Finite Group Databasebeta

← #22619309

Subgroups — #22619309

Order 9344.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#17989423C211yesmaximal, characteristic
2C2×#14556622C212yesmaximal, complemented
3#17988769C214yesmaximal, complemented
4#17988764C214yesmaximal, complemented
5#17988605C212yesmaximal, complemented
6#17988571C211yesmaximal, characteristic, complemented
7#536×C73C211yesmaximal, characteristic, complemented
8#14556639C2212yes—
9#14556638—22no—
10#14556637—22no—
11#14556636—22no—
12#14556636C2212yes—
13C2×#11924028—22no—
14C2×#11924028C2212yes—
15#41×D73—24no—
16#14556622—24no—
17#14556622C2218yescomplemented
18C2×#11924018—22no—
19C2×#11924015—22no—
20C2×#11924012C2212yescomplemented
21#14556519C2218yescomplemented
22#14556510C2212yescomplemented
23#14556508C2212yescomplemented
24C2×#41×C73C2211yescharacteristic
25C22×Q8×C73C2211yescharacteristic, complemented
26C4⋊Q8×C73C2212yescomplemented
27#124×C73C2212yescomplemented
28#123×C73—22no—
29C22⋊Q8×C73—22no—
30#117×C73C2211yescharacteristic, complemented
31#11924029—24no—
32#11924028—24no—
33#11924028—24no—
34#11924028—22no—
35#11924028—24no—
36#11924028—22no—
37#11924028D414yes—
38#11924028C2314yes—
39#11924027—28no—
40#11924027—24no—
41C2×Dic146—22no—
42C2×Dic146D412yes—
43Q8×D73—24no—
44C2×D292—22no—
45C2×D292D412yes—
46C2×C22⋊D73—24no—
47C2×C22⋊D73—22no—
48D4×D73—28no—
49D4×D73—24no—
50C2×C4×D73—24no—
51C2×C4×D73—22no—
52#11924018—28no—
53#11924015—28no—
54#11924012—22no—
55#11924012C2314yes—
56C2×#10596831—22no—
57#41×C73—22no—
58#41×C73—22no—
59C2×Q8×C73—21no—
60C2×Q8×C73—22no—
61C2×Q8×C73—22no—
62C2×Q8×C73C2312yes—
63C2×Q8×C73C2311yescharacteristic
64C2×D4×C73D412yes—
65C2×D4×C73C2311yescharacteristic
66C22×C4×C73—21no—
67C22×C4×C73D412yes—
68C22×C4×C73C2311yescharacteristic
69C4⋊C4×C73—21no—
70C4⋊C4×C73—24no—
71C22⋊C4×C73—21no—
72C22⋊C4×C73—24no—
73C42×C73C2312yes—
74C23×C73C2×D412yes—
75C23×C73C2×D411yescharacteristic
76C22×D73—24no—
77C22×D73—22no—
78Q8×C73—22no—
79Q8×C73—24no—
80Q8×C73—41no—
81Q8×C73—41no—
82D4×C73—24no—
83D4×C73—22no—
84C2×C4×C73—22no—
85C2×C4×C73—22no—
86C2×C4×C73—22no—
87C2×C4×C73—22no—
88C2×C4×C73—42no—
89C2×C4×C73C2×D412yes—
90C2×C4×C73C2×D412yes—
91C2×C4×C73C2×D412yes—
92C2×C4×C73C2×D411yescharacteristic
93C2×C4×C73C2411yescharacteristic
94C22⋊D73—28no—
95C22⋊D73—24no—
96C22⋊D73—24no—
97C2×Dic73—24no—
98C2×Dic73—22no—
99D292—24no—
100D292—22no—
101D292C2×D414yes—
102C4×D73—28no—
103C4×D73—24no—
104C4×D73—24no—
105Dic146—24no—
106Dic146—22no—
107Dic146C2×D414yes—
108#10596831—24no—
109C22×C73—22no—
110C22×C73—41no—
111C22×C73—42no—
112C22×C73#12012yes—
113C22×C73C22×D411yescharacteristic
114C2×D73—24no—
115C2×D73—82no—
116C2×D73—84no—
117C4×C73—22no—
118C4×C73—42no—
119C4×C73—42no—
120C4×C73#12011yescharacteristic
121C4×C73#12011yescharacteristic
122C4×C73C22×D412yes—
123Dic73—24no—
124C2×C73—21no—
125C2×C73—22no—
126C2×C73—42no—
127C2×C73C2×#12011yescharacteristic
128D73—44no—
129#3022—731nomaximal, Sylow
130C73#302211yescharacteristic, complemented, Sylow
131#585—731no—
132C2×#137—732no—
133#536D7311yescharacteristic, complemented
134#456—734no—
135#455—734no—
136#421—732no—
137#413—731no—
1382−1+4—732no—
1392−1+4—1462no—
140H(5,𝔽2)—732no—
141H(5,𝔽2)—1462no—
142H(5,𝔽2)—1462no—
143C2×#41—732no—
144C2×#41—1464no—
145C2×#41—1462no—
146C2×#41C2×D7311yescharacteristic
147C22×Q8C2×D7311yescharacteristic, complemented
148#137—738no—
149#137—1464no—
150C2×Q16—1462no—
151C2×SD16—1462no—
152C2×M16—732no—
153C4⋊Q8C2×D7312yescomplemented
154#124C2×D7312yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D7311yescharacteristic, complemented
158#104—738no—
159#101—732no—
160#100—732no—
161#41—22no—
162#41—22no—
163#41—734no—
164#41—734no—
165#41—1464no—
166#41—1464no—
167#41—1464no—
168#41—1468no—
169#41—1464no—
170#41—1462no—
171#41—1462no—
172#41—1464no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—732no—
177C2×Q8—1462no—
178C2×Q8—1464no—
179C2×Q8C22×D7312yes—
180C2×Q8C22×D7311yescharacteristic
181C2×D4—732no—
182C2×D4—1462no—
183C2×D4—1464no—
184C2×D4—1468no—
185C2×D4—1462no—
186C2×D4—1464no—
187C2×D4C22⋊D7312yes—
188C2×D4C22×D7311yescharacteristic
189C22×C4—21no—
190C22×C4—1464no—
191C22×C4—1462no—
192C22×C4C22⋊D7312yes—
193C22×C4C22×D7311yescharacteristic
194Q16—1468no—
195SD16—1468no—
196M16—734no—
197M16—1462no—
198C2×C8—1462no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D7312yes—
204C23—1464no—
205C23—1462no—
206C23C2×C22⋊D7312yes—
207C23C2×C22⋊D7311yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—734no—
213Q8—1464no—
214Q8—1462no—
215D4—24no—
216D4—22no—
217D4—734no—
218D4—1464no—
219D4—1468no—
220D4—1464no—
221D4—1464no—
222D4—1462no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—1468no—
229C2×C4—1464no—
230C2×C4—1462no—
231C2×C4—1464no—
232C2×C4—1464no—
233C2×C4D4×D7312yes—
234C2×C4D4×D7312yes—
235C2×C4C2×C22⋊D7312yes—
236C2×C4C2×C22⋊D7311yescharacteristic
237C2×C4C23×D7311yescharacteristic
238C8—1464no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—5842no—
243C22—5844no—
244C22—1464no—
245C22#1455655912yes—
246C22C22×C22⋊D7311yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—1464no—
251C4#1455655911yescharacteristic
252C4#1455655911yescharacteristic
253C4C2×D4×D7312yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—2924no—
258C2C2×#1455655911yescharacteristic
subgroupG/H
centerC2C2×#14556559
commutatorC2×C4×C73C24
FrattiniC2×C4C23×D73
Fitting#536×C73C2
socleC2×C73C2×#120
hypercenter#536D73
hypocenterC73#3022
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#14556622C2semidirect
3. #17988769C2semidirect
4. #17988764C2semidirect
5. #17988605C2semidirect
6. #17988571C2semidirect
7. #536×C73C2semidirect
17. #14556622C22semidirect
20. C2×#11924012C22semidirect
21. #14556519C22semidirect
22. #14556510C22semidirect
23. #14556508C22semidirect
25. C22×Q8×C73C22semidirect
26. C4⋊Q8×C73C22semidirect
27. #124×C73C22semidirect
30. #117×C73C22semidirect
130. C73#3022semidirect
133. #536D73semidirect
147. C22×Q8C2×D73semidirect
153. C4⋊Q8C2×D73semidirect
154. #124C2×D73semidirect
157. #117C2×D73semidirect