Finite Group Databasebeta

← #32985129

Subgroups — #32985129

Order 18688.

650 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22609911C211yesmaximal, characteristic, complemented
2C2×#17988324C212yesmaximal, complemented
3C2×#17988321C212yesmaximal
4#22608158C211yesmaximal, characteristic, complemented
5C73×#2850C211yesmaximal, characteristic
6C2×C4⋊#10596831—21no—
7C2×C22⋊#10596831C2212yes—
8C2×#14556498—21no—
9C2×#14556498—21no—
10C2×#14556498C2212yes—
11#17988342—21no—
12#17988324—24no—
13#17988321—24no—
14C2×#124×C73C411yescharacteristic, complemented
15C2×C22⋊Q8×C73C2211yescharacteristic
16C2×#121×C73C2211yescharacteristic
17C22×C42×C73C2211yescharacteristic
18#391×C73C411yescharacteristic, complemented
19#390×C73C411yescharacteristic, complemented
20#388×C73—21no—
21#388×C73—21no—
22#386×C73—21no—
23#386×C73—21no—
24#382×C73C411yescharacteristic, complemented
25C23×Dic73—22no—
26C22×#10596831—22no—
27C4⋊#10596831—42no—
28C22⋊#10596831—22no—
29C22⋊#10596831—22no—
30#14556498—22no—
31#14556498—42no—
32#14556498—42no—
33#14556498D414yes—
34C22×Q8×C73C2×C411yescharacteristic
35C22×D4×C73—21no—
36C22×D4×C73C2×C411yescharacteristic
37C23×C4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73C2311yescharacteristic
40#124×C73—42no—
41C22⋊Q8×C73—22no—
42C22⋊Q8×C73—22no—
43#121×C73—22no—
44#121×C73—22no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73C2×C411yescharacteristic
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C42×C73—42no—
55C2×C42×C73C2×C411yescharacteristic
56C2×C42×C73C2×C411yescharacteristic
57C2×C42×C73D411yescharacteristic
58C2×C42×C73D411yescharacteristic
59#95×C73—21no—
60#95×C73—21no—
61#95×C73—21no—
62#95×C73—21no—
63C22×Dic73—24no—
64C22×Dic73—42no—
65C22×Dic73—42no—
66C22×Dic73—42no—
67C22×Dic73—42no—
68C22×Dic73—42no—
69C22×Dic73—42no—
70C2×#10596831—42no—
71C2×#10596831—42no—
72C2×#10596831—42no—
73C24×C73—21no—
74C24×C73C2×D411yescharacteristic
75C2×Q8×C73—22no—
76C2×Q8×C73—41no—
77C2×Q8×C73—41no—
78C2×D4×C73—22no—
79C2×D4×C73—41no—
80C2×D4×C73—41no—
81C2×D4×C73—42no—
82C2×D4×C73—42no—
83C2×D4×C73—41no—
84C2×D4×C73—41no—
85C22×C4×C73—22no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—22no—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—41no—
93C22×C4×C73—41no—
94C22×C4×C73—41no—
95C22×C4×C73—41no—
96C22×C4×C73—41no—
97C22×C4×C73C22⋊C411yescharacteristic
98C22×C4×C73C22⋊C411yescharacteristic
99C22×C4×C73C22⋊C411yescharacteristic
100C22×C4×C73C22⋊C411yescharacteristic
101C22×C4×C73C22×C411yescharacteristic
102C22×C4×C73C2×D412yes—
103C22×C4×C73C2×D411yescharacteristic
104C22×C4×C73#4112yes—
105C4⋊C4×C73—22no—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—41no—
108C4⋊C4×C73—41no—
109C22⋊C4×C73—42no—
110C22⋊C4×C73—42no—
111C22⋊C4×C73—42no—
112C22⋊C4×C73—42no—
113C22⋊C4×C73—42no—
114C22⋊C4×C73—41no—
115C22⋊C4×C73—41no—
116C42×C73—22no—
117C42×C73—22no—
118C42×C73—41no—
119C42×C73—41no—
120C23×C73—21no—
121C23×C73—21no—
122C23×C73—21no—
123C23×C73—21no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73—41no—
129C23×C73—41no—
130C23×C73—41no—
131C23×C73C2×C22⋊C411yescharacteristic
132C23×C73#12112yes—
133C23×C73#13512yes—
134C23×C73AGL(1,ℤ/8)11yescharacteristic
135C23×C73#13711yescharacteristic
136Q8×C73—82no—
137D4×C73—82no—
138D4×C73—82no—
139D4×C73—82no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—21no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—21no—
147C2×C4×C73—21no—
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73—21no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—42no—
156C2×C4×C73—42no—
157C2×C4×C73—44no—
158C2×C4×C73—42no—
159C2×C4×C73—42no—
160C2×C4×C73—41no—
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—41no—
168C2×C4×C73—41no—
169C2×C4×C73#9912yes—
170C2×C4×C73C4×D412yes—
171C2×C4×C73#12411yescharacteristic
172C2×C4×C73#12711yescharacteristic
173C2×Dic73—44no—
174C2×Dic73—44no—
175C2×Dic73—44no—
176C2×Dic73—82no—
177C2×Dic73—82no—
178C2×Dic73—82no—
179C2×Dic73—82no—
180#10596831—82no—
181C22×C73—22no—
182C22×C73—22no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—21no—
186C22×C73—21no—
187C22×C73—22no—
188C22×C73—22no—
189C22×C73—81no—
190C22×C73—81no—
191C22×C73—81no—
192C22×C73—81no—
193C22×C73—81no—
194C22×C73—81no—
195C22×C73—81no—
196C22×C73#39011yescharacteristic
197C22×C73C2×#9911yescharacteristic
198C22×C73#41011yescharacteristic
199C22×C73#41611yescharacteristic
200C22×C73#41711yescharacteristic
201C22×C73#42111yescharacteristic
202C22×C73#42111yescharacteristic
203C22×C73#46312yes—
204C22×C73#46412yes—
205C4×C73—21no—
206C4×C73—22no—
207C4×C73—21no—
208C4×C73—22no—
209C4×C73—41no—
210C4×C73—41no—
211C4×C73—41no—
212C4×C73—81no—
213C4×C73—81no—
214C4×C73—81no—
215Dic73—84no—
216#9716—731nomaximal, Sylow
217C2×C73—22no—
218C2×C73—22no—
219C2×C73—161no—
220C2×C73#151211yescharacteristic
221C2×C73#151311yescharacteristic
222C2×C73#151611yescharacteristic
223C2×C73#152111yescharacteristic
224C2×C73#196311yescharacteristic
225C2×C73#196411yescharacteristic
226C2×C73#198311yescharacteristic
227#2850D7311yescharacteristic
228#1797—731no—
229C2×#328—732no—
230C2×#327—732no—
231#1434—731no—
232C73#971611yescharacteristic, complemented, Sylow
233C2×#124Dic7311yescharacteristic, complemented
234C2×C22⋊Q8C2×D7311yescharacteristic
235C2×#121C2×D7311yescharacteristic
236C22×C42C2×D7311yescharacteristic
237C2×C4⋊C8—1461no—
238C2×C22⋊C8—732no—
239#391Dic7311yescharacteristic, complemented
240#390Dic7311yescharacteristic, complemented
241#388—21no—
242#388—21no—
243#386—21no—
244#386—21no—
245#382Dic7311yescharacteristic, complemented
246C2×#95—732no—
247C2×#95—1461no—
248C2×#95—1461no—
249#336—1461no—
250#328—1464no—
251#327—1464no—
252C22×Q8C2×Dic7311yescharacteristic
253C22×D4—21no—
254C22×D4C2×Dic7311yescharacteristic
255C23×C4—1462no—
256C23×C4C22⋊D7311yescharacteristic
257C23×C4C22⋊D7311yescharacteristic
258C23×C4C22×D7311yescharacteristic
259C22×C8—1462no—
260#124—42no—
261C22⋊Q8—22no—
262C22⋊Q8—22no—
263#121—22no—
264#121—22no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C2×Dic7311yescharacteristic
267C2×C4⋊C4C2×Dic7311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—42no—
275C2×C42C2×Dic7311yescharacteristic
276C2×C42C2×Dic7311yescharacteristic
277C2×C42C22⋊D7311yescharacteristic
278C2×C42C22⋊D7311yescharacteristic
279C4⋊C8—2922no—
280C22⋊C8—1462no—
281C22⋊C8—1462no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—2922no—
287#95—2922no—
288#95—734no—
289#95—1462no—
290C24—21no—
291C24C2×C22⋊D7311yescharacteristic
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×D4—22no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—2922no—
315C22×C4—2922no—
316C22×C4—2922no—
317C22×C4—2922no—
318C22×C4—2922no—
319C22×C4—2922no—
320C22×C4—1464no—
321C22×C4C22⋊Dic7311yescharacteristic
322C22×C4C22⋊Dic7311yescharacteristic
323C22×C4C22⋊Dic7311yescharacteristic
324C22×C4C22⋊Dic7311yescharacteristic
325C22×C4C22×Dic7311yescharacteristic
326C22×C4D4×D7312yes—
327C22×C4C2×C22⋊D7311yescharacteristic
328C22×C4#1192402712yes—
329C2×C8—2922no—
330C2×C8—2922no—
331C2×C8—2922no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C42—22no—
344C42—22no—
345C42—41no—
346C42—41no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23C2×C22⋊Dic7311yescharacteristic
359C23#1455656212yes—
360C23#1455661112yes—
361C23#1455661611yescharacteristic
362C23#1455662111yescharacteristic
363Q8—82no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—2924no—
397C2×C4—2924no—
398C2×C4—2924no—
399C2×C4—5842no—
400C2×C4—5842no—
401C2×C4—5842no—
402C2×C4—5842no—
403C2×C4#1455650612yes—
404C2×C4D4×Dic7312yes—
405C2×C4#1455657811yescharacteristic
406C2×C4#1455658711yescharacteristic
407C8—5842no—
408C22—22no—
409C22—22no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22#1798849011yescharacteristic
424C22C2×#1455650611yescharacteristic
425C22#1798856111yescharacteristic
426C22#1798858311yescharacteristic
427C22#1798858811yescharacteristic
428C22#1798860411yescharacteristic
429C22#1798860411yescharacteristic
430C22#1798879912yes—
431C22#1798880612yes—
432C4—21no—
433C4—22no—
434C4—21no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—81no—
440C4—81no—
441C4—81no—
442C4—5844no—
443C2—22no—
444C2—22no—
445C2—161no—
446C2#2260846911yescharacteristic
447C2#2260847411yescharacteristic
448C2#2260848711yescharacteristic
449C2#2260850811yescharacteristic
450C2#2261076411yescharacteristic
451C2#2261076811yescharacteristic
452C2#2261086411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic73
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2850C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#121×C73C22
cosocleC23×C4×C73C23
hypercenter#2850D73
hypocenterC73#9716
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
1. #22609911C2semidirect
2. C2×#17988324C2semidirect
4. #22608158C2semidirect
14. C2×#124×C73C4semidirect
18. #391×C73C4semidirect
19. #390×C73C4semidirect
24. #382×C73C4semidirect
232. C73#9716semidirect
233. C2×#124Dic73semidirect
239. #391Dic73semidirect
240. #390Dic73semidirect
245. #382Dic73semidirect

Maximal subgroups

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

subgroupindex
#226099112
C2×#179883242
C2×#179883212
#226081582
C73×#28502
#971673