Finite Group Databasebeta

← #33095246

Subgroups — #33095246

Order 18688.

1538 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989149C211yesmaximal, characteristic
2#22612529C211yesmaximal, characteristic, complemented
3#22610767C214yesmaximal, complemented
4#22610766C214yesmaximal, complemented
5#22609845C214yesmaximal, complemented
6C73×#2312C211yesmaximal, characteristic, complemented
7C23×C22⋊D73C2212yes—
8#17989149—24no—
9#17989149—24no—
10C2×#14556575C414yescomplemented
11C2×#14556565C2212yes—
12C2×#14556565C2212yes—
13C2×#14556561C414yescomplemented
14C2×#14556505C22116yescomplemented
15#17988480—24no—
16#17988447C2212yescomplemented
17#17988443C2212yes—
18C4×C22⋊Dic73C2212yescomplemented
19C22×C22⋊C4×C73C2211yescharacteristic
20#388×C73—24no—
21#380×C73C2212yescomplemented
22#379×C73C2212yes—
23C4×C22⋊C4×C73C2212yescomplemented
24C24×D73C2×C412yes—
25C22×D292C2×C412yes—
26C22×D292C2×C412yes—
27C22×C22⋊D73—24no—
28C22×C22⋊D73—24no—
29C22×C22⋊D73—48no—
30C22×C22⋊D73C2×C414yes—
31C22×C22⋊D73C2314yes—
32C23×Dic73C2×C412yes—
33#14556575—216no—
34#14556565—24no—
35#14556565—24no—
36#14556565—24no—
37#14556565—24no—
38#14556561—216no—
39C2×C4⋊Dic73—22no—
40C2×C4⋊Dic73C2×C412yes—
41C2×C4⋊Dic73C2×C412yes—
42C2×C22⋊Dic73—22no—
43C2×C22⋊Dic73—22no—
44C2×C22⋊Dic73C2314yes—
45C2×#11924005C2×C414yes—
46C2×#11924005C2×C414yes—
47C2×C4×Dic73—22no—
48#14556505—416no—
49#14556498—22no—
50#14556498—22no—
51C25×C73C2311yescharacteristic
52C23×C4×C73C2311yescharacteristic
53C23×C4×C73C2311yescharacteristic
54C2×C4⋊C4×C73—22no—
55C2×C22⋊C4×C73—22no—
56C2×C22⋊C4×C73—22no—
57C2×C22⋊C4×C73C2×C414yes—
58C2×C22⋊C4×C73D414yes—
59C2×C22⋊C4×C73D414yes—
60C2×C22⋊C4×C73C2314yes—
61C2×C42×C73—22no—
62#95×C73—22no—
63#95×C73—22no—
64C23×D73—24no—
65C23×D73—44no—
66C23×D73—82no—
67C23×D73—82no—
68C23×D73C22×C414yes—
69C2×D292—24no—
70C2×D292—24no—
71C2×D292—82no—
72C2×D292—82no—
73C2×C22⋊D73—28no—
74C2×C22⋊D73—24no—
75C2×C22⋊D73—24no—
76C2×C22⋊D73—216no—
77C2×C22⋊D73—48no—
78C2×C22⋊D73—416no—
79C2×C22⋊D73—416no—
80C22×Dic73—24no—
81C22×Dic73—24no—
82C22×Dic73—44no—
83C22×Dic73C22×C414yes—
84C4⋊Dic73—24no—
85C4⋊Dic73—24no—
86C4⋊Dic73—82no—
87C22⋊Dic73—28no—
88C22⋊Dic73—82no—
89C22⋊Dic73—82no—
90#11924005—28no—
91#11924005—28no—
92C4×Dic73—82no—
93C24×C73—24no—
94C24×C73—81no—
95C24×C73—81no—
96C24×C73C22×C412yes—
97C24×C73C2×D412yes—
98C24×C73C2×D412yes—
99C24×C73C2411yescharacteristic
100C22×C4×C73—24no—
101C22×C4×C73—81no—
102C22×C4×C73—81no—
103C22×C4×C73C22×C412yes—
104C22×C4×C73C22×C412yes—
105C22×C4×C73C2×D412yes—
106C22×C4×C73C2×D412yes—
107C22×C4×C73C2×D412yes—
108C22×C4×C73C2×D412yes—
109C4⋊C4×C73—82no—
110C22⋊C4×C73—28no—
111C22⋊C4×C73—82no—
112C22⋊C4×C73—82no—
113C22⋊C4×C73C22⋊C4116yes—
114C42×C73—82no—
115C23×C73—24no—
116C23×C73—24no—
117C23×C73—24no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—42no—
121C23×C73—42no—
122C23×C73—42no—
123C23×C73—42no—
124C23×C73—42no—
125C23×C73—42no—
126C23×C73—88no—
127C23×C73C2×C22⋊C414yes—
128C23×C73C23×C411yescharacteristic
129C23×C73C22×D411yescharacteristic
130C23×C73C22×D411yescharacteristic
131C23×C73H(5,𝔽2)14yes—
132C22×D73—24no—
133C22×D73—28no—
134C22×D73—48no—
135C22×D73—44no—
136C22×D73—44no—
137C22×D73—84no—
138C22×D73—84no—
139C22×D73—816no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—42no—
143C2×C4×C73—42no—
144C2×C4×C73—42no—
145C2×C4×C73—42no—
146C2×C4×C73—44no—
147C2×C4×C73—44no—
148C2×C4×C73—44no—
149C2×C4×C73C2×C22⋊C414yes—
150C2×C4×C73C2×C22⋊C414yes—
151C22⋊D73—416no—
152C22⋊D73—48no—
153C22⋊D73—48no—
154C2×Dic73—24no—
155C2×Dic73—28no—
156C2×Dic73—48no—
157C2×Dic73—44no—
158C2×Dic73—44no—
159C2×Dic73—44no—
160D292—84no—
161D292—84no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—24no—
169C22×C73—24no—
170C22×C73—24no—
171C22×C73—44no—
172C22×C73—48no—
173C22×C73—84no—
174C22×C73—84no—
175C22×C73C22×C22⋊C411yescharacteristic
176C22×C73#51812yes—
177C22×C73#53512yes—
178C22×C73#53512yes—
179C2×D73—44no—
180C2×D73—44no—
181C2×D73—44no—
182C2×D73—816no—
183C2×D73—88no—
184C4×C73—24no—
185C4×C73—24no—
186C4×C73—84no—
187Dic73—44no—
188Dic73—84no—
189#21795—731nomaximal, Sylow
190C2×C73—22no—
191C2×C73—22no—
192C2×C73—22no—
193C2×C73—44no—
194C2×C73#230911yescharacteristic
195C2×C73#288311yescharacteristic
196C2×C73#288311yescharacteristic
197D73—84no—
198C2×#534—731no—
199#2312—731no—
200#2312D7311yescharacteristic, complemented
201#1964—734no—
202#1964—734no—
203#1781—734no—
204C73#2179511yescharacteristic, complemented, Sylow
205C23×D4—732no—
206#534—1464no—
207#534—1464no—
208C2×#123—734no—
209C2×#121—732no—
210C2×#121—732no—
211C2×#120—734no—
212C22×C22⋊C4C2×D7311yescharacteristic
213C2×#99—7316no—
214#388—24no—
215#388—1464no—
216#380—732no—
217#380C2×D7312yescomplemented
218#379—732no—
219#379C2×D7312yes—
220C4×C22⋊C4—732no—
221C4×C22⋊C4C2×D7312yescomplemented
222C25—732no—
223C25C22×D7311yescharacteristic
224C22×D4—2928no—
225C22×D4—734no—
226C22×D4—734no—
227C22×D4—732no—
228C22×D4—732no—
229C22×D4—1464no—
230C22×D4—1464no—
231C23×C4—732no—
232C23×C4C22×D7311yescharacteristic
233C23×C4C22×D7311yescharacteristic
234#123—14616no—
235#121—1464no—
236#121—1464no—
237#121—1464no—
238#121—1464no—
239#120—14616no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—732no—
242C2×C4⋊C4—732no—
243C2×C4⋊C4—1462no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—734no—
247C2×C22⋊C4—734no—
248C2×C22⋊C4—734no—
249C2×C22⋊C4—1462no—
250C2×C22⋊C4—1462no—
251C2×C22⋊C4C4×D7314yes—
252C2×C22⋊C4D29214yes—
253C2×C22⋊C4C22⋊D7314yes—
254C2×C22⋊C4C22×D7314yes—
255C2×C42—22no—
256C2×C42—1462no—
257#99—29216no—
258#95—22no—
259#95—22no—
260#95—1462no—
261#95—1462no—
262C24—24no—
263C24—81no—
264C24—81no—
265C24—2924no—
266C24—5842no—
267C24—5842no—
268C24—734no—
269C24—1464no—
270C24C2×C4×D7312yes—
271C24C2×C22⋊D7312yes—
272C24C2×D29212yes—
273C24C23×D7311yescharacteristic
274C2×D4—29216no—
275C2×D4—29216no—
276C2×D4—2928no—
277C2×D4—5842no—
278C2×D4—5842no—
279C2×D4—1468no—
280C2×D4—1464no—
281C2×D4—1464no—
282C2×D4—14616no—
283C2×D4—1464no—
284C2×D4—1464no—
285C22×C4—24no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—2924no—
289C22×C4—734no—
290C22×C4—1464no—
291C22×C4—1464no—
292C22×C4C2×C4×D7312yes—
293C22×C4C2×C4×D7312yes—
294C22×C4C2×C22⋊D7312yes—
295C22×C4C2×C22⋊D7312yes—
296C22×C4C2×D29212yes—
297C22×C4C2×D29212yes—
298C4⋊C4—82no—
299C4⋊C4—5842no—
300C4⋊C4—1464no—
301C4⋊C4—1464no—
302C22⋊C4—28no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—5842no—
306C22⋊C4—5842no—
307C22⋊C4—1468no—
308C22⋊C4—1468no—
309C22⋊C4—1468no—
310C22⋊C4#11924005116yes—
311C42—82no—
312C42—5842no—
313C23—24no—
314C23—24no—
315C23—24no—
316C23—41no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—88no—
325C23—2928no—
326C23—2924no—
327C23—2924no—
328C23—5844no—
329C23—5844no—
330C23—58416no—
331C23—1464no—
332C23—1468no—
333C23C2×#1192400514yes—
334C23C22×C4×D7311yescharacteristic
335C23C22×C22⋊D7311yescharacteristic
336C23C22×D29211yescharacteristic
337C23#1455663714yes—
338D4—29216no—
339D4—2928no—
340D4—2928no—
341D4—5844no—
342D4—5844no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—2928no—
353C2×C4—2924no—
354C2×C4—2924no—
355C2×C4—2924no—
356C2×C4—1464no—
357C2×C4—1468no—
358C2×C4C2×#1192400514yes—
359C2×C4C2×#1192400514yes—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—24no—
367C22—24no—
368C22—24no—
369C22—44no—
370C22—48no—
371C22—84no—
372C22—84no—
373C22—2924no—
374C22—2924no—
375C22—2924no—
376C22—5848no—
377C22—58416no—
378C22C22×#1192400511yescharacteristic
379C22#1798906112yes—
380C22#1798915112yes—
381C22#1798915612yes—
382C4—24no—
383C4—24no—
384C4—84no—
385C4—2924no—
386C4—5844no—
387C2—22no—
388C2—22no—
389C2—22no—
390C2—44no—
391C2—5844no—
392C2#2261251111yescharacteristic
393C2#2261839111yescharacteristic
394C2#2261839111yescharacteristic
subgroupG/H
centerC22C22×#11924005
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC73×#2312C2
socleC22×C73C22×C22⋊C4
prime radicalC2×#17989149C2
cosocleC24×C73C24
hypercenter#2312D73
hypocenterC73#21795
LCSG ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C22⋊C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #22612529C2semidirect
3. #22610767C2semidirect
4. #22610766C2semidirect
5. #22609845C2semidirect
6. C73×#2312C2semidirect
10. C2×#14556575C4semidirect
13. C2×#14556561C4semidirect
14. C2×#14556505C22semidirect
16. #17988447C22semidirect
18. C4×C22⋊Dic73C22semidirect
21. #380×C73C22semidirect
23. C4×C22⋊C4×C73C22semidirect
200. #2312D73semidirect
204. C73#21795semidirect
217. #380C2×D73semidirect
221. C4×C22⋊C4C2×D73semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179891492
#226125292
#226107672
#226107662
#226098452
C73×#23122
#2179573