Finite Group Databasebeta

← #22620736

Subgroups — #22620736

Order 9344.

462 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 115 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⋊Q8×D73C211yesmaximal, characteristic, complemented
2C2×#14556566C211yesmaximal, characteristic
3#17988922C211yesmaximal, characteristic, complemented
4#17988902C212yesmaximal, complemented
5#17988836C212yesmaximal, complemented
6#17988834C212yesmaximal, complemented
7#17988740C211yesmaximal, characteristic, complemented
8#17988737C211yesmaximal, characteristic, complemented
9#17988727C212yesmaximal, complemented
10#17988549C211yesmaximal, characteristic, complemented
11#484×C73C211yesmaximal, characteristic, complemented
12C2×Q8×D73C2211yescharacteristic, complemented
13C22×D292—21no—
14C22×D292C2211yescharacteristic
15C22×C4×D73C2211yescharacteristic
16C2×#11924017C2212yescomplemented
17C2×#11924015C2212yescomplemented
18C2×#11924011C2212yescomplemented
19#14556571C2211yescharacteristic, complemented
20#14556570C2211yescharacteristic
21#14556569—21no—
22#14556568—21no—
23#14556567C2211yescharacteristic, complemented
24#14556566—21no—
25#14556566—21no—
26#14556566C2212yes—
27#14556566C2212yes—
28C4⋊C4×D73—21no—
29C4⋊C4×D73C2211yescharacteristic, complemented
30C2×#11924005—21no—
31C22⋊C4×D73—21no—
32#14556524C2212yescomplemented
33#14556514C2212yescomplemented
34#14556513C2212yescomplemented
35#14556511C2212yescomplemented
36C22⋊#10596831C2211yescharacteristic
37#14556503C2212yescomplemented
38C22⋊Q8×C73C2211yescharacteristic
39C2×C4⋊C4×C73C2211yescharacteristic
40#107×C73C2212yescomplemented
41#103×C73C2212yescomplemented
42C22⋊C8×C73C2211yescharacteristic
43C23×D73—21no—
44C23×D73D411yescharacteristic
45C2×Dic146—21no—
46C2×Dic146C2311yescharacteristic
47Q8×D73—42no—
48C2×D292—22no—
49C2×D292—22no—
50C2×D292—21no—
51C2×D292—42no—
52C2×D292—44no—
53C2×D292C2312yes—
54C22×Dic73D411yescharacteristic
55C2×C4×D73—21no—
56C2×C4×D73—21no—
57C2×C4×D73—42no—
58C2×C4×D73D411yescharacteristic
59C2×C4×D73D411yescharacteristic
60C2×C4×D73C2311yescharacteristic
61C2×C4×D73C2311yescharacteristic
62#11924017—42no—
63#11924015—42no—
64#11924011—42no—
65C2×#10596831C2312yes—
66C4⋊Dic73—21no—
67C4⋊Dic73C2311yescharacteristic
68#11924007—21no—
69#11924007—21no—
70#11924007—21no—
71C22⋊Dic73—21no—
72#11924005—21no—
73#11924005—21no—
74#11924005—22no—
75#11924005—22no—
76C2×Q8×C73C2311yescharacteristic
77C22×C4×C73—21no—
78C22×C4×C73C2311yescharacteristic
79C2×C8×C73C2312yes—
80C4⋊C4×C73—21no—
81C4⋊C4×C73—21no—
82C4⋊C4×C73C2312yes—
83C4⋊C4×C73C2311yescharacteristic
84C22⋊C4×C73—21no—
85C23×C73C2×D411yescharacteristic
86C22×D73—22no—
87C22×D73—22no—
88C22×D73—22no—
89C22×D73—41no—
90C22×D73—42no—
91C22×D73—42no—
92C22×D73—42no—
93C22×D73C2×D411yescharacteristic
94C22×D73C2×D411yescharacteristic
95Q8×C73—22no—
96C2×C4×C73—22no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—22no—
100C2×C4×C73—42no—
101C2×C4×C73C2×D411yescharacteristic
102C2×C4×C73C2411yescharacteristic
103C2×Dic73—21no—
104C2×Dic73—21no—
105C2×Dic73—41no—
106C2×Dic73C2×D411yescharacteristic
107C2×Dic73C2×D411yescharacteristic
108D292—22no—
109D292—22no—
110D292—42no—
111D292—44no—
112D292—44no—
113C4×D73—22no—
114C4×D73—22no—
115C4×D73—41no—
116C4×D73—41no—
117C4×D73—42no—
118C4×D73—42no—
119Dic146—22no—
120Dic146—42no—
121C8×C73—22no—
122#10596831—22no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73#13712yes—
126C22×C73C22×D411yescharacteristic
127C2×D73—21no—
128C2×D73—21no—
129C2×D73—21no—
130C2×D73—21no—
131C2×D73—21no—
132C2×D73—44no—
133C2×D73—42no—
134C2×D73—42no—
135C2×D73—42no—
136C2×D73—84no—
137C2×D73AGL(1,ℤ/8)12yes—
138C4×C73—21no—
139C4×C73—41no—
140C4×C73—41no—
141C4×C73—42no—
142C4×C73H(5,𝔽2)12yes—
143Dic73—21no—
144Dic73—21no—
145Dic73—41no—
146Dic73—41no—
147C2×C73—22no—
148C2×C73#53411yescharacteristic
149C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
150C2×C73C2×#13711yescharacteristic
151D73—22no—
152D73—41no—
153D73—82no—
154#3193—731nomaximal, Sylow
155C73#319311yescharacteristic, complemented, Sylow
156C2×C22⋊Q8—731no—
157C2×#121—731no—
158#484—731no—
159#484D7311yescharacteristic, complemented
160#480—732no—
161#468—732no—
162#468—732no—
163#450—731no—
164#450—731no—
165#448—732no—
166#407—731no—
167C22×Q8—731no—
168C22×D4—731no—
169C22×D4—1461no—
170C23×C4—731no—
171C2×SD16—732no—
172C2×SD16—732no—
173C2×M16—732no—
174C22⋊Q8—731no—
175C22⋊Q8—731no—
176C22⋊Q8—731no—
177C22⋊Q8—1461no—
178C22⋊Q8—1461no—
179C22⋊Q8C2×D7311yescharacteristic
180#121—732no—
181#121—732no—
182#121—1461no—
183#121—1461no—
184C2×C4⋊C4—731no—
185C2×C4⋊C4—1461no—
186C2×C4⋊C4C2×D7311yescharacteristic
187C2×C22⋊C4—1461no—
188C2×C22⋊C4—1461no—
189#107—732no—
190#107C2×D7312yescomplemented
191#103—732no—
192#103C2×D7312yescomplemented
193#102—732no—
194#102—732no—
195C22⋊C8—731no—
196C22⋊C8—732no—
197C22⋊C8C2×D7311yescharacteristic
198C24—731no—
199C24—1461no—
200C2×Q8—2922no—
201C2×Q8—731no—
202C2×Q8—1461no—
203C2×Q8C22×D7311yescharacteristic
204C2×D4—2922no—
205C2×D4—2924no—
206C2×D4—732no—
207C2×D4—1462no—
208C2×D4—1462no—
209C2×D4—1461no—
210C22×C4—21no—
211C22×C4—2922no—
212C22×C4—731no—
213C22×C4—731no—
214C22×C4—731no—
215C22×C4—731no—
216C22×C4—731no—
217C22×C4—1461no—
218C22×C4—1461no—
219C22×C4C22×D7311yescharacteristic
220SD16—2922no—
221SD16—2922no—
222M16—2922no—
223C2×C8—732no—
224C2×C8C22×D7312yes—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—731no—
228C4⋊C4—1461no—
229C4⋊C4—1461no—
230C4⋊C4—1461no—
231C4⋊C4—1461no—
232C4⋊C4C22×D7311yescharacteristic
233C4⋊C4C22×D7312yes—
234C22⋊C4—21no—
235C22⋊C4—1462no—
236C22⋊C4—1462no—
237C22⋊C4—1461no—
238C22⋊C4—1461no—
239C22⋊C4—1461no—
240C23—2921no—
241C23—2922no—
242C23—2922no—
243C23—2922no—
244C23—731no—
245C23—731no—
246C23—1462no—
247C23—1462no—
248C23—1462no—
249C23D4×D7311yescharacteristic
250Q8—22no—
251Q8—2922no—
252Q8—1462no—
253D4—2922no—
254D4—2924no—
255D4—2924no—
256D4—1462no—
257D4—1462no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—42no—
263C2×C4—2921no—
264C2×C4—2921no—
265C2×C4—2922no—
266C2×C4—2922no—
267C2×C4—2921no—
268C2×C4—731no—
269C2×C4—731no—
270C2×C4—1461no—
271C2×C4—1461no—
272C2×C4—1462no—
273C2×C4—1462no—
274C2×C4D4×D7311yescharacteristic
275C2×C4C23×D7311yescharacteristic
276C8—22no—
277C8—1462no—
278C22—21no—
279C22—21no—
280C22—2924no—
281C22—2922no—
282C22—2922no—
283C22—2922no—
284C22—5844no—
285C22—732no—
286C22—1461no—
287C22—1461no—
288C22—1461no—
289C22—1461no—
290C22—1461no—
291C22#1455662312yes—
292C22C2×D4×D7311yescharacteristic
293C4—21no—
294C4—41no—
295C4—42no—
296C4—41no—
297C4—2921no—
298C4—2921no—
299C4—1461no—
300C4—1461no—
301C4#1455663712yes—
302C2—22no—
303C2—2921no—
304C2—5842no—
305C2—1462no—
306C2#1798914811yescharacteristic
307C2AGL(1,ℤ/8)×D7311yescharacteristic
308C2C2×#1455662311yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#484×C73C2
socleC22×C73C22×D4
hypercenter#484D73
hypocenterC73#3193
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. C22⋊Q8×D73C2semidirect
3. #17988922C2semidirect
4. #17988902C2semidirect
5. #17988836C2semidirect
6. #17988834C2semidirect
7. #17988740C2semidirect
8. #17988737C2semidirect
9. #17988727C2semidirect
10. #17988549C2semidirect
11. #484×C73C2semidirect
12. C2×Q8×D73C22semidirect
16. C2×#11924017C22semidirect
17. C2×#11924015C22semidirect
18. C2×#11924011C22semidirect
19. #14556571C22semidirect
23. #14556567C22semidirect
29. C4⋊C4×D73C22semidirect
32. #14556524C22semidirect
33. #14556514C22semidirect
34. #14556513C22semidirect
35. #14556511C22semidirect
37. #14556503C22semidirect
40. #107×C73C22semidirect
41. #103×C73C22semidirect
155. C73#3193semidirect
159. #484D73semidirect
190. #107C2×D73semidirect
192. #103C2×D73semidirect