Finite Group Databasebeta

← #32985704

Subgroups — #32985704

Order 18688.

510 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609969C211yesmaximal, characteristic, complemented
2#22609707C211yesmaximal, characteristic, complemented
3#22608451C211yesmaximal, characteristic, complemented
4#22608447C211yesmaximal, characteristic
5#22608295C211yesmaximal, characteristic, complemented
6#22608291C211yesmaximal, characteristic
7C2×#465×C73C211yesmaximal, characteristic
8C2×#14556527—21no—
9#17988551C2211yescharacteristic
10#17988551C2211yescharacteristic
11C2×C22⋊#10596831C2211yescharacteristic
12C2×#14556501—21no—
13C4×C22⋊Dic73C2211yescharacteristic
14#17988352—21no—
15#17988341—21no—
16#17988324—22no—
17#17988321—22no—
18#17988313—22no—
19#17988311—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
31C2×#11924012—21no—
32C2×#11924012—21no—
33C22×#10596831—21no—
34C2×C22⋊Dic73—21no—
35C2×C4×Dic73—21no—
36#14556527—42no—
37C22⋊#10596831—21no—
38C22⋊#10596831—21no—
39C22⋊#10596831—21no—
40C22⋊#10596831—21no—
41C22⋊#10596831—21no—
42C22⋊#10596831D412yes—
43#14556501—42no—
44#14556498D412yes—
45C22×Q8×C73C2×C411yescharacteristic
46C22×D4×C73—21no—
47C22×D4×C73C2×C411yescharacteristic
48C23×C4×C73C2311yescharacteristic
49C2×SD16×C73—21no—
50C2×SD16×C73—21no—
51C2×SD16×C73—42no—
52C22×C8×C73—42no—
53C22×C8×C73C2×C411yescharacteristic
54C22×C8×C73C2×C411yescharacteristic
55C22×C8×C73D411yescharacteristic
56C22×C8×C73D411yescharacteristic
57C22×C8×C73D411yescharacteristic
58C22×C8×C73D411yescharacteristic
59C22⋊Q8×C73—21no—
60C22⋊Q8×C73—21no—
61C22⋊Q8×C73—21no—
62C22⋊Q8×C73—21no—
63#121×C73—21no—
64#121×C73—21no—
65#121×C73—21no—
66#121×C73—21no—
67C2×C4⋊C4×C73—21no—
68C2×C4⋊C4×C73C2×C411yescharacteristic
69C2×C4⋊C4×C73C2×C411yescharacteristic
70C2×C22⋊C4×C73—21no—
71C2×C22⋊C4×C73—21no—
72#106×C73—21no—
73#106×C73—21no—
74#103×C73—21no—
75#103×C73—21no—
76#102×C73—21no—
77#102×C73—21no—
78C22×Dic73—22no—
79#11924012—41no—
80#11924012—41no—
81#11924012—41no—
82#11924012—41no—
83C2×#10596831—22no—
84C2×#10596831—41no—
85C2×#10596831—41no—
86C2×#10596831—41no—
87C2×#10596831—41no—
88C22⋊Dic73—42no—
89C4×Dic73—41no—
90C4×Dic73—41no—
91C24×C73—21no—
92C24×C73#4111yescharacteristic
93C2×Q8×C73—21no—
94C2×Q8×C73—21no—
95C2×Q8×C73—41no—
96C2×Q8×C73—41no—
97C2×D4×C73—21no—
98C2×D4×C73—21no—
99C2×D4×C73—41no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C2×D4×C73—41no—
103C2×D4×C73—82no—
104C22×C4×C73—21no—
105C22×C4×C73—21no—
106C22×C4×C73—21no—
107C22×C4×C73—41no—
108C22×C4×C73—41no—
109C22×C4×C73C22×C411yescharacteristic
110C22×C4×C73C2×D411yescharacteristic
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73C2×D411yescharacteristic
114C22×C4×C73#4111yescharacteristic
115SD16×C73—42no—
116SD16×C73—42no—
117C2×C8×C73—22no—
118C2×C8×C73—22no—
119C2×C8×C73—22no—
120C2×C8×C73—41no—
121C2×C8×C73—41no—
122C2×C8×C73—41no—
123C2×C8×C73C22⋊C411yescharacteristic
124C2×C8×C73C22⋊C411yescharacteristic
125C2×C8×C73C22⋊C411yescharacteristic
126C2×C8×C73C22⋊C411yescharacteristic
127C4⋊C4×C73—21no—
128C4⋊C4×C73—21no—
129C4⋊C4×C73—21no—
130C4⋊C4×C73—21no—
131C4⋊C4×C73—41no—
132C4⋊C4×C73—41no—
133C22⋊C4×C73—41no—
134C22⋊C4×C73—41no—
135C22⋊C4×C73—41no—
136C22⋊C4×C73—41no—
137C23×C73—22no—
138C23×C73—22no—
139C23×C73—41no—
140C23×C73—41no—
141C23×C73—42no—
142C23×C73—81no—
143C23×C73—81no—
144C23×C73C4×D411yescharacteristic
145C23×C73#12111yescharacteristic
146C23×C73#12411yescharacteristic
147Q8×C73—41no—
148Q8×C73—41no—
149Q8×C73—81no—
150D4×C73—41no—
151D4×C73—41no—
152D4×C73—81no—
153D4×C73—82no—
154D4×C73—82no—
155C2×C4×C73—21no—
156C2×C4×C73—21no—
157C2×C4×C73—21no—
158C2×C4×C73—21no—
159C2×C4×C73—21no—
160C2×C4×C73—21no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—42no—
168C2×C4×C73—81no—
169C2×C4×C73—81no—
170C2×C4×C73C2×C22⋊C411yescharacteristic
171C2×C4×C73C4×D411yescharacteristic
172C2×C4×C73#12111yescharacteristic
173C2×C4×C73#12711yescharacteristic
174C2×Dic73—42no—
175C2×Dic73—42no—
176C2×Dic73—42no—
177C8×C73—21no—
178C8×C73—21no—
179C8×C73—21no—
180C8×C73#9912yes—
181#10596831—42no—
182#10596831—81no—
183#10596831—81no—
184C22×C73—21no—
185C22×C73—21no—
186C22×C73—21no—
187C22×C73—21no—
188C22×C73—21no—
189C22×C73—21no—
190C22×C73—21no—
191C22×C73—44no—
192C22×C73—81no—
193C22×C73—81no—
194C22×C73—81no—
195C22×C73—162no—
196C22×C73#39011yescharacteristic
197C22×C73#44412yes—
198C22×C73#47112yes—
199C4×C73—21no—
200C4×C73—21no—
201C4×C73—21no—
202C4×C73—81no—
203C4×C73—81no—
204C4×C73—81no—
205C4×C73C2×#9911yescharacteristic
206C4×C73#41011yescharacteristic
207Dic73—82no—
208#9816—731nomaximal, Sylow
209C2×C73—22no—
210C2×C73—22no—
211C2×C73—41no—
212C2×C73—161no—
213C2×C73#196411yescharacteristic
214C2×C73#197511yescharacteristic
215C2×C73#197711yescharacteristic
216C2×#465D7311yescharacteristic
217#1810—731no—
218#1752—731no—
219#1508—731no—
220#1507—731no—
221#1472—731no—
222#1471—731no—
223C73#981611yescharacteristic, complemented, Sylow
224C22×SD16Dic7311yescharacteristic, complemented
225C23×C8C2×D7311yescharacteristic
226C2×C22⋊Q8C2×D7311yescharacteristic
227C2×#121C2×D7311yescharacteristic
228#465—21no—
229#465—21no—
230#465—21no—
231#465—21no—
232C2×#108—1461no—
233C2×#106Dic7311yescharacteristic, complemented
234C2×#103Dic7311yescharacteristic, complemented
235C2×#102Dic7311yescharacteristic, complemented
236#407—731no—
237#407—731no—
238C2×C22⋊C8—731no—
239C2×#97—1461no—
240C4×C22⋊C4—731no—
241#341—1461no—
242#336—1461no—
243#328—1462no—
244#327—1462no—
245#324—1462no—
246#323—1462no—
247C22×Q8C2×Dic7311yescharacteristic
248C22×D4—21no—
249C22×D4C2×Dic7311yescharacteristic
250C23×C4C22×D7311yescharacteristic
251C2×SD16—21no—
252C2×SD16—21no—
253C2×SD16—42no—
254C2×M16—1461no—
255C2×M16—1461no—
256C22×C8—42no—
257C22×C8—1461no—
258C22×C8C2×Dic7311yescharacteristic
259C22×C8C2×Dic7311yescharacteristic
260C22×C8C22⋊D7311yescharacteristic
261C22×C8C22⋊D7311yescharacteristic
262C22×C8C22⋊D7311yescharacteristic
263C22×C8C22⋊D7311yescharacteristic
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C2×Dic7311yescharacteristic
274C2×C4⋊C4C2×Dic7311yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—1461no—
278C2×C42—1461no—
279#108—2922no—
280#106—21no—
281#106—21no—
282#103—21no—
283#103—21no—
284#102—21no—
285#102—21no—
286C22⋊C8—732no—
287C22⋊C8—1461no—
288C22⋊C8—1461no—
289C22⋊C8—1461no—
290C22⋊C8—1461no—
291C22⋊C8—1461no—
292#97—2922no—
293#95—732no—
294C24—21no—
295C24#1192402711yescharacteristic
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—82no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—1462no—
313C22×C4C22×Dic7311yescharacteristic
314C22×C4D4×D7311yescharacteristic
315C22×C4D4×D7311yescharacteristic
316C22×C4C2×C22⋊D7311yescharacteristic
317C22×C4C2×C22⋊D7311yescharacteristic
318C22×C4#1192402711yescharacteristic
319SD16—42no—
320SD16—42no—
321M16—2921no—
322M16—2921no—
323M16—2921no—
324M16—2921no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—2921no—
332C2×C8—2921no—
333C2×C8—2921no—
334C2×C8—2921no—
335C2×C8—1462no—
336C2×C8C22⋊Dic7311yescharacteristic
337C2×C8C22⋊Dic7311yescharacteristic
338C2×C8C22⋊Dic7311yescharacteristic
339C2×C8C22⋊Dic7311yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—2922no—
351C42—2921no—
352C42—2921no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23D4×Dic7311yescharacteristic
361C23#1455656211yescharacteristic
362C23#1455657811yescharacteristic
363Q8—41no—
364Q8—41no—
365Q8—81no—
366D4—41no—
367D4—41no—
368D4—81no—
369D4—82no—
370D4—82no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—2922no—
387C2×C4—2922no—
388C2×C4—2922no—
389C2×C4C2×C22⋊Dic7311yescharacteristic
390C2×C4D4×Dic7311yescharacteristic
391C2×C4#1455656211yescharacteristic
392C2×C4#1455658711yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8—2922no—
397C8—5841no—
398C8—5841no—
399C8#1455650612yes—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—44no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—162no—
412C22#1798849011yescharacteristic
413C22#1798870912yes—
414C22#1798884712yes—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—5842no—
422C4C2×#1455650611yescharacteristic
423C4#1798856111yescharacteristic
424C2—22no—
425C2—22no—
426C2—41no—
427C2—161no—
428C2#2261076811yescharacteristic
429C2#2261082511yescharacteristic
430C2#2261083611yescharacteristic
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#9816
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. #22609969C2semidirect
2. #22609707C2semidirect
3. #22608451C2semidirect
5. #22608295C2semidirect
20. C22×SD16×C73C4semidirect
28. C2×#106×C73C4semidirect
29. C2×#103×C73C4semidirect
30. C2×#102×C73C4semidirect
223. C73#9816semidirect
224. C22×SD16Dic73semidirect
233. C2×#106Dic73semidirect
234. C2×#103Dic73semidirect
235. C2×#102Dic73semidirect

Maximal subgroups

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

subgroupindex
#226099692
#226097072
#226084512
#226084472
#226082952
#226082912
C2×#465×C732
#981673