Finite Group Databasebeta

← #13121856

Subgroups — #13121856

Order 1664.

528 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88178C211yesmaximal, characteristic, complemented
2C2×#88132C211yesmaximal, characteristic, complemented
3C13×#386C211yesmaximal, characteristic
4#11718882C211yesmaximal, characteristic
5#11718881C211yesmaximal, characteristic
6#11718871C211yesmaximal, characteristic, complemented
7#11718610C211yesmaximal, characteristic, complemented
8#11718610C211yesmaximal, characteristic, complemented
9#11718609C211yesmaximal, characteristic, complemented
10#11718609C211yesmaximal, characteristic, complemented
11#11718607C211yesmaximal, characteristic, complemented
12#11718322C211yesmaximal, characteristic, complemented
13#11718318C211yesmaximal, characteristic, complemented
14#11718308C211yesmaximal, characteristic, complemented
15#11718307C211yesmaximal, characteristic, complemented
16C22×D4×C13C2211yescharacteristic
17C23×C4×C13C2211yescharacteristic
18C23×Dic13C2211yescharacteristic
19C22×Dic26C2211yescharacteristic
20C2×C13×C22⋊C4C2211yescharacteristic
21C2×C13×C22⋊C4C2211yescharacteristic
22C2×C13×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic13C2211yescharacteristic
24C2×C22⋊Dic13C2211yescharacteristic
25C2×C22⋊Dic13C2211yescharacteristic
26C2×C22⋊Dic13C2211yescharacteristic
27C2×C22⋊Dic13C2211yescharacteristic
28C2×C22⋊Dic13C2211yescharacteristic
29C2×C22⋊Dic13C2211yescharacteristic
30C2×C4⋊Dic13C2211yescharacteristic
31C2×C4⋊Dic13C2211yescharacteristic
32C2×C4⋊Dic13C2211yescharacteristic
33C2×C4⋊Dic13C2211yescharacteristic
34#88178—24no—
35C2×#6101C2211yescharacteristic
36C2×#6101C2211yescharacteristic
37C2×#6101C2211yescharacteristic
38C2×#6101C2211yescharacteristic
39C2×#6101C2211yescharacteristic
40C2×#6101C2211yescharacteristic
41C2×#6101C2211yescharacteristic
42C2×#6101C2211yescharacteristic
43C2×C4×Dic13C2211yescharacteristic
44#88132—24no—
45C13×#95C2211yescharacteristic
46C13×#95C2211yescharacteristic
47#88071C2211yescharacteristic
48#88071C2211yescharacteristic
49#88071C2211yescharacteristic
50#88071C2211yescharacteristic
51#88071C2211yescharacteristic
52#88071C2211yescharacteristic
53C24×C13C2311yescharacteristic
54C24×C13C2311yescharacteristic
55C2×D4×C13—41no—
56C2×D4×C13—41no—
57C2×D4×C13Q814yescomplemented
58C22×C4×C13—22no—
59C22×C4×C13—41no—
60C22×C4×C13—41no—
61C22×C4×C13C2311yescharacteristic
62C22×C4×C13C2311yescharacteristic
63C22×C4×C13C2311yescharacteristic
64C22×C4×C13C2311yescharacteristic
65C22×C4×C13C2311yescharacteristic
66C22×Dic13—22no—
67C22×Dic13—41no—
68C22×Dic13—41no—
69C22×Dic13C2311yescharacteristic
70C22×Dic13C2311yescharacteristic
71C22×Dic13C2311yescharacteristic
72C22×Dic13C2311yescharacteristic
73C22×Dic13C2311yescharacteristic
74C22×Dic13C2311yescharacteristic
75C22×Dic13C2311yescharacteristic
76C22×Dic13C2311yescharacteristic
77C2×Dic26—41no—
78C2×Dic26—41no—
79C2×Dic26D414yescomplemented
80C13×C22⋊C4—22no—
81C13×C22⋊C4—41no—
82C13×C22⋊C4—41no—
83C22⋊Dic13—22no—
84C22⋊Dic13—22no—
85C22⋊Dic13—22no—
86C22⋊Dic13—22no—
87C22⋊Dic13—22no—
88C22⋊Dic13—41no—
89C22⋊Dic13—41no—
90C4⋊Dic13—22no—
91C4⋊Dic13—22no—
92C4⋊Dic13—41no—
93C4⋊Dic13—41no—
94#6101—22no—
95#6101—22no—
96#6101—22no—
97#6101—22no—
98#6101—22no—
99#6101—41no—
100#6101—41no—
101#6101—41no—
102C4×Dic13—41no—
103#2732—131nomaximal, Sylow
104C23×C13—21no—
105C23×C13—21no—
106C23×C13—21no—
107C23×C13—21no—
108C23×C13—21no—
109C23×C13—21no—
110C23×C13—21no—
111C23×C13—21no—
112C23×C13—21no—
113C23×C13—21no—
114C23×C13—21no—
115C23×C13—21no—
116C23×C13C2×Q812yes—
117C23×C13C2×Q812yes—
118C23×C13C2411yescharacteristic
119D4×C13—44no—
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—
134C2×C4×C13C2×D412yes—
135C2×C4×C13C2×Q812yes—
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—21no—
154C2×Dic13—21no—
155C2×Dic13—21no—
156C2×Dic13—21no—
157C2×Dic13—44no—
158C2×Dic13C2×D412yes—
159C2×Dic13C2×D412yes—
160C2×Dic13#4112yes—
161Dic26—44no—
162C2×C22⋊Q8—131no—
163C2×C22⋊Q8—131no—
164#399—131no—
165#399—131no—
166#398—131no—
167#398—131no—
168#396—131no—
169#396—131no—
170#391—131no—
171#390—131no—
172#389—131no—
173#386—131no—
174#386D1311yescharacteristic
175#385—131no—
176#385—131no—
177C22×C13—22no—
178C22×C13—22no—
179C22×C13—22no—
180C22×C13—22no—
181C22×C13—22no—
182C22×C13—22no—
183C22×C13—41no—
184C22×C13—41no—
185C22×C13—41no—
186C22×C13—41no—
187C22×C13—41no—
188C22×C13—41no—
189C22×C13—41no—
190C22×C13—41no—
191C22×C13C22×D411yescharacteristic
192C22×C13C22×Q811yescharacteristic
193C22×C13C2×#4111yescharacteristic
194C22×C13H(5,𝔽2)11yescharacteristic
195C22×C13H(5,𝔽2)11yescharacteristic
196C22×C13H(5,𝔽2)11yescharacteristic
197C22×C132−1+411yescharacteristic
198C4×C13—22no—
199C4×C13—22no—
200C4×C13—41no—
201C4×C13—41no—
202C4×C13—41no—
203Dic13—22no—
204Dic13—22no—
205Dic13—22no—
206Dic13—41no—
207Dic13—41no—
208Dic13—41no—
209Dic13—41no—
210Dic13—41no—
211C22×Q8—131no—
212C22×D4C2×D1311yescharacteristic
213C23×C4—131no—
214C23×C4C2×D1311yescharacteristic
215C22⋊Q8—264no—
216C22⋊Q8—264no—
217C2×C4⋊C4—131no—
218C2×C4⋊C4—131no—
219C2×C4⋊C4—131no—
220C2×C4⋊C4—131no—
221C2×C4⋊C4—131no—
222C2×C4⋊C4—131no—
223C2×C4⋊C4—131no—
224C2×C4⋊C4—131no—
225C2×C4⋊C4—131no—
226C2×C4⋊C4—131no—
227C2×C4⋊C4—131no—
228C2×C4⋊C4—131no—
229C2×C22⋊C4—131no—
230C2×C22⋊C4—131no—
231C2×C22⋊C4—131no—
232C2×C22⋊C4—131no—
233C2×C22⋊C4—131no—
234C2×C22⋊C4—131no—
235C2×C22⋊C4—131no—
236C2×C22⋊C4C2×D1311yescharacteristic
237C2×C22⋊C4C2×D1311yescharacteristic
238C2×C22⋊C4C2×D1311yescharacteristic
239C2×C42—131no—
240#95—131no—
241#95—131no—
242#95—131no—
243#95—131no—
244#95—131no—
245#95—131no—
246#95C2×D1311yescharacteristic
247#95C2×D1311yescharacteristic
248C2×C13—42no—
249C2×C13—42no—
250C2×C13#53411yescharacteristic
251C2×C13#53911yescharacteristic
252C2×C13#54311yescharacteristic
253C2×C13D4×Q811yescharacteristic
254C2×C13#55011yescharacteristic
255C2×C13#55411yescharacteristic
256C2×C13#56211yescharacteristic
257C24C22×D1311yescharacteristic
258C24C22×D1311yescharacteristic
259C2×Q8—134no—
260C2×Q8—521no—
261C2×Q8—521no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4Dic2614yescomplemented
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—131no—
269C22×C4—131no—
270C22×C4—131no—
271C22×C4—131no—
272C22×C4—131no—
273C22×C4—131no—
274C22×C4—131no—
275C22×C4—131no—
276C22×C4—262no—
277C22×C4—521no—
278C22×C4—521no—
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—262no—
290C4⋊C4—262no—
291C4⋊C4—521no—
292C4⋊C4—521no—
293C4⋊C4—521no—
294C4⋊C4—521no—
295C4⋊C4—521no—
296C22⋊C4—22no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—262no—
300C22⋊C4—262no—
301C22⋊C4—262no—
302C22⋊C4—262no—
303C22⋊C4—262no—
304C22⋊C4—521no—
305C22⋊C4—521no—
306C42—521no—
307C13#273211yescharacteristic, complemented, Sylow
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C2×Dic2612yes—
321C23C2×Dic2612yes—
322C23C23×D1311yescharacteristic
323Q8—524no—
324D4—44no—
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—21no—
337C2×C4—21no—
338C2×C4—44no—
339C2×C4—132no—
340C2×C4—132no—
341C2×C4—132no—
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—261no—
358C2×C4—261no—
359C2×C4—261no—
360C2×C4—261no—
361C2×C4—261no—
362C2×C4—261no—
363C2×C4—524no—
364C2×C4C2×Dic2612yes—
365C2×C4D4×D1312yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22C22×Dic2611yescharacteristic
381C22C2×D4×D1311yescharacteristic
382C22#8825111yescharacteristic
383C22#8825111yescharacteristic
384C22#8825111yescharacteristic
385C22#41×D1311yescharacteristic
386C22#8825711yescharacteristic
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—262no—
393C4—262no—
394C4—262no—
395C4—521no—
396C4—521no—
397C4—521no—
398C4—521no—
399C4—521no—
400C2—42no—
401C2—42no—
402C2#1171918811yescharacteristic
403C2D4×Dic2611yescharacteristic
404C2#1171924411yescharacteristic
405C2#1171929411yescharacteristic
406C2#1171930411yescharacteristic
407C2#1171932611yescharacteristic
408C2#1171936511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#386C2
prime radicalC22×D4×C13C22
hypercenter#386D13
hypocenterC13#2732
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#88178C2semidirect
2. C2×#88132C2semidirect
6. #11718871C2semidirect
7. #11718610C2semidirect
8. #11718610C2semidirect
9. #11718609C2semidirect
10. #11718609C2semidirect
11. #11718607C2semidirect
12. #11718322C2semidirect
13. #11718318C2semidirect
14. #11718308C2semidirect
15. #11718307C2semidirect
57. C2×D4×C13Q8semidirect
79. C2×Dic26D4semidirect
264. C2×D4Dic26semidirect
307. C13#2732semidirect

Maximal subgroups

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

subgroupindex
C2×#881782
C2×#881322
C13×#3862
#117188822
#117188812
#117188712
#117186102
#117186092
#117186072
#117183222
#117183182
#117183082
#117183072
#273213