Finite Group Databasebeta

← #33240373

Subgroups — #33240373

Order 18688.

696 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623065C212yesmaximal, complemented
2C2×#17988705C211yesmaximal, characteristic, complemented
3#22612009C212yesmaximal, complemented
4#22612004C212yesmaximal, complemented
5#22611997C214yesmaximal, complemented
6C2×#17988408C211yesmaximal, characteristic, complemented
7#22611680C212yesmaximal, complemented
8C73×#3147C211yesmaximal, characteristic, complemented
9C2×#14556613C2211yescharacteristic
10C2×#14556597—21no—
11#17989028C2214yes—
12#17989027—22no—
13#17989023C2214yes—
14#17989022—22no—
15#17989017—24no—
16C2×#14556531C2212yes—
17#17988705—21no—
18#17988705—21no—
19#17988705C2212yes—
20#17988705C2212yes—
21C2×#14556527C2211yescharacteristic
22C2×#14556519—21no—
23#17988408C2212yescomplemented
24#17988408C2212yescomplemented
25#17988403C2214yescomplemented
26#17988400C2214yescomplemented
27C2×#135×C73—21no—
28C2×#135×C73C2212yes—
29#532×C73—21no—
30#495×C73—21no—
31#495×C73C2212yescomplemented
32#494×C73C2211yescharacteristic, complemented
33#493×C73C2211yescharacteristic, complemented
34#492×C73—21no—
35C2×C4×C8×C73C2211yescharacteristic
36C2×#11924028—21no—
37#14556613—21no—
38#14556613—21no—
39#14556613D412yescomplemented
40#14556613C2312yes—
41C2×Dic292D411yescharacteristic, complemented
42C2×#11924017—21no—
43C2×D584D411yescharacteristic, complemented
44#14556597—22no—
45#14556597—22no—
46#14556597—41no—
47#14556597—41no—
48C2×#11924012—21no—
49C2×#11924011—41no—
50C2×C8×D73—41no—
51#14556537—24no—
52#14556535—24no—
53#14556534—24no—
54#14556532—24no—
55#14556531—22no—
56#14556531C2314yes—
57C2×#11924003—22no—
58#14556527—21no—
59#14556527C2312yes—
60#14556519—22no—
61#14556519—22no—
62C2×#41×C73—22no—
63#135×C73—22no—
64#135×C73—22no—
65#135×C73—22no—
66#135×C73—22no—
67#135×C73—41no—
68#135×C73—41no—
69#135×C73—41no—
70#135×C73—41no—
71C2×Q16×C73—21no—
72C2×Q16×C73C2312yes—
73C2×SD16×C73—22no—
74C2×SD16×C73—22no—
75C2×D8×C73—21no—
76C2×D8×C73C2312yes—
77C22×C8×C73D411yescharacteristic
78C22×C8×C73C2311yescharacteristic
79C4⋊Q8×C73—21no—
80#127×C73—21no—
81#124×C73—22no—
82#121×C73—42no—
83C4×D4×C73—42no—
84C2×C42×C73D411yescharacteristic
85C4×C8×C73D411yescharacteristic
86C4×C8×C73D411yescharacteristic
87C4×C8×C73C2311yescharacteristic
88C4×C8×C73C2311yescharacteristic
89#11924028—22no—
90#11924028—22no—
91#11924028—41no—
92#11924028—41no—
93C2×Dic146—21no—
94C2×D292—21no—
95C2×C22⋊D73—41no—
96C2×C4×D73—41no—
97Dic292—21no—
98Dic292C2×D412yes—
99#11924017—22no—
100#11924017—22no—
101D584—21no—
102D584C2×D412yes—
103#11924012—22no—
104#11924012—41no—
105#11924011—42no—
106#11924011—42no—
107C2×#10596831—41no—
108C8×D73—42no—
109C8×D73—42no—
110#11924003—24no—
111#41×C73—42no—
112#41×C73—42no—
113#41×C73—42no—
114#41×C73—42no—
115C2×Q8×C73—22no—
116C2×D4×C73—22no—
117C2×D4×C73—42no—
118C2×D4×C73—42no—
119C22×C4×C73—21no—
120C22×C4×C73—42no—
121C22×C4×C73C2×D411yescharacteristic
122Q16×C73—22no—
123Q16×C73—22no—
124Q16×C73—41no—
125Q16×C73—41no—
126SD16×C73—42no—
127SD16×C73—42no—
128SD16×C73—42no—
129SD16×C73—42no—
130D8×C73—22no—
131D8×C73—22no—
132D8×C73—41no—
133D8×C73—41no—
134C2×C8×C73D811yescharacteristic
135C2×C8×C73D811yescharacteristic
136C2×C8×C73D811yescharacteristic
137C2×C8×C73D811yescharacteristic
138C2×C8×C73C2×D411yescharacteristic
139C2×C8×C73C2×D411yescharacteristic
140C2×C8×C73C2×D411yescharacteristic
141C2×C8×C73C2×D411yescharacteristic
142C2×C8×C73C2×D411yescharacteristic
143C2×C8×C73#4111yescharacteristic
144C2×C8×C73#4111yescharacteristic
145C2×C8×C73C2411yescharacteristic
146C4⋊C4×C73—42no—
147C22⋊C4×C73—44no—
148C42×C73—21no—
149C42×C73C2×D411yescharacteristic
150C42×C73C2×D411yescharacteristic
151C23×C73—42no—
152C23×C73AGL(1,ℤ/8)11yescharacteristic
153C22×D73—41no—
154Q8×C73—42no—
155Q8×C73—42no—
156D4×C73—42no—
157D4×C73—42no—
158D4×C73—84no—
159D4×C73—82no—
160D4×C73—82no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—42no—
164C2×C4×C73—42no—
165C2×C4×C73—82no—
166C2×C4×C73—82no—
167C2×C4×C73#12111yescharacteristic
168C2×C4×C73#12111yescharacteristic
169C2×C4×C73C2×D811yescharacteristic
170C2×C4×C73C2×D811yescharacteristic
171C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C73C22×D411yescharacteristic
173C22⋊D73—42no—
174C22⋊D73—42no—
175C2×Dic73—41no—
176D292—22no—
177D292—41no—
178C4×D73—42no—
179C4×D73—42no—
180Dic146—22no—
181Dic146—41no—
182C8×C73#12111yescharacteristic
183C8×C73#12111yescharacteristic
184C8×C73C2×D811yescharacteristic
185C8×C73C2×D811yescharacteristic
186C8×C73C2×D811yescharacteristic
187C8×C73C2×D811yescharacteristic
188C8×C73C22×D411yescharacteristic
189C8×C73C2×#4111yescharacteristic
190#10596831—42no—
191C22×C73—41no—
192C22×C73—84no—
193C22×C73—82no—
194C22×C73#45911yescharacteristic
195C22×C73#45911yescharacteristic
196C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
197C2×D73—42no—
198C2×D73—82no—
199C4×C73—22no—
200C4×C73—41no—
201C4×C73—82no—
202C4×C73#45911yescharacteristic
203C4×C73#45911yescharacteristic
204C4×C73C2×#12111yescharacteristic
205C4×C73C22×D811yescharacteristic
206Dic73—42no—
207#33594—731nomaximal, Sylow
208C2×C73—21no—
209C2×C73—21no—
210C2×C73—21no—
211C2×C73—162no—
212C2×C73C2×#45911yescharacteristic
213D73—82no—
214#3414—732no—
215#3147D7311yescharacteristic, complemented
216C2×#443—731no—
217#2214—732no—
218#2213—732no—
219#2212—734no—
220C2×#364—731no—
221#2148—732no—
222C73#3359411yescharacteristic, complemented, Sylow
223C2×#135—21no—
224C2×#135—731no—
225C2×#135C2×D7312yes—
226C2×#131—1461no—
227#532—21no—
228#510—734no—
229#510—1462no—
230#509—734no—
231#509—1462no—
232#508—1464no—
233C2×M32—732no—
234#495—21no—
235#495C2×D7312yescomplemented
236#494C2×D7311yescharacteristic, complemented
237#493C2×D7311yescharacteristic, complemented
238#492—21no—
239#443—732no—
240#443—732no—
241#443—1461no—
242#443—1461no—
243C2×#108—731no—
244C2×#104—1461no—
245C2×C4×C8C2×D7311yescharacteristic
246#364—732no—
247#364—732no—
248#362—734no—
249#361—734no—
250C2×#41—22no—
251C2×#41—1461no—
252#135—22no—
253#135—22no—
254#135—22no—
255#135—22no—
256#135—41no—
257#135—41no—
258#135—41no—
259#135—41no—
260#135—732no—
261#135—732no—
262#135—1461no—
263#135—1461no—
264C2×Q16—21no—
265C2×Q16—731no—
266C2×Q16C22×D7312yes—
267C2×SD16—22no—
268C2×SD16—22no—
269C2×SD16—1461no—
270C2×D8—21no—
271C2×D8—731no—
272C2×D8C22×D7312yes—
273#131—2921no—
274#131—2921no—
275#131—1462no—
276#131—1462no—
277C2×M16—2921no—
278C2×M16—1461no—
279C22×C8—2921no—
280C22×C8C22⋊D7311yescharacteristic
281C22×C8C22×D7311yescharacteristic
282C4⋊Q8—21no—
283#127—21no—
284#124—22no—
285#121—42no—
286C4×D4—42no—
287C2×C42C22⋊D7311yescharacteristic
288Q32—1464no—
289SD32—1464no—
290SD32—1464no—
291D16—1464no—
292M32—734no—
293M32—1462no—
294C2×C16—1462no—
295#108—732no—
296#108—1461no—
297#104—1462no—
298#104—1462no—
299C4×C8C22⋊D7311yescharacteristic
300C4×C8C22⋊D7311yescharacteristic
301C4×C8C22×D7311yescharacteristic
302C4×C8C22×D7311yescharacteristic
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
307#41—2921no—
308#41—2921no—
309#41—1462no—
310#41—1462no—
311C2×Q8—22no—
312C2×Q8—1461no—
313C2×D4—22no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—2921no—
317C2×D4—1461no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—2921no—
321C22×C4C2×C22⋊D7311yescharacteristic
322Q16—22no—
323Q16—22no—
324Q16—41no—
325Q16—41no—
326Q16—732no—
327Q16—1461no—
328SD16—42no—
329SD16—42no—
330SD16—42no—
331SD16—42no—
332SD16—1462no—
333SD16—1462no—
334D8—22no—
335D8—22no—
336D8—41no—
337D8—41no—
338D8—732no—
339D8—1461no—
340M16—2921no—
341M16—2922no—
342M16—2922no—
343M16—1462no—
344C2×C8—2921no—
345C2×C8—2922no—
346C2×C8—2922no—
347C2×C8#1192401311yescharacteristic
348C2×C8#1192401311yescharacteristic
349C2×C8#1192401311yescharacteristic
350C2×C8#1192401311yescharacteristic
351C2×C8D4×D7311yescharacteristic
352C2×C8D4×D7311yescharacteristic
353C2×C8C2×C22⋊D7311yescharacteristic
354C2×C8C2×C22⋊D7311yescharacteristic
355C2×C8C2×C22⋊D7311yescharacteristic
356C2×C8#1192402711yescharacteristic
357C2×C8#1192402711yescharacteristic
358C2×C8C23×D7311yescharacteristic
359C4⋊C4—42no—
360C22⋊C4—44no—
361C42—21no—
362C42C2×C22⋊D7311yescharacteristic
363C42C2×C22⋊D7311yescharacteristic
364C16—1464no—
365C23—42no—
366C23—2921no—
367C23#1455661711yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—2921no—
371Q8—1462no—
372D4—42no—
373D4—42no—
374D4—84no—
375D4—82no—
376D4—82no—
377D4—2922no—
378D4—2922no—
379D4—2921no—
380D4—1462no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—2921no—
388C2×C4—2922no—
389C2×C4—2922no—
390C2×C4#1455656211yescharacteristic
391C2×C4#1455656211yescharacteristic
392C2×C4C2×#1192401311yescharacteristic
393C2×C4C2×#1192401311yescharacteristic
394C2×C4#1455661711yescharacteristic
395C2×C4C22×C22⋊D7311yescharacteristic
396C8—2922no—
397C8#1455656211yescharacteristic
398C8#1455656211yescharacteristic
399C8C2×#1192401311yescharacteristic
400C8C2×#1192401311yescharacteristic
401C8C2×#1192401311yescharacteristic
402C8C2×#1192401311yescharacteristic
403C8C2×D4×D7311yescharacteristic
404C8C2×#1192402711yescharacteristic
405C22—41no—
406C22—84no—
407C22—82no—
408C22—2922no—
409C22—5842no—
410C22#1798877911yescharacteristic
411C22#1798877911yescharacteristic
412C22C2×#1455661711yescharacteristic
413C4—22no—
414C4—41no—
415C4—82no—
416C4—2922no—
417C4#1798877911yescharacteristic
418C4#1798877911yescharacteristic
419C4C2×#1455656211yescharacteristic
420C4C22×#1192401311yescharacteristic
421C2—21no—
422C2—21no—
423C2—21no—
424C2—162no—
425C2—5842no—
426C2C2×#1798877911yescharacteristic
subgroupG/H
centerC4#17988779
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3147C2
socleC2×C73C2×#459
hypercenter#3147D73
hypocenterC73#33594
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #22623065C2semidirect
2. C2×#17988705C2semidirect
3. #22612009C2semidirect
4. #22612004C2semidirect
5. #22611997C2semidirect
6. C2×#17988408C2semidirect
7. #22611680C2semidirect
8. C73×#3147C2semidirect
23. #17988408C22semidirect
24. #17988408C22semidirect
25. #17988403C22semidirect
26. #17988400C22semidirect
31. #495×C73C22semidirect
32. #494×C73C22semidirect
33. #493×C73C22semidirect
39. #14556613D4semidirect
41. C2×Dic292D4semidirect
43. C2×D584D4semidirect
215. #3147D73semidirect
222. C73#33594semidirect
235. #495C2×D73semidirect
236. #494C2×D73semidirect
237. #493C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226230652
C2×#179887052
#226120092
#226120042
#226119972
C2×#179884082
#226116802
C73×#31472
#3359473