Finite Group Databasebeta

← #22620945

Subgroups — #22620945

Order 9344.

490 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 113 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
1#127×D73C211yesmaximal, characteristic, complemented
2#17989116C211yesmaximal, characteristic, complemented
3#17988977C211yesmaximal, characteristic, complemented
4#17988952C211yesmaximal, characteristic, complemented
5#17988951C211yesmaximal, characteristic, complemented
6#17988785C212yesmaximal, complemented
7#17988780C212yesmaximal, complemented
8#17988736C212yesmaximal, complemented
9#17988719C212yesmaximal, complemented
10#17988642C211yesmaximal, characteristic, complemented
11#496×C73C211yesmaximal, characteristic, complemented
12C2×D4×D73—21no—
13C2×D4×D73—21no—
14C2×D4×D73C2212yes—
15C2×#11924015C2212yescomplemented
16C2×#11924013C2212yescomplemented
17#14556588C2211yescharacteristic
18#14556587—21no—
19#14556587—21no—
20#14556587C2212yes—
21#14556584—21no—
22#14556583C2211yescharacteristic, complemented
23#14556573—21no—
24#14556567C2211yescharacteristic
25#14556562C2211yescharacteristic
26Q8×Dic73C2211yescharacteristic, complemented
27D4×Dic73C2211yescharacteristic, complemented
28#14556551—21no—
29#14556547—21no—
30C42×D73C2211yescharacteristic
31#14556520C2212yescomplemented
32#14556513C2212yescomplemented
33#14556513C2212yescomplemented
34#14556512C2212yescomplemented
35#14556512C2212yescomplemented
36#14556503C2212yes—
37#14556502C2211yescharacteristic, complemented
38C2×SD16×C73C2212yescomplemented
39C2×D8×C73C2212yescomplemented
40#127×C73C2211yescharacteristic
41#124×C73C2211yescharacteristic, complemented
42#97×C73C2211yescharacteristic, complemented
43C23×D73—22no—
44C2×D292—21no—
45C2×D292—21no—
46C2×D292C2312yes—
47C2×C22⋊D73—21no—
48C2×C22⋊D73—22no—
49C2×C22⋊D73—22no—
50C2×C22⋊D73—22no—
51D4×D73—42no—
52D4×D73—42no—
53D4×D73—44no—
54D4×D73—44no—
55C22×Dic73—21no—
56C2×C4×D73D411yescharacteristic
57C2×C4×D73D411yescharacteristic
58C2×C4×D73C2311yescharacteristic
59#11924015—42no—
60#11924013—42no—
61C2×#10596831C2312yes—
62C4⋊Dic73—21no—
63C4⋊Dic73C2311yescharacteristic
64C4⋊Dic73C2311yescharacteristic
65#11924007—21no—
66#11924007—21no—
67#11924007—21no—
68C22⋊Dic73—21no—
69C22⋊Dic73—21no—
70#11924005—21no—
71C4×Dic73—21no—
72C4×Dic73D411yescharacteristic
73C4×Dic73D411yescharacteristic
74C4×Dic73C2311yescharacteristic
75C2×Q8×C73C2311yescharacteristic
76C2×D4×C73—21no—
77C2×D4×C73—21no—
78C2×D4×C73C2311yescharacteristic
79C2×D4×C73C2312yes—
80SD16×C73—42no—
81D8×C73—42no—
82C2×C8×C73C2312yes—
83C22⋊C4×C73—21no—
84C22⋊C4×C73—21no—
85C42×C73C2311yescharacteristic
86C23×C73—21no—
87C23×C73—22no—
88C22×D73—22no—
89C22×D73—44no—
90C22×D73—44no—
91C22×D73—44no—
92C22×D73C2×D411yescharacteristic
93Q8×C73—22no—
94D4×C73—22no—
95D4×C73—22no—
96D4×C73—22no—
97D4×C73—42no—
98D4×C73—42no—
99C2×C4×C73—21no—
100C2×C4×C73C2×D411yescharacteristic
101C2×C4×C73C2×D411yescharacteristic
102C2×C4×C73C2411yescharacteristic
103C22⋊D73—44no—
104C22⋊D73—44no—
105C22⋊D73—42no—
106C22⋊D73—44no—
107C2×Dic73—21no—
108C2×Dic73—21no—
109C2×Dic73—42no—
110C2×Dic73C2×D411yescharacteristic
111C2×Dic73C2×D411yescharacteristic
112C2×Dic73C2×D411yescharacteristic
113D292—22no—
114D292—22no—
115D292—42no—
116D292—42no—
117C4×D73—22no—
118C4×D73—22no—
119C4×D73—22no—
120C8×C73—22no—
121#10596831—22no—
122C22×C73—42no—
123C22×C73—41no—
124C22×C73—41no—
125C22×C73—42no—
126C22×C73—42no—
127C22×C73—41no—
128C22×C73C22×D411yescharacteristic
129C2×D73—21no—
130C2×D73—21no—
131C2×D73—42no—
132C2×D73—42no—
133C2×D73—42no—
134C2×D73—88no—
135C2×D73AGL(1,ℤ/8)12yes—
136C4×C73—21no—
137C4×C73—21no—
138C4×C73—41no—
139C4×C73H(5,𝔽2)12yes—
140Dic73—21no—
141Dic73—21no—
142Dic73—41no—
143Dic73—41no—
144Dic73AGL(1,ℤ/8)12yes—
145C2×C73—82no—
146C2×C73—81no—
147C2×C73#53511yescharacteristic
148C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
149C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
150D73—22no—
151D73—82no—
152#3217—731nomaximal, Sylow
153C73#321711yescharacteristic, complemented, Sylow
154C2×#127—731no—
155#529—731no—
156#496—731no—
157#496D7311yescharacteristic, complemented
158#490—731no—
159#490—731no—
160#460—732no—
161#459—732no—
162#450—732no—
163#447—732no—
164#432—731no—
165C22×D4—732no—
166C22×D4—1461no—
167C22×D4—1461no—
168C2×SD16—732no—
169C2×SD16C2×D7312yescomplemented
170C2×D8—732no—
171C2×D8C2×D7312yescomplemented
172#127—732no—
173#127—731no—
174#127—1461no—
175#127—1461no—
176#127C2×D7311yescharacteristic
177#126—1461no—
178#125—731no—
179#124C2×D7311yescharacteristic, complemented
180#123—1461no—
181C22⋊Q8—731no—
182#121—731no—
183C4×Q8—731no—
184C4×D4—731no—
185C4×D4—1461no—
186#117—1461no—
187C2×C42—731no—
188C4⋊C8—732no—
189#102—732no—
190#102—732no—
191#102—732no—
192#102—732no—
193C22⋊C8—732no—
194#97—731no—
195#97C2×D7311yescharacteristic, complemented
196C24—1462no—
197C2×Q8C22×D7311yescharacteristic
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—2922no—
201C2×D4—2922no—
202C2×D4—2924no—
203C2×D4—2924no—
204C2×D4—732no—
205C2×D4—1462no—
206C2×D4—1462no—
207C2×D4—1462no—
208C2×D4—1461no—
209C2×D4—1461no—
210C2×D4—1461no—
211C2×D4C22×D7312yes—
212C2×D4C22×D7311yescharacteristic
213C22×C4—731no—
214C22×C4—731no—
215C22×C4—731no—
216C22×C4—1461no—
217SD16—42no—
218SD16—2922no—
219D8—42no—
220D8—2922no—
221C2×C8—732no—
222C2×C8C22×D7312yes—
223C4⋊C4—731no—
224C4⋊C4—731no—
225C4⋊C4—1461no—
226C4⋊C4—1461no—
227C4⋊C4—1461no—
228C4⋊C4—1461no—
229C22⋊C4—21no—
230C22⋊C4—21no—
231C22⋊C4—1461no—
232C22⋊C4—1461no—
233C22⋊C4—1461no—
234C42—731no—
235C42—731no—
236C42—731no—
237C42—1461no—
238C42C22×D7311yescharacteristic
239C23—21no—
240C23—22no—
241C23—2924no—
242C23—2924no—
243C23—2924no—
244C23—731no—
245C23—1462no—
246Q8—22no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—2922no—
253D4—2922no—
254D4—2922no—
255D4—2924no—
256D4—2924no—
257D4—2924no—
258D4—1462no—
259D4—1462no—
260C2×C4—21no—
261C2×C4—2922no—
262C2×C4—731no—
263C2×C4—731no—
264C2×C4—731no—
265C2×C4—1462no—
266C2×C4—1462no—
267C2×C4—1461no—
268C2×C4—1461no—
269C2×C4—1462no—
270C2×C4D4×D7311yescharacteristic
271C2×C4D4×D7311yescharacteristic
272C2×C4C23×D7311yescharacteristic
273C8—22no—
274C8—1462no—
275C22—42no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—41no—
281C22—2922no—
282C22—2922no—
283C22—2922no—
284C22—5848no—
285C22—732no—
286C22—1461no—
287C22—1461no—
288C22C2×D4×D7311yescharacteristic
289C4—21no—
290C4—21no—
291C4—41no—
292C4—2921no—
293C4—2921no—
294C4—732no—
295C4—1461no—
296C4—1461no—
297C4#1455663612yes—
298C2—82no—
299C2—81no—
300C2—5842no—
301C2—1462no—
302C2#1798915011yescharacteristic
303C2AGL(1,ℤ/8)×D7311yescharacteristic
304C2AGL(1,ℤ/8)×D7311yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#496×C73C2
socleC22×C73C22×D4
hypercenter#496D73
hypocenterC73#3217
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #127×D73C2semidirect
2. #17989116C2semidirect
3. #17988977C2semidirect
4. #17988952C2semidirect
5. #17988951C2semidirect
6. #17988785C2semidirect
7. #17988780C2semidirect
8. #17988736C2semidirect
9. #17988719C2semidirect
10. #17988642C2semidirect
11. #496×C73C2semidirect
15. C2×#11924015C22semidirect
16. C2×#11924013C22semidirect
22. #14556583C22semidirect
26. Q8×Dic73C22semidirect
27. D4×Dic73C22semidirect
31. #14556520C22semidirect
32. #14556513C22semidirect
33. #14556513C22semidirect
34. #14556512C22semidirect
35. #14556512C22semidirect
37. #14556502C22semidirect
38. C2×SD16×C73C22semidirect
39. C2×D8×C73C22semidirect
41. #124×C73C22semidirect
42. #97×C73C22semidirect
153. C73#3217semidirect
157. #496D73semidirect
169. C2×SD16C2×D73semidirect
171. C2×D8C2×D73semidirect
179. #124C2×D73semidirect
195. #97C2×D73semidirect