Finite Group Databasebeta

← #16822639

Subgroups — #16822639

Order 3712.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11745256C211yesmaximal, characteristic
2#14271065C211yesmaximal, characteristic
3#14271062C211yesmaximal, characteristic, complemented
4#14271060C211yesmaximal, characteristic, complemented
5#14271055C211yesmaximal, characteristic, complemented
6#14271053C211yesmaximal, characteristic, complemented
7#14271052C211yesmaximal, characteristic, complemented
8#14271052C211yesmaximal, characteristic, complemented
9#14271047C211yesmaximal, characteristic, complemented
10#14271041C211yesmaximal, characteristic
11#14271037C211yesmaximal, characteristic, complemented
12#14271037C211yesmaximal, characteristic, complemented
13#14271022C211yesmaximal, characteristic, complemented
14#14271020C211yesmaximal, characteristic, complemented
15C29×#387C211yesmaximal, characteristic, complemented
16C22×D116C2211yescharacteristic
17C22×C4×D29C2211yescharacteristic
18C22×C4×D29C2211yescharacteristic
19C2×C4⋊C4×C29C2211yescharacteristic
20C2×C4⋊C4×C29C2211yescharacteristic
21C2×C42×C29C2211yescharacteristic
22C2×#90781C2211yescharacteristic
23C2×#90781C2211yescharacteristic
24C2×#90781C2211yescharacteristic
25C2×#90781C2211yescharacteristic
26C2×#90781C2211yescharacteristic
27C2×#90781C2211yescharacteristic
28C2×#90781C2211yescharacteristic
29C2×#90781C2211yescharacteristic
30C2×#90781C2211yescharacteristic
31C2×#90781C2211yescharacteristic
32C2×C4⋊Dic29C2211yescharacteristic
33C2×C4⋊Dic29C2211yescharacteristic
34C2×C4⋊Dic29C2211yescharacteristic
35C2×#90779C2211yescharacteristic
36C2×#90779C2211yescharacteristic
37C2×#90779C2211yescharacteristic
38C2×#90779C2211yescharacteristic
39C2×C4×Dic29C2211yescharacteristic
40C2×C4×Dic29C2211yescharacteristic
41#11745256C418yescomplemented
42C29×#95C2211yescharacteristic
43C29×#95C2211yescharacteristic
44C29×#95C2211yescharacteristic
45C29×#95C2211yescharacteristic
46#11745179C2211yescharacteristic
47#11745179C2211yescharacteristic
48#11745179C2211yescharacteristic
49#11745179C2211yescharacteristic
50#11745179C2211yescharacteristic
51#11745179C2211yescharacteristic
52C23×D29C2311yescharacteristic
53C23×D29C2311yescharacteristic
54C22×C4×C29C2311yescharacteristic
55C22×C4×C29C2311yescharacteristic
56C22×C4×C29C2311yescharacteristic
57C22×C4×C29C2311yescharacteristic
58C22×C4×C29C2311yescharacteristic
59C22×C4×C29C2311yescharacteristic
60C22×C4×C29C2311yescharacteristic
61C22×Dic29C2311yescharacteristic
62C22×Dic29C2311yescharacteristic
63C22×Dic29C2311yescharacteristic
64C22×Dic29C2311yescharacteristic
65C22×Dic29C2311yescharacteristic
66C22×Dic29C2311yescharacteristic
67C2×D116—41no—
68C2×D116—41no—
69C2×D116C2×C414yes—
70C2×C4×D29—41no—
71C2×C4×D29—41no—
72C2×C4×D29—41no—
73C2×C4×D29—41no—
74C2×C4×D29—41no—
75C2×C4×D29C2×C414yescomplemented
76C4⋊C4×C29—41no—
77C4⋊C4×C29C2×C414yescomplemented
78C42×C29—41no—
79#90781—41no—
80#90781—41no—
81#90781—41no—
82#90781—41no—
83#90781—41no—
84#90781—41no—
85#90781—41no—
86#90781C2×C414yes—
87#90781C2×C414yes—
88#90781C2×C414yescomplemented
89C4⋊Dic29—41no—
90C4⋊Dic29—41no—
91C4⋊Dic29—41no—
92#90779—41no—
93#90779—41no—
94#90779—41no—
95#90779C2×C414yescomplemented
96C4×Dic29—41no—
97C4×Dic29—41no—
98C23×C29C2411yescharacteristic
99C22×D29—21no—
100C22×D29—21no—
101C22×D29—21no—
102C22×D29—21no—
103C22×D29—21no—
104C22×D29—21no—
105C22×D29—21no—
106C22×D29—21no—
107C22×D29—21no—
108C22×D29—21no—
109C22×D29—21no—
110C22×D29—21no—
111C22×D29C22×C412yes—
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—21no—
129C2×C4×C29—21no—
130C2×C4×C29—21no—
131C2×C4×C29C22×C412yes—
132C2×C4×C29C22×C412yes—
133C2×C4×C29C22×C412yes—
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—
152D116—44no—
153C4×D29—44no—
154C4×D29—44no—
155#2376—291nomaximal, Sylow
156C22×C29C23×C411yescharacteristic
157C22×C29H(5,𝔽2)11yescharacteristic
158C22×C29H(5,𝔽2)11yescharacteristic
159C22×C29H(5,𝔽2)11yescharacteristic
160C22×C29H(5,𝔽2)11yescharacteristic
161C22×C292−1+411yescharacteristic
162C22×C292−1+411yescharacteristic
163C2×D29—22no—
164C2×D29—22no—
165C2×D29—22no—
166C2×D29—22no—
167C2×D29—22no—
168C2×D29—22no—
169C2×D29—41no—
170C2×D29—41no—
171C2×D29—41no—
172C2×D29—41no—
173C2×D29—41no—
174C2×D29—41no—
175C2×D29—41no—
176C2×D29—41no—
177C4×C29—22no—
178C4×C29—22no—
179C4×C29—22no—
180C4×C29—41no—
181C4×C29—41no—
182C4×C29—41no—
183C4×C29—41no—
184Dic29—22no—
185Dic29—22no—
186Dic29—41no—
187Dic29—41no—
188Dic29—41no—
189Dic29—41no—
190C2×#123—291no—
191#390—291no—
192#389—291no—
193#389—291no—
194#388—291no—
195#388—291no—
196#388—291no—
197#388—291no—
198#387—291no—
199#387D2911yescharacteristic, complemented
200#386—291no—
201#385—291no—
202#385—291no—
203#380—291no—
204#380—291no—
205C2×C29#51811yescharacteristic
206C2×C29#52011yescharacteristic
207C2×C29#52011yescharacteristic
208C2×C29#56011yescharacteristic
209C2×C29#56211yescharacteristic
210C2×C29#56211yescharacteristic
211C2×C29#56311yescharacteristic
212D29—42no—
213D29—42no—
214C22×D4—291no—
215C23×C4—291no—
216C23×C4—291no—
217#123—298no—
218C2×C4⋊C4—291no—
219C2×C4⋊C4—291no—
220C2×C4⋊C4—291no—
221C2×C4⋊C4—291no—
222C2×C4⋊C4—291no—
223C2×C4⋊C4—291no—
224C2×C4⋊C4—291no—
225C2×C4⋊C4C2×D2911yescharacteristic
226C2×C4⋊C4C2×D2911yescharacteristic
227C2×C22⋊C4—291no—
228C2×C22⋊C4—291no—
229C2×C22⋊C4—291no—
230C2×C22⋊C4—291no—
231C2×C22⋊C4—291no—
232C2×C22⋊C4—291no—
233C2×C22⋊C4—291no—
234C2×C22⋊C4—291no—
235C2×C22⋊C4—291no—
236C2×C22⋊C4—291no—
237C2×C42—291no—
238C2×C42—291no—
239C2×C42C2×D2911yescharacteristic
240#95—291no—
241#95—291no—
242#95—291no—
243#95—291no—
244#95—291no—
245#95—291no—
246#95C2×D2911yescharacteristic
247#95C2×D2911yescharacteristic
248#95C2×D2911yescharacteristic
249#95C2×D2911yescharacteristic
250C29#237611yescharacteristic, complemented, Sylow
251C24—291no—
252C24—291no—
253C2×D4—294no—
254C2×D4—1161no—
255C2×D4—1161no—
256C22×C4—291no—
257C22×C4—291no—
258C22×C4—294no—
259C22×C4—291no—
260C22×C4—291no—
261C22×C4—291no—
262C22×C4—291no—
263C22×C4—1161no—
264C22×C4—1161no—
265C22×C4—1161no—
266C22×C4—1161no—
267C22×C4—1161no—
268C22×C4C22×D2911yescharacteristic
269C22×C4C22×D2911yescharacteristic
270C22×C4C22×D2911yescharacteristic
271C22×C4C22×D2911yescharacteristic
272C22×C4C22×D2911yescharacteristic
273C22×C4C22×D2911yescharacteristic
274C22×C4C22×D2911yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—294no—
277C4⋊C4—1161no—
278C4⋊C4—1161no—
279C4⋊C4—1161no—
280C4⋊C4—1161no—
281C4⋊C4—1161no—
282C4⋊C4—1161no—
283C4⋊C4C4×D2914yescomplemented
284C22⋊C4—294no—
285C22⋊C4—294no—
286C22⋊C4—294no—
287C22⋊C4—1161no—
288C22⋊C4—1161no—
289C22⋊C4—1161no—
290C22⋊C4—1161no—
291C22⋊C4—1161no—
292C22⋊C4—1161no—
293C22⋊C4—1161no—
294C42—41no—
295C42—1161no—
296C42—1161no—
297C23—292no—
298C23—292no—
299C23—581no—
300C23—581no—
301C23—581no—
302C23—581no—
303C23—581no—
304C23—581no—
305C23—581no—
306C23—581no—
307C23—581no—
308C23—581no—
309C23—581no—
310C23—581no—
311C23C23×D2911yescharacteristic
312D4—1164no—
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—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—292no—
332C2×C4—292no—
333C2×C4—581no—
334C2×C4—581no—
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—1164no—
350C2×C4—1164no—
351C2×C4C2×C4×D2912yes—
352C2×C4C2×C4×D2912yes—
353C2×C4C2×C4×D2912yes—
354C22—582no—
355C22—582no—
356C22—582no—
357C22—582no—
358C22—582no—
359C22—582no—
360C22—1161no—
361C22—1161no—
362C22—1161no—
363C22—1161no—
364C22—1161no—
365C22—1161no—
366C22—1161no—
367C22—1161no—
368C22C22×C4×D2911yescharacteristic
369C22#1174535911yescharacteristic
370C22#1174536211yescharacteristic
371C22#1174536411yescharacteristic
372C22#1174536411yescharacteristic
373C22#1174536411yescharacteristic
374C22#1174536511yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—582no—
383C4—582no—
384C4—1161no—
385C4—1161no—
386C4—1161no—
387C4—1161no—
388C2—1162no—
389C2—1162no—
390C2#1427163511yescharacteristic
391C2#1427164311yescharacteristic
392C2#1427164511yescharacteristic
393C2#1427190411yescharacteristic
394C2#1427190911yescharacteristic
395C2#1427191011yescharacteristic
396C2#1427191711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D29
commutatorC22×C29C23×C4
FittingC29×#387C2
socle, cosocleC23×C29C24
prime radicalC22×D116C22
hypercenter#387D29
hypocenterC29#2376
LCSG ▷ C22×C29 ▷ C29
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #14271062C2semidirect
4. #14271060C2semidirect
5. #14271055C2semidirect
6. #14271053C2semidirect
7. #14271052C2semidirect
8. #14271052C2semidirect
9. #14271047C2semidirect
11. #14271037C2semidirect
12. #14271037C2semidirect
13. #14271022C2semidirect
14. #14271020C2semidirect
15. C29×#387C2semidirect
41. #11745256C4semidirect
75. C2×C4×D29C2×C4semidirect
77. C4⋊C4×C29C2×C4semidirect
88. #90781C2×C4semidirect
95. #90779C2×C4semidirect
199. #387D29semidirect
250. C29#2376semidirect
283. C4⋊C4C4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452562
#142710652
#142710622
#142710602
#142710552
#142710532
#142710522
#142710472
#142710412
#142710372
#142710222
#142710202
C29×#3872
#237629