Finite Group Databasebeta

← #33071252

Subgroups — #33071252

Order 18688.

628 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621086C211yesmaximal, characteristic, complemented
2#22619939C211yesmaximal, characteristic
3#22618955C211yesmaximal, characteristic
4#22609380C212yesmaximal, complemented
5#22609352C212yesmaximal, complemented
6#22609318C212yesmaximal, complemented
7#22609304C212yesmaximal, complemented
8#22608712C211yesmaximal, characteristic, complemented
9#22608642C211yesmaximal, characteristic, complemented
10#22608596C211yesmaximal, characteristic, complemented
11C73×#3110C211yesmaximal, characteristic
12#17989144C2211yescharacteristic
13#17988990C2211yescharacteristic, complemented
14#17988983C2211yescharacteristic, complemented
15#17988937C2211yescharacteristic, complemented
16#17988937C2211yescharacteristic, complemented
17#17988925—22no—
18#17988917—21no—
19#17988894—22no—
20#17988877C2212yes—
21#17988792C2212yes—
22#17988738—21no—
23C4×#11924016—22no—
24D4×#10596831—22no—
25D4×#10596831C2212yes—
26D4×#10596831C2212yes—
27#17988640C2211yescharacteristic
28#17988640C2211yescharacteristic
29#17988640C2211yescharacteristic
30C2×C22⋊#10596831—22no—
31#17988339C2212yescomplemented
32#17988337C2212yescomplemented
33#17988335C2212yescomplemented
34#17988334C2212yescomplemented
35#17988333C2212yescomplemented
36#17988333C2212yescomplemented
37#533×C73C2211yescharacteristic
38C2×C4×D4×C73C2211yescharacteristic
39#484×C73—21no—
40#474×C73C2212yes—
41#461×C73C2212yes—
42#450×C73—21no—
43#440×C73—22no—
44#432×C73C2211yescharacteristic
45C2×#11924016—22no—
46C22×#10596831—22no—
47C22×#10596831—22no—
48#14556591C2311yescharacteristic
49#14556591C2311yescharacteristic
50#14556583—21no—
51#14556570—21no—
52#14556570—21no—
53C4×Dic146—21no—
54C4×Dic146—21no—
55#14556550—21no—
56#14556524—22no—
57#14556522—22no—
58#14556522—22no—
59C4⋊#10596831D412yes—
60C4⋊#10596831C2312yes—
61C4⋊#10596831C2312yes—
62#14556515—22no—
63#14556515—22no—
64#14556514—22no—
65#14556512—22no—
66C22⋊#10596831—22no—
67C22⋊#10596831—22no—
68C22⋊#10596831—21no—
69C22⋊#10596831—21no—
70C22⋊#10596831D412yes—
71C4×#10596831—21no—
72C4×#10596831—21no—
73C4×#10596831C2311yescharacteristic
74C4×#10596831C2311yescharacteristic
75C22×D4×C73D411yescharacteristic
76C23×C4×C73—21no—
77C2×SD16×C73—22no—
78C4⋊Q8×C73C2311yescharacteristic
79C4⋊Q8×C73C2311yescharacteristic
80#125×C73—21no—
81C22⋊Q8×C73—21no—
82C22⋊Q8×C73—21no—
83C4×Q8×C73—21no—
84C4×Q8×C73—21no—
85C4×D4×C73—21no—
86C4×D4×C73—21no—
87C4×D4×C73D412yes—
88C4×D4×C73C2312yes—
89#117×C73—21no—
90C2×C4⋊C4×C73D411yescharacteristic
91C2×C22⋊C4×C73—21no—
92C2×C42×C73C2311yescharacteristic
93#107×C73—22no—
94C4⋊C8×C73C2312yes—
95#103×C73—22no—
96#102×C73—22no—
97C22⋊C8×C73—21no—
98#97×C73—21no—
99C2×Dic146—21no—
100C2×Dic146—21no—
101#11924016—44no—
102C2×#10596831—22no—
103C2×#10596831—22no—
104C2×#10596831—82no—
105C2×#10596831—82no—
106C2×#10596831C2×D412yes—
107C2×#10596831C2×D412yes—
108C4⋊Dic73—21no—
109C4⋊Dic73—21no—
110C4⋊Dic73—21no—
111C4⋊Dic73—21no—
112#11924007—41no—
113#11924007—41no—
114C22⋊Dic73—41no—
115C4×Dic73—41no—
116C24×C73—21no—
117C2×Q8×C73—21no—
118C2×Q8×C73—21no—
119C2×D4×C73—24no—
120C2×D4×C73—21no—
121C2×D4×C73C2×D412yes—
122C22×C4×C73—21no—
123C22×C4×C73—22no—
124C22×C4×C73—22no—
125C22×C4×C73—82no—
126C22×C4×C73C2×D411yescharacteristic
127C22×C4×C73C2×D411yescharacteristic
128C22×C4×C73#4111yescharacteristic
129SD16×C73—44no—
130C2×C8×C73—22no—
131C4⋊C4×C73—21no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73—21no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—21no—
136C4⋊C4×C73—41no—
137C4⋊C4×C73—41no—
138C4⋊C4×C73C2×D412yes—
139C22⋊C4×C73—22no—
140C22⋊C4×C73—22no—
141C22⋊C4×C73—41no—
142C42×C73—41no—
143C42×C73C2×D411yescharacteristic
144C42×C73C2×D411yescharacteristic
145C42×C73#4111yescharacteristic
146C42×C73C2411yescharacteristic
147C23×C73—24no—
148C23×C73—22no—
149C23×C73—81no—
150C23×C73—81no—
151C23×C73#12111yescharacteristic
152Q8×C73—42no—
153Q8×C73—42no—
154D4×C73—24no—
155D4×C73—22no—
156D4×C73#13514yes—
157C2×C4×C73—22no—
158C2×C4×C73—22no—
159C2×C4×C73—22no—
160C2×C4×C73—41no—
161C2×C4×C73—42no—
162C2×C4×C73—42no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—82no—
166C2×C4×C73—81no—
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12111yescharacteristic
169C2×C4×C73#12111yescharacteristic
170C2×C4×C73C22×D411yescharacteristic
171C2×C4×C73C22×D411yescharacteristic
172C2×C4×C73C2×#4111yescharacteristic
173C2×Dic73—41no—
174C2×Dic73—41no—
175Dic146—42no—
176Dic146—81no—
177C8×C73—42no—
178#10596831—22no—
179#10596831—22no—
180#10596831—42no—
181#10596831—42no—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—24no—
185C22×C73—21no—
186C22×C73—42no—
187C22×C73—42no—
188C22×C73—84no—
189C22×C73C2×#12111yescharacteristic
190C4×C73—21no—
191C4×C73—21no—
192C4×C73—21no—
193C4×C73—21no—
194C4×C73—42no—
195C4×C73—81no—
196C4×C73—81no—
197C4×C73C2×#13512yes—
198C4×C73#57712yes—
199Dic73—81no—
200Dic73—81no—
201#19395—731nomaximal, Sylow
202C2×C73—41no—
203C2×C73—44no—
204C2×C73#304211yescharacteristic
205C2×C73#305011yescharacteristic
206C2×C73#306911yescharacteristic
207#3231—731no—
208#3110D7311yescharacteristic
209#3101—731no—
210#2972—731no—
211#1683—732no—
212#1679—732no—
213#1673—732no—
214#1671—732no—
215#1572—731no—
216#1554—731no—
217#1544—731no—
218C73#1939511yescharacteristic, complemented, Sylow
219#533—731no—
220#533C2×D7311yescharacteristic
221C2×C4×D4C2×D7311yescharacteristic
222#500—731no—
223#498—731no—
224#487—731no—
225#487—731no—
226#485—1462no—
227#484—21no—
228#483—1461no—
229#479—1462no—
230#476—732no—
231#474C2×D7312yes—
232#461—732no—
233#461C2×D7312yes—
234#450—21no—
235#450—1461no—
236#440—22no—
237C4×SD16—1462no—
238C8×D4—732no—
239C8×D4—732no—
240C8×D4—1462no—
241#432C2×D7311yescharacteristic
242#431—731no—
243#431—731no—
244#431—731no—
245C2×C22⋊C8—1462no—
246#335—732no—
247#334—732no—
248#333—732no—
249#332—732no—
250#332—732no—
251#332—732no—
252C22×D4C22⋊D7311yescharacteristic
253C23×C4—21no—
254C2×SD16—22no—
255C2×SD16—1462no—
256C22×C8—1462no—
257C22×C8—1462no—
258C4⋊Q8—731no—
259C4⋊Q8—731no—
260C4⋊Q8C22×D7311yescharacteristic
261C4⋊Q8C22×D7311yescharacteristic
262#125—21no—
263#125—1461no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—1461no—
267C22⋊Q8—1461no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—1461no—
271C4×Q8—1461no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4C22⋊D7312yes—
275C4×D4C22×D7312yes—
276#117—21no—
277#117—1461no—
278C2×C4⋊C4C22⋊D7311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C42C22×D7311yescharacteristic
281#107—22no—
282#107—1462no—
283#106—1462no—
284#106—1462no—
285C4⋊C8—732no—
286C4⋊C8—732no—
287C4⋊C8—732no—
288C4⋊C8C22×D7312yes—
289#103—22no—
290#103—1462no—
291#103—1462no—
292#103—1462no—
293#102—22no—
294#102—1462no—
295C22⋊C8—21no—
296C22⋊C8—732no—
297C22⋊C8—1462no—
298C22⋊C8—1462no—
299C22⋊C8—1461no—
300C22⋊C8—1461no—
301#97—21no—
302C4×C8—731no—
303C4×C8—731no—
304C4×C8—1461no—
305C4×C8—1461no—
306C24—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—1461no—
310C2×Q8—1461no—
311C2×D4—24no—
312C2×D4—21no—
313C2×D4C2×C22⋊D7312yes—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—82no—
318C22×C4D4×D7311yescharacteristic
319C22×C4C2×C22⋊D7311yescharacteristic
320C22×C4#1192402711yescharacteristic
321SD16—44no—
322SD16—2924no—
323C2×C8—22no—
324C2×C8—5842no—
325C2×C8—5842no—
326C2×C8—732no—
327C2×C8—732no—
328C2×C8—1462no—
329C2×C8—1462no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—2921no—
338C4⋊C4—2921no—
339C4⋊C4—1461no—
340C4⋊C4—1461no—
341C4⋊C4—1461no—
342C4⋊C4—1461no—
343C4⋊C4C2×C22⋊D7312yes—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—41no—
347C22⋊C4—2921no—
348C42—41no—
349C42—2921no—
350C42D4×D7311yescharacteristic
351C42C2×C22⋊D7311yescharacteristic
352C42#1192402711yescharacteristic
353C42C23×D7311yescharacteristic
354C23—24no—
355C23—22no—
356C23—81no—
357C23—81no—
358C23#1455656311yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—2922no—
362Q8—5841no—
363D4—24no—
364D4—22no—
365D4#1455661114yes—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—2921no—
377C2×C4—2921no—
378C2×C4#1455656311yescharacteristic
379C2×C4#1455656311yescharacteristic
380C2×C4#1455656311yescharacteristic
381C2×C4C2×D4×D7311yescharacteristic
382C2×C4C22×C22⋊D7311yescharacteristic
383C2×C4C2×#1192402711yescharacteristic
384C8—42no—
385C8—2922no—
386C8—2922no—
387C8—1462no—
388C8—1462no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—84no—
396C22C2×#1455656311yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—5841no—
405C4—5841no—
406C4C2×#1455661112yes—
407C4#1798939712yes—
408C2—41no—
409C2—44no—
410C2#2261944511yescharacteristic
411C2#2261950911yescharacteristic
412C2#2261966511yescharacteristic
subgroupG/H
centerC22C2×#14556563
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3110C2
socleC22×C73C2×#121
prime radicalC22×D4×C73D4
hypercenter#3110D73
hypocenterC73#19395
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #22621086C2semidirect
4. #22609380C2semidirect
5. #22609352C2semidirect
6. #22609318C2semidirect
7. #22609304C2semidirect
8. #22608712C2semidirect
9. #22608642C2semidirect
10. #22608596C2semidirect
13. #17988990C22semidirect
14. #17988983C22semidirect
15. #17988937C22semidirect
16. #17988937C22semidirect
31. #17988339C22semidirect
32. #17988337C22semidirect
33. #17988335C22semidirect
34. #17988334C22semidirect
35. #17988333C22semidirect
36. #17988333C22semidirect
218. C73#19395semidirect

Maximal subgroups

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

subgroupindex
#226210862
#226199392
#226189552
#226093802
#226093522
#226093182
#226093042
#226087122
#226086422
#226085962
C73×#31102
#1939573