Finite Group Databasebeta

← #22619189

Subgroups — #22619189

Order 9344.

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

normal subgroups only

This group has 754 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×#11924028C211yesmaximal, characteristic
2C2×#14556622C212yesmaximal, complemented
3#17988747C214yesmaximal, complemented
4#17988742C212yesmaximal, complemented
5#17988737C212yesmaximal, complemented
6#17988586C212yesmaximal, complemented
7#17988551C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C73C211yesmaximal, characteristic, complemented
9C2×#11924028—28no—
10C2×#11924028—21no—
11C2×#11924028—21no—
12C2×#11924028C2212yes—
13C2×#11924028C2212yes—
14C22×Dic146C2211yescharacteristic
15C22×D292C2211yescharacteristic
16C22×C22⋊D73—21no—
17C22×C4×D73—21no—
18#14556622—28no—
19C2×#11924018C2214yescomplemented
20C2×#11924015C2214yescomplemented
21C2×#11924012C2212yescomplemented
22#14556514C2214yescomplemented
23#14556511C2214yescomplemented
24C22⋊#10596831C2212yes—
25C22⋊#10596831C2212yescomplemented
26C22×Q8×C73C2211yescharacteristic, complemented
27C23×C4×C73C2211yescharacteristic
28C22⋊Q8×C73—21no—
29C22⋊Q8×C73—21no—
30C22⋊Q8×C73C2212yes—
31C22⋊Q8×C73C2212yescomplemented
32C2×C4⋊C4×C73—21no—
33C2×C4⋊C4×C73C2211yescharacteristic, complemented
34C2×C22⋊C4×C73—21no—
35C23×D73—21no—
36#11924028—28no—
37#11924028—28no—
38#11924028—24no—
39#11924028—24no—
40#11924028—24no—
41#11924028D418yes—
42C2×Dic146—21no—
43C2×Dic146—24no—
44C2×Dic146C2312yes—
45C2×D292—21no—
46C2×D292—24no—
47C2×D292C2312yes—
48C2×C22⋊D73—28no—
49C2×C22⋊D73—22no—
50C2×C22⋊D73—22no—
51C22×Dic73—21no—
52C2×C4×D73—28no—
53C2×C4×D73—22no—
54C2×C4×D73—22no—
55#11924018—28no—
56#11924015—28no—
57#11924012—24no—
58C2×#10596831C2314yes—
59C24×C73D411yescharacteristic
60C2×Q8×C73—21no—
61C2×Q8×C73—24no—
62C2×Q8×C73C2312yes—
63C22×C4×C73—24no—
64C22×C4×C73—21no—
65C22×C4×C73—21no—
66C22×C4×C73D411yescharacteristic
67C22×C4×C73D411yescharacteristic
68C22×C4×C73D411yescharacteristic
69C22×C4×C73C2311yescharacteristic
70C22×C4×C73C2311yescharacteristic
71C22×C4×C73C2311yescharacteristic
72C4⋊C4×C73—22no—
73C4⋊C4×C73—22no—
74C4⋊C4×C73—21no—
75C4⋊C4×C73C2312yes—
76C22⋊C4×C73—22no—
77C22⋊C4×C73—22no—
78C23×C73—22no—
79C23×C73—22no—
80C23×C73—41no—
81C23×C73C2×D411yescharacteristic
82C23×C73C2×D411yescharacteristic
83C23×C73C2×D411yescharacteristic
84C22×D73—24no—
85C22×D73—22no—
86C22×D73—42no—
87C22×D73—42no—
88Q8×C73—24no—
89Q8×C73—42no—
90C2×C4×C73—22no—
91C2×C4×C73—22no—
92C2×C4×C73—22no—
93C2×C4×C73—22no—
94C2×C4×C73—22no—
95C2×C4×C73—22no—
96C2×C4×C73—22no—
97C2×C4×C73—42no—
98C2×C4×C73—42no—
99C2×C4×C73C2×D414yes—
100C2×C4×C73C2×D411yescharacteristic
101C2×C4×C73C2×D411yescharacteristic
102C2×C4×C73C2×D411yescharacteristic
103C2×C4×C73C2411yescharacteristic
104C22⋊D73—28no—
105C22⋊D73—28no—
106C2×Dic73—24no—
107C2×Dic73—22no—
108D292—24no—
109D292—22no—
110D292C2×D414yes—
111C4×D73—28no—
112C4×D73—28no—
113Dic146—24no—
114Dic146—22no—
115Dic146C2×D414yes—
116#10596831—24no—
117C22×C73—21no—
118C22×C73—21no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73—21no—
123C22×C73—21no—
124C22×C73—44no—
125C22×C73#12012yes—
126C22×C73#13712yescomplemented
127C22×C73C22×D411yescharacteristic
128C2×D73—24no—
129C2×D73—48no—
130C2×D73—42no—
131C2×D73—42no—
132C4×C73—21no—
133C4×C73—21no—
134C4×C73—42no—
135C4×C73—42no—
136C4×C73#12012yes—
137C4×C73C22×D412yes—
138Dic73—24no—
139C2×C73—22no—
140C2×C73—22no—
141C2×C73—41no—
142C2×C73C2×#12011yescharacteristic
143C2×C73C2×#13711yescharacteristic
144C2×C73#57511yescharacteristic
145D73—44no—
146#3007—731nomaximal, Sylow
147C73#300711yescharacteristic, complemented, Sylow
148C22×#41—731no—
149C2×#137—732no—
150C2×C22⋊Q8D7311yescharacteristic, complemented
151#452—734no—
152#451—732no—
153#450—732no—
154#417—732no—
155#407—731no—
156C2×#41—732no—
157C2×#41—732no—
158C2×#41—1468no—
159C2×#41—1461no—
160C2×#41—1461no—
161C22×Q8—731no—
162C22×Q8C2×D7311yescharacteristic, complemented
163C22×D4—731no—
164C22×D4—1461no—
165C23×C4—1461no—
166C23×C4C2×D7311yescharacteristic
167#137—1468no—
168C2×Q16—734no—
169C2×SD16—734no—
170C2×M16—732no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D7312yes—
174C22⋊Q8C2×D7312yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D7311yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—734no—
179#102—734no—
180C22⋊C8—732no—
181C22⋊C8—732no—
182C24—1461no—
183C24C22⋊D7311yescharacteristic
184#41—738no—
185#41—1464no—
186#41—1464no—
187#41—1468no—
188#41—1468no—
189#41—1464no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—732no—
193C2×Q8—1464no—
194C2×Q8—1461no—
195C2×Q8C22×D7312yes—
196C2×D4—732no—
197C2×D4—1464no—
198C2×D4—1468no—
199C2×D4—1462no—
200C2×D4—1462no—
201C2×D4—1461no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—1468no—
206C22×C4—1461no—
207C22×C4—1462no—
208C22×C4—1462no—
209C22×C4C22⋊D7311yescharacteristic
210C22×C4C22⋊D7311yescharacteristic
211C22×C4C22⋊D7311yescharacteristic
212C22×C4C22×D7311yescharacteristic
213C22×C4C22×D7311yescharacteristic
214C22×C4C22×D7311yescharacteristic
215Q16—1468no—
216SD16—1468no—
217M16—1464no—
218C2×C8—734no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D7312yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—2922no—
229C23—2922no—
230C23—1464no—
231C23—1462no—
232C23C2×C22⋊D7311yescharacteristic
233C23C2×C22⋊D7311yescharacteristic
234C23C2×C22⋊D7311yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—734no—
238Q8—1464no—
239Q8—1462no—
240D4—734no—
241D4—1464no—
242D4—1468no—
243D4—1468no—
244D4—1462no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—1468no—
255C2×C4—1464no—
256C2×C4—1462no—
257C2×C4—1468no—
258C2×C4D4×D7314yes—
259C2×C4C2×C22⋊D7311yescharacteristic
260C2×C4C2×C22⋊D7311yescharacteristic
261C2×C4C2×C22⋊D7311yescharacteristic
262C2×C4C23×D7311yescharacteristic
263C8—1464no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—2928no—
273C22—2922no—
274C22—2922no—
275C22—1464no—
276C22#1455655912yes—
277C22#1455662212yescomplemented
278C22C22×C22⋊D7311yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—1464no—
284C4#1455655912yes—
285C4C2×D4×D7312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—2924no—
290C2C2×#1455655911yescharacteristic
291C2C2×#1455662211yescharacteristic
292C2#1798938511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D73
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
FittingC2×C22⋊Q8×C73C2
socleC22×C73C22×D4
hypercenterC2×C22⋊Q8D73
hypocenterC73#3007
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#14556622C2semidirect
3. #17988747C2semidirect
4. #17988742C2semidirect
5. #17988737C2semidirect
6. #17988586C2semidirect
7. #17988551C2semidirect
8. C2×C22⋊Q8×C73C2semidirect
19. C2×#11924018C22semidirect
20. C2×#11924015C22semidirect
21. C2×#11924012C22semidirect
22. #14556514C22semidirect
23. #14556511C22semidirect
25. C22⋊#10596831C22semidirect
26. C22×Q8×C73C22semidirect
31. C22⋊Q8×C73C22semidirect
33. C2×C4⋊C4×C73C22semidirect
126. C22×C73#137semidirect
147. C73#3007semidirect
150. C2×C22⋊Q8D73semidirect
162. C22×Q8C2×D73semidirect
174. C22⋊Q8C2×D73semidirect
176. C2×C4⋊C4C2×D73semidirect
277. C22#14556622semidirect