Finite Group Databasebeta

← #22615742

Subgroups — #22615742

Order 9344.

668 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#14556568C211yesmaximal, characteristic, complemented
2C22×#11924005C211yesmaximal, characteristic
3#17988522C212yesmaximal, complemented
4#17988511C212yesmaximal, complemented
5#17988509C212yesmaximal, complemented
6#17988500C212yesmaximal, complemented
7#17988460C211yesmaximal, characteristic, complemented
8#17988445C211yesmaximal, characteristic, complemented
9#17988442C211yesmaximal, characteristic
10#17988442C211yesmaximal, characteristic
11#385×C73C211yesmaximal, characteristic, complemented
12C24×D73C2211yescharacteristic
13C22×Dic146C2211yescharacteristic, complemented
14C23×Dic73C2211yescharacteristic
15#14556568—24no—
16C2×C4⋊Dic73C2211yescharacteristic, complemented
17C2×#11924007C2211yescharacteristic, complemented
18C2×#11924007C2211yescharacteristic, complemented
19C2×#11924007C2212yescomplemented
20C2×#11924007C2212yescomplemented
21C2×C22⋊Dic73C2211yescharacteristic
22C2×C22⋊Dic73C2211yescharacteristic
23C2×#11924005—24no—
24C2×#11924005C2212yes—
25C2×#11924005C2212yes—
26C2×#11924005C2212yes—
27C2×#11924005C2212yes—
28C2×#11924005C2212yes—
29C2×#11924005C2212yes—
30#14556498C2212yescomplemented
31#14556498C2212yescomplemented
32C23×C4×C73C2211yescharacteristic
33C2×C4⋊C4×C73C2212yescomplemented
34C2×C22⋊C4×C73C2211yescharacteristic
35C2×C22⋊C4×C73C2211yescharacteristic
36#95×C73C2212yescomplemented
37C23×D73—22no—
38C23×D73—22no—
39C23×D73—42no—
40C23×D73—42no—
41C23×D73Q814yescomplemented
42C23×D73C2312yes—
43C2×Dic146—22no—
44C2×Dic146—42no—
45C22×Dic73—42no—
46C22×Dic73D414yes—
47C22×Dic73C2312yes—
48C22×Dic73C2311yescharacteristic
49C22×Dic73C2311yescharacteristic
50C22×Dic73C2311yescharacteristic
51C22×Dic73C2311yescharacteristic
52C4⋊Dic73—22no—
53#11924007—22no—
54#11924007—22no—
55#11924007—42no—
56#11924007—42no—
57C22⋊Dic73—22no—
58C22⋊Dic73—22no—
59#11924005—24no—
60#11924005—24no—
61#11924005—42no—
62#11924005—42no—
63#11924005—42no—
64#11924005—42no—
65C24×C73C2311yescharacteristic
66C22×C4×C73—22no—
67C22×C4×C73—42no—
68C22×C4×C73C2311yescharacteristic
69C22×C4×C73C2312yes—
70C22×C4×C73C2312yes—
71C22×C4×C73C2311yescharacteristic
72C4⋊C4×C73—42no—
73C22⋊C4×C73—22no—
74C22⋊C4×C73—22no—
75C23×C73—22no—
76C23×C73—22no—
77C23×C73C2×D412yes—
78C23×C73C2×Q812yes—
79C23×C73#4112yes—
80C23×C73C2411yescharacteristic
81C22×D73—28no—
82C22×D73—22no—
83C22×D73—22no—
84C22×D73—22no—
85C22×D73—22no—
86C22×D73—22no—
87C22×D73—22no—
88C22×D73—48no—
89C22×D73—44no—
90C22×D73—44no—
91C22×D73—48no—
92C22×D73C2×Q814yes—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—22no—
96C2×C4×C73—22no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—22no—
101C2×C4×C73—22no—
102C2×C4×C73—22no—
103C2×C4×C73—44no—
104C2×Dic73—22no—
105C2×Dic73—22no—
106C2×Dic73—21no—
107C2×Dic73—21no—
108C2×Dic73—22no—
109C2×Dic73—22no—
110C2×Dic73—22no—
111C2×Dic73—22no—
112C2×Dic73—22no—
113C2×Dic73—44no—
114C2×Dic73C2×D412yes—
115C2×Dic73C2×D412yes—
116Dic146—44no—
117C22×C73—22no—
118C22×C73—22no—
119C22×C73—24no—
120C22×C73—24no—
121C22×C73C22⋊Q814yes—
122C22×C73C22×D411yescharacteristic
123C22×C73C22×Q811yescharacteristic
124C22×C73C2×#4111yescharacteristic
125C22×C73H(5,𝔽2)12yes—
126C22×C73H(5,𝔽2)12yes—
127C2×D73—24no—
128C2×D73—24no—
129C2×D73—24no—
130C2×D73—416no—
131C2×D73—42no—
132C2×D73—42no—
133C2×D73—42no—
134C2×D73—42no—
135C4×C73—41no—
136C4×C73—41no—
137C4×C73—42no—
138C4×C73—42no—
139Dic73—22no—
140Dic73—22no—
141Dic73—41no—
142Dic73—41no—
143Dic73—42no—
144C2×C73—24no—
145C2×C73C2×C22⋊Q811yescharacteristic
146C2×C73#53411yescharacteristic
147C2×C73#53511yescharacteristic
148C2×C73#53812yes—
149C2×C73#54312yes—
150D73—44no—
151#2626—731nomaximal, Sylow
152C73#262611yescharacteristic, complemented, Sylow
153C2×C22⋊Q8—731no—
154C22×C22⋊C4—731no—
155#399—732no—
156#396—732no—
157#396—732no—
158#393—732no—
159#385—731no—
160#385D7311yescharacteristic, complemented
161#380—731no—
162#379—731no—
163#379—731no—
164C25—731no—
165C22×Q8—731no—
166C23×C4—731no—
167C23×C4C2×D7311yescharacteristic
168C22⋊Q8—1464no—
169C2×C4⋊C4—732no—
170C2×C4⋊C4—732no—
171C2×C4⋊C4—731no—
172C2×C4⋊C4—731no—
173C2×C4⋊C4—731no—
174C2×C4⋊C4C2×D7312yescomplemented
175C2×C22⋊C4—732no—
176C2×C22⋊C4—732no—
177C2×C22⋊C4—732no—
178C2×C22⋊C4—732no—
179C2×C22⋊C4—731no—
180C2×C22⋊C4—731no—
181C2×C22⋊C4—732no—
182C2×C22⋊C4—732no—
183C2×C22⋊C4—1464no—
184C2×C22⋊C4C2×D7311yescharacteristic
185C2×C22⋊C4C2×D7311yescharacteristic
186#95—732no—
187#95—732no—
188#95C2×D7312yescomplemented
189C24—2922no—
190C24—2922no—
191C24—734no—
192C24—732no—
193C24—1462no—
194C24—1462no—
195C24C22×D7311yescharacteristic
196C2×Q8—2922no—
197C2×Q8—1462no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—2922no—
201C22×C4—734no—
202C22×C4—731no—
203C22×C4—731no—
204C22×C4—732no—
205C22×C4—731no—
206C22×C4—731no—
207C22×C4C22×D7312yes—
208C22×C4C22×D7312yes—
209C22×C4C22×D7311yescharacteristic
210C22×C4C22×D7311yescharacteristic
211C4⋊C4—42no—
212C4⋊C4—2922no—
213C4⋊C4—2922no—
214C4⋊C4—1462no—
215C4⋊C4—1462no—
216C4⋊C4—1462no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—2922no—
220C22⋊C4—2922no—
221C22⋊C4—2922no—
222C22⋊C4—2922no—
223C22⋊C4—1464no—
224C22⋊C4—1464no—
225C22⋊C4—1462no—
226C22⋊C4—1462no—
227C23—22no—
228C23—22no—
229C23—2924no—
230C23—2924no—
231C23—2928no—
232C23—2928no—
233C23—734no—
234C23—1468no—
235C23—1462no—
236C23—1462no—
237C23—1462no—
238C23—1462no—
239C23—1462no—
240C23—1462no—
241C23D4×D7312yes—
242C23Q8×D7312yes—
243C23#1192402812yes—
244C23C23×D7311yescharacteristic
245Q8—2924no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—44no—
257C2×C4—2924no—
258C2×C4—732no—
259C2×C4—732no—
260C2×C4—1461no—
261C2×C4—1461no—
262C2×C4—1462no—
263C2×C4—1462no—
264C2×C4—1462no—
265C2×C4—1462no—
266C2×C4—1462no—
267C2×C4—1462no—
268C2×C4—1462no—
269C22—22no—
270C22—22no—
271C22—24no—
272C22—24no—
273C22—29216no—
274C22—2922no—
275C22—2922no—
276C22—2922no—
277C22—2922no—
278C22—1464no—
279C22—1464no—
280C22—1464no—
281C22#1455656914yes—
282C22C2×D4×D7311yescharacteristic
283C22C2×Q8×D7311yescharacteristic
284C22C2×#1192402811yescharacteristic
285C22#1455663612yes—
286C22#1455663712yes—
287C4—41no—
288C4—42no—
289C4—41no—
290C4—42no—
291C4—2921no—
292C4—2921no—
293C4—2922no—
294C4—1462no—
295C4—1462no—
296C2—24no—
297C2—2924no—
298C2C2×#1455656911yescharacteristic
299C2#1798914811yescharacteristic
300C2#1798915211yescharacteristic
301C2#1798917012yes—
302C2#1798921212yes—
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
Fitting#385×C73C2
hypercenter#385D73
hypocenterC73#2626
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×#14556568C2semidirect
3. #17988522C2semidirect
4. #17988511C2semidirect
5. #17988509C2semidirect
6. #17988500C2semidirect
7. #17988460C2semidirect
8. #17988445C2semidirect
11. #385×C73C2semidirect
13. C22×Dic146C22semidirect
16. C2×C4⋊Dic73C22semidirect
17. C2×#11924007C22semidirect
18. C2×#11924007C22semidirect
19. C2×#11924007C22semidirect
20. C2×#11924007C22semidirect
30. #14556498C22semidirect
31. #14556498C22semidirect
33. C2×C4⋊C4×C73C22semidirect
36. #95×C73C22semidirect
41. C23×D73Q8semidirect
152. C73#2626semidirect
160. #385D73semidirect
174. C2×C4⋊C4C2×D73semidirect
188. #95C2×D73semidirect