Finite Group Databasebeta

← #16822828

Subgroups — #16822828

Order 3712.

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

normal subgroups only

This group has 864 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
2C2×#11745256C212yesmaximal, complemented
3C2×C4⋊C4×D29C211yesmaximal, characteristic, complemented
4#14271082C211yesmaximal, characteristic, complemented
5#14271081C211yesmaximal, characteristic, complemented
6#14271080C211yesmaximal, characteristic, complemented
7#14271041C212yesmaximal, complemented
8#14271037C212yesmaximal, complemented
9#14271034C212yesmaximal, complemented
10#14271025C211yesmaximal, characteristic, complemented
11C29×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C29C2211yescharacteristic
13C2×#90809—28no—
14C2×#90809—24no—
15C22×D116C2211yescharacteristic
16C22×D116C2212yes—
17C22×C4×D29C2212yes—
18C22×C4×D29C2211yescharacteristic
19C22×C4×D29C2211yescharacteristic
20C22×C4×D29C2211yescharacteristic
21C2×C4⋊C4×C29C2212yes—
22C2×C4⋊C4×C29C2211yescharacteristic
23C2×#90781C2212yes—
24C2×#90781C2212yes—
25C2×#90781C2212yes—
26C2×#90781C2212yes—
27C2×#90781C2212yes—
28C2×C4⋊Dic29C2211yescharacteristic
29C2×#90779C2211yescharacteristic
30C2×#90779C2212yes—
31C2×#90779C2211yescharacteristic
32#11745256—28no—
33C4⋊C4×D29—24no—
34C29×#95C2211yescharacteristic
35C29×#95C2212yes—
36#11745179C2212yes—
37#11745179C2211yescharacteristic
38#11745179C2212yes—
39C23×D29C2312yes—
40C23×D29C2311yescharacteristic
41C2×Q8×C29—24no—
42C2×Q8×C29—22no—
43C22×C4×C29C2311yescharacteristic
44C22×C4×C29C2312yes—
45C22×C4×C29C2312yes—
46C22×C4×C29C2311yescharacteristic
47C22×C4×C29C2311yescharacteristic
48C22×Dic29C2311yescharacteristic
49C22×Dic29C2312yes—
50C22×Dic29C2311yescharacteristic
51C22×Dic29C2311yescharacteristic
52#90809—232no—
53C2×D116—24no—
54C2×D116—24no—
55C2×D116—24no—
56C2×D116—22no—
57C2×D116D418yes—
58C2×C4×D29—24no—
59C2×C4×D29—24no—
60C2×C4×D29—22no—
61C2×C4×D29—24no—
62C2×C4×D29—22no—
63C2×C4×D29—24no—
64C2×C4×D29—22no—
65C2×C4×D29—22no—
66C2×C4×D29—24no—
67C2×C4×D29D414yes—
68C4⋊C4×C29—22no—
69C4⋊C4×C29—24no—
70#90781—24no—
71#90781—24no—
72#90781—24no—
73#90781—24no—
74#90781—24no—
75C4⋊Dic29—22no—
76#90779—22no—
77#90779—24no—
78#90779—22no—
79C23×C29C2411yescharacteristic
80C22×D29—21no—
81C22×D29—21no—
82C22×D29—22no—
83C22×D29—22no—
84C22×D29—22no—
85C22×D29—22no—
86C22×D29—22no—
87C22×D29—22no—
88C22×D29—22no—
89C22×D29C2×D414yes—
90C22×D29C2×D414yes—
91C22×D29C2×D412yes—
92Q8×C29—28no—
93C2×C4×C29—21no—
94C2×C4×C29—22no—
95C2×C4×C29—22no—
96C2×C4×C29—22no—
97C2×C4×C29—21no—
98C2×C4×C29—21no—
99C2×C4×C29—22no—
100C2×C4×C29—22no—
101C2×C4×C29—22no—
102C2×C4×C29—22no—
103C2×C4×C29C2×D414yes—
104C2×C4×C29C2×D412yes—
105C2×C4×C29#4112yescomplemented
106C2×Dic29—21no—
107C2×Dic29—22no—
108C2×Dic29—21no—
109C2×Dic29—21no—
110C2×Dic29—22no—
111C2×Dic29—22no—
112C2×Dic29—22no—
113C2×Dic29—22no—
114C2×Dic29C2×D412yes—
115C2×Dic29#4112yescomplemented
116D116—28no—
117D116—216no—
118C4×D29—28no—
119C4×D29—216no—
120C4×D29—28no—
121C4×D29—44no—
122#2406—291nomaximal, Sylow
123C22×C29C22×D411yescharacteristic
124C22×C29C22×D412yes—
125C22×C29C2×#4111yescharacteristic
126C22×C29C2×#4111yescharacteristic
127C22×C292−1+412yes—
128C2×D29—22no—
129C2×D29—22no—
130C2×D29—24no—
131C2×D29—24no—
132C2×D29—24no—
133C2×D29—24no—
134C2×D29—24no—
135C2×D29—42no—
136C2×D29—42no—
137C2×D29—42no—
138C2×D29—42no—
139C2×D29#12014yes—
140C4×C29—24no—
141C4×C29—22no—
142C4×C29—22no—
143C4×C29—41no—
144C4×C29—42no—
145Dic29—22no—
146Dic29—22no—
147Dic29—41no—
148Dic29—42no—
149C22×#41—291no—
150C2×#123—292no—
151C22×C4⋊C4—291no—
152#395—291no—
153#395D2911yescharacteristic, complemented
154#394—291no—
155#394—291no—
156#386—292no—
157#385—292no—
158#385—292no—
159#381—291no—
160C2×C29C2×#12011yescharacteristic
161C2×C29#52511yescharacteristic
162C2×C29#53611yescharacteristic
163C2×C29#54712yes—
164C2×C29#54712yes—
165D29—24no—
166D29—44no—
167C2×#41—584no—
168C2×#41—588no—
169C22×Q8C2×D2911yescharacteristic
170C22×D4—292no—
171C22×D4—291no—
172C23×C4—291no—
173C23×C4—291no—
174C23×C4—291no—
175C23×C4—292no—
176#123—588no—
177C2×C4⋊C4—291no—
178C2×C4⋊C4—291no—
179C2×C4⋊C4—291no—
180C2×C4⋊C4—292no—
181C2×C4⋊C4—584no—
182C2×C4⋊C4C2×D2911yescharacteristic
183C2×C4⋊C4C2×D2912yes—
184C2×C22⋊C4—292no—
185C2×C22⋊C4—292no—
186C2×C22⋊C4—292no—
187C2×C22⋊C4—292no—
188C2×C22⋊C4—292no—
189#95—292no—
190#95—291no—
191#95—292no—
192#95C2×D2911yescharacteristic
193#95C2×D2912yes—
194C29#240611yescharacteristic, complemented, Sylow
195C24—291no—
196C24—292no—
197#41—5832no—
198C2×Q8—22no—
199C2×Q8—24no—
200C2×D4—298no—
201C2×D4—584no—
202C2×D4—584no—
203C2×D4—584no—
204C2×D4—582no—
205C22×C4—291no—
206C22×C4—291no—
207C22×C4—294no—
208C22×C4—292no—
209C22×C4—291no—
210C22×C4—584no—
211C22×C4—582no—
212C22×C4—582no—
213C22×C4—584no—
214C22×C4—584no—
215C22×C4—582no—
216C22×C4—584no—
217C22×C4—584no—
218C22×C4—582no—
219C22×C4C22×D2912yes—
220C22×C4C22×D2911yescharacteristic
221C22×C4C22×D2911yescharacteristic
222C22×C4C22×D2912yes—
223C22×C4C22×D2911yescharacteristic
224C4⋊C4—22no—
225C4⋊C4—24no—
226C4⋊C4—582no—
227C4⋊C4—582no—
228C4⋊C4—582no—
229C4⋊C4—584no—
230C22⋊C4—584no—
231C22⋊C4—584no—
232C22⋊C4—584no—
233C22⋊C4—584no—
234C22⋊C4—584no—
235C23—294no—
236C23—292no—
237C23—294no—
238C23—581no—
239C23—582no—
240C23—581no—
241C23—582no—
242C23—582no—
243C23—582no—
244C23—582no—
245C23—582no—
246C23—582no—
247C23C23×D2911yescharacteristic
248Q8—28no—
249D4—5816no—
250D4—588no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—292no—
262C2×C4—292no—
263C2×C4—588no—
264C2×C4—5816no—
265C2×C4—588no—
266C2×C4—581no—
267C2×C4—582no—
268C2×C4—581no—
269C2×C4—582no—
270C2×C4—582no—
271C2×C4—581no—
272C2×C4—582no—
273C2×C4—582no—
274C2×C4—1164no—
275C2×C4D4×D2912yes—
276C2×C4D4×D2914yes—
277C2×C4#9080912yescomplemented
278C22—294no—
279C22—584no—
280C22—584no—
281C22—582no—
282C22—582no—
283C22—584no—
284C22—584no—
285C22—584no—
286C22—1162no—
287C22—1162no—
288C22—1162no—
289C22—1162no—
290C22C2×D4×D2912yes—
291C22C2×D4×D2911yescharacteristic
292C22C2×#9080911yescharacteristic
293C22#1174536212yes—
294C22#41×D2911yescharacteristic
295C4—22no—
296C4—24no—
297C4—22no—
298C4—42no—
299C4—41no—
300C4—582no—
301C4—582no—
302C4—1162no—
303C4—1161no—
304C2—584no—
305C2—1164no—
306C2#120×D2911yescharacteristic
307C2#1427167611yescharacteristic
308C2#1427173811yescharacteristic
309C2#1427181512yes—
310C2#1427181612yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC29×#395C2
prime radicalC22×#90809C2
hypercenter#395D29
hypocenterC29#2406
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#11745256C2semidirect
3. C2×C4⋊C4×D29C2semidirect
4. #14271082C2semidirect
5. #14271081C2semidirect
6. #14271080C2semidirect
7. #14271041C2semidirect
8. #14271037C2semidirect
9. #14271034C2semidirect
10. #14271025C2semidirect
11. C29×#395C2semidirect
105. C2×C4×C29#41semidirect
115. C2×Dic29#41semidirect
153. #395D29semidirect
194. C29#2406semidirect
277. C2×C4#90809semidirect

Maximal subgroups

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

subgroupindex
C22×#908092
C2×#117452562
C2×C4⋊C4×D292
#142710822
#142710812
#142710802
#142710412
#142710372
#142710342
#142710252
C29×#3952
#240629