Finite Group Databasebeta

← #33089650

Subgroups — #33089650

Order 18688.

1726 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989137C211yesmaximal, characteristic
2C2×#17988586C214yesmaximal, complemented
3C2×C4×#11924012C211yesmaximal, characteristic, complemented
4#22610745C214yesmaximal, complemented
5#22610736C212yesmaximal, complemented
6#22610732C212yesmaximal, complemented
7C2×#384×C73C211yesmaximal, characteristic, complemented
8C22×#11924028C2212yes—
9C22×#11924012C2212yes—
10#17989137—24no—
11#17989137C418yescomplemented
12C2×#14556591C2211yescharacteristic
13C2×#14556588C2211yescharacteristic
14C2×#14556579C2212yes—
15C2×#14556565—22no—
16C2×C4×D292—22no—
17#17988586—216no—
18C2×#14556514C2218yes—
19C2×#14556511C2218yes—
20C4×#11924012—24no—
21C2×#14556502C2212yes—
22C2×C4×#10596831C2212yes—
23C22×C4⋊C4×C73—22no—
24C22×C4⋊C4×C73C2212yes—
25C22×C42×C73C2211yescharacteristic
26#384×C73—22no—
27#384×C73C2212yes—
28#384×C73C2212yes—
29C2×#11924028—24no—
30C2×#11924028—28no—
31C2×#11924028—28no—
32C2×#11924028C2×C418yes—
33C22×Dic146C2312yes—
34C22×D292—22no—
35C22×D292C2312yes—
36C22×C22⋊D73—22no—
37C22×C4×D73—22no—
38C2×#11924012—24no—
39C2×#11924012—24no—
40C2×#11924012D418yescomplemented
41C22×#10596831C2314yes—
42#14556591—22no—
43#14556591C2×C414yescomplemented
44#14556588—22no—
45#14556588C2×C414yescomplemented
46#14556579—24no—
47#14556579C2×C418yescomplemented
48#14556565—216no—
49C4×D292—216no—
50C2×C4⋊Dic73—22no—
51C2×#11924005—24no—
52#14556514—216no—
53#14556511—216no—
54#14556502—24no—
55C4×#10596831—24no—
56C23×C4×C73—22no—
57C23×C4×C73D412yes—
58C23×C4×C73C2311yescharacteristic
59C2×C4⋊C4×C73—28no—
60C2×C4⋊C4×C73—22no—
61C2×C4⋊C4×C73—24no—
62C2×C4⋊C4×C73—24no—
63C2×C4⋊C4×C73—48no—
64C2×C4⋊C4×C73C2314yes—
65C2×C42×C73C2×C414yes—
66C2×C42×C73D412yes—
67C2×C42×C73D414yes—
68C2×C42×C73C2311yescharacteristic
69C2×C42×C73C2311yescharacteristic
70C23×D73—22no—
71#11924028—232no—
72#11924028—416no—
73C2×Dic146—24no—
74C2×Dic146—24no—
75C2×Dic146—22no—
76C2×Dic146C22×C414yes—
77C2×D292—24no—
78C2×D292—24no—
79C2×D292—28no—
80C2×D292—22no—
81C2×D292—44no—
82C2×D292—44no—
83C2×D292C22×C414yes—
84C2×C22⋊D73—28no—
85C2×C22⋊D73—44no—
86C2×C22⋊D73—44no—
87C22×Dic73—22no—
88C2×C4×D73—28no—
89C2×C4×D73—44no—
90C2×C4×D73—44no—
91#11924012—216no—
92C2×#10596831—24no—
93C2×#10596831—24no—
94C2×#10596831C2×D418yes—
95C4⋊Dic73—28no—
96#11924005—216no—
97C24×C73C2×D411yescharacteristic
98C22×C4×C73—24no—
99C22×C4×C73—24no—
100C22×C4×C73—24no—
101C22×C4×C73—48no—
102C22×C4×C73—44no—
103C22×C4×C73C22⋊C418yes—
104C22×C4×C73C22×C412yes—
105C22×C4×C73C2×D414yes—
106C22×C4×C73C2×D412yes—
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73C2×D412yes—
109C22×C4×C73C2×D412yes—
110C22×C4×C73#4114yes—
111C22×C4×C73C2411yescharacteristic
112C4⋊C4×C73—28no—
113C4⋊C4×C73—48no—
114C4⋊C4×C73—44no—
115C4⋊C4×C73—48no—
116C42×C73C22⋊C418yes—
117C42×C73C22×C412yes—
118C42×C73C22×C412yes—
119C42×C73C2×D412yes—
120C42×C73C2×D412yes—
121C23×C73—22no—
122C23×C73—22no—
123C23×C73C2×C22⋊C412yes—
124C23×C73#12412yes—
125C23×C73#12712yes—
126C23×C73C22×D411yescharacteristic
127C22×D73—24no—
128C22×D73—42no—
129C22×D73—42no—
130C22×D73—44no—
131C22×D73—44no—
132C2×C4×C73—22no—
133C2×C4×C73—22no—
134C2×C4×C73—216no—
135C2×C4×C73—44no—
136C2×C4×C73—48no—
137C2×C4×C73—88no—
138C2×C4×C73C2×C22⋊C414yes—
139C2×C4×C73C2×C22⋊C412yes—
140C2×C4×C73C2×C22⋊C414yes—
141C2×C4×C73C4×D418yes—
142C2×C4×C73#12118yes—
143C2×C4×C73#12412yes—
144C2×C4×C73#12712yes—
145C2×C4×C73AGL(1,ℤ/8)14yes—
146C2×C4×C73#13714yes—
147C2×C4×C73C23×C411yescharacteristic
148C2×C4×C73C22×D412yes—
149C2×C4×C73C22×D411yescharacteristic
150C2×C4×C73C2×#4112yes—
151C22⋊D73—416no—
152C2×Dic73—24no—
153C2×Dic73—42no—
154C2×Dic73—42no—
155D292—28no—
156D292—44no—
157D292—416no—
158C4×D73—416no—
159Dic146—28no—
160Dic146—44no—
161#10596831—28no—
162C22×C73—24no—
163C22×C73—24no—
164C22×C73—22no—
165C22×C73—22no—
166C22×C73#39014yes—
167C22×C73C22×C22⋊C411yescharacteristic
168C22×C73C2×#12411yescharacteristic
169C22×C73C2×#12711yescharacteristic
170C22×C73C2×AGL(1,ℤ/8)12yes—
171C22×C73C2×#13712yes—
172C2×D73—48no—
173C2×D73—44no—
174C2×D73—84no—
175C2×D73—82no—
176C2×D73—82no—
177C4×C73—24no—
178C4×C73—84no—
179C4×C73#39014yes—
180C4×C73#41718yescomplemented
181C4×C73C2×C4×D412yes—
182C4×C73C2×#12112yes—
183Dic73—44no—
184#21200—731nomaximal, Sylow
185C2×C73—24no—
186C2×C73C2×#39011yescharacteristic
187C2×C73C2×#41712yes—
188C2×C73#314112yes—
189C2×C73#315111yescharacteristic
190C2×C73#315211yescharacteristic
191D73—84no—
192C2×#532—731no—
193C2×#417—734no—
194C2×C4×M16—731no—
195C2×#384D7311yescharacteristic, complemented
196#1959—734no—
197#1957—732no—
198#1956—732no—
199C73#2120011yescharacteristic, complemented, Sylow
200C22×#41—732no—
201C22×M16—732no—
202#532—738no—
203#532—1464no—
204C2×C4⋊Q8—731no—
205C2×#127—731no—
206C2×#124—732no—
207C2×#121—1462no—
208C2×C4×D4—1462no—
209C22×C4⋊C4—22no—
210C22×C4⋊C4C2×D7312yes—
211C22×C42C2×D7311yescharacteristic
212#417—14616no—
213C2×#103—738no—
214C2×#102—738no—
215C4×M16—1464no—
216C2×#97—732no—
217C2×C4×C8—732no—
218#384—22no—
219#384C2×D7312yes—
220#384C2×D7312yes—
221C2×#41—738no—
222C2×#41—1468no—
223C2×#41—1468no—
224C2×#41—1464no—
225C22×Q8—732no—
226C22×D4—732no—
227C22×D4—1462no—
228C22×D4—1462no—
229C23×C4—22no—
230C23×C4—1462no—
231C23×C4C22⋊D7312yes—
232C23×C4C22×D7311yescharacteristic
233C2×M16—738no—
234C2×M16—1464no—
235C2×M16—1464no—
236C22×C8—734no—
237C4⋊Q8—734no—
238C4⋊Q8—1462no—
239#127—734no—
240#127—1462no—
241#124—738no—
242#124—1464no—
243#121—14616no—
244C4×D4—14616no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4—1462no—
251C2×C4⋊C4C22×D7314yes—
252C2×C22⋊C4—1464no—
253C2×C42C4×D7314yes—
254C2×C42D29214yes—
255C2×C42C22⋊D7312yes—
256C2×C42C22×D7311yescharacteristic
257C2×C42C22×D7311yescharacteristic
258#103—14616no—
259#102—14616no—
260#97—1464no—
261C4×C8—1464no—
262C24—1462no—
263C24C2×C22⋊D7311yescharacteristic
264#41—29216no—
265#41—14632no—
266C2×Q8—734no—
267C2×Q8—1464no—
268C2×Q8—1464no—
269C2×Q8—1462no—
270C2×D4—2924no—
271C2×D4—2924no—
272C2×D4—2924no—
273C2×D4—2924no—
274C2×D4—734no—
275C2×D4—1468no—
276C2×D4—1468no—
277C2×D4—1464no—
278C2×D4—1462no—
279C2×D4—1464no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—48no—
284C22×C4—44no—
285C22×C4—2924no—
286C22×C4—2924no—
287C22×C4—1462no—
288C22×C4—1468no—
289C22×C4#1192400518yes—
290C22×C4C2×C4×D7312yes—
291C22×C4D4×D7314yes—
292C22×C4C2×C22⋊D7312yes—
293C22×C4C2×C22⋊D7311yescharacteristic
294C22×C4C2×C22⋊D7312yes—
295C22×C4C2×D29212yes—
296C22×C4#1192402914yes—
297C22×C4C23×D7311yescharacteristic
298M16—14616no—
299C2×C8—738no—
300C2×C8—1464no—
301C2×C8—1464no—
302C4⋊C4—28no—
303C4⋊C4—48no—
304C4⋊C4—44no—
305C4⋊C4—48no—
306C4⋊C4—1468no—
307C22⋊C4—14616no—
308C42#1192400518yes—
309C42C2×C4×D7312yes—
310C42C2×C4×D7312yes—
311C42C2×D29212yes—
312C42C2×D29212yes—
313C23—22no—
314C23—22no—
315C23—2924no—
316C23—2922no—
317C23—2922no—
318C23—2924no—
319C23—1464no—
320C23C2×#1192400512yes—
321C23#1455658012yes—
322C23#1455658712yes—
323C23C22×C22⋊D7311yescharacteristic
324Q8—2924no—
325Q8—1468no—
326D4—29216no—
327D4—2924no—
328D4—29216no—
329D4—1468no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—216no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—88no—
336C2×C4—2922no—
337C2×C4—2922no—
338C2×C4—29216no—
339C2×C4—1464no—
340C2×C4C2×#1192400514yes—
341C2×C4C2×#1192400512yes—
342C2×C4C2×#1192400514yes—
343C2×C4#1455655318yes—
344C2×C4#1455656618yes—
345C2×C4#1455658012yes—
346C2×C4#1455658712yes—
347C2×C4#1455661714yes—
348C2×C4#1455662214yes—
349C2×C4C22×C4×D7311yescharacteristic
350C2×C4C2×D4×D7312yes—
351C2×C4C22×D29211yescharacteristic
352C2×C4C2×#1192402912yes—
353C8—1468no—
354C22—24no—
355C22—24no—
356C22—22no—
357C22—22no—
358C22—2928no—
359C22—2924no—
360C22—5844no—
361C22—5842no—
362C22—5842no—
363C22#1798848814yes—
364C22C22×#1192400511yescharacteristic
365C22C2×#1455658011yescharacteristic
366C22C2×#1455658711yescharacteristic
367C22C2×#1455661712yes—
368C22C2×#1455662212yes—
369C4—24no—
370C4—84no—
371C4—2924no—
372C4#1798848814yes—
373C4#1798858618yescomplemented
374C4C2×#1455655312yes—
375C4C2×#1455656612yes—
376C2—24no—
377C2—5844no—
378C2C2×#1798848811yescharacteristic
379C2C2×#1798858612yes—
380C2#2262032712yes—
381C2#2262038111yescharacteristic
382C2#2262038811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#384×C73C2
socleC23×C73C22×D4
prime radicalC2×#17989137C2
cosocleC22×C4×C73C24
hypercenterC2×#384D73
hypocenterC73#21200
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
2. C2×#17988586C2semidirect
3. C2×C4×#11924012C2semidirect
4. #22610745C2semidirect
5. #22610736C2semidirect
6. #22610732C2semidirect
7. C2×#384×C73C2semidirect
11. #17989137C4semidirect
40. C2×#11924012D4semidirect
43. #14556591C2×C4semidirect
45. #14556588C2×C4semidirect
47. #14556579C2×C4semidirect
180. C4×C73#417semidirect
195. C2×#384D73semidirect
199. C73#21200semidirect
373. C4#17988586semidirect

Maximal subgroups

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

subgroupindex
C2×#179891372
C2×#179885862
C2×C4×#119240122
#226107452
#226107362
#226107322
C2×#384×C732
#2120073