Finite Group Databasebeta

← #16822802

Subgroups — #16822802

Order 3712.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 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
1C23×C22⋊D29C211yesmaximal, characteristic, complemented
2C2×#11745315C212yesmaximal, complemented
3C2×#11745299C212yesmaximal, complemented
4#14271071C214yesmaximal, complemented
5#14271043C212yesmaximal, complemented
6#14271042C212yesmaximal, complemented
7#14271019C211yesmaximal, characteristic, complemented
8C23×D4×C29C211yesmaximal, characteristic, complemented
9C25×C29C2211yescharacteristic
10C25×C29C2211yescharacteristic, complemented
11C24×D29C2211yescharacteristic, complemented
12C22×D4×C29—24no—
13C22×D4×C29—28no—
14C22×D4×C29C2212yes—
15C22×D4×C29C2212yescomplemented
16C23×C4×C29C2211yescharacteristic, complemented
17C22×C22⋊D29—24no—
18C22×C22⋊D29—28no—
19C22×C22⋊D29C2212yes—
20C22×C22⋊D29C2212yescomplemented
21C22×C22⋊D29C2214yescomplemented
22C23×Dic29C2211yescharacteristic, complemented
23#11745315—28no—
24C2×C22⋊Dic29C2212yescomplemented
25C2×C22⋊Dic29C2212yescomplemented
26C2×C22⋊Dic29C2214yescomplemented
27C2×C22⋊Dic29C2212yescomplemented
28#11745299—28no—
29C2×#90781C2214yescomplemented
30#11745179C2214yescomplemented
31C24×C29—24no—
32C24×C29—24no—
33C24×C29—22no—
34C24×C29—24no—
35C24×C29—24no—
36C24×C29—42no—
37C24×C29—42no—
38C24×C29D414yescomplemented
39C24×C29C2311yescharacteristic
40C24×C29C2312yes—
41C24×C29C2311yescharacteristic
42C24×C29C2311yescharacteristic
43C23×D29—22no—
44C23×D29—24no—
45C23×D29—24no—
46C23×D29—42no—
47C23×D29C2312yes—
48C2×D4×C29—232no—
49C2×D4×C29—24no—
50C2×D4×C29—24no—
51C2×D4×C29—24no—
52C2×D4×C29—24no—
53C2×D4×C29—24no—
54C2×D4×C29D418yescomplemented
55C22×C4×C29—22no—
56C22×C4×C29—24no—
57C22×C4×C29C2312yes—
58C2×C22⋊D29—232no—
59C2×C22⋊D29—24no—
60C2×C22⋊D29—24no—
61C2×C22⋊D29—24no—
62C2×C22⋊D29—24no—
63C2×C22⋊D29—28no—
64C2×C22⋊D29—28no—
65C2×C22⋊D29—24no—
66C2×C22⋊D29—44no—
67C2×C22⋊D29D418yescomplemented
68C22×Dic29—22no—
69C22×Dic29—24no—
70C22×Dic29C2314yes—
71C22×Dic29C2312yes—
72C22⋊Dic29—24no—
73C22⋊Dic29—24no—
74C22⋊Dic29—28no—
75C22⋊Dic29—42no—
76#90781—28no—
77C23×C29—22no—
78C23×C29—22no—
79C23×C29—22no—
80C23×C29—22no—
81C23×C29—28no—
82C23×C29—28no—
83C23×C29—28no—
84C23×C29—24no—
85C23×C29—22no—
86C23×C29—22no—
87C23×C29—22no—
88C23×C29—22no—
89C23×C29—22no—
90C23×C29—24no—
91C23×C29—44no—
92C23×C29—416no—
93C23×C29—416no—
94C23×C29—44no—
95C23×C29—44no—
96C23×C29C2×D414yes—
97C23×C29C2×D412yes—
98C23×C29C2×D414yes—
99C23×C29C2×D412yes—
100C23×C29C2×D412yes—
101C23×C29C2×D414yescomplemented
102C23×C29C2411yescharacteristic
103C22×D29—28no—
104C22×D29—24no—
105C22×D29—22no—
106C22×D29—22no—
107C22×D29—24no—
108C22×D29—416no—
109C22×D29—44no—
110C22×D29—44no—
111C22×D29C2×D414yescomplemented
112D4×C29—216no—
113D4×C29—216no—
114C2×C4×C29—28no—
115C2×C4×C29—22no—
116C2×C4×C29—22no—
117C2×C4×C29C2×D414yescomplemented
118C22⋊D29—216no—
119C22⋊D29—216no—
120C22⋊D29—416no—
121C2×Dic29—28no—
122C2×Dic29—22no—
123C2×Dic29—22no—
124C2×Dic29—24no—
125C2×Dic29—24no—
126C2×Dic29—24no—
127C2×Dic29C2×D414yescomplemented
128#2403—291nomaximal, Sylow
129C22×C29—22no—
130C22×C29—22no—
131C22×C29—22no—
132C22×C29—22no—
133C22×C29—28no—
134C22×C29—28no—
135C22×C29—22no—
136C22×C29—22no—
137C22×C29—24no—
138C22×C29—22no—
139C22×C29—28no—
140C22×C29—416no—
141C22×C29—416no—
142C22×C29—42no—
143C22×C29—42no—
144C22×C29—44no—
145C22×C29—44no—
146C22×C29#12014yescomplemented
147C22×C29#12014yescomplemented
148C22×C29C22×D412yes—
149C22×C29C22×D411yescharacteristic
150C22×C29C22×D412yes—
151C22×C29H(5,𝔽2)12yes—
152C2×D29—24no—
153C2×D29—28no—
154C2×D29—416no—
155C2×D29—42no—
156C2×D29—42no—
157C2×D29—44no—
158C4×C29—24no—
159Dic29—24no—
160Dic29—44no—
161C23×D4—291no—
162C23×D4D2911yescharacteristic, complemented
163C2×#120—292no—
164C2×#120—292no—
165#392—294no—
166#386—292no—
167#386—292no—
168#379—291no—
169C2×C29—24no—
170C2×C29—24no—
171C2×C29—42no—
172C2×C29—44no—
173C2×C29C2×#12011yescharacteristic
174C2×C29C2×#12011yescharacteristic
175C2×C29#53411yescharacteristic
176C2×C29D4214yes—
177D29—44no—
178C25—291no—
179C25C2×D2911yescharacteristic
180C25C2×D2911yescharacteristic, complemented
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—292no—
184C22×D4—292no—
185C22×D4—294no—
186C22×D4—588no—
187C22×D4—584no—
188C22×D4C2×D2912yes—
189C22×D4C2×D2912yescomplemented
190C23×C4—291no—
191C23×C4C2×D2911yescharacteristic, complemented
192#120—588no—
193#120—588no—
194C2×C22⋊C4—292no—
195C2×C22⋊C4—292no—
196C2×C22⋊C4—294no—
197C2×C22⋊C4—294no—
198C2×C22⋊C4—292no—
199#95—294no—
200C29#240311yescharacteristic, complemented, Sylow
201C24—24no—
202C24—24no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—42no—
207C24—42no—
208C24—292no—
209C24—584no—
210C24—582no—
211C24—584no—
212C24—1162no—
213C24C22⋊D2914yescomplemented
214C24C22×D2911yescharacteristic
215C24C22×D2912yes—
216C24C22×D2911yescharacteristic
217C24C22×D2911yescharacteristic
218C2×D4—232no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—298no—
225C2×D4—584no—
226C2×D4—584no—
227C2×D4—5832no—
228C2×D4—584no—
229C2×D4—584no—
230C2×D4—584no—
231C2×D4—588no—
232C2×D4—588no—
233C2×D4—1164no—
234C2×D4C22⋊D2918yescomplemented
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—292no—
238C22×C4—294no—
239C22×C4—584no—
240C22×C4—582no—
241C22×C4C22×D2912yes—
242C22⋊C4—588no—
243C22⋊C4—584no—
244C22⋊C4—584no—
245C22⋊C4—588no—
246C22⋊C4—1162no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—28no—
253C23—28no—
254C23—24no—
255C23—22no—
256C23—22no—
257C23—24no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—44no—
262C23—416no—
263C23—416no—
264C23—44no—
265C23—44no—
266C23—294no—
267C23—588no—
268C23—582no—
269C23—584no—
270C23—584no—
271C23—582no—
272C23—11616no—
273C23—1164no—
274C23—1164no—
275C23D4×D2914yes—
276C23C2×C22⋊D2912yes—
277C23C2×C22⋊D2912yes—
278C23C2×C22⋊D2914yes—
279C23C2×C22⋊D2912yes—
280C23C2×C22⋊D2914yescomplemented
281C23C23×D2911yescharacteristic
282D4—216no—
283D4—216no—
284D4—5816no—
285D4—5816no—
286D4—11616no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—294no—
291C2×C4—588no—
292C2×C4—582no—
293C2×C4—582no—
294C2×C4—584no—
295C2×C4—584no—
296C2×C4—584no—
297C2×C4C2×C22⋊D2914yescomplemented
298C22—28no—
299C22—28no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—28no—
305C22—22no—
306C22—22no—
307C22—24no—
308C22—22no—
309C22—416no—
310C22—416no—
311C22—44no—
312C22—44no—
313C22—42no—
314C22—42no—
315C22—588no—
316C22—584no—
317C22—11616no—
318C22—1164no—
319C22—1162no—
320C22—1162no—
321C22#1174529914yescomplemented
322C22#1174531514yescomplemented
323C22C2×D4×D2912yes—
324C22#1174535912yes—
325C22C22×C22⋊D2912yes—
326C22C22×C22⋊D2911yescharacteristic
327C4—24no—
328C4—584no—
329C4—1164no—
330C2—24no—
331C2—24no—
332C2—44no—
333C2—42no—
334C2—1164no—
335C2C2×#1174529911yescharacteristic
336C2C2×#1174531511yescharacteristic
337C2#1427172211yescharacteristic
338C2D4×C22⋊D2914yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC23×D4×C29C2
hypercenterC23×D4D29
hypocenterC29#2403
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D29C2semidirect
2. C2×#11745315C2semidirect
3. C2×#11745299C2semidirect
4. #14271071C2semidirect
5. #14271043C2semidirect
6. #14271042C2semidirect
7. #14271019C2semidirect
8. C23×D4×C29C2semidirect
10. C25×C29C22semidirect
11. C24×D29C22semidirect
15. C22×D4×C29C22semidirect
16. C23×C4×C29C22semidirect
20. C22×C22⋊D29C22semidirect
21. C22×C22⋊D29C22semidirect
22. C23×Dic29C22semidirect
24. C2×C22⋊Dic29C22semidirect
25. C2×C22⋊Dic29C22semidirect
26. C2×C22⋊Dic29C22semidirect
27. C2×C22⋊Dic29C22semidirect
29. C2×#90781C22semidirect
30. #11745179C22semidirect
38. C24×C29D4semidirect
54. C2×D4×C29D4semidirect
67. C2×C22⋊D29D4semidirect
101. C23×C29C2×D4semidirect
111. C22×D29C2×D4semidirect
117. C2×C4×C29C2×D4semidirect
127. C2×Dic29C2×D4semidirect
146. C22×C29#120semidirect
147. C22×C29#120semidirect
162. C23×D4D29semidirect
180. C25C2×D29semidirect
189. C22×D4C2×D29semidirect
191. C23×C4C2×D29semidirect
200. C29#2403semidirect
213. C24C22⋊D29semidirect
234. C2×D4C22⋊D29semidirect
280. C23C2×C22⋊D29semidirect
297. C2×C4C2×C22⋊D29semidirect
321. C22#11745299semidirect
322. C22#11745315semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D292
C2×#117453152
C2×#117452992
#142710712
#142710432
#142710422
#142710192
C23×D4×C292
#240329