Finite Group Databasebeta

← #33240631

Subgroups — #33240631

Order 18688.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620442C211yesmaximal, characteristic, complemented
2C4×#14556610C211yesmaximal, characteristic
3#22612098C212yesmaximal, complemented
4#22612090C212yesmaximal, complemented
5#22612080C211yesmaximal, characteristic, complemented
6#22612074C211yesmaximal, characteristic, complemented
7#22612068C211yesmaximal, characteristic, complemented
8#22611714C212yesmaximal, complemented
9C4⋊C16×D73C211yesmaximal, characteristic, complemented
10#22611635C212yesmaximal
11C73×#2226C211yesmaximal, characteristic, complemented
12C2×#14556610C2211yescharacteristic
13#17989142—21no—
14C4×#11924027—21no—
15C2×C16×D73C2212yes—
16#17988992C2211yescharacteristic
17#17988988—21no—
18#17988988C2212yes—
19#17988983—21no—
20C4×Dic292C2211yescharacteristic
21#17988681C2211yescharacteristic
22#17988678—21no—
23C4×#11924016—21no—
24D8×Dic73C2211yescharacteristic
25#17988624—21no—
26#17988624C2211yescharacteristic
27#17988624C2212yes—
28#17988580—21no—
29C4×C8×D73C2211yescharacteristic
30#17988415C2212yes—
31#17988412C2212yes—
32#17988412C2212yes—
33C4⋊#11924003C2211yescharacteristic
34#17988405C2212yes—
35#17988392C2212yes—
36#17988390C2212yes—
37#17988388C2212yes—
38#500×C73C2211yescharacteristic
39C4×D8×C73C2211yescharacteristic
40#367×C73C2212yes—
41C4⋊C16×C73C2211yescharacteristic
42#357×C73C2212yes—
43C2×#11924027—21no—
44#14556610—22no—
45#14556610Q814yescomplemented
46C2×Dic292C2311yescharacteristic
47C2×#11924016—21no—
48C2×C8×D73D411yescharacteristic
49C2×C8×D73C2311yescharacteristic
50#14556591—21no—
51#14556582—22no—
52#14556571—41no—
53#14556567—41no—
54Q8×Dic73—41no—
55C4×Dic146—21no—
56#14556556—21no—
57#14556556—41no—
58D4×Dic73—21no—
59C4×C22⋊D73—41no—
60#14556551—41no—
61#14556549—22no—
62#14556549—21no—
63#14556547—41no—
64C2×C4×Dic73—41no—
65C42×D73D411yescharacteristic
66C2×#11924003C2312yes—
67C16×D73—24no—
68#14556525—21no—
69#14556525C2312yes—
70#14556525C2311yescharacteristic
71#14556522—22no—
72#14556522—21no—
73#14556522—22no—
74#14556516—21no—
75#14556514—21no—
76#14556512—21no—
77C4×#10596831D411yescharacteristic
78C8×Dic73D411yescharacteristic
79C8×Dic73C2311yescharacteristic
80C2×D8×C73C2311yescharacteristic
81C4⋊Q8×C73—21no—
82C4×D4×C73—21no—
83C2×C16×C73C2312yes—
84#107×C73—21no—
85#107×C73C2312yes—
86#107×C73C2311yescharacteristic
87#102×C73—21no—
88C4×C8×C73C2311yescharacteristic
89#11924027—24no—
90#11924027—42no—
91C2×Dic146—21no—
92C2×Dic146—41no—
93C2×C22⋊D73—41no—
94C22×Dic73—41no—
95C22×Dic73—41no—
96C2×C4×D73D811yescharacteristic
97C2×C4×D73D811yescharacteristic
98C2×C4×D73C2×D411yescharacteristic
99Dic292—21no—
100Dic292C2×Q812yes—
101#11924016—24no—
102C2×#10596831C2×D411yescharacteristic
103C2×#10596831C2×D411yescharacteristic
104C8×D73—22no—
105C8×D73C2×Q812yes—
106C8×D73#4112yes—
107C4⋊Dic73—21no—
108C4⋊Dic73—22no—
109C4⋊Dic73—41no—
110C4⋊Dic73—42no—
111#11924007—41no—
112#11924007—41no—
113#11924007—42no—
114C22⋊Dic73—41no—
115#11924005—41no—
116#11924005—42no—
117C4×Dic73—41no—
118C4×Dic73—41no—
119C4×Dic73—41no—
120C4×Dic73—42no—
121C4×Dic73D811yescharacteristic
122C4×Dic73D811yescharacteristic
123C4×Dic73C2×D411yescharacteristic
124#11924003—22no—
125C2×Q8×C73—41no—
126C2×D4×C73—21no—
127C22×C4×C73—41no—
128D8×C73—21no—
129D8×C73C2×Q812yes—
130C2×C8×C73C2×D411yescharacteristic
131C2×C8×C73C2411yescharacteristic
132C4⋊C4×C73—21no—
133C4⋊C4×C73—22no—
134C4⋊C4×C73—41no—
135C22⋊C4×C73—41no—
136C42×C73C2×D411yescharacteristic
137C16×C73—22no—
138C23×C73—41no—
139C22×D73C2×D811yescharacteristic
140Q8×C73—82no—
141D4×C73—22no—
142D4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—42no—
145C2×C4×C73—82no—
146C2×C4×C73C2×D811yescharacteristic
147C2×C4×C73C2×D811yescharacteristic
148C2×C4×C73C22×D411yescharacteristic
149C22⋊D73—44no—
150C2×Dic73—41no—
151C2×Dic73—44no—
152C2×Dic73—41no—
153C2×Dic73—42no—
154C2×Dic73—82no—
155C2×Dic73C2×D811yescharacteristic
156C2×Dic73C2×D811yescharacteristic
157C2×Dic73C2×D811yescharacteristic
158C4×D73—22no—
159C4×D73—41no—
160C4×D73C22⋊Q812yes—
161C4×D73#13712yes—
162Dic146—22no—
163Dic146—41no—
164Dic146—82no—
165C8×C73—21no—
166C8×C73C22×Q811yescharacteristic
167C8×C73C2×#4111yescharacteristic
168#10596831—21no—
169#10596831C22⋊Q812yes—
170C22×C73—42no—
171C22×C73—81no—
172C22×C73—81no—
173C22×C73C22×D811yescharacteristic
174C2×D73—21no—
175C2×D73—21no—
176C2×D73#47412yes—
177C4×C73—21no—
178C4×C73—81no—
179C4×C73—82no—
180C4×C73#50812yes—
181C4×C73C2×C22⋊Q811yescharacteristic
182C4×C73C2×#13711yescharacteristic
183Dic73—21no—
184Dic73—21no—
185Dic73—42no—
186Dic73—82no—
187Dic73—81no—
188Dic73#47412yes—
189#33626—731nomaximal, Sylow
190C2×C73—82no—
191C2×C73C2×#47411yescharacteristic
192C2×C73C2×#50811yescharacteristic
193C2×C73#341411yescharacteristic
194D73—22no—
195#3160—731no—
196C4×#135—731no—
197#2229—732no—
198#2228—732no—
199#2227—731no—
200#2226D7311yescharacteristic, complemented
201#2225—731no—
202#2224—731no—
203#2158—732no—
204C2×C4⋊C16—731no—
205#2138—732no—
206C73#3362611yescharacteristic, complemented, Sylow
207C2×#135—731no—
208#533—1461no—
209C4×#41—1461no—
210C22×C16—732no—
211#500—731no—
212#500C2×D7311yescharacteristic
213#499—732no—
214#499—1461no—
215#498—1461no—
216C4×Q16—731no—
217C4×Q16—731no—
218C4×SD16—1461no—
219C4×SD16—1461no—
220C4×D8—731no—
221C4×D8C2×D7311yescharacteristic
222#427—732no—
223#427—731no—
224#427—1461no—
225#416—1461no—
226C2×C4×C8—731no—
227#367—732no—
228#367C2×D7312yes—
229#366—732no—
230#366—732no—
231C4⋊C16—732no—
232C4⋊C16—731no—
233C4⋊C16C2×D7311yescharacteristic
234#358—732no—
235#358—732no—
236#357—732no—
237#357C2×D7312yes—
238C2×#41—1461no—
239#135—734no—
240#135—1462no—
241C2×Q16—731no—
242C2×SD16—1461no—
243C2×D8C22×D7311yescharacteristic
244C22×C8—731no—
245C22×C8—731no—
246C4⋊Q8—21no—
247C4⋊Q8—1461no—
248#125—1462no—
249C22⋊Q8—2921no—
250C22⋊Q8—2921no—
251C4×Q8—2921no—
252C4×Q8—2921no—
253C4×Q8—1461no—
254C4×Q8—1461no—
255C4×D4—21no—
256C4×D4—2921no—
257C4×D4—2921no—
258C4×D4—1461no—
259#117—2921no—
260#117—1461no—
261#117—1462no—
262C2×C42—2921no—
263C2×C42—731no—
264C2×C16—732no—
265C2×C16—1464no—
266C2×C16C22×D7312yes—
267#107—21no—
268#107—732no—
269#107—731no—
270#107—1461no—
271#107C22×D7312yes—
272#107C22×D7311yescharacteristic
273#106—1461no—
274#106—1462no—
275#106—1462no—
276#103—1461no—
277#103—1461no—
278#102—21no—
279#102—1461no—
280C4×C8—731no—
281C4×C8—731no—
282C4×C8—731no—
283C4×C8C22×D7311yescharacteristic
284#41—2922no—
285#41—1464no—
286C2×Q8—41no—
287C2×Q8—2921no—
288C2×Q8—1461no—
289C2×D4—21no—
290C2×D4—2921no—
291C22×C4—41no—
292C22×C4—2921no—
293C22×C4—2921no—
294C22×C4—731no—
295C22×C4—731no—
296C22×C4—731no—
297Q16—732no—
298Q16—1461no—
299SD16—1464no—
300D8—21no—
301D8Q8×D7312yes—
302C2×C8—731no—
303C2×C8—732no—
304C2×C8—732no—
305C2×C8—731no—
306C2×C8—1462no—
307C2×C8D4×D7311yescharacteristic
308C2×C8C23×D7311yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—41no—
312C4⋊C4—2921no—
313C4⋊C4—2922no—
314C4⋊C4—2922no—
315C4⋊C4—2921no—
316C4⋊C4—2921no—
317C4⋊C4—1461no—
318C4⋊C4—1462no—
319C22⋊C4—41no—
320C22⋊C4—2921no—
321C22⋊C4—2922no—
322C22⋊C4—2921no—
323C42—2921no—
324C42—2921no—
325C42—2922no—
326C42—2921no—
327C42—731no—
328C42—731no—
329C42—731no—
330C42D4×D7311yescharacteristic
331C16—22no—
332C16—1462no—
333C23—41no—
334C23—731no—
335Q8—82no—
336Q8—2921no—
337Q8—5842no—
338Q8—1462no—
339D4—22no—
340D4—41no—
341D4—2924no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—2921no—
346C2×C4—2921no—
347C2×C4—2924no—
348C2×C4—2921no—
349C2×C4—2922no—
350C2×C4—5842no—
351C2×C4—731no—
352C2×C4—732no—
353C2×C4—732no—
354C2×C4—731no—
355C2×C4—731no—
356C2×C4—1462no—
357C2×C4D8×D7311yescharacteristic
358C2×C4D8×D7311yescharacteristic
359C2×C4C2×D4×D7311yescharacteristic
360C8—21no—
361C8—732no—
362C8—1461no—
363C8C2×Q8×D7311yescharacteristic
364C8#41×D7311yescharacteristic
365C22—42no—
366C22—81no—
367C22—81no—
368C22—732no—
369C22—1461no—
370C22—1461no—
371C22C2×D8×D7311yescharacteristic
372C4—21no—
373C4—81no—
374C4—82no—
375C4—2922no—
376C4—5842no—
377C4—5841no—
378C4—732no—
379C4—1461no—
380C4—1461no—
381C4#1798901812yes—
382C4C22⋊Q8×D7311yescharacteristic
383C4#137×D7311yescharacteristic
384C2—82no—
385C2—1462no—
386C2#474×D7311yescharacteristic
387C2C2×#1798901811yescharacteristic
388C2#2262306311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2226C2
socleC22×C73C22×D8
prime radicalC2×#14556610C22
hypercenter#2226D73
hypocenterC73#33626
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #22620442C2semidirect
3. #22612098C2semidirect
4. #22612090C2semidirect
5. #22612080C2semidirect
6. #22612074C2semidirect
7. #22612068C2semidirect
8. #22611714C2semidirect
9. C4⋊C16×D73C2semidirect
11. C73×#2226C2semidirect
45. #14556610Q8semidirect
200. #2226D73semidirect
206. C73#33626semidirect

Maximal subgroups

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

subgroupindex
#226204422
C4×#145566102
#226120982
#226120902
#226120802
#226120742
#226120682
#226117142
C4⋊C16×D732
#226116352
C73×#22262
#3362673