Finite Group Databasebeta

← #32985495

Subgroups — #32985495

Order 18688.

564 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22610917C211yesmaximal, characteristic, complemented
2#22610768C211yesmaximal, characteristic
3#22609956C211yesmaximal, characteristic, complemented
4C8×C22⋊Dic73C211yesmaximal, characteristic, complemented
5#22608503C211yesmaximal, characteristic, complemented
6#22608499C211yesmaximal, characteristic
7C2×#465×C73C211yesmaximal, characteristic
8C2×#14556523—21no—
9C2×#14556506—22no—
10C2×C22⋊#10596831C2211yescharacteristic
11C2×C8×Dic73—21no—
12#17988447C2211yescharacteristic
13#17988447C2211yescharacteristic
14C4×C22⋊Dic73C2211yescharacteristic
15#17988354—22no—
16#17988350—21no—
17#17988341—21no—
18#17988324—22no—
19#17988321—22no—
20C22×SD16×C73C411yescharacteristic, complemented
21C23×C8×C73C2211yescharacteristic
22C2×C22⋊Q8×C73C2211yescharacteristic
23C2×#121×C73C2211yescharacteristic
24#465×C73—21no—
25#465×C73—21no—
26#465×C73—21no—
27#465×C73—21no—
28C2×#106×C73C411yescharacteristic, complemented
29C2×#103×C73C411yescharacteristic, complemented
30C2×#102×C73C411yescharacteristic, complemented
31C22×#10596831—21no—
32C2×C4⋊Dic73—21no—
33C2×C4⋊Dic73—21no—
34C2×C22⋊Dic73—21no—
35C2×C22⋊Dic73—21no—
36C2×C22⋊Dic73D412yes—
37C2×C4×Dic73—21no—
38#14556523—42no—
39#14556506—44no—
40C22⋊#10596831—21no—
41C22⋊#10596831D412yes—
42C8×Dic73—42no—
43#14556498—21no—
44#14556498—21no—
45#14556498—21no—
46C22×Q8×C73C2×C411yescharacteristic
47C22×D4×C73—21no—
48C22×D4×C73C2×C411yescharacteristic
49C23×C4×C73C2311yescharacteristic
50C2×SD16×C73—21no—
51C2×SD16×C73—21no—
52C2×SD16×C73—42no—
53C22×C8×C73—42no—
54C22×C8×C73C2×C411yescharacteristic
55C22×C8×C73C2×C411yescharacteristic
56C22×C8×C73D411yescharacteristic
57C22×C8×C73D411yescharacteristic
58C22×C8×C73D411yescharacteristic
59C22×C8×C73D411yescharacteristic
60C22⋊Q8×C73—21no—
61C22⋊Q8×C73—21no—
62C22⋊Q8×C73—21no—
63C22⋊Q8×C73—21no—
64#121×C73—21no—
65#121×C73—21no—
66#121×C73—21no—
67#121×C73—21no—
68C2×C4⋊C4×C73—21no—
69C2×C4⋊C4×C73C2×C411yescharacteristic
70C2×C4⋊C4×C73C2×C411yescharacteristic
71C2×C22⋊C4×C73—21no—
72C2×C22⋊C4×C73—21no—
73#106×C73—21no—
74#106×C73—21no—
75#103×C73—21no—
76#103×C73—21no—
77#102×C73—21no—
78#102×C73—21no—
79C22×Dic73—22no—
80C22×Dic73—41no—
81C22×Dic73—41no—
82C2×#10596831—22no—
83C2×#10596831—41no—
84C2×#10596831—41no—
85C4⋊Dic73—41no—
86C4⋊Dic73—41no—
87C4⋊Dic73—41no—
88C4⋊Dic73—41no—
89C22⋊Dic73—22no—
90C22⋊Dic73—42no—
91C22⋊Dic73—42no—
92C22⋊Dic73SD1614yescomplemented
93C4×Dic73—41no—
94C4×Dic73—41no—
95C24×C73—21no—
96C24×C73C2×D411yescharacteristic
97C2×Q8×C73—21no—
98C2×Q8×C73—21no—
99C2×Q8×C73—41no—
100C2×Q8×C73—41no—
101C2×D4×C73—21no—
102C2×D4×C73—21no—
103C2×D4×C73—41no—
104C2×D4×C73—41no—
105C2×D4×C73—41no—
106C2×D4×C73—41no—
107C2×D4×C73—42no—
108C2×D4×C73—42no—
109C22×C4×C73—21no—
110C22×C4×C73—21no—
111C22×C4×C73—21no—
112C22×C4×C73—41no—
113C22×C4×C73—41no—
114C22×C4×C73C22×C411yescharacteristic
115C22×C4×C73C2×D411yescharacteristic
116C22×C4×C73C2×D411yescharacteristic
117C22×C4×C73C2×D411yescharacteristic
118C22×C4×C73#4111yescharacteristic
119C22×C4×C73#4111yescharacteristic
120SD16×C73—42no—
121SD16×C73—42no—
122C2×C8×C73—22no—
123C2×C8×C73—22no—
124C2×C8×C73—22no—
125C2×C8×C73—41no—
126C2×C8×C73—41no—
127C2×C8×C73—41no—
128C2×C8×C73C22⋊C411yescharacteristic
129C2×C8×C73C22⋊C411yescharacteristic
130C2×C8×C73C22⋊C411yescharacteristic
131C2×C8×C73C22⋊C411yescharacteristic
132C4⋊C4×C73—21no—
133C4⋊C4×C73—21no—
134C4⋊C4×C73—21no—
135C4⋊C4×C73—21no—
136C4⋊C4×C73—41no—
137C4⋊C4×C73—41no—
138C22⋊C4×C73—41no—
139C22⋊C4×C73—41no—
140C22⋊C4×C73—41no—
141C22⋊C4×C73—41no—
142C23×C73—41no—
143C23×C73—41no—
144C23×C73—41no—
145C23×C73—42no—
146C23×C73—81no—
147C23×C73—81no—
148C23×C73C4×D411yescharacteristic
149C23×C73#12111yescharacteristic
150C23×C73#12711yescharacteristic
151C23×C73C2×SD1612yes—
152C23×C73#13512yes—
153Q8×C73—41no—
154Q8×C73—41no—
155Q8×C73—81no—
156D4×C73—41no—
157D4×C73—41no—
158D4×C73—81no—
159D4×C73—82no—
160D4×C73—81no—
161D4×C73—81no—
162C2×C4×C73—21no—
163C2×C4×C73—21no—
164C2×C4×C73—21no—
165C2×C4×C73—21no—
166C2×C4×C73—21no—
167C2×C4×C73—21no—
168C2×C4×C73—41no—
169C2×C4×C73—41no—
170C2×C4×C73—41no—
171C2×C4×C73—41no—
172C2×C4×C73—41no—
173C2×C4×C73—41no—
174C2×C4×C73—42no—
175C2×C4×C73—81no—
176C2×C4×C73—81no—
177C2×C4×C73C2×C22⋊C411yescharacteristic
178C2×C4×C73C4×D411yescharacteristic
179C2×C4×C73#12111yescharacteristic
180C2×C4×C73#12411yescharacteristic
181C2×Dic73—24no—
182C2×Dic73—42no—
183C2×Dic73—42no—
184C2×Dic73—81no—
185C2×Dic73—81no—
186C2×Dic73—81no—
187C2×Dic73—81no—
188C2×Dic73—81no—
189C2×Dic73—81no—
190C8×C73—21no—
191C8×C73—21no—
192C8×C73—21no—
193C8×C73#9912yescomplemented
194#10596831—42no—
195C22×C73—21no—
196C22×C73—21no—
197C22×C73—21no—
198C22×C73—21no—
199C22×C73—21no—
200C22×C73—21no—
201C22×C73—42no—
202C22×C73—42no—
203C22×C73—81no—
204C22×C73—81no—
205C22×C73—81no—
206C22×C73—162no—
207C22×C73#39011yescharacteristic
208C22×C73C4×SD1612yes—
209C22×C73#46512yes—
210C22×C73#49211yescharacteristic
211C22×C73#49511yescharacteristic
212C4×C73—21no—
213C4×C73—21no—
214C4×C73—21no—
215C4×C73—81no—
216C4×C73—81no—
217C4×C73—81no—
218C4×C73C2×#9911yescharacteristic
219C4×C73#41011yescharacteristic
220Dic73—44no—
221Dic73—161no—
222Dic73—161no—
223#9781—731nomaximal, Sylow
224C2×C73—22no—
225C2×C73—21no—
226C2×C73—21no—
227C2×C73—22no—
228C2×C73—161no—
229C2×C73#196411yescharacteristic
230C2×C73#196811yescharacteristic
231C2×C73#197611yescharacteristic
232C2×#465D7311yescharacteristic
233#1996—731no—
234#1964—731no—
235#1807—731no—
236C8×C22⋊C4—731no—
237#1520—731no—
238#1519—731no—
239C73#978111yescharacteristic, complemented, Sylow
240C22×SD16Dic7311yescharacteristic, complemented
241C23×C8C2×D7311yescharacteristic
242C2×C22⋊Q8C2×D7311yescharacteristic
243C2×#121C2×D7311yescharacteristic
244#465—21no—
245#465—21no—
246#465—21no—
247#465—21no—
248C2×#106—1461no—
249C2×#106Dic7311yescharacteristic, complemented
250C2×#103Dic7311yescharacteristic, complemented
251C2×#102Dic7311yescharacteristic, complemented
252C2×#99—1462no—
253C2×C22⋊C8—731no—
254C2×C4×C8—1461no—
255#380—731no—
256#380—731no—
257C4×C22⋊C4—731no—
258#342—1462no—
259#340—1461no—
260#336—1461no—
261#328—1462no—
262#327—1462no—
263C22×Q8C2×Dic7311yescharacteristic
264C22×D4—21no—
265C22×D4C2×Dic7311yescharacteristic
266C23×C4C22×D7311yescharacteristic
267C2×SD16—21no—
268C2×SD16—21no—
269C2×SD16—42no—
270C22×C8—42no—
271C22×C8—1461no—
272C22×C8C2×Dic7311yescharacteristic
273C22×C8C2×Dic7311yescharacteristic
274C22×C8C22⋊D7311yescharacteristic
275C22×C8C22⋊D7311yescharacteristic
276C22×C8C22⋊D7311yescharacteristic
277C22×C8C22⋊D7311yescharacteristic
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—21no—
283#121—21no—
284#121—21no—
285#121—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—1461no—
288C2×C4⋊C4—1461no—
289C2×C4⋊C4C2×Dic7311yescharacteristic
290C2×C4⋊C4C2×Dic7311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—732no—
294C2×C22⋊C4—1461no—
295C2×C22⋊C4—1461no—
296C2×C42—1461no—
297#106—21no—
298#106—21no—
299#106—2922no—
300#103—21no—
301#103—21no—
302#102—21no—
303#102—21no—
304#99—2924no—
305C22⋊C8—732no—
306C22⋊C8—1461no—
307C4×C8—2922no—
308#95—1461no—
309#95—1461no—
310#95—1461no—
311C24—21no—
312C24D4×D7311yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—2921no—
331C22×C4—2921no—
332C22×C4—1462no—
333C22×C4C22×Dic7311yescharacteristic
334C22×C4D4×D7311yescharacteristic
335C22×C4C2×C22⋊D7311yescharacteristic
336C22×C4C2×C22⋊D7311yescharacteristic
337C22×C4#1192402711yescharacteristic
338C22×C4#1192402711yescharacteristic
339SD16—42no—
340SD16—42no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—2921no—
348C2×C8—2921no—
349C2×C8—1462no—
350C2×C8C22⋊Dic7311yescharacteristic
351C2×C8C22⋊Dic7311yescharacteristic
352C2×C8C22⋊Dic7311yescharacteristic
353C2×C8C22⋊Dic7311yescharacteristic
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—2921no—
361C4⋊C4—2921no—
362C4⋊C4—2921no—
363C4⋊C4—2921no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—2922no—
369C22⋊C4—2922no—
370C22⋊C4—734no—
371C22⋊C4—1462no—
372C42—2921no—
373C42—2921no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23D4×Dic7311yescharacteristic
381C23#1455656211yescharacteristic
382C23#1455658711yescharacteristic
383C23SD16×D7312yes—
384C23#1455661212yes—
385Q8—41no—
386Q8—41no—
387Q8—81no—
388D4—41no—
389D4—41no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—82no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—2922no—
410C2×C4—2922no—
411C2×C4—5841no—
412C2×C4—5841no—
413C2×C4—5841no—
414C2×C4—5841no—
415C2×C4—5841no—
416C2×C4—5841no—
417C2×C4—1464no—
418C2×C4C2×C22⋊Dic7311yescharacteristic
419C2×C4D4×Dic7311yescharacteristic
420C2×C4#1455656211yescharacteristic
421C2×C4#1455657811yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—2922no—
426C8#1455650612yescomplemented
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—162no—
439C22#1798849011yescharacteristic
440C22SD16×Dic7312yes—
441C22#1798881312yes—
442C22#1798895811yescharacteristic
443C22#1798896811yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—2924no—
451C4—11681no—
452C4—11681no—
453C4C2×#1455650611yescharacteristic
454C4#1798856111yescharacteristic
455C2—22no—
456C2—21no—
457C2—21no—
458C2—22no—
459C2—161no—
460C2#2261076811yescharacteristic
461C2#2261078711yescharacteristic
462C2#2261083011yescharacteristic
subgroupG/H
centerC22#17988490
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#465×C73C2
socleC22×C73#390
prime radicalC2×#121×C73C22
cosocleC23×C4×C73C23
hypercenterC2×#465D73
hypocenterC73#9781
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
1. #22610917C2semidirect
3. #22609956C2semidirect
4. C8×C22⋊Dic73C2semidirect
5. #22608503C2semidirect
20. C22×SD16×C73C4semidirect
28. C2×#106×C73C4semidirect
29. C2×#103×C73C4semidirect
30. C2×#102×C73C4semidirect
92. C22⋊Dic73SD16semidirect
193. C8×C73#99semidirect
239. C73#9781semidirect
240. C22×SD16Dic73semidirect
249. C2×#106Dic73semidirect
250. C2×#103Dic73semidirect
251. C2×#102Dic73semidirect
426. C8#14556506semidirect

Maximal subgroups

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

subgroupindex
#226109172
#226107682
#226099562
C8×C22⋊Dic732
#226085032
#226084992
C2×#465×C732
#978173