Finite Group Databasebeta

← #16822306

Subgroups — #16822306

Order 3712.

540 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745247C211yesmaximal, characteristic
2#14271068C211yesmaximal, characteristic, complemented
3#14271067C211yesmaximal, characteristic, complemented
4#14271056C211yesmaximal, characteristic, complemented
5#14271053C211yesmaximal, characteristic, complemented
6#14271053C211yesmaximal, characteristic, complemented
7#14271051C211yesmaximal, characteristic, complemented
8#14271042C211yesmaximal, characteristic
9#14271039C211yesmaximal, characteristic
10#14271037C211yesmaximal, characteristic, complemented
11#14271030C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic29C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic29C211yesmaximal, characteristic, complemented
14#14271010C211yesmaximal, characteristic, complemented
15C29×#385C211yesmaximal, characteristic, complemented
16C23×C4×C29C2211yescharacteristic
17C22×C22⋊D29C2211yescharacteristic
18C22×C4×D29C2211yescharacteristic
19C22×Dic58C2211yescharacteristic
20C2×C4⋊C4×C29C2211yescharacteristic
21C2×C4⋊C4×C29C2211yescharacteristic
22C2×C22⋊C4×C29C2211yescharacteristic
23C2×C22⋊C4×C29C2211yescharacteristic
24C2×C22⋊Dic29C2211yescharacteristic
25C2×C22⋊Dic29C2211yescharacteristic
26C2×C22⋊Dic29C2211yescharacteristic
27C2×#90781C2211yescharacteristic
28C2×#90781C2211yescharacteristic
29C2×#90781C2211yescharacteristic
30C2×#90781C2211yescharacteristic
31C2×#90781C2211yescharacteristic
32C2×C4⋊Dic29C2211yescharacteristic
33C2×#90779C2211yescharacteristic
34C2×C4×Dic29C2211yescharacteristic
35C2×C4×Dic29C2211yescharacteristic
36C2×C4×Dic29C2211yescharacteristic
37C2×C4×Dic29C2211yescharacteristic
38C2×C4×Dic29C2211yescharacteristic
39#11745247C418yescomplemented
40C29×#95C2211yescharacteristic
41C29×#95C2211yescharacteristic
42#11745179C2211yescharacteristic
43#11745179C2211yescharacteristic
44#11745179C2211yescharacteristic
45#11745179C2211yescharacteristic
46#11745179C2211yescharacteristic
47#11745179C2211yescharacteristic
48#11745179C2211yescharacteristic
49#11745179C2211yescharacteristic
50#11745179C2211yescharacteristic
51#11745179C2211yescharacteristic
52C24×C29C2311yescharacteristic
53C23×D29C2311yescharacteristic
54C22×C4×C29—22no—
55C22×C4×C29—41no—
56C22×C4×C29—41no—
57C22×C4×C29C2311yescharacteristic
58C22×C4×C29C2311yescharacteristic
59C22×C4×C29C2311yescharacteristic
60C22×C4×C29C2311yescharacteristic
61C22×C4×C29C2311yescharacteristic
62C22×C4×C29C2311yescharacteristic
63C2×C22⋊D29—41no—
64C2×C22⋊D29—41no—
65C2×C22⋊D29C2×C414yes—
66C22×Dic29C2311yescharacteristic
67C22×Dic29C2311yescharacteristic
68C22×Dic29C2311yescharacteristic
69C22×Dic29C2311yescharacteristic
70C22×Dic29C2311yescharacteristic
71C22×Dic29C2311yescharacteristic
72C22×Dic29C2311yescharacteristic
73C2×C4×D29—22no—
74C2×C4×D29—41no—
75C2×C4×D29—41no—
76C2×Dic58—41no—
77C2×Dic58—41no—
78C2×Dic58C2×C414yescomplemented
79C4⋊C4×C29—22no—
80C4⋊C4×C29—41no—
81C22⋊C4×C29—41no—
82C22⋊C4×C29C2×C414yescomplemented
83C22⋊Dic29—22no—
84C22⋊Dic29—41no—
85C22⋊Dic29C2×C414yescomplemented
86#90781—22no—
87#90781—41no—
88#90781—41no—
89#90781C2×C414yescomplemented
90#90781C2×C414yescomplemented
91C4⋊Dic29—22no—
92#90779—41no—
93C4×Dic29—41no—
94C4×Dic29—41no—
95C4×Dic29—41no—
96C4×Dic29—41no—
97C4×Dic29C2×C414yescomplemented
98C23×C29—21no—
99C23×C29—21no—
100C23×C29—21no—
101C23×C29—21no—
102C23×C29—21no—
103C23×C29—21no—
104C23×C29C22×C412yes—
105C23×C29C2411yescharacteristic
106C22×D29—21no—
107C22×D29—21no—
108C22×D29—21no—
109C22×D29—21no—
110C22×D29—21no—
111C22×D29—21no—
112C22×D29C22×C412yes—
113C2×C4×C29—21no—
114C2×C4×C29—21no—
115C2×C4×C29—21no—
116C2×C4×C29—21no—
117C2×C4×C29—21no—
118C2×C4×C29—21no—
119C2×C4×C29—21no—
120C2×C4×C29—21no—
121C2×C4×C29—21no—
122C2×C4×C29—21no—
123C2×C4×C29—21no—
124C2×C4×C29—21no—
125C2×C4×C29—21no—
126C2×C4×C29—21no—
127C2×C4×C29—21no—
128C2×C4×C29—44no—
129C2×C4×C29C22×C412yes—
130C2×C4×C29C22×C412yes—
131C2×C4×C29#4112yes—
132C22⋊D29—44no—
133C2×Dic29—21no—
134C2×Dic29—21no—
135C2×Dic29—21no—
136C2×Dic29—21no—
137C2×Dic29—21no—
138C2×Dic29—21no—
139C2×Dic29—21no—
140C2×Dic29—21no—
141C2×Dic29—21no—
142C2×Dic29—21no—
143C2×Dic29—21no—
144C2×Dic29—21no—
145C2×Dic29—21no—
146C2×Dic29—21no—
147C2×Dic29—21no—
148C2×Dic29—21no—
149C2×Dic29—21no—
150C2×Dic29C22×C412yes—
151C2×Dic29C22×C412yes—
152C2×Dic29C22×C412yes—
153C2×Dic29#4112yes—
154C4×D29—44no—
155Dic58—44no—
156#2337—291nomaximal, Sylow
157C22×C29—22no—
158C22×C29—22no—
159C22×C29—22no—
160C22×C29—41no—
161C22×C29—41no—
162C22×C29—41no—
163C22×C29—41no—
164C22×C29C23×C411yescharacteristic
165C22×C29C2×#4111yescharacteristic
166C22×C29C2×#4111yescharacteristic
167C22×C29H(5,𝔽2)11yescharacteristic
168C22×C29H(5,𝔽2)11yescharacteristic
169C22×C29H(5,𝔽2)11yescharacteristic
170C22×C292−1+411yescharacteristic
171C2×D29—22no—
172C2×D29—22no—
173C2×D29—22no—
174C2×D29—41no—
175C2×D29—41no—
176C2×D29—41no—
177C2×D29—41no—
178C4×C29—22no—
179C4×C29—22no—
180C4×C29—22no—
181C4×C29—41no—
182C4×C29—41no—
183C4×C29—41no—
184Dic29—22no—
185Dic29—22no—
186Dic29—22no—
187Dic29—22no—
188Dic29—41no—
189Dic29—41no—
190Dic29—41no—
191C2×#124—291no—
192#391—291no—
193#391—291no—
194#388—291no—
195#388—291no—
196#388—291no—
197#388—291no—
198#386—291no—
199#386—291no—
200#385—291no—
201#385D2911yescharacteristic, complemented
202#383—291no—
203C4×C4⋊C4—291no—
204C4×C22⋊C4—291no—
205C4×C22⋊C4—291no—
206C2×C29—42no—
207C2×C29C4×#4111yescharacteristic
208C2×C29#51811yescharacteristic
209C2×C29#52011yescharacteristic
210C2×C29#53811yescharacteristic
211C2×C29#53811yescharacteristic
212C2×C29#54211yescharacteristic
213C2×C29#54211yescharacteristic
214D29—42no—
215C22×Q8—291no—
216C22×D4—291no—
217C23×C4—291no—
218C23×C4C2×D2911yescharacteristic
219#124—298no—
220C2×C4⋊C4—291no—
221C2×C4⋊C4—291no—
222C2×C4⋊C4C2×D2911yescharacteristic
223C2×C4⋊C4C2×D2911yescharacteristic
224C2×C22⋊C4—291no—
225C2×C22⋊C4—291no—
226C2×C22⋊C4—291no—
227C2×C22⋊C4—291no—
228C2×C22⋊C4—291no—
229C2×C22⋊C4—291no—
230C2×C22⋊C4—291no—
231C2×C22⋊C4—291no—
232C2×C22⋊C4C2×D2911yescharacteristic
233C2×C22⋊C4C2×D2911yescharacteristic
234C2×C42—291no—
235C2×C42—291no—
236C2×C42—291no—
237C2×C42—291no—
238C2×C42—291no—
239#95—291no—
240#95—291no—
241#95—291no—
242#95—291no—
243#95—291no—
244#95—291no—
245#95—291no—
246#95—291no—
247#95—291no—
248#95—291no—
249#95C2×D2911yescharacteristic
250#95C2×D2911yescharacteristic
251C29#233711yescharacteristic, complemented, Sylow
252C24—291no—
253C24C22×D2911yescharacteristic
254C2×Q8—294no—
255C2×Q8—1161no—
256C2×Q8—1161no—
257C2×D4—294no—
258C2×D4—1161no—
259C2×D4—1161no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—291no—
264C22×C4—291no—
265C22×C4—291no—
266C22×C4—291no—
267C22×C4—291no—
268C22×C4—291no—
269C22×C4—291no—
270C22×C4—582no—
271C22×C4—1161no—
272C22×C4—1161no—
273C22×C4C22×D2911yescharacteristic
274C22×C4C22×D2911yescharacteristic
275C22×C4C22×D2911yescharacteristic
276C22×C4C22×D2911yescharacteristic
277C22×C4C22×D2911yescharacteristic
278C22×C4C22×D2911yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—41no—
281C4⋊C4—582no—
282C4⋊C4—1161no—
283C22⋊C4—41no—
284C22⋊C4—294no—
285C22⋊C4—294no—
286C22⋊C4—294no—
287C22⋊C4—582no—
288C22⋊C4—582no—
289C22⋊C4—1161no—
290C22⋊C4—1161no—
291C22⋊C4—1161no—
292C22⋊C4C4×D2914yescomplemented
293C42—294no—
294C42—1161no—
295C42—1161no—
296C42—1161no—
297C42—1161no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—292no—
305C23—581no—
306C23—581no—
307C23—581no—
308C23—581no—
309C23—581no—
310C23—581no—
311C23C2×C4×D2912yes—
312C23C23×D2911yescharacteristic
313Q8—1164no—
314D4—1164no—
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—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—292no—
332C2×C4—292no—
333C2×C4—292no—
334C2×C4—292no—
335C2×C4—581no—
336C2×C4—581no—
337C2×C4—581no—
338C2×C4—581no—
339C2×C4—581no—
340C2×C4—581no—
341C2×C4—581no—
342C2×C4—581no—
343C2×C4—581no—
344C2×C4—581no—
345C2×C4—581no—
346C2×C4—581no—
347C2×C4—581no—
348C2×C4—581no—
349C2×C4—581no—
350C2×C4—581no—
351C2×C4—581no—
352C2×C4—1164no—
353C2×C4C2×C4×D2912yes—
354C2×C4C2×C4×D2912yes—
355C2×C4#9080712yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—582no—
364C22—582no—
365C22—582no—
366C22—1161no—
367C22—1161no—
368C22—1161no—
369C22—1161no—
370C22C22×C4×D2911yescharacteristic
371C22C2×#9080711yescharacteristic
372C22#1174535911yescharacteristic
373C22#1174535911yescharacteristic
374C22#1174536211yescharacteristic
375C22#41×D2911yescharacteristic
376C22#1174536411yescharacteristic
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—582no—
384C4—582no—
385C4—582no—
386C4—582no—
387C4—1161no—
388C4—1161no—
389C4—1161no—
390C2—42no—
391C2—1162no—
392C2C4×#9080711yescharacteristic
393C2#1427163511yescharacteristic
394C2#1427164411yescharacteristic
395C2#1427174811yescharacteristic
396C2#1427174911yescharacteristic
397C2#1427178311yescharacteristic
398C2#1427178611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutatorC22×C29C23×C4
FittingC29×#385C2
socle, cosocleC23×C29C24
prime radicalC22×C22⋊D29C22
hypercenter#385D29
hypocenterC29#2337
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #14271068C2semidirect
3. #14271067C2semidirect
4. #14271056C2semidirect
5. #14271053C2semidirect
6. #14271053C2semidirect
7. #14271051C2semidirect
10. #14271037C2semidirect
11. #14271030C2semidirect
12. C4⋊C4×Dic29C2semidirect
13. C4×C22⋊Dic29C2semidirect
14. #14271010C2semidirect
15. C29×#385C2semidirect
39. #11745247C4semidirect
78. C2×Dic58C2×C4semidirect
82. C22⋊C4×C29C2×C4semidirect
85. C22⋊Dic29C2×C4semidirect
89. #90781C2×C4semidirect
90. #90781C2×C4semidirect
97. C4×Dic29C2×C4semidirect
201. #385D29semidirect
251. C29#2337semidirect
292. C22⋊C4C4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452472
#142710682
#142710672
#142710562
#142710532
#142710512
#142710422
#142710392
#142710372
#142710302
C4⋊C4×Dic292
C4×C22⋊Dic292
#142710102
C29×#3852
#233729