Finite Group Databasebeta

← #22615659

Subgroups — #22615659

Order 9344.

816 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 153 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×#11924027C211yesmaximal, characteristic
2C2×#14556569C211yesmaximal, characteristic, complemented
3C2×#14556568C212yesmaximal, complemented
4#17988508C211yesmaximal, characteristic, complemented
5#17988507C211yesmaximal, characteristic, complemented
6#17988503C212yesmaximal, complemented
7#17988502C212yesmaximal, complemented
8#17988501C211yesmaximal, characteristic, complemented
9#17988446C211yesmaximal, characteristic, complemented
10#17988445C212yesmaximal, complemented
11#394×C73C211yesmaximal, characteristic, complemented
12C2×#11924027—28no—
13C2×#11924027—24no—
14C22×Dic146C2211yescharacteristic
15C22×Dic146C2211yescharacteristic
16C22×C22⋊D73C2212yes—
17C23×Dic73C2212yes—
18C22×C4×D73C2211yescharacteristic
19#14556569—24no—
20#14556568—28no—
21C2×C4⋊Dic73C2212yes—
22C2×#11924007C2211yescharacteristic
23C2×#11924007C2212yes—
24C2×#11924007C2211yescharacteristic
25C2×#11924007C2212yes—
26C2×C22⋊Dic73C2212yes—
27C2×C22⋊Dic73C2212yes—
28C2×#11924005C2211yescharacteristic
29C2×#11924005C2211yescharacteristic
30C2×#11924005C2212yes—
31#14556498C2212yes—
32#14556498C2212yes—
33#14556498C2211yescharacteristic
34C22×D4×C73C2211yescharacteristic
35C2×C4⋊C4×C73C2211yescharacteristic
36C2×C22⋊C4×C73C2212yes—
37C2×C22⋊C4×C73C2212yes—
38#95×C73C2211yescharacteristic
39C23×D73C2311yescharacteristic
40#11924027—232no—
41C2×Dic146—22no—
42C2×Dic146—24no—
43C2×Dic146—24no—
44C2×Dic146—22no—
45C2×C22⋊D73—24no—
46C2×C22⋊D73—24no—
47C2×C22⋊D73—24no—
48C22×Dic73—24no—
49C22×Dic73—24no—
50C22×Dic73D418yes—
51C22×Dic73C2311yescharacteristic
52C22×Dic73C2312yes—
53C22×Dic73C2311yescharacteristic
54C22×Dic73C2312yes—
55C22×Dic73C2311yescharacteristic
56C2×C4×D73—24no—
57C2×C4×D73D414yes—
58C4⋊Dic73—24no—
59#11924007—22no—
60#11924007—24no—
61#11924007—24no—
62#11924007—22no—
63C22⋊Dic73—24no—
64C22⋊Dic73—42no—
65#11924005—22no—
66#11924005—22no—
67#11924005—42no—
68C24×C73C2312yes—
69C2×D4×C73—24no—
70C2×D4×C73—22no—
71C22×C4×C73C2312yes—
72C22×C4×C73C2311yescharacteristic
73C22×C4×C73C2311yescharacteristic
74C22×C4×C73C2311yescharacteristic
75C4⋊C4×C73—22no—
76C22⋊C4×C73—24no—
77C22⋊C4×C73—42no—
78C23×C73—22no—
79C23×C73—22no—
80C23×C73—22no—
81C23×C73—22no—
82C23×C73—22no—
83C23×C73—22no—
84C23×C73C2×D414yes—
85C23×C73C2411yescharacteristic
86C22×D73—22no—
87C22×D73—21no—
88C22×D73—22no—
89C22×D73—21no—
90C22×D73C2×D412yes—
91D4×C73—28no—
92C2×C4×C73—22no—
93C2×C4×C73—22no—
94C2×C4×C73—22no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—21no—
101C2×C4×C73C2×D412yes—
102C22⋊D73—216no—
103C2×Dic73—22no—
104C2×Dic73—22no—
105C2×Dic73—216no—
106C2×Dic73—21no—
107C2×Dic73—21no—
108C2×Dic73—22no—
109C2×Dic73—22no—
110C2×Dic73—22no—
111C2×Dic73—21no—
112C2×Dic73—22no—
113C2×Dic73—21no—
114C2×Dic73C2×D414yes—
115C2×Dic73C2×D414yes—
116C2×Dic73C2×D412yes—
117C4×D73—28no—
118Dic146—28no—
119Dic146—44no—
120C22×C73—24no—
121C22×C73—24no—
122C22×C73—24no—
123C22×C73—42no—
124C22×C73—42no—
125C22×C73—42no—
126C22×C73—42no—
127C22×C73C22×D411yescharacteristic
128C22×C73C22×D412yes—
129C22×C73H(5,𝔽2)11yescharacteristic
130C22×C732−1+412yes—
131C22×C732−1+411yescharacteristic
132C2×D73—22no—
133C2×D73—22no—
134C2×D73—22no—
135C2×D73—42no—
136C2×D73—42no—
137C4×C73—22no—
138C4×C73—41no—
139C4×C73—42no—
140C4×C73—41no—
141Dic73—24no—
142Dic73—41no—
143Dic73—42no—
144Dic73—41no—
145Dic73#12014yes—
146C2×C73—44no—
147C2×C73C2×#12011yescharacteristic
148C2×C73#53611yescharacteristic
149C2×C73#53612yes—
150C2×C73#53712yes—
151C2×C73#53711yescharacteristic
152D73—42no—
153#2616—731nomaximal, Sylow
154C73#261611yescharacteristic, complemented, Sylow
155C22×#41—731no—
156C2×C22⋊Q8—731no—
157C2×C22⋊Q8—732no—
158#395—731no—
159#395—731no—
160#394—731no—
161#394—732no—
162#394—732no—
163#394D7311yescharacteristic, complemented
164#380—732no—
165#380—731no—
166C2×#41—1468no—
167C2×#41—1464no—
168C22×Q8—731no—
169C22×Q8—731no—
170C22×D4—732no—
171C22×D4C2×D7311yescharacteristic
172C23×C4—732no—
173C23×C4—731no—
174C22⋊Q8—1468no—
175C22⋊Q8—1464no—
176C2×C4⋊C4—731no—
177C2×C4⋊C4—732no—
178C2×C4⋊C4—732no—
179C2×C4⋊C4—731no—
180C2×C4⋊C4—732no—
181C2×C4⋊C4C2×D7311yescharacteristic
182C2×C22⋊C4—732no—
183C2×C22⋊C4—732no—
184C2×C22⋊C4—732no—
185C2×C22⋊C4—731no—
186C2×C22⋊C4—731no—
187C2×C22⋊C4C2×D7312yes—
188C2×C22⋊C4C2×D7312yes—
189#95—732no—
190#95—732no—
191#95—731no—
192#95C2×D7311yescharacteristic
193C24—731no—
194C24C22×D7312yes—
195#41—14632no—
196C2×Q8—1464no—
197C2×Q8—1462no—
198C2×Q8—1464no—
199C2×Q8—1462no—
200C2×D4—24no—
201C2×D4—22no—
202C2×D4—1464no—
203C2×D4—1464no—
204C2×D4—1464no—
205C22×C4—731no—
206C22×C4—738no—
207C22×C4—732no—
208C22×C4—734no—
209C22×C4—731no—
210C22×C4—732no—
211C22×C4—731no—
212C22×C4—1464no—
213C22×C4—1464no—
214C22×C4—1464no—
215C22×C4C22×D7311yescharacteristic
216C22×C4C22×D7312yes—
217C22×C4C22×D7311yescharacteristic
218C22×C4C22×D7311yescharacteristic
219C4⋊C4—22no—
220C4⋊C4—1464no—
221C4⋊C4—1464no—
222C4⋊C4—1462no—
223C4⋊C4—1464no—
224C4⋊C4—1462no—
225C22⋊C4—24no—
226C22⋊C4—42no—
227C22⋊C4—2922no—
228C22⋊C4—2922no—
229C22⋊C4—1462no—
230C22⋊C4—1462no—
231C22⋊C4—1464no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—732no—
239C23—1462no—
240C23—1461no—
241C23—1461no—
242C23—1462no—
243C23D4×D7314yes—
244C23C23×D7311yescharacteristic
245Q8—2924no—
246Q8—1468no—
247D4—28no—
248D4—14616no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—732no—
259C2×C4—734no—
260C2×C4—734no—
261C2×C4—1468no—
262C2×C4—1462no—
263C2×C4—1461no—
264C2×C4—1461no—
265C2×C4—1462no—
266C2×C4—1462no—
267C2×C4—1461no—
268C2×C4—1462no—
269C2×C4—1461no—
270C2×C4—14616no—
271C2×C4—1462no—
272C2×C4—1462no—
273C2×C4D4×D7312yes—
274C22—24no—
275C22—24no—
276C22—24no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—2922no—
282C22—2922no—
283C22—1462no—
284C22—1462no—
285C22—1462no—
286C22C2×D4×D7311yescharacteristic
287C22C2×D4×D7312yes—
288C22#1455663611yescharacteristic
289C22#1455663812yes—
290C22#1455663811yescharacteristic
291C4—22no—
292C4—42no—
293C4—41no—
294C4—41no—
295C4—2922no—
296C4—2921no—
297C4—2921no—
298C4—734no—
299C4—1464no—
300C2—44no—
301C2—2922no—
302C2#120×D7311yescharacteristic
303C2#1798915711yescharacteristic
304C2#1798915912yes—
305C2#1798916412yes—
306C2#1798916611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
Fitting#394×C73C2
hypercenter#394D73
hypocenterC73#2616
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#14556569C2semidirect
3. C2×#14556568C2semidirect
4. #17988508C2semidirect
5. #17988507C2semidirect
6. #17988503C2semidirect
7. #17988502C2semidirect
8. #17988501C2semidirect
9. #17988446C2semidirect
10. #17988445C2semidirect
11. #394×C73C2semidirect
154. C73#2616semidirect
163. #394D73semidirect