Finite Group Databasebeta

← #72246823

Subgroups — #72246823

Order 4032.

980 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72246811C211yesmaximal, characteristic, complemented
2C3×#10620493C212yesmaximal, complemented
3C7×#63559C211yesmaximal, characteristic, complemented
4#72245722C212yesmaximal, complemented
5C2×#11759568C211yesmaximal, characteristic
6#13053603—32nomaximal
7C22×D7×C3⋊D3C412yes—
8C2×C4×C3⋊D21C2211yescharacteristic
9C2×C7×C4⋊(C3⋊D3)C2211yescharacteristic
10C2×C32×D28C2211yescharacteristic
11C2×#92676C412yes—
12#11759568—22no—
13#11759568—22no—
14C3×#66627C2212yescomplemented
15C3×C7×#208C2212yescomplemented
16#11759193—22no—
17#11759190—22no—
18C2×#66681—32no—
19C2×#66681—61no—
20M16×D21—32no—
21#10620854—32no—
22C7×#823—32no—
23C3×#6397—32no—
24#10620494—32no—
25#10620493D312yescomplemented
26#10591503—71nomaximal, Hall
27C22×C3⋊D21C2×C411yescharacteristic
28C22×C7×C3⋊D3C2×C412yes—
29C22×C32×D7C2×C412yes—
30C2×D7×C3⋊D3—22no—
31C2×D7×C3⋊D3—22no—
32C2×D7×C3⋊D3—44no—
33C2×C32×C4×C7C2311yescharacteristic
34C2×C3⋊Dic21C2×C411yescharacteristic
35C4×C3⋊D21D412yes—
36C4×C3⋊D21D412yes—
37C7×C4⋊(C3⋊D3)—22no—
38C32×D28—22no—
39#92676—24no—
40C3×C3⋊(C7⋊C8)—22no—
41C3×C7×C3⋊C8—22no—
42D7×#100—91noHall
43C22×D3×D7—34no—
44C22×D3×D7—62no—
45C2×C4×D21—32no—
46C2×C4×D21—61no—
47C2×C7×D12—32no—
48C2×C7×D12—61no—
49C2×C3×D28—21no—
50C2×C3×D28C2×D312yes—
51C2×#4245—34no—
52C2×#4245—62no—
53#66681—62no—
54#66681—62no—
55#66681—62no—
56#66681—62no—
57#66681—62no—
58#66681—62no—
59#66681—62no—
60#66681—62no—
61C3×C7×M16—32no—
62#66627C2×D312yescomplemented
63#66622—62no—
64C8×D21—62no—
65C7×#208C2×D312yescomplemented
66C3×#1136—32no—
67#66559—62no—
68D21⋊C8—62no—
69C2×D4×C3⋊D3—71no—
70#63818—72no—
71C3×#833—72no—
72#63559—71no—
73#63559D711yescharacteristic, complemented
74C22×C32×C7C22×C411yescharacteristic
75C2×C3⋊D21—22no—
76C2×C3⋊D21C22⋊C411yescharacteristic
77C2×C3⋊D21C22⋊C411yescharacteristic
78C2×C7×C3⋊D3—22no—
79C2×C7×C3⋊D3—42no—
80C2×C32×D7—22no—
81C2×C32×D7—42no—
82D7×C3⋊D3—44no—
83D7×C3⋊D3—44no—
84C32×C4×C7C2×D412yes—
85C3⋊Dic21C22⋊C412yes—
86C2×D4×D7—91no—
87D7×M16—92no—
88C7×#100—91no—
89#6407—91no—
90#6397—32no—
91D3×#100—212no—
92C22×D21—32no—
93C22×D21—61no—
94C22×D3×C7—34no—
95C22×D3×C7—62no—
96C22×C3×D7—22no—
97C22×C3×D7C4×D314yes—
98C2×D3×D7—64no—
99C2×D3×D7—64no—
100C2×D3×D7—62no—
101C2×D3×D7—62no—
102C2×D3×D7—62no—
103C2×D3×D7—62no—
104C2×D3×D7—124no—
105C2×D3×D7—128no—
106C2×D3×D7—124no—
107C2×C3×C4×C7—21no—
108C2×C3×C4×C7C22×D312yes—
109C2×Dic21—32no—
110C2×Dic21—61no—
111C4×D21—32no—
112C4×D21—32no—
113C4×D21—32no—
114C4×D21—32no—
115C4×D21—62no—
116C4×D21—62no—
117C7×D12—62no—
118C7×D12—62no—
119C7×D12—62no—
120C7×D12—62no—
121C3×D28—22no—
122C3×D28—22no—
123C3×D28—22no—
124C3×D28—22no—
125#4245—64no—
126#4245—64no—
127#4245—64no—
128#4245—64no—
129C3×C7×C8—62no—
130C3⋊(C7⋊C8)—22no—
131C3×C7⋊C8—62no—
132C7×C3⋊C8—22no—
133C23×C3⋊D3—72no—
134C2×C32×D4—71no—
135C2×#638—72no—
136D4×C3⋊D3—142no—
137D4×C3⋊D3—142no—
138C2×C4⋊(C3⋊D3)C2×D711yescharacteristic
139C2×C4×C3⋊D3—71no—
140C3×#208—72no—
141C3×#208C2×D712yescomplemented
142#3712—142no—
143#3710—142no—
144C2×C32×C7—21no—
145C2×C32×C7C2×C22⋊C411yescharacteristic
146C3⋊D21—21no—
147C3⋊D21#10012yescomplemented
148C7×C3⋊D3—42no—
149C32×D7—42no—
150C23×D7—92no—
151C2×C7×D4—91no—
152C2×C22⋊D7—92no—
153D4×D7—182no—
154D4×D7—182no—
155C2×D28D3211yescharacteristic
156C2×C4×D7—91no—
157C7×M16—32no—
158#1136—32no—
159#1131—182no—
160C8×D7—182no—
161C2×D3×D4—212no—
162C2×D3×D4—421no—
163D3×M16—212no—
164D3×M16—212no—
165C3×#100—212no—
166#833—72no—
167#823—32no—
168#823—212no—
169C22×C3×C7—21no—
170C22×C3×C7C2×C4×D312yes—
171C2×D21—32no—
172C2×D21—32no—
173C2×D21—61no—
174C2×D21—61no—
175C2×D21—64no—
176C2×D21—62no—
177C2×D21—62no—
178C2×D3×C7—62no—
179C2×D3×C7—62no—
180C2×D3×C7—64no—
181C2×D3×C7—122no—
182C2×D3×C7—124no—
183C2×D3×C7—122no—
184C2×C3×D7—22no—
185C2×C3×D7—22no—
186C2×C3×D7—24no—
187C2×C3×D7—42no—
188C2×C3×D7—42no—
189C2×C3×D7—44no—
190D3×D7—128no—
191D3×D7—124no—
192D3×D7—124no—
193D3×D7—124no—
194D3×D7—124no—
195D3×D7—128no—
196C3×C4×C7—22no—
197C3×C4×C7D3×D412yes—
198C3×C4×C7D3×D412yes—
199Dic21—32no—
200Dic21—32no—
201Dic21—62no—
202C23×C32—72no—
203C22×C3⋊D3—71no—
204C22×C3⋊D3—142no—
205C22×C3⋊D3—142no—
206C22×C3⋊D3—284no—
207C22×C3⋊D3C4×D712yes—
208C32×D4—142no—
209C2×C32×C4C22×D711yescharacteristic
210#638—144no—
211C2×C3⋊Dic3—71no—
212C4⋊(C3⋊D3)—22no—
213C4×C3⋊D3—72no—
214C4×C3⋊D3—72no—
215C3×C3⋊C8—22no—
216C3×C3⋊C8—142no—
217C2×#100—631noSylow
218C32×C7C2×#10011yescharacteristic, complemented, Hall
219C23×C7—92no—
220C22×D7—91no—
221C22×D7—182no—
222C22×D7—182no—
223C22×D7—364no—
224C22×D7#62412yes—
225C7×D4—182no—
226C2×C4×C7C2×D3211yescharacteristic
227C22⋊D7—184no—
228C2×Dic7—91no—
229D28—22no—
230C4×D7—92no—
231C4×D7—92no—
232C7×C8—62no—
233C7⋊C8—62no—
234C23×D3—214no—
235C23×D3—422no—
236C2×C3×D4—72no—
237C2×C3×D4—141no—
238C2×#67—214no—
239C2×#67—422no—
240D3×D4—422no—
241D3×D4—422no—
242D3×D4—422no—
243D3×D4—422no—
244D3×D4—422no—
245D3×D4—422no—
246D3×D4—422no—
247D3×D4—422no—
248C2×D12—32no—
249C2×D12—61no—
250C2×C4×D3—212no—
251C2×C4×D3—421no—
252C3×M16—32no—
253C3×M16—212no—
254#208—72no—
255#208D3×D712yescomplemented
256ΓL(1,𝔽25)—422no—
257ΓL(1,𝔽25)—422no—
258D3×C8—422no—
259D3×C8—422no—
260C2×C3×C7—21no—
261C2×C3×C7—21no—
262C2×C3×C7—22no—
263C2×C3×C7—21no—
264C2×C3×C7D3×C22⋊C412yes—
265D21—34no—
266D21—62no—
267D21—61no—
268D21—61no—
269D21—62no—
270C3×D7—42no—
271C3×D7—42no—
272C3×D7—44no—
273D3×C7—122no—
274D3×C7—122no—
275D3×C7—124no—
276C22×C32—142no—
277C22×C32—282no—
278C22×C32C2×C4×D711yescharacteristic
279C2×C3⋊D3—22no—
280C2×C3⋊D3—42no—
281C2×C3⋊D3—71no—
282C2×C3⋊D3—71no—
283C2×C3⋊D3—142no—
284C2×C3⋊D3—284no—
285C2×C3⋊D3—284no—
286C32×C4D4×D712yes—
287C3⋊Dic3—72no—
288C22×D4—631no—
289C2×M16—632no—
290#100—91no—
291#100—212no—
292#100—631no—
293C22×C7—182no—
294C22×C7—362no—
295C22×C7C2×#62411yescharacteristic
296C2×D7—22no—
297C2×D7—42no—
298C2×D7—91no—
299C2×D7—91no—
300C2×D7—182no—
301C2×D7—364no—
302C2×D7—364no—
303C4×C7#381112yes—
304Dic7—92no—
305C23×C3—74no—
306C23×C3—142no—
307C22×D3—34no—
308C22×D3—62no—
309C22×D3—212no—
310C22×D3—421no—
311C22×D3—424no—
312C22×D3—424no—
313C22×D3—422no—
314C22×D3—422no—
315C22×D3—422no—
316C22×D3—422no—
317C22×D3—844no—
318C22×D3—848no—
319C22×D3—844no—
320C3×D4—142no—
321C3×D4—142no—
322C3×D4—142no—
323C3×D4—142no—
324C2×C3×C4—21no—
325C2×C3×C4C2×D3×D712yes—
326#67—424no—
327#67—424no—
328#67—424no—
329#67—424no—
330C2×Dic3—212no—
331C2×Dic3—421no—
332D12—62no—
333D12—62no—
334D12—62no—
335D12—62no—
336C4×D3—212no—
337C4×D3—212no—
338C4×D3—212no—
339C4×D3—212no—
340C4×D3—422no—
341C4×D3—422no—
342C3×C8—62no—
343C3×C8—422no—
344C3⋊C8—22no—
345C3⋊C8—142no—
346C3×C7—21no—
347C3×C7D3×#10012yescomplemented
348C2×C32—21no—
349C2×C32—282no—
350C2×C32D7×C22⋊C411yescharacteristic
351C3⋊D3—42no—
352C3⋊D3—72no—
353C3⋊D3—141no—
354C24—632no—
355C2×D4—71no—
356C2×D4—91no—
357C2×D4—632no—
358C2×D4—1262no—
359C2×D4—1262no—
360C22×C4—631no—
361M16—32no—
362M16—212no—
363M16—1262no—
364C2×C8—1262no—
365C2×C7—21no—
366C2×C7—362no—
367C2×C7#6397411yescharacteristic
368D7—42no—
369D7—92no—
370D7—181no—
371C22×C3—21no—
372C22×C3—142no—
373C22×C3—142no—
374C22×C3—144no—
375C22×C3—284no—
376C22×C3—282no—
377C22×C3—282no—
378C22×C3C4×D3×D712yes—
379C2×D3—62no—
380C2×D3—62no—
381C2×D3—64no—
382C2×D3—124no—
383C2×D3—122no—
384C2×D3—122no—
385C2×D3—212no—
386C2×D3—212no—
387C2×D3—421no—
388C2×D3—421no—
389C2×D3—424no—
390C2×D3—422no—
391C2×D3—422no—
392C2×D3—848no—
393C2×D3—844no—
394C2×D3—844no—
395C2×D3—848no—
396C2×D3—844no—
397C2×D3—844no—
398C3×C4—22no—
399C3×C4#6668112yes—
400C3×C4#6669412yes—
401Dic3—212no—
402Dic3—212no—
403Dic3—422no—
404C32D7×#10011yescharacteristic, complemented, Sylow
405C23—72no—
406C23—92no—
407C23—631no—
408C23—1262no—
409C23—1262no—
410C23—2524no—
411D4—142no—
412D4—182no—
413D4—1264no—
414C2×C4—631no—
415C2×C4—632no—
416C2×C4—632no—
417C2×C4D32×D711yescharacteristic
418C8—62no—
419C8—422no—
420C7#1059150311yescharacteristic, complemented, Sylow
421C2×C3—21no—
422C2×C3—21no—
423C2×C3—22no—
424C2×C3—21no—
425C2×C3—282no—
426C2×C3—282no—
427C2×C3—284no—
428C2×C3#1062103512yes—
429D3—122no—
430D3—122no—
431D3—124no—
432D3—214no—
433D3—422no—
434D3—421no—
435D3—421no—
436D3—422no—
437C22—142no—
438C22—182no—
439C22—282no—
440C22—362no—
441C22—631no—
442C22—631no—
443C22—1262no—
444C22—2524no—
445C22—2524no—
446C22D7×#62411yescharacteristic
447C4—632no—
448C4#1175942712yes—
449C3—21no—
450C3#1305360312yescomplemented
451C2—21no—
452C2—282no—
453C2—362no—
454C2—632no—
455C2—1261no—
456C2#7224682211yescharacteristic
subgroupG/H
centerC2#72246822
commutatorC22×C32×C7C22×C4
Frattini, hypercenterC2×C4D32×D7
Fitting, cosocleC2×C32×C4×C7C23
socleC2×C32×C7C2×C22⋊C4
prime radicalC2×#11759568C2
hypocenterC32×C7C2×#100
LCSG ▷ C22×C32×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #72246811C2semidirect
2. C3×#10620493C2semidirect
3. C7×#63559C2semidirect
4. #72245722C2semidirect
14. C3×#66627C22semidirect
15. C3×C7×#208C22semidirect
25. #10620493D3semidirect
62. #66627C2×D3semidirect
65. C7×#208C2×D3semidirect
73. #63559D7semidirect
141. C3×#208C2×D7semidirect
147. C3⋊D21#100semidirect
218. C32×C7C2×#100semidirect
255. #208D3×D7semidirect
347. C3×C7D3×#100semidirect
404. C32D7×#100semidirect
420. C7#10591503semidirect
450. C3#13053603semidirect

Maximal subgroups

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

subgroupindex
#722468112
C3×#106204932
C7×#635592
#722457222
C2×#117595682
#130536033
#105915037