Finite Group Databasebeta

← #13121153

Subgroups — #13121153

Order 1664.

544 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88151C211yesmaximal, characteristic
2C2×#88150C211yesmaximal, characteristic
3C2×#88142C211yesmaximal, characteristic, complemented
4C2×#88142C211yesmaximal, characteristic, complemented
5C13×#398C211yesmaximal, characteristic, complemented
6#11718653C211yesmaximal, characteristic
7#11718650C211yesmaximal, characteristic
8#11718640C211yesmaximal, characteristic, complemented
9#11718639C211yesmaximal, characteristic, complemented
10#11718334C211yesmaximal, characteristic
11#11718331C211yesmaximal, characteristic
12#11718330C211yesmaximal, characteristic
13#11718318C211yesmaximal, characteristic, complemented
14#11718317C211yesmaximal, characteristic, complemented
15#11718315C211yesmaximal, characteristic, complemented
16C22×C4×D13C2211yescharacteristic
17C22×Dic26C2211yescharacteristic
18C22×Dic26C2211yescharacteristic
19C22×Dic26C2211yescharacteristic
20C2×C13×C4⋊C4C2211yescharacteristic
21C2×C13×C4⋊C4C2211yescharacteristic
22C2×C13×C4⋊C4C2211yescharacteristic
23C2×C13×C4⋊C4C2211yescharacteristic
24C2×#6103C2211yescharacteristic
25C2×#6103C2211yescharacteristic
26C2×#6103C2211yescharacteristic
27C2×#6103C2211yescharacteristic
28C2×#6103C2211yescharacteristic
29C2×#6103C2211yescharacteristic
30C2×C4⋊Dic13C2211yescharacteristic
31C2×C4⋊Dic13C2211yescharacteristic
32C2×C4⋊Dic13C2211yescharacteristic
33C2×#6101C2211yescharacteristic
34C2×#6101C2211yescharacteristic
35C2×#6101C2211yescharacteristic
36C2×#6101C2211yescharacteristic
37C2×#6101C2211yescharacteristic
38C2×#6101C2211yescharacteristic
39C2×C4×Dic13C2211yescharacteristic
40C2×C4×Dic13C2211yescharacteristic
41#88151—24no—
42#88150—24no—
43#88142—24no—
44#88142—24no—
45C13×#95C2211yescharacteristic
46C13×#95C2211yescharacteristic
47C13×#95C2211yescharacteristic
48#88071C2211yescharacteristic
49#88071C2211yescharacteristic
50#88071C2211yescharacteristic
51#88071C2211yescharacteristic
52#88071C2211yescharacteristic
53#88071C2211yescharacteristic
54#88071C2211yescharacteristic
55C23×D13C2311yescharacteristic
56C22×C4×C13C2311yescharacteristic
57C22×C4×C13C2311yescharacteristic
58C22×C4×C13C2311yescharacteristic
59C22×C4×C13C2311yescharacteristic
60C22×C4×C13C2311yescharacteristic
61C22×C4×C13C2311yescharacteristic
62C22×C4×C13C2311yescharacteristic
63C22×Dic13C2311yescharacteristic
64C22×Dic13C2311yescharacteristic
65C22×Dic13C2311yescharacteristic
66C22×Dic13C2311yescharacteristic
67C22×Dic13C2311yescharacteristic
68C22×Dic13C2311yescharacteristic
69C22×Dic13C2311yescharacteristic
70C2×C4×D13—22no—
71C2×C4×D13—22no—
72C2×C4×D13—41no—
73C2×Dic26—22no—
74C2×Dic26—22no—
75C2×Dic26—22no—
76C2×Dic26—22no—
77C2×Dic26—22no—
78C2×Dic26—22no—
79C2×Dic26—41no—
80C2×Dic26—41no—
81C2×Dic26—41no—
82C13×C4⋊C4—22no—
83C13×C4⋊C4—22no—
84C13×C4⋊C4—22no—
85C13×C4⋊C4—22no—
86#6103—22no—
87#6103—22no—
88#6103—22no—
89#6103—22no—
90#6103—41no—
91#6103Q814yescomplemented
92C4⋊Dic13—22no—
93C4⋊Dic13—22no—
94C4⋊Dic13D414yes—
95#6101—22no—
96#6101—22no—
97#6101—22no—
98#6101—22no—
99#6101—41no—
100#6101D414yes—
101C4×Dic13—22no—
102C4×Dic13—22no—
103#2677—131nomaximal, Sylow
104C23×C13C2411yescharacteristic
105C22×D13—21no—
106C22×D13—21no—
107C22×D13—21no—
108C22×D13—21no—
109C22×D13—21no—
110C22×D13—21no—
111C22×D13C2×Q812yes—
112C2×C4×C13—21no—
113C2×C4×C13—21no—
114C2×C4×C13—21no—
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×C13C2×D412yes—
131C2×C4×C13C2×D412yes—
132C2×C4×C13C2×Q812yes—
133C2×Dic13—21no—
134C2×Dic13—21no—
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×Dic13C2×D412yes—
150C2×Dic13C2×D412yes—
151C2×Dic13C2×D412yes—
152C2×Dic13C2×D412yes—
153C2×Dic13C2×Q812yes—
154C4×D13—44no—
155Dic26—44no—
156Dic26—44no—
157Dic26—44no—
158C2×C4⋊Q8—131no—
159C2×C4⋊Q8—131no—
160C2×C22⋊Q8—131no—
161C2×C22⋊Q8—131no—
162#399—131no—
163#398—131no—
164#398D1311yescharacteristic, complemented
165#395—131no—
166#395—131no—
167#393—131no—
168#388—131no—
169#388—131no—
170#387—131no—
171#387—131no—
172#385—131no—
173C22×C13C22×D411yescharacteristic
174C22×C13C22×D411yescharacteristic
175C22×C13C22×Q811yescharacteristic
176C22×C13H(5,𝔽2)11yescharacteristic
177C22×C132−1+411yescharacteristic
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—
187C4×C13—22no—
188C4×C13—22no—
189C4×C13—22no—
190C4×C13—41no—
191C4×C13—41no—
192C4×C13—41no—
193C4×C13—41no—
194Dic13—22no—
195Dic13—22no—
196Dic13—22no—
197Dic13—22no—
198Dic13—22no—
199Dic13—41no—
200Dic13—41no—
201C22×Q8—131no—
202C22×Q8—131no—
203C22×Q8—131no—
204C23×C4—131no—
205C4⋊Q8—264no—
206C4⋊Q8—264no—
207C22⋊Q8—264no—
208C22⋊Q8—264no—
209C2×C4⋊C4—131no—
210C2×C4⋊C4—131no—
211C2×C4⋊C4—131no—
212C2×C4⋊C4—131no—
213C2×C4⋊C4—131no—
214C2×C4⋊C4—131no—
215C2×C4⋊C4—131no—
216C2×C4⋊C4—131no—
217C2×C4⋊C4—131no—
218C2×C4⋊C4C2×D1311yescharacteristic
219C2×C4⋊C4C2×D1311yescharacteristic
220C2×C4⋊C4C2×D1311yescharacteristic
221C2×C4⋊C4C2×D1311yescharacteristic
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×C42—131no—
229C2×C42—131no—
230#95—131no—
231#95—131no—
232#95—131no—
233#95—131no—
234#95—131no—
235#95—131no—
236#95—131no—
237#95C2×D1311yescharacteristic
238#95C2×D1311yescharacteristic
239#95C2×D1311yescharacteristic
240C2×C13#53611yescharacteristic
241C2×C13#53611yescharacteristic
242C2×C13#54411yescharacteristic
243C2×C13D4211yescharacteristic
244C2×C13D4×Q811yescharacteristic
245C2×C13D4×Q811yescharacteristic
246C2×C13#56311yescharacteristic
247D13—42no—
248C24—131no—
249C2×Q8—262no—
250C2×Q8—262no—
251C2×Q8—262no—
252C2×Q8—262no—
253C2×Q8—262no—
254C2×Q8—262no—
255C2×Q8—521no—
256C2×Q8—521no—
257C2×Q8—521no—
258C22×C4—131no—
259C22×C4—131no—
260C22×C4—131no—
261C22×C4—131no—
262C22×C4—131no—
263C22×C4—131no—
264C22×C4—131no—
265C22×C4—262no—
266C22×C4—262no—
267C22×C4—521no—
268C22×C4C22×D1311yescharacteristic
269C22×C4C22×D1311yescharacteristic
270C22×C4C22×D1311yescharacteristic
271C22×C4C22×D1311yescharacteristic
272C22×C4C22×D1311yescharacteristic
273C22×C4C22×D1311yescharacteristic
274C22×C4C22×D1311yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—134no—
280C4⋊C4—134no—
281C4⋊C4—262no—
282C4⋊C4—262no—
283C4⋊C4—262no—
284C4⋊C4—262no—
285C4⋊C4—262no—
286C4⋊C4—262no—
287C4⋊C4—521no—
288C22⋊C4—134no—
289C22⋊C4—262no—
290C22⋊C4—262no—
291C22⋊C4—262no—
292C22⋊C4—262no—
293C22⋊C4—521no—
294C42—262no—
295C42—262no—
296C13#267711yescharacteristic, complemented, Sylow
297C23—132no—
298C23—261no—
299C23—261no—
300C23—261no—
301C23—261no—
302C23—261no—
303C23—261no—
304C23C23×D1311yescharacteristic
305Q8—524no—
306Q8—524no—
307Q8—524no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
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—132no—
327C2×C4—132no—
328C2×C4—132no—
329C2×C4—132no—
330C2×C4—132no—
331C2×C4—261no—
332C2×C4—261no—
333C2×C4—261no—
334C2×C4—261no—
335C2×C4—261no—
336C2×C4—261no—
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—524no—
348C2×C4D4×D1312yes—
349C2×C4D4×D1312yes—
350C2×C4Q8×D1312yes—
351C22—262no—
352C22—262no—
353C22—262no—
354C22—521no—
355C22—521no—
356C22—521no—
357C22—521no—
358C22C2×D4×D1311yescharacteristic
359C22C2×D4×D1311yescharacteristic
360C22C2×Q8×D1311yescharacteristic
361C22#8825411yescharacteristic
362C22#8825611yescharacteristic
363C22#8825711yescharacteristic
364C22#8825711yescharacteristic
365C4—22no—
366C4—22no—
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—262no—
373C4—262no—
374C4—262no—
375C4—262no—
376C4—262no—
377C4—521no—
378C4—521no—
379C2—522no—
380C2#1171931411yescharacteristic
381C2#1171933311yescharacteristic
382C2#1171933711yescharacteristic
383C2#1171935311yescharacteristic
384C2#1171939411yescharacteristic
385C2#1171940411yescharacteristic
386C2#1171943211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#398C2
prime radicalC23×D13C23
hypercenter#398D13
hypocenterC13#2677
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
3. C2×#88142C2semidirect
4. C2×#88142C2semidirect
5. C13×#398C2semidirect
8. #11718640C2semidirect
9. #11718639C2semidirect
13. #11718318C2semidirect
14. #11718317C2semidirect
15. #11718315C2semidirect
91. #6103Q8semidirect
164. #398D13semidirect
296. C13#2677semidirect

Maximal subgroups

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

subgroupindex
C2×#881512
C2×#881502
C2×#881422
C13×#3982
#117186532
#117186502
#117186402
#117186392
#117183342
#117183312
#117183302
#117183182
#117183172
#117183152
#267713