Finite Group Databasebeta

← #22615737

Subgroups — #22615737

Order 9344.

668 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 143 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×#14556571C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×D73C211yesmaximal, characteristic
3#17988522C211yesmaximal, characteristic, complemented
4#17988510C212yesmaximal, complemented
5#17988509C212yesmaximal, complemented
6#17988500C211yesmaximal, characteristic, complemented
7#17988499C211yesmaximal, characteristic, complemented
8#17988458C212yesmaximal, complemented
9#17988446C211yesmaximal, characteristic, complemented
10#17988442C212yesmaximal
11#399×C73C211yesmaximal, characteristic, complemented
12C24×D73C2211yescharacteristic
13C22×Dic146C2211yescharacteristic, complemented
14C22×C4×D73C2211yescharacteristic
15C22×C4×D73C2211yescharacteristic
16#14556571—24no—
17C2×C4⋊Dic73C2212yescomplemented
18C2×#11924007C2212yescomplemented
19C2×#11924007C2212yescomplemented
20C2×C22⋊Dic73C2211yescharacteristic
21C2×C22⋊Dic73C2212yes—
22C2×#11924005C2212yes—
23C2×#11924005C2212yes—
24C2×#11924005C2211yescharacteristic
25C2×#11924005C2211yescharacteristic
26C2×#11924005C2212yes—
27C2×#11924005C2212yes—
28C22⋊C4×D73—24no—
29#14556498C2211yescharacteristic, complemented
30#14556498C2212yescomplemented
31#14556498C2212yescomplemented
32C2×C4⋊C4×C73C2211yescharacteristic, complemented
33C2×C4⋊C4×C73C2212yescomplemented
34C2×C22⋊C4×C73C2211yescharacteristic
35C2×C22⋊C4×C73C2212yes—
36#95×C73C2211yescharacteristic, complemented
37C23×D73—22no—
38C23×D73—22no—
39C23×D73—42no—
40C23×D73—42no—
41C23×D73Q814yescomplemented
42C23×D73C2311yescharacteristic
43C23×D73C2311yescharacteristic
44C2×Dic146—22no—
45C2×Dic146—42no—
46C22×Dic73C2312yes—
47C22×Dic73C2312yes—
48C22×Dic73C2311yescharacteristic
49C22×Dic73C2311yescharacteristic
50C2×C4×D73—22no—
51C2×C4×D73—42no—
52C2×C4×D73—42no—
53C2×C4×D73D414yes—
54C4⋊Dic73—24no—
55#11924007—42no—
56#11924007—42no—
57C22⋊Dic73—22no—
58C22⋊Dic73—42no—
59#11924005—24no—
60#11924005—24no—
61#11924005—22no—
62#11924005—22no—
63#11924005—42no—
64#11924005—42no—
65C24×C73C2311yescharacteristic
66C22×C4×C73C2312yes—
67C22×C4×C73C2311yescharacteristic
68C22×C4×C73C2312yes—
69C22×C4×C73C2311yescharacteristic
70C4⋊C4×C73—22no—
71C4⋊C4×C73—42no—
72C22⋊C4×C73—22no—
73C22⋊C4×C73—42no—
74C23×C73—21no—
75C23×C73—21no—
76C23×C73—22no—
77C23×C73—22no—
78C23×C73C2×Q812yes—
79C23×C73C2411yescharacteristic
80C22×D73—22no—
81C22×D73—22no—
82C22×D73—21no—
83C22×D73—21no—
84C22×D73—22no—
85C22×D73—22no—
86C22×D73—28no—
87C22×D73—48no—
88C22×D73—44no—
89C22×D73—44no—
90C22×D73—48no—
91C22×D73C2×D412yes—
92C22×D73C2×Q812yes—
93C22×D73C2×Q812yes—
94C22×D73#4112yes—
95C2×C4×C73—22no—
96C2×C4×C73—21no—
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—22no—
104C2×C4×C73C2×D412yes—
105C2×Dic73—22no—
106C2×Dic73—21no—
107C2×Dic73—22no—
108C2×Dic73—22no—
109C2×Dic73—22no—
110C2×Dic73—22no—
111C2×Dic73—22no—
112C2×Dic73—22no—
113C2×Dic73—22no—
114C2×Dic73C2×D412yes—
115C4×D73—44no—
116C4×D73—44no—
117Dic146—44no—
118C22×C73—22no—
119C22×C73—24no—
120C22×C73—41no—
121C22×C73—41no—
122C22×C73—42no—
123C22×C73C22×D411yescharacteristic
124C22×C73C22×Q811yescharacteristic
125C22×C73C2×#4111yescharacteristic
126C22×C73H(5,𝔽2)12yes—
127C22×C73H(5,𝔽2)12yes—
128C2×D73—22no—
129C2×D73—22no—
130C2×D73—24no—
131C2×D73—24no—
132C2×D73—22no—
133C2×D73—24no—
134C2×D73—416no—
135C2×D73—41no—
136C2×D73—41no—
137C2×D73—42no—
138C2×D73C22⋊Q814yes—
139C4×C73—22no—
140C4×C73—41no—
141C4×C73—42no—
142C4×C73—42no—
143Dic73—22no—
144Dic73—42no—
145Dic73—41no—
146Dic73—42no—
147C2×C73—42no—
148C2×C73C2×C22⋊Q811yescharacteristic
149C2×C73#53411yescharacteristic
150C2×C73#53511yescharacteristic
151C2×C73#53812yes—
152C2×C73#54312yes—
153D73—24no—
154D73—42no—
155#2626—731nomaximal, Sylow
156C73#262611yescharacteristic, complemented, Sylow
157C2×C22⋊Q8—731no—
158C22×C22⋊C4—731no—
159#399—731no—
160#399D7311yescharacteristic, complemented
161#396—732no—
162#396—732no—
163#393—731no—
164#393—731no—
165#385—732no—
166#380—731no—
167#379—732no—
168C25—731no—
169C22×Q8—731no—
170C23×C4—731no—
171C23×C4—731no—
172C22⋊Q8—1464no—
173C2×C4⋊C4—732no—
174C2×C4⋊C4—732no—
175C2×C4⋊C4—732no—
176C2×C4⋊C4C2×D7311yescharacteristic, complemented
177C2×C4⋊C4C2×D7312yescomplemented
178C2×C22⋊C4—732no—
179C2×C22⋊C4—732no—
180C2×C22⋊C4—732no—
181C2×C22⋊C4—731no—
182C2×C22⋊C4—731no—
183C2×C22⋊C4—731no—
184C2×C22⋊C4—732no—
185C2×C22⋊C4—732no—
186C2×C22⋊C4—1464no—
187C2×C22⋊C4C2×D7311yescharacteristic
188C2×C22⋊C4C2×D7312yes—
189#95—731no—
190#95—732no—
191#95—732no—
192#95C2×D7311yescharacteristic, complemented
193C24—2922no—
194C24—2922no—
195C24—734no—
196C24—731no—
197C24—731no—
198C24—1462no—
199C24—1462no—
200C24C22×D7311yescharacteristic
201C2×Q8—2922no—
202C2×Q8—1462no—
203C22×C4—2922no—
204C22×C4—2922no—
205C22×C4—734no—
206C22×C4—731no—
207C22×C4—732no—
208C22×C4—732no—
209C22×C4—731no—
210C22×C4—1462no—
211C22×C4C22×D7311yescharacteristic
212C22×C4C22×D7312yes—
213C22×C4C22×D7312yes—
214C22×C4C22×D7311yescharacteristic
215C4⋊C4—22no—
216C4⋊C4—42no—
217C4⋊C4—2922no—
218C4⋊C4—2922no—
219C4⋊C4—1464no—
220C22⋊C4—22no—
221C22⋊C4—42no—
222C22⋊C4—2922no—
223C22⋊C4—2922no—
224C22⋊C4—2922no—
225C22⋊C4—1462no—
226C22⋊C4—1464no—
227C22⋊C4—1464no—
228C22⋊C4—1462no—
229C22⋊C4—1462no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—22no—
234C23—2928no—
235C23—2924no—
236C23—2924no—
237C23—2928no—
238C23—732no—
239C23—732no—
240C23—732no—
241C23—732no—
242C23—1468no—
243C23—1461no—
244C23—1461no—
245C23—1462no—
246C23—1462no—
247C23—1462no—
248C23—1462no—
249C23Q8×D7312yes—
250C23C23×D7311yescharacteristic
251Q8—2924no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—2924no—
262C2×C4—2924no—
263C2×C4—732no—
264C2×C4—1461no—
265C2×C4—1462no—
266C2×C4—1462no—
267C2×C4—1462no—
268C2×C4—1462no—
269C2×C4—1462no—
270C2×C4—1462no—
271C2×C4—1462no—
272C2×C4—1462no—
273C2×C4D4×D7312yes—
274C22—24no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—29216no—
280C22—2921no—
281C22—2921no—
282C22—2922no—
283C22—734no—
284C22—1464no—
285C22—1462no—
286C22—1462no—
287C22—1462no—
288C22—1464no—
289C22—1464no—
290C22C2×D4×D7311yescharacteristic
291C22C2×Q8×D7311yescharacteristic
292C22#41×D7311yescharacteristic
293C22#1455663612yes—
294C22#1455663712yes—
295C4—22no—
296C4—41no—
297C4—42no—
298C4—42no—
299C4—2922no—
300C4—2921no—
301C4—2922no—
302C4—1462no—
303C2—42no—
304C2—2922no—
305C2—1464no—
306C2C22⋊Q8×D7311yescharacteristic
307C2#1798914811yescharacteristic
308C2#1798915011yescharacteristic
309C2#1798917312yes—
310C2#1798921212yes—
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
Fitting#399×C73C2
hypercenter#399D73
hypocenterC73#2626
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#14556571C2semidirect
3. #17988522C2semidirect
4. #17988510C2semidirect
5. #17988509C2semidirect
6. #17988500C2semidirect
7. #17988499C2semidirect
8. #17988458C2semidirect
9. #17988446C2semidirect
11. #399×C73C2semidirect
13. C22×Dic146C22semidirect
17. C2×C4⋊Dic73C22semidirect
18. C2×#11924007C22semidirect
19. C2×#11924007C22semidirect
29. #14556498C22semidirect
30. #14556498C22semidirect
31. #14556498C22semidirect
32. C2×C4⋊C4×C73C22semidirect
33. C2×C4⋊C4×C73C22semidirect
36. #95×C73C22semidirect
41. C23×D73Q8semidirect
156. C73#2626semidirect
160. #399D73semidirect
176. C2×C4⋊C4C2×D73semidirect
177. C2×C4⋊C4C2×D73semidirect
192. #95C2×D73semidirect