392C2—522no—
393C2—1041no—
394C2#1171929611yescharacteristic
395C2#1171933511yescharacteristic
396C2#1171934811yescharacteristic
397C2#1171935211yescharacteristic
398C2#1171942911yescharacteristic
399C2#1171943711yescharacteristic
400C2#1171943811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#395C2
prime radicalC22×D52C22
hypercenter#395D13
hypocenterC13#2700
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#88151C2semidirect
3. C13×#395C2semidirect
4. #11718891C2semidirect
5. #11718655C2semidirect
6. #11718654C2semidirect
8. #11718651C2semidirect
9. #11718642C2semidirect
10. #11718335C2semidirect
11. #11718334C2semidirect
12. #11718330C2semidirect
14. #11718325C2semidirect
15. #11718307C2semidirect
163. #395D13semidirect
299. C13#2700semidirect

Maximal subgroups

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

subgroupindex
C2×#882012
C2×#881512
C13×#3952
#117188912
#117186552
#117186542
#117186522
#117186512
#117186422
#117183352
#117183342
#117183302
#117183262
#117183252
#117183072
#270013