Finite Group Databasebeta

← #22620913

Subgroups — #22620913

Order 9344.

444 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 111 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#17989135C211yesmaximal, characteristic, complemented
2#124×D73C211yesmaximal, characteristic, complemented
3#17988974C211yesmaximal, characteristic, complemented
4#17988947C211yesmaximal, characteristic, complemented
5#17988946C211yesmaximal, characteristic, complemented
6#17988804C212yesmaximal, complemented
7#17988800C212yesmaximal, complemented
8#17988724C212yesmaximal, complemented
9#17988719C212yesmaximal, complemented
10#17988642C211yesmaximal, characteristic, complemented
11#496×C73C211yesmaximal, characteristic, complemented
12C2×#11924027—21no—
13C2×Q8×D73C2211yescharacteristic
14C2×D4×D73C2211yescharacteristic
15C2×#11924016C2212yescomplemented
16C2×#11924013C2212yes—
17#14556591C2211yescharacteristic, complemented
18#14556580C2211yescharacteristic
19#14556579C2211yescharacteristic
20#14556578—21no—
21#14556578C2211yescharacteristic
22#14556577—21no—
23#14556577—21no—
24#14556562—21no—
25#14556562C2212yes—
26D4×Dic73—21no—
27D4×Dic73C2212yescomplemented
28C22⋊C4×D73—21no—
29C22⋊C4×D73—21no—
30C42×D73C2211yescharacteristic
31#14556520C2212yescomplemented
32#14556516C2212yescomplemented
33#14556515C2212yescomplemented
34#14556513C2212yes—
35#14556512C2212yes—
36#14556503C2212yes—
37#14556502C2211yescharacteristic, complemented
38C2×SD16×C73C2212yescomplemented
39C2×D8×C73C2212yes—
40#127×C73C2211yescharacteristic, complemented
41#124×C73C2211yescharacteristic
42#97×C73C2211yescharacteristic, complemented
43C23×D73—21no—
44#11924027—44no—
45C2×Dic146—21no—
46C2×Dic146—21no—
47C2×Dic146C2311yescharacteristic
48Q8×D73—42no—
49C2×D292C2311yescharacteristic
50C2×C22⋊D73—22no—
51C2×C22⋊D73—21no—
52D4×D73—42no—
53C22×Dic73—22no—
54C2×C4×D73—21no—
55C2×C4×D73D411yescharacteristic
56C2×C4×D73D411yescharacteristic
57C2×C4×D73C2311yescharacteristic
58#11924016—42no—
59#11924013—42no—
60C2×#10596831C2312yes—
61C4⋊Dic73—21no—
62C4⋊Dic73C2312yes—
63C22⋊Dic73—21no—
64C22⋊Dic73—21no—
65C22⋊Dic73—22no—
66C22⋊Dic73—22no—
67#11924005—21no—
68#11924005—21no—
69#11924005—21no—
70#11924005—21no—
71C4×Dic73D411yescharacteristic
72C4×Dic73D411yescharacteristic
73C4×Dic73C2311yescharacteristic
74C2×Q8×C73C2311yescharacteristic
75C2×D4×C73—21no—
76C2×D4×C73—21no—
77C2×D4×C73C2311yescharacteristic
78C2×D4×C73C2312yes—
79SD16×C73—42no—
80D8×C73—42no—
81C2×C8×C73C2312yes—
82C22⋊C4×C73—21no—
83C22⋊C4×C73—21no—
84C42×C73C2311yescharacteristic
85C23×C73—21no—
86C23×C73—22no—
87C22×D73—21no—
88C22×D73—42no—
89C22×D73—42no—
90C22×D73—42no—
91C22×D73C2×D411yescharacteristic
92Q8×C73—22no—
93D4×C73—22no—
94D4×C73—22no—
95D4×C73—22no—
96D4×C73—42no—
97D4×C73—42no—
98C2×C4×C73—21no—
99C2×C4×C73C2×D411yescharacteristic
100C2×C4×C73C2×D411yescharacteristic
101C2×C4×C73C2411yescharacteristic
102C22⋊D73—42no—
103C22⋊D73—44no—
104C2×Dic73—21no—
105C2×Dic73—22no—
106C2×Dic73—44no—
107C2×Dic73C2×D411yescharacteristic
108C2×Dic73C2×D411yescharacteristic
109C2×Dic73C2×D411yescharacteristic
110D292—22no—
111C4×D73—22no—
112C4×D73—22no—
113C4×D73—41no—
114C4×D73—42no—
115Dic146—22no—
116Dic146—42no—
117Dic146—42no—
118C8×C73—22no—
119#10596831—22no—
120C22×C73—41no—
121C22×C73—42no—
122C22×C73—42no—
123C22×C73—41no—
124C22×C73—42no—
125C22×C73—41no—
126C22×C73C22×D411yescharacteristic
127C2×D73—21no—
128C2×D73—21no—
129C2×D73—41no—
130C2×D73—41no—
131C2×D73—41no—
132C2×D73—84no—
133C2×D73AGL(1,ℤ/8)12yes—
134C4×C73—21no—
135C4×C73—21no—
136C4×C73—41no—
137C4×C73H(5,𝔽2)12yes—
138Dic73—21no—
139Dic73—21no—
140Dic73—21no—
141Dic73—41no—
142Dic73—42no—
143C2×C73—81no—
144C2×C73—82no—
145C2×C73#53511yescharacteristic
146C2×C73C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C73#57511yescharacteristic
148D73—22no—
149D73—81no—
150#3213—731nomaximal, Sylow
151C73#321311yescharacteristic, complemented, Sylow
152#532—731no—
153C2×#124—731no—
154#496—731no—
155#496D7311yescharacteristic, complemented
156#489—731no—
157#489—731no—
158#463—732no—
159#463—732no—
160#448—732no—
161#447—732no—
162#432—731no—
163C2×#41—1461no—
164C22×Q8—731no—
165C22×D4—731no—
166C2×SD16—732no—
167C2×SD16C2×D7312yescomplemented
168C2×D8—732no—
169C2×D8C2×D7312yes—
170C4⋊Q8—731no—
171#127C2×D7311yescharacteristic, complemented
172#124—731no—
173#124—731no—
174#124—731no—
175#124—1461no—
176#124—1461no—
177#124—1461no—
178#124C2×D7311yescharacteristic
179#121—732no—
180#121—1461no—
181C4×D4—732no—
182C4×D4—1461no—
183C2×C22⋊C4—1461no—
184C2×C22⋊C4—1461no—
185C2×C42—731no—
186C4⋊C8—732no—
187#103—732no—
188#103—732no—
189#102—732no—
190#102—732no—
191C22⋊C8—732no—
192#97—731no—
193#97C2×D7311yescharacteristic, complemented
194C24—1461no—
195#41—2924no—
196C2×Q8—2922no—
197C2×Q8—731no—
198C2×Q8—1461no—
199C2×Q8—1461no—
200C2×Q8C22×D7311yescharacteristic
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—2922no—
204C2×D4—731no—
205C2×D4—1461no—
206C2×D4—1462no—
207C2×D4C22×D7311yescharacteristic
208C2×D4C22×D7312yes—
209C22×C4—731no—
210C22×C4—731no—
211C22×C4—731no—
212C22×C4—1461no—
213C22×C4—1462no—
214SD16—42no—
215SD16—2922no—
216D8—42no—
217D8—2922no—
218C2×C8—732no—
219C2×C8C22×D7312yes—
220C4⋊C4—732no—
221C4⋊C4—1461no—
222C22⋊C4—21no—
223C22⋊C4—21no—
224C22⋊C4—1461no—
225C22⋊C4—1461no—
226C22⋊C4—1461no—
227C22⋊C4—1461no—
228C22⋊C4—1462no—
229C22⋊C4—1462no—
230C22⋊C4—1461no—
231C22⋊C4—1461no—
232C42—731no—
233C42—731no—
234C42—731no—
235C42C22×D7311yescharacteristic
236C23—21no—
237C23—22no—
238C23—2922no—
239C23—2922no—
240C23—2922no—
241C23—731no—
242C23—1461no—
243Q8—22no—
244Q8—2922no—
245Q8—2922no—
246Q8—1462no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—42no—
251D4—42no—
252D4—2924no—
253D4—2922no—
254D4—1462no—
255C2×C4—21no—
256C2×C4—2921no—
257C2×C4—2922no—
258C2×C4—2924no—
259C2×C4—731no—
260C2×C4—731no—
261C2×C4—731no—
262C2×C4—1462no—
263C2×C4—1461no—
264C2×C4—1462no—
265C2×C4—1462no—
266C2×C4D4×D7311yescharacteristic
267C2×C4D4×D7311yescharacteristic
268C2×C4C23×D7311yescharacteristic
269C8—22no—
270C8—1462no—
271C22—42no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—41no—
277C22—2921no—
278C22—2921no—
279C22—2921no—
280C22—5844no—
281C22—732no—
282C22—1461no—
283C22—1461no—
284C22C2×D4×D7311yescharacteristic
285C4—21no—
286C4—21no—
287C4—41no—
288C4—2921no—
289C4—2922no—
290C4—1461no—
291C4—1461no—
292C4—1461no—
293C4#1455663612yes—
294C2—81no—
295C2—82no—
296C2—5841no—
297C2—1462no—
298C2#1798915011yescharacteristic
299C2AGL(1,ℤ/8)×D7311yescharacteristic
300C2#1798938311yescharacteristic
subgroupG/H
centerC22C2×D4×D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#496×C73C2
socleC22×C73C22×D4
hypercenter#496D73
hypocenterC73#3213
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. #17989135C2semidirect
2. #124×D73C2semidirect
3. #17988974C2semidirect
4. #17988947C2semidirect
5. #17988946C2semidirect
6. #17988804C2semidirect
7. #17988800C2semidirect
8. #17988724C2semidirect
9. #17988719C2semidirect
10. #17988642C2semidirect
11. #496×C73C2semidirect
15. C2×#11924016C22semidirect
17. #14556591C22semidirect
27. D4×Dic73C22semidirect
31. #14556520C22semidirect
32. #14556516C22semidirect
33. #14556515C22semidirect
37. #14556502C22semidirect
38. C2×SD16×C73C22semidirect
40. #127×C73C22semidirect
42. #97×C73C22semidirect
151. C73#3213semidirect
155. #496D73semidirect
167. C2×SD16C2×D73semidirect
171. #127C2×D73semidirect
193. #97C2×D73semidirect