Finite Group Databasebeta

← #16822812

Subgroups — #16822812

Order 3712.

1056 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 1056 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#90809C211yesmaximal, characteristic, complemented
2C2×#11745244C212yesmaximal, complemented
3C2×C22⋊C4×D29C211yesmaximal, characteristic, complemented
4#14271074C211yesmaximal, characteristic, complemented
5#14271073C211yesmaximal, characteristic, complemented
6#14271072C211yesmaximal, characteristic, complemented
7#14271041C212yesmaximal, complemented
8#14271041C212yesmaximal, complemented
9#14271039C212yesmaximal, complemented
10#14271025C211yesmaximal, characteristic, complemented
11C29×#393C211yesmaximal, characteristic, complemented
12C24×D29C2211yescharacteristic
13C22×Q8×C29C2211yescharacteristic, complemented
14C22×C22⋊D29C2212yes—
15C2×#90809—28no—
16C2×#90809—24no—
17C22×D116C2212yes—
18C22×D116C2212yes—
19C22×D116C2211yescharacteristic, complemented
20C22×C4×D29C2211yescharacteristic
21C22×C4×D29C2211yescharacteristic
22C22×C4×D29C2212yescomplemented
23C2×C22⋊C4×C29C2212yes—
24C2×C22⋊C4×C29C2211yescharacteristic
25C2×C22⋊Dic29C2211yescharacteristic
26C2×#90781C2212yes—
27C2×#90781C2211yescharacteristic
28C2×#90781C2212yes—
29C2×#90781C2211yescharacteristic
30C2×#90781C2212yescomplemented
31C2×#90781C2212yescomplemented
32#11745244—28no—
33C22⋊C4×D29—24no—
34C29×#95C2211yescharacteristic, complemented
35C29×#95C2212yescomplemented
36#11745179C2212yescomplemented
37#11745179C2211yescharacteristic, complemented
38#11745179C2212yescomplemented
39C24×C29C2311yescharacteristic
40C23×D29—22no—
41C23×D29—22no—
42C23×D29—22no—
43C23×D29—24no—
44C23×D29—42no—
45C23×D29C2312yes—
46C23×D29C2311yescharacteristic
47C23×D29C2311yescharacteristic
48C2×Q8×C29—24no—
49C2×Q8×C29—22no—
50C22×C4×C29C2311yescharacteristic
51C22×C4×C29C2312yes—
52C22×C4×C29C2311yescharacteristic
53C22×C4×C29C2312yes—
54C2×C22⋊D29—24no—
55C2×C22⋊D29—24no—
56C2×C22⋊D29—42no—
57C22×Dic29C2312yes—
58C22×Dic29C2311yescharacteristic
59C22×Dic29C2311yescharacteristic
60#90809—232no—
61C2×D116—24no—
62C2×D116—24no—
63C2×D116—24no—
64C2×D116—22no—
65C2×D116—24no—
66C2×D116—24no—
67C2×D116—42no—
68C2×D116D418yescomplemented
69C2×C4×D29—24no—
70C2×C4×D29—24no—
71C2×C4×D29—24no—
72C2×C4×D29—24no—
73C2×C4×D29—22no—
74C2×C4×D29—42no—
75C2×C4×D29D414yescomplemented
76C22⋊C4×C29—22no—
77C22⋊C4×C29—24no—
78C22⋊Dic29—22no—
79#90781—24no—
80#90781—22no—
81#90781—24no—
82#90781—22no—
83#90781—42no—
84#90781—42no—
85C23×C29—21no—
86C23×C29—22no—
87C23×C29—21no—
88C23×C29—22no—
89C23×C29#4112yescomplemented
90C23×C29C2411yescharacteristic
91C22×D29—21no—
92C22×D29—21no—
93C22×D29—22no—
94C22×D29—28no—
95C22×D29—21no—
96C22×D29—22no—
97C22×D29—22no—
98C22×D29—22no—
99C22×D29—22no—
100C22×D29—22no—
101C22×D29—21no—
102C22×D29—22no—
103C22×D29—22no—
104C22×D29—22no—
105C22×D29—48no—
106C22×D29—48no—
107C22×D29—44no—
108C22×D29—44no—
109C22×D29C2×D414yes—
110C22×D29C2×D414yes—
111C22×D29C2×D412yes—
112C22×D29#4112yescomplemented
113Q8×C29—28no—
114C2×C4×C29—22no—
115C2×C4×C29—22no—
116C2×C4×C29—22no—
117C2×C4×C29—21no—
118C2×C4×C29—22no—
119C2×C4×C29—22no—
120C2×C4×C29—22no—
121C2×C4×C29—22no—
122C2×C4×C29C2×D414yes—
123C2×C4×C29C2×D412yes—
124C22⋊D29—48no—
125C2×Dic29—22no—
126C2×Dic29—22no—
127C2×Dic29—21no—
128C2×Dic29—22no—
129C2×Dic29—22no—
130C2×Dic29—22no—
131C2×Dic29C2×D412yes—
132D116—28no—
133D116—216no—
134D116—48no—
135C4×D29—216no—
136C4×D29—28no—
137C4×D29—44no—
138#2404—291nomaximal, Sylow
139C22×C29—22no—
140C22×C29—24no—
141C22×C29—41no—
142C22×C29—41no—
143C22×C29—42no—
144C22×C29C22×D411yescharacteristic
145C22×C29C22×D412yes—
146C22×C29C2×#4111yescharacteristic
147C22×C29C2×#4111yescharacteristic
148C22×C29H(5,𝔽2)12yes—
149C2×D29—22no—
150C2×D29—22no—
151C2×D29—24no—
152C2×D29—24no—
153C2×D29—24no—
154C2×D29—22no—
155C2×D29—24no—
156C2×D29—24no—
157C2×D29—24no—
158C2×D29—416no—
159C2×D29—41no—
160C2×D29—42no—
161C2×D29—42no—
162C2×D29—41no—
163C2×D29—42no—
164C2×D29—42no—
165C2×D29—42no—
166C2×D29#12014yes—
167C4×C29—24no—
168C4×C29—22no—
169C4×C29—41no—
170C4×C29—42no—
171Dic29—22no—
172Dic29—41no—
173Dic29—42no—
174C22×#41—291no—
175C2×#120—292no—
176C22×C22⋊C4—291no—
177#393—291no—
178#393D2911yescharacteristic, complemented
179#392—291no—
180#392—291no—
181#386—292no—
182#386—292no—
183#386—292no—
184#381—291no—
185C2×C29—42no—
186C2×C29C2×#12011yescharacteristic
187C2×C29#52511yescharacteristic
188C2×C29#53511yescharacteristic
189C2×C29#54612yes—
190C2×C29#54612yes—
191D29—24no—
192D29—44no—
193D29—42no—
194C25—291no—
195C2×#41—584no—
196C2×#41—588no—
197C22×Q8C2×D2911yescharacteristic, complemented
198C22×D4—292no—
199C22×D4—292no—
200C22×D4—291no—
201C22×D4—292no—
202C23×C4—291no—
203C23×C4—292no—
204C23×C4—291no—
205#120—588no—
206C2×C22⋊C4—291no—
207C2×C22⋊C4—291no—
208C2×C22⋊C4—291no—
209C2×C22⋊C4—292no—
210C2×C22⋊C4—292no—
211C2×C22⋊C4—292no—
212C2×C22⋊C4—292no—
213C2×C22⋊C4—584no—
214C2×C22⋊C4C2×D2911yescharacteristic
215C2×C22⋊C4C2×D2912yes—
216#95—292no—
217#95—291no—
218#95—292no—
219#95C2×D2911yescharacteristic, complemented
220#95C2×D2912yescomplemented
221C29#240411yescharacteristic, complemented, Sylow
222C24—291no—
223C24—291no—
224C24—292no—
225C24—582no—
226C24—582no—
227C24—584no—
228C24—582no—
229C24—1162no—
230C24C22×D2911yescharacteristic
231#41—5832no—
232C2×Q8—22no—
233C2×Q8—24no—
234C2×D4—298no—
235C2×D4—584no—
236C2×D4—584no—
237C2×D4—584no—
238C2×D4—584no—
239C2×D4—584no—
240C2×D4—584no—
241C2×D4—584no—
242C2×D4—582no—
243C2×D4—1162no—
244C2×D4—1162no—
245C22×C4—291no—
246C22×C4—294no—
247C22×C4—292no—
248C22×C4—291no—
249C22×C4—584no—
250C22×C4—584no—
251C22×C4—584no—
252C22×C4—582no—
253C22×C4—584no—
254C22×C4—1162no—
255C22×C4C22×D2912yes—
256C22×C4C22×D2911yescharacteristic
257C22×C4C22×D2911yescharacteristic
258C22×C4C22×D2912yes—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—582no—
262C22⋊C4—584no—
263C22⋊C4—582no—
264C22⋊C4—582no—
265C22⋊C4—584no—
266C22⋊C4—1162no—
267C22⋊C4—1162no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—294no—
273C23—292no—
274C23—294no—
275C23—292no—
276C23—588no—
277C23—581no—
278C23—581no—
279C23—582no—
280C23—581no—
281C23—582no—
282C23—582no—
283C23—582no—
284C23—582no—
285C23—581no—
286C23—582no—
287C23—582no—
288C23—582no—
289C23—582no—
290C23—1168no—
291C23—1164no—
292C23—1168no—
293C23—1164no—
294C23#9080912yescomplemented
295C23C23×D2911yescharacteristic
296Q8—28no—
297D4—5816no—
298D4—588no—
299D4—1168no—
300D4—1168no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—292no—
310C2×C4—588no—
311C2×C4—5816no—
312C2×C4—582no—
313C2×C4—581no—
314C2×C4—582no—
315C2×C4—582no—
316C2×C4—582no—
317C2×C4—582no—
318C2×C4—1164no—
319C2×C4D4×D2912yes—
320C2×C4D4×D2914yes—
321C22—24no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—294no—
327C22—584no—
328C22—584no—
329C22—582no—
330C22—582no—
331C22—584no—
332C22—584no—
333C22—584no—
334C22—584no—
335C22—582no—
336C22—11616no—
337C22—1162no—
338C22—1162no—
339C22—1162no—
340C22—1161no—
341C22—1162no—
342C22—1161no—
343C22—1162no—
344C22C2×D4×D2912yes—
345C22C2×D4×D2911yescharacteristic
346C22C2×#9080911yescharacteristic
347C22#41×D2911yescharacteristic
348C22#1174536412yes—
349C4—22no—
350C4—24no—
351C4—42no—
352C4—41no—
353C4—582no—
354C4—1162no—
355C4—1161no—
356C2—42no—
357C2—584no—
358C2—1162no—
359C2—1164no—
360C2#120×D2911yescharacteristic
361C2#1427167611yescharacteristic
362C2#1427173111yescharacteristic
363C2#1427180412yes—
364C2#1427180612yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#393C2
hypercenter#393D29
hypocenterC29#2404
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×#90809C2semidirect
2. C2×#11745244C2semidirect
3. C2×C22⋊C4×D29C2semidirect
4. #14271074C2semidirect
5. #14271073C2semidirect
6. #14271072C2semidirect
7. #14271041C2semidirect
8. #14271041C2semidirect
9. #14271039C2semidirect
10. #14271025C2semidirect
11. C29×#393C2semidirect
13. C22×Q8×C29C22semidirect
19. C22×D116C22semidirect
22. C22×C4×D29C22semidirect
30. C2×#90781C22semidirect
31. C2×#90781C22semidirect
34. C29×#95C22semidirect
35. C29×#95C22semidirect
36. #11745179C22semidirect
37. #11745179C22semidirect
38. #11745179C22semidirect
68. C2×D116D4semidirect
75. C2×C4×D29D4semidirect
89. C23×C29#41semidirect
112. C22×D29#41semidirect
178. #393D29semidirect
197. C22×Q8C2×D29semidirect
219. #95C2×D29semidirect
220. #95C2×D29semidirect
221. C29#2404semidirect
294. C23#90809semidirect

Maximal subgroups

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

subgroupindex
C22×#908092
C2×#117452442
C2×C22⋊C4×D292
#142710742
#142710732
#142710722
#142710412
#142710392
#142710252
C29×#3932
#240429