Finite Group Databasebeta

← #32989429

Subgroups — #32989429

Order 18688.

556 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614880C211yesmaximal, characteristic
2#22610892C211yesmaximal, characteristic, complemented
3#22610694C211yesmaximal, characteristic, complemented
4#22609932C211yesmaximal, characteristic
5C2×#17988332C211yesmaximal, characteristic, complemented
6C2×#17988316C211yesmaximal, characteristic
7C73×#1279C211yesmaximal, characteristic, complemented
8C2×#14556591C2211yescharacteristic
9C2×#14556571—21no—
10C2×#14556571—21no—
11C2×C4×D292C2211yescharacteristic
12C2×#14556522—21no—
13C2×C4⋊#10596831C2211yescharacteristic
14C2×#14556513—21no—
15#17988524—22no—
16#17988489—21no—
17#17988457—21no—
18#17988457C2211yescharacteristic
19#17988350—21no—
20#17988350—22no—
21#17988349—21no—
22#17988332—22no—
23#17988316—22no—
24C2×C4⋊C8×C73C2211yescharacteristic
25C2×C4×C8×C73C2211yescharacteristic
26C4×C4⋊C4×C73C2211yescharacteristic
27C22×Dic146—21no—
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C22×#10596831—21no—
31#14556591—21no—
32#14556591—21no—
33#14556591—21no—
34#14556591—21no—
35#14556571—42no—
36#14556571—42no—
37#14556571—42no—
38#14556571—42no—
39C4×D292—21no—
40C4×D292—21no—
41C4×D292—21no—
42C4×D292—21no—
43C2×C4⋊Dic73—22no—
44C2×C4⋊Dic73—22no—
45C2×C4⋊Dic73—21no—
46C2×C4⋊Dic73—21no—
47C2×C4⋊Dic73—21no—
48C2×C4⋊Dic73—21no—
49C2×C4⋊Dic73D412yes—
50C2×C4⋊Dic73D411yescharacteristic
51C2×#11924005—22no—
52C2×#11924005—22no—
53C2×#11924005—21no—
54#14556522—42no—
55C4⋊#10596831—21no—
56C4⋊#10596831D412yes—
57#14556513—42no—
58C22×C8×C73—21no—
59C22×C8×C73—21no—
60C2×C4⋊C4×C73D411yescharacteristic
61C2×C4⋊C4×C73D411yescharacteristic
62C2×C42×C73—21no—
63C2×C42×C73C2311yescharacteristic
64C4⋊C8×C73—21no—
65C4⋊C8×C73—21no—
66C4×C8×C73—21no—
67C4×C8×C73—21no—
68#95×C73—21no—
69C23×D73—21no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C2×Dic146—41no—
73C2×Dic146—41no—
74C2×Dic146—41no—
75C2×Dic146—41no—
76C2×D292—21no—
77C2×D292—21no—
78C2×D292—21no—
79C2×D292—21no—
80C2×D292—21no—
81C2×D292SD1612yes—
82C22×Dic73—22no—
83C22×Dic73—21no—
84C22×Dic73—41no—
85C2×C4×D73—41no—
86C2×C4×D73—41no—
87C2×C4×D73—41no—
88C2×C4×D73—41no—
89C2×C4×D73—41no—
90C2×C4×D73—41no—
91C2×#10596831—22no—
92C2×#10596831—41no—
93C2×#10596831—41no—
94C4⋊Dic73—21no—
95C4⋊Dic73—22no—
96C4⋊Dic73—21no—
97C4⋊Dic73—22no—
98C4⋊Dic73—44no—
99C4⋊Dic73—44no—
100C4⋊Dic73—41no—
101C4⋊Dic73—41no—
102C4⋊Dic73—42no—
103C4⋊Dic73—41no—
104C4⋊Dic73—41no—
105C4⋊Dic73—41no—
106C4⋊Dic73—41no—
107#11924005—44no—
108#11924005—44no—
109#11924005—41no—
110#11924005—41no—
111C22×C4×C73—21no—
112C22×C4×C73—21no—
113C22×C4×C73C2×D411yescharacteristic
114C22×C4×C73C2×D411yescharacteristic
115C22×C4×C73C2×D411yescharacteristic
116C2×C8×C73—22no—
117C2×C8×C73—41no—
118C2×C8×C73—41no—
119C2×C8×C73—41no—
120C2×C8×C73—41no—
121C2×C8×C73—41no—
122C4⋊C4×C73—21no—
123C4⋊C4×C73—21no—
124C4⋊C4×C73D812yes—
125C4⋊C4×C73SD1612yes—
126C42×C73—41no—
127C42×C73—41no—
128C42×C73C2×D411yescharacteristic
129C42×C73#4111yescharacteristic
130C42×C73#4111yescharacteristic
131C42×C73#4111yescharacteristic
132C23×C73#12011yescharacteristic
133C22×D73—22no—
134C22×D73—41no—
135C22×D73—41no—
136C22×D73—41no—
137C22×D73—41no—
138C22×D73—41no—
139C22×D73—41no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—22no—
143C2×C4×C73—22no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73#12111yescharacteristic
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73#12111yescharacteristic
151C2×C4×C73#12311yescharacteristic
152C2×C4×C73#12311yescharacteristic
153C2×C4×C73#12411yescharacteristic
154C2×C4×C73C2×D811yescharacteristic
155C2×C4×C73C2×SD1611yescharacteristic
156C2×C4×C73C2×SD1611yescharacteristic
157C2×C4×C73#13511yescharacteristic
158C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C73#13711yescharacteristic
161C2×C4×C73#13711yescharacteristic
162C2×Dic73—41no—
163C2×Dic73—42no—
164C2×Dic73—42no—
165C2×Dic73—41no—
166C2×Dic73—42no—
167C2×Dic73—41no—
168C2×Dic73—81no—
169C2×Dic73—81no—
170C2×Dic73—81no—
171D292—22no—
172D292—22no—
173D292—41no—
174D292—41no—
175C4×D73—81no—
176C4×D73—81no—
177C4×D73—81no—
178C4×D73—81no—
179Dic146—81no—
180Dic146—81no—
181Dic146—81no—
182Dic146—81no—
183C8×C73—42no—
184C8×C73—81no—
185#10596831—42no—
186C22×C73#39411yescharacteristic
187C22×C73#44711yescharacteristic
188C22×C73#44811yescharacteristic
189C22×C73#45011yescharacteristic
190C22×C73#45211yescharacteristic
191C22×C73#45411yescharacteristic
192C22×C73#45611yescharacteristic
193C2×D73—42no—
194C2×D73—42no—
195C2×D73—42no—
196C2×D73—81no—
197C2×D73—81no—
198C2×D73—81no—
199C2×D73—81no—
200C4×C73—21no—
201C4×C73—21no—
202C4×C73—42no—
203C4×C73—42no—
204C4×C73#45911yescharacteristic
205C4×C73#46011yescharacteristic
206C4×C73#46511yescharacteristic
207C4×C73#47011yescharacteristic
208C4×C73#48311yescharacteristic
209C4×C73#48511yescharacteristic
210C4×C73#48611yescharacteristic
211C4×C73#49011yescharacteristic
212Dic73—82no—
213Dic73—81no—
214Dic73—161no—
215#10420—731nomaximal, Sylow
216C2×C73#162711yescharacteristic
217C2×C73#163511yescharacteristic
218C2×C73#164711yescharacteristic
219C2×C73#165111yescharacteristic
220C2×C73#203511yescharacteristic
221C2×C73#203811yescharacteristic
222C2×C73#204111yescharacteristic
223D73—82no—
224#2542—731no—
225#1990—731no—
226#1948—731no—
227#1802—731no—
228C2×#332—731no—
229C2×D4⋊C8—731no—
230#1279D7311yescharacteristic, complemented
231C73#1042011yescharacteristic, complemented, Sylow
232C2×C4⋊Q8—731no—
233C2×C22⋊Q8—1461no—
234C2×C22⋊Q8—1461no—
235C2×C4×D4—731no—
236C2×#106—1461no—
237C2×C4⋊C8—731no—
238C2×C4⋊C8C2×D7311yescharacteristic
239C2×#102—1461no—
240C2×C4×C8C2×D7311yescharacteristic
241#399—1462no—
242#390—1461no—
243#384—731no—
244#384—1461no—
245C4×C4⋊C4C2×D7311yescharacteristic
246#340—1461no—
247#340—1462no—
248#340—1461no—
249#332—1462no—
250D4⋊C8—1462no—
251C22×Q8—1461no—
252C22×D4—731no—
253C23×C4—1461no—
254C22×C8—21no—
255C22×C8—21no—
256C22×C8—1461no—
257C4⋊Q8—1461no—
258C4⋊Q8—1461no—
259C4⋊Q8—1461no—
260C4⋊Q8—1461no—
261C22⋊Q8—2922no—
262C22⋊Q8—2922no—
263C22⋊Q8—2922no—
264C22⋊Q8—2922no—
265C4×D4—1461no—
266C4×D4—1461no—
267C4×D4—1461no—
268C4×D4—1461no—
269C2×C4⋊C4—731no—
270C2×C4⋊C4—732no—
271C2×C4⋊C4—1462no—
272C2×C4⋊C4—1462no—
273C2×C4⋊C4—1461no—
274C2×C4⋊C4—1461no—
275C2×C4⋊C4—1461no—
276C2×C4⋊C4—1461no—
277C2×C4⋊C4D29211yescharacteristic
278C2×C4⋊C4D29211yescharacteristic
279C2×C22⋊C4—1462no—
280C2×C22⋊C4—1462no—
281C2×C22⋊C4—1461no—
282C2×C42—21no—
283C2×C42C22×D7311yescharacteristic
284#106—2922no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—732no—
288C4⋊C8—1461no—
289#102—2922no—
290C4×C8—21no—
291C4×C8—21no—
292#95—21no—
293C24—1461no—
294C2×Q8—2921no—
295C2×Q8—2921no—
296C2×Q8—2921no—
297C2×Q8—2921no—
298C2×Q8—2921no—
299C2×Q8—2921no—
300C2×D4—732no—
301C2×D4—1461no—
302C2×D4—1461no—
303C2×D4—1461no—
304C2×D4—1461no—
305C2×D4—1461no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—2921no—
309C22×C4—2921no—
310C22×C4—2921no—
311C22×C4—2921no—
312C22×C4—2921no—
313C22×C4—2921no—
314C22×C4—2921no—
315C22×C4—1462no—
316C22×C4—1461no—
317C22×C4D4×D7311yescharacteristic
318C22×C4D4×D7311yescharacteristic
319C22×C4C2×D29211yescharacteristic
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—2921no—
327C2×C8—2921no—
328C2×C8—1462no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—2924no—
332C4⋊C4—2924no—
333C4⋊C4—2922no—
334C4⋊C4—2921no—
335C4⋊C4—2921no—
336C4⋊C4—2921no—
337C4⋊C4—2921no—
338C4⋊C4—2921no—
339C4⋊C4—2921no—
340C4⋊C4—1461no—
341C4⋊C4—1462no—
342C4⋊C4—1461no—
343C4⋊C4—1462no—
344C4⋊C4D58412yes—
345C4⋊C4#1192401712yes—
346C22⋊C4—2924no—
347C22⋊C4—2924no—
348C22⋊C4—2921no—
349C22⋊C4—2921no—
350C42—41no—
351C42—41no—
352C42D4×D7311yescharacteristic
353C42#1192402811yescharacteristic
354C42#1192402811yescharacteristic
355C42#1192402911yescharacteristic
356C23—2921no—
357C23—2921no—
358C23—2921no—
359C23—2921no—
360C23—2921no—
361C23—2921no—
362C23—1462no—
363C23#1455656111yescharacteristic
364Q8—5841no—
365Q8—5841no—
366Q8—5841no—
367Q8—5841no—
368D4—2921no—
369D4—2921no—
370D4—1462no—
371D4—1462no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—2922no—
381C2×C4—2921no—
382C2×C4—2922no—
383C2×C4—2921no—
384C2×C4—2921no—
385C2×C4—2922no—
386C2×C4—5841no—
387C2×C4—5841no—
388C2×C4—5841no—
389C2×C4—5841no—
390C2×C4—5841no—
391C2×C4—5841no—
392C2×C4—5841no—
393C2×C4#1455656411yescharacteristic
394C2×C4#1455656411yescharacteristic
395C2×C4#1455656611yescharacteristic
396C2×C4#1455657611yescharacteristic
397C2×C4#1455657611yescharacteristic
398C2×C4#1455657911yescharacteristic
399C2×C4C2×D58411yescharacteristic
400C2×C4SD16×D7311yescharacteristic
401C2×C4C2×#1192401711yescharacteristic
402C2×C4#1455661011yescharacteristic
403C2×C4#1455661911yescharacteristic
404C2×C4#1455661911yescharacteristic
405C2×C4#1455662011yescharacteristic
406C2×C4#1455662311yescharacteristic
407C8—42no—
408C8—81no—
409C8—2922no—
410C22—2922no—
411C22—2922no—
412C22—2922no—
413C22—5841no—
414C22—5841no—
415C22—5841no—
416C22—5841no—
417C22#1798850511yescharacteristic
418C22#1798872311yescharacteristic
419C22#1798872711yescharacteristic
420C22#1798874011yescharacteristic
421C22#1798875011yescharacteristic
422C22#1798876111yescharacteristic
423C22#1798877111yescharacteristic
424C4—21no—
425C4—21no—
426C4—42no—
427C4—42no—
428C4—5841no—
429C4—5842no—
430C4—11681no—
431C4#1798878311yescharacteristic
432C4#1798878811yescharacteristic
433C4#1798881511yescharacteristic
434C4#1798884411yescharacteristic
435C4#1798891911yescharacteristic
436C4#1798893011yescharacteristic
437C4#1798893411yescharacteristic
438C4#1798895211yescharacteristic
439C2—5842no—
440C2#2260903611yescharacteristic
441C2#2260908411yescharacteristic
442C2#2260916811yescharacteristic
443C2#2260919711yescharacteristic
444C2#2261113011yescharacteristic
445C2#2261115111yescharacteristic
446C2#2261116611yescharacteristic
subgroupG/H
centerC23#14556561
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1279C2
socleC23×C73#120
prime radicalC22×D292D4
hypercenter#1279D73
hypocenterC73#10420
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
2. #22610892C2semidirect
3. #22610694C2semidirect
5. C2×#17988332C2semidirect
7. C73×#1279C2semidirect
230. #1279D73semidirect
231. C73#10420semidirect

Maximal subgroups

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

subgroupindex
#226148802
#226108922
#226106942
#226099322
C2×#179883322
C2×#179883162
C73×#12792
#1042073