Finite Group Databasebeta

← #33071245

Subgroups — #33071245

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#22620026C211yesmaximal, characteristic
2#22619935C211yesmaximal, characteristic
3#22618954C211yesmaximal, characteristic
4#22609379C212yesmaximal, complemented
5#22609351C212yesmaximal, complemented
6#22609317C212yesmaximal, complemented
7#22609303C212yesmaximal, complemented
8#22608711C211yesmaximal, characteristic, complemented
9#22608641C211yesmaximal, characteristic, complemented
10#22608596C211yesmaximal, characteristic, complemented
11C73×#3231C211yesmaximal, characteristic, complemented
12C2×C4×D292C2211yescharacteristic
13#17988920—21no—
14#17988915—21no—
15#17988874C2212yes—
16#17988864C2212yes—
17#17988789C2212yes—
18#17988789C2212yes—
19#17988737—21no—
20#17988737—21no—
21#17988685—22no—
22C4×#11924015—22no—
23C8×D292C2212yes—
24C8×D292C2212yes—
25#17988652—22no—
26#17988644C2211yescharacteristic
27#17988640C2211yescharacteristic
28#17988640C2211yescharacteristic
29C22⋊C8×D73—22no—
30#17988340C2212yescomplemented
31#17988338C2212yescomplemented
32#17988336C2212yescomplemented
33#17988333C2212yescomplemented
34#17988333C2212yescomplemented
35#17988333C2212yescomplemented
36#533×C73C2211yescharacteristic
37#533×C73C2211yescharacteristic
38#500×C73C2211yescharacteristic, complemented
39#498×C73C2211yescharacteristic, complemented
40#487×C73C2211yescharacteristic, complemented
41#487×C73C2211yescharacteristic, complemented
42#485×C73—22no—
43#479×C73—22no—
44#431×C73C2211yescharacteristic
45C22×D292D411yescharacteristic
46C22×C4×D73—21no—
47C2×#11924015—22no—
48C2×#11924015—22no—
49C2×C8×D73—22no—
50C2×C8×D73—22no—
51C4×D292—21no—
52C4×D292—21no—
53C4×D292D412yes—
54C4×D292C2312yes—
55C2×C4⋊Dic73D411yescharacteristic
56C2×#11924005—21no—
57#14556524—22no—
58#14556522—22no—
59C4⋊#10596831C2312yes—
60C4⋊#10596831C2312yes—
61C4⋊#10596831C2312yes—
62#14556515—22no—
63#14556515—22no—
64#14556511—22no—
65#14556511—22no—
66C22⋊#10596831—21no—
67C22⋊#10596831—21no—
68C22⋊#10596831—21no—
69#14556503—22no—
70#14556503—22no—
71#14556502—21no—
72C4×#10596831—21no—
73C4×#10596831—21no—
74C4⋊Q8×C73C2311yescharacteristic
75C4⋊Q8×C73C2311yescharacteristic
76C4⋊Q8×C73C2311yescharacteristic
77C4⋊Q8×C73C2311yescharacteristic
78#125×C73—21no—
79#125×C73—21no—
80C22⋊Q8×C73—21no—
81C22⋊Q8×C73—21no—
82C22⋊Q8×C73—21no—
83C22⋊Q8×C73—21no—
84C4×Q8×C73—21no—
85C4×Q8×C73—21no—
86C4×Q8×C73—21no—
87C4×Q8×C73—21no—
88#117×C73—21no—
89#117×C73—21no—
90C2×C42×C73C2311yescharacteristic
91#107×C73—22no—
92#106×C73—22no—
93C4⋊C8×C73D412yes—
94#103×C73—22no—
95#103×C73—22no—
96C22⋊C8×C73D412yes—
97C4×C8×C73C2311yescharacteristic
98C4×C8×C73C2311yescharacteristic
99C23×D73—21no—
100C2×D292—21no—
101C2×D292—24no—
102C2×D292C2×D412yes—
103C22×Dic73—21no—
104C2×C4×D73—22no—
105C2×C4×D73—22no—
106C2×C4×D73—82no—
107#11924015—44no—
108#11924015—44no—
109C2×#10596831—22no—
110C2×#10596831—22no—
111C2×#10596831—22no—
112C8×D73—82no—
113C8×D73—82no—
114C4⋊Dic73—21no—
115C4⋊Dic73C2×D412yes—
116#11924005—22no—
117#11924005—22no—
118C2×Q8×C73—21no—
119C2×Q8×C73—21no—
120C2×Q8×C73—21no—
121C2×Q8×C73—21no—
122C22×C4×C73C2×D411yescharacteristic
123C22×C4×C73C2×D411yescharacteristic
124C22×C4×C73#4111yescharacteristic
125C2×C8×C73C2×D412yes—
126C2×C8×C73C2×D412yes—
127C4⋊C4×C73—21no—
128C4⋊C4×C73—21no—
129C4⋊C4×C73—21no—
130C4⋊C4×C73—21no—
131C4⋊C4×C73—21no—
132C4⋊C4×C73—21no—
133C4⋊C4×C73—21no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—41no—
136C4⋊C4×C73—41no—
137C4⋊C4×C73—41no—
138C4⋊C4×C73—41no—
139C22⋊C4×C73—41no—
140C22⋊C4×C73—41no—
141C42×C73—41no—
142C42×C73—41no—
143C42×C73C2×D411yescharacteristic
144C42×C73C2×D411yescharacteristic
145C42×C73#4111yescharacteristic
146C42×C73C2411yescharacteristic
147C23×C73#12111yescharacteristic
148C22×D73—22no—
149C22×D73—24no—
150C22×D73—81no—
151C22×D73—81no—
152Q8×C73—42no—
153Q8×C73—42no—
154Q8×C73—42no—
155Q8×C73—81no—
156C2×C4×C73—22no—
157C2×C4×C73—22no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73#12111yescharacteristic
164C2×C4×C73#12111yescharacteristic
165C2×C4×C73#12111yescharacteristic
166C2×C4×C73C22×D411yescharacteristic
167C2×C4×C73C22×D411yescharacteristic
168C2×C4×C73C2×#4111yescharacteristic
169C2×Dic73—22no—
170C2×Dic73—81no—
171D292—24no—
172D292—22no—
173D292#13514yes—
174C4×D73—42no—
175C4×D73—42no—
176C4×D73—82no—
177C8×C73—22no—
178C8×C73—22no—
179#10596831—42no—
180#10596831—42no—
181#10596831—42no—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73C2×#12111yescharacteristic
186C2×D73—24no—
187C2×D73—42no—
188C2×D73—42no—
189C2×D73—84no—
190C4×C73—21no—
191C4×C73—21no—
192C4×C73—21no—
193C4×C73—21no—
194C4×C73—81no—
195C4×C73—81no—
196C4×C73—81no—
197C4×C73—81no—
198C4×C73C2×#13512yes—
199C4×C73#57712yes—
200Dic73—42no—
201#19395—731nomaximal, Sylow
202C2×C73—41no—
203C2×C73#304211yescharacteristic
204C2×C73#305011yescharacteristic
205C2×C73#306911yescharacteristic
206D73—44no—
207#3231D7311yescharacteristic, complemented
208#3110—731no—
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#533C2×D7311yescharacteristic
220#533C2×D7311yescharacteristic
221C2×C4×D4—731no—
222#500C2×D7311yescharacteristic, complemented
223#498C2×D7311yescharacteristic, complemented
224#487C2×D7311yescharacteristic, complemented
225#487C2×D7311yescharacteristic, complemented
226#485—22no—
227#484—1461no—
228#483—1461no—
229#479—22no—
230#476—732no—
231#474—732no—
232#461—732no—
233#461—732no—
234#450—1461no—
235#450—1461no—
236#440—1462no—
237C4×SD16—1462no—
238C8×D4—732no—
239C8×D4—732no—
240C8×D4—1462no—
241#432—731no—
242#431—731no—
243#431—731no—
244#431C2×D7311yescharacteristic
245C2×C22⋊C8—1462no—
246#335—732no—
247#334—732no—
248#333—732no—
249#332—732no—
250#332—732no—
251#332—732no—
252C22×D4—731no—
253C23×C4—1461no—
254C2×SD16—1462no—
255C2×SD16—1462no—
256C22×C8—1462no—
257C22×C8—1462no—
258C4⋊Q8C22×D7311yescharacteristic
259C4⋊Q8C22×D7311yescharacteristic
260C4⋊Q8C22×D7311yescharacteristic
261C4⋊Q8C22×D7311yescharacteristic
262#125—21no—
263#125—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—732no—
273C4×D4—732no—
274C4×D4—1461no—
275C4×D4—1461no—
276#117—21no—
277#117—21no—
278C2×C4⋊C4—731no—
279C2×C22⋊C4—1461no—
280C2×C42C22×D7311yescharacteristic
281#107—22no—
282#107—1462no—
283#106—22no—
284#106—1462no—
285C4⋊C8—732no—
286C4⋊C8—732no—
287C4⋊C8—732no—
288C4⋊C8C22⋊D7312yes—
289#103—22no—
290#103—22no—
291#103—1462no—
292#103—1462no—
293#102—1462no—
294#102—1462no—
295C22⋊C8—1461no—
296C22⋊C8—1461no—
297C22⋊C8—1461no—
298C22⋊C8—1462no—
299C22⋊C8—1462no—
300C22⋊C8C22⋊D7312yes—
301#97—1461no—
302C4×C8—1461no—
303C4×C8—1461no—
304C4×C8C22×D7311yescharacteristic
305C4×C8C22×D7311yescharacteristic
306C24—1461no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×D4—732no—
312C2×D4—1464no—
313C2×D4—1461no—
314C22×C4—5842no—
315C22×C4—1462no—
316C22×C4—1461no—
317C22×C4—1462no—
318C22×C4D4×D7311yescharacteristic
319C22×C4C2×C22⋊D7311yescharacteristic
320C22×C4#1192402711yescharacteristic
321SD16—2924no—
322SD16—2924no—
323C2×C8—5842no—
324C2×C8—5842no—
325C2×C8—1462no—
326C2×C8—1462no—
327C2×C8—1462no—
328C2×C8C2×C22⋊D7312yes—
329C2×C8C2×C22⋊D7312yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—732no—
343C4⋊C4—1461no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—1462no—
347C22⋊C4—1462no—
348C42—41no—
349C42—41no—
350C42D4×D7311yescharacteristic
351C42C2×C22⋊D7311yescharacteristic
352C42#1192402711yescharacteristic
353C42C23×D7311yescharacteristic
354C23—5841no—
355C23—5841no—
356C23—1464no—
357C23—1462no—
358C23#1455656211yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—81no—
363D4—734no—
364D4—1464no—
365D4—1462no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—2922no—
374C2×C4—2922no—
375C2×C4—5842no—
376C2×C4—5841no—
377C2×C4—1462no—
378C2×C4#1455656211yescharacteristic
379C2×C4#1455656211yescharacteristic
380C2×C4#1455656211yescharacteristic
381C2×C4C2×D4×D7311yescharacteristic
382C2×C4C22×C22⋊D7311yescharacteristic
383C2×C4C2×#1192402711yescharacteristic
384C8—22no—
385C8—22no—
386C8—2922no—
387C8—2922no—
388C8—2922no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—2922no—
393C22—2922no—
394C22—5844no—
395C22—1464no—
396C22C2×#1455656211yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—2922no—
406C4#135×D7312yes—
407C4#1798939512yes—
408C2—41no—
409C2—2924no—
410C2#2261943911yescharacteristic
411C2#2261950811yescharacteristic
412C2#2261966411yescharacteristic
subgroupG/H
centerC22C2×#14556562
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#3231C2
socleC22×C73C2×#121
prime radicalC22×D292D4
hypercenter#3231D73
hypocenterC73#19395
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3231

Complements

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

NH ≅ G/Ntype
4. #22609379C2semidirect
5. #22609351C2semidirect
6. #22609317C2semidirect
7. #22609303C2semidirect
8. #22608711C2semidirect
9. #22608641C2semidirect
10. #22608596C2semidirect
11. C73×#3231C2semidirect
30. #17988340C22semidirect
31. #17988338C22semidirect
32. #17988336C22semidirect
33. #17988333C22semidirect
34. #17988333C22semidirect
35. #17988333C22semidirect
38. #500×C73C22semidirect
39. #498×C73C22semidirect
40. #487×C73C22semidirect
41. #487×C73C22semidirect
207. #3231D73semidirect
218. C73#19395semidirect
222. #500C2×D73semidirect
223. #498C2×D73semidirect
224. #487C2×D73semidirect
225. #487C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226200262
#226199352
#226189542
#226093792
#226093512
#226093172
#226093032
#226087112
#226086412
#226085962
C73×#32312
#1939573