Finite Group Databasebeta

← #72246817

Subgroups — #72246817

Order 4032.

980 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 980 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C3×#10620672C211yesmaximal, characteristic, complemented
2#72246807C212yesmaximal, complemented
3C3×C7×#833C211yesmaximal, characteristic, complemented
4#72245722C212yesmaximal, complemented
5C2×D4×C3⋊D21C211yesmaximal, characteristic
6#100×D21—31nomaximal
7#13054493—31nomaximal
8C23×C3⋊D21C412yes—
9C2×C32×C7×D4C2211yescharacteristic
10C2×#92726C412yes—
11D4×C3⋊D21—22no—
12D4×C3⋊D21—22no—
13C2×C4⋊(C3⋊D21)C2211yescharacteristic
14C2×C4×C3⋊D21C2211yescharacteristic
15C3×#66627C2212yescomplemented
16C3×C7×#208C2212yescomplemented
17#11759193—22no—
18#11759190—22no—
19C2×D4×D21—31no—
20C2×D4×D21—31no—
21C2×D4×D21—61no—
22M16×D21—32no—
23#10620854—32no—
24C3×C7×#100—31no—
25#10620672D311yescharacteristic, complemented
26#10620662—32no—
27C7×#833D311yescharacteristic, complemented
28C3×#6407—31no—
29#10620495—32no—
30#10591503—71nomaximal, Hall
31C23×C32×C7C2×C412yes—
32C22×C3⋊D21—22no—
33C22×C3⋊D21—22no—
34C22×C3⋊D21—44no—
35C22×C3⋊D21C2×C411yescharacteristic
36C22×C3⋊D21C2×C412yes—
37C32×C7×D4—22no—
38C2×C32×C4×C7C2311yescharacteristic
39#92726—24no—
40C2×C3⋊Dic21C2×C411yescharacteristic
41C4⋊(C3⋊D21)—22no—
42C4×C3⋊D21D412yes—
43C4×C3⋊D21D412yes—
44C3×C3⋊(C7⋊C8)—22no—
45C3×C7×C3⋊C8—22no—
46D7×#100—91noHall
47C23×D21—32no—
48C23×D21—32no—
49C23×D21—62no—
50C2×C3×C7×D4—21no—
51C2×C3×C7×D4C2×D311yescharacteristic
52C2×C3×C7×D4C2×D311yescharacteristic
53C2×#4268—32no—
54C2×#4268—32no—
55C2×#4268—62no—
56D4×D21—62no—
57D4×D21—62no—
58D4×D21—62no—
59D4×D21—62no—
60D4×D21—62no—
61D4×D21—62no—
62D4×D21—62no—
63D4×D21—62no—
64C2×D84—31no—
65C2×D84—31no—
66C2×D84—61no—
67C2×C4×D21—31no—
68C2×C4×D21—31no—
69C2×C4×D21—61no—
70C3×C7×M16—32no—
71#66627C2×D312yescomplemented
72#66622—62no—
73C8×D21—62no—
74C7×#208C2×D312yescomplemented
75C3×#1136—32no—
76#66559—62no—
77D21⋊C8—62no—
78C2×D4×C3⋊D3—71no—
79#63818—72no—
80C3×#833—71no—
81C3×#833D711yescharacteristic, complemented
82#63559—72no—
83C22×C32×C7—22no—
84C22×C32×C7—42no—
85C22×C32×C7C22×C411yescharacteristic
86C2×C3⋊D21—22no—
87C2×C3⋊D21—22no—
88C2×C3⋊D21—44no—
89C2×C3⋊D21—44no—
90C2×C3⋊D21—42no—
91C2×C3⋊D21C22⋊C411yescharacteristic
92C2×C3⋊D21C22⋊C411yescharacteristic
93C32×C4×C7C2×D412yes—
94C3⋊Dic21C22⋊C412yes—
95C2×D4×D7—91no—
96D7×M16—92no—
97C7×#100—31no—
98#6407—31no—
99#6397—92no—
100D3×#100—211no—
101D3×#100—211no—
102C23×C3×C7—22no—
103C23×C3×C7C4×D312yes—
104C23×C3×C7C4×D312yes—
105C22×D21—31no—
106C22×D21—31no—
107C22×D21—32no—
108C22×D21—32no—
109C22×D21—61no—
110C22×D21—62no—
111C22×D21—62no—
112C22×D21—62no—
113C22×D21—62no—
114C22×D21—62no—
115C22×D21—62no—
116C22×D21—62no—
117C22×D21—62no—
118C22×D21—62no—
119C22×D21—124no—
120C22×D21—124no—
121C22×D21—124no—
122C22×D21—124no—
123C3×C7×D4—22no—
124C3×C7×D4—22no—
125C3×C7×D4—22no—
126C3×C7×D4—22no—
127C2×C3×C4×C7—21no—
128C2×C3×C4×C7C22×D311yescharacteristic
129C2×C3×C4×C7C22×D311yescharacteristic
130#4268—64no—
131#4268—64no—
132#4268—64no—
133#4268—64no—
134C2×Dic21—31no—
135C2×Dic21—31no—
136C2×Dic21—61no—
137D84—62no—
138D84—62no—
139D84—62no—
140D84—62no—
141C4×D21—32no—
142C4×D21—32no—
143C4×D21—32no—
144C4×D21—32no—
145C4×D21—62no—
146C4×D21—62no—
147C3×C7×C8—62no—
148C3⋊(C7⋊C8)—22no—
149C3×C7⋊C8—62no—
150C7×C3⋊C8—22no—
151C23×C3⋊D3—72no—
152C2×C32×D4C2×D711yescharacteristic
153C2×#638—72no—
154D4×C3⋊D3—142no—
155D4×C3⋊D3—142no—
156C2×C4⋊(C3⋊D3)—71no—
157C2×C4×C3⋊D3—71no—
158C3×#208—72no—
159C3×#208C2×D712yescomplemented
160#3712—142no—
161#3710—142no—
162C2×C32×C7—21no—
163C2×C32×C7—42no—
164C2×C32×C7C2×C22⋊C411yescharacteristic
165C3⋊D21—21no—
166C3⋊D21—42no—
167C3⋊D21#10012yescomplemented
168C23×D7—92no—
169C2×C7×D4D3211yescharacteristic
170C2×C22⋊D7—92no—
171D4×D7—182no—
172D4×D7—182no—
173C2×D28—91no—
174C2×C4×D7—91no—
175C7×M16—32no—
176#1136—32no—
177#1131—182no—
178C8×D7—182no—
179C2×D3×D4—211no—
180C2×D3×D4—211no—
181C2×D3×D4—421no—
182D3×M16—212no—
183D3×M16—212no—
184C3×#100—31no—
185C3×#100—211no—
186#833—71no—
187#833D2111yescharacteristic, complemented
188#823—212no—
189#823—212no—
190C22×C3×C7—22no—
191C22×C3×C7—22no—
192C22×C3×C7—22no—
193C22×C3×C7—22no—
194C22×C3×C7—21no—
195C22×C3×C7—42no—
196C22×C3×C7—42no—
197C22×C3×C7—42no—
198C22×C3×C7—42no—
199C22×C3×C7C2×C4×D311yescharacteristic
200C22×C3×C7C2×C4×D311yescharacteristic
201C2×D21—31no—
202C2×D21—31no—
203C2×D21—31no—
204C2×D21—31no—
205C2×D21—61no—
206C2×D21—61no—
207C2×D21—62no—
208C2×D21—62no—
209C2×D21—62no—
210C2×D21—62no—
211C2×D21—62no—
212C2×D21—62no—
213C2×D21—62no—
214C2×D21—62no—
215C2×D21—124no—
216C2×D21—124no—
217C2×D21—124no—
218C2×D21—124no—
219C2×D21—124no—
220C2×D21—124no—
221C2×D21—124no—
222C2×D21—124no—
223C2×D21—122no—
224C2×D21—122no—
225C2×D21—122no—
226C2×D21—122no—
227C3×C4×C7—22no—
228C3×C4×C7D3×D412yes—
229C3×C4×C7D3×D412yes—
230Dic21—32no—
231Dic21—32no—
232Dic21—62no—
233C23×C32C4×D712yes—
234C22×C3⋊D3—71no—
235C22×C3⋊D3—72no—
236C22×C3⋊D3—142no—
237C22×C3⋊D3—142no—
238C22×C3⋊D3—284no—
239C32×D4—22no—
240C2×C32×C4C22×D711yescharacteristic
241#638—144no—
242C2×C3⋊Dic3—71no—
243C4⋊(C3⋊D3)—142no—
244C4×C3⋊D3—72no—
245C4×C3⋊D3—72no—
246C3×C3⋊C8—22no—
247C3×C3⋊C8—142no—
248C2×#100—631noSylow
249C32×C7C2×#10011yescharacteristic, complemented, Hall
250C23×C7#62412yes—
251C22×D7—91no—
252C22×D7—92no—
253C22×D7—182no—
254C22×D7—182no—
255C22×D7—364no—
256C7×D4—22no—
257C2×C4×C7C2×D3211yescharacteristic
258C22⋊D7—184no—
259C2×Dic7—91no—
260D28—182no—
261C4×D7—92no—
262C4×D7—92no—
263C7×C8—62no—
264C7⋊C8—62no—
265C23×D3—212no—
266C23×D3—212no—
267C23×D3—422no—
268C2×C3×D4—21no—
269C2×C3×D4D3×D711yescharacteristic
270C2×C3×D4C2×D2111yescharacteristic
271C2×#67—212no—
272C2×#67—212no—
273C2×#67—422no—
274D3×D4—422no—
275D3×D4—422no—
276D3×D4—422no—
277D3×D4—422no—
278D3×D4—422no—
279D3×D4—422no—
280D3×D4—422no—
281D3×D4—422no—
282C2×D12—211no—
283C2×D12—211no—
284C2×D12—421no—
285C2×C4×D3—211no—
286C2×C4×D3—211no—
287C2×C4×D3—421no—
288C3×M16—32no—
289C3×M16—212no—
290#208—72no—
291#208C2×D2112yescomplemented
292ΓL(1,𝔽25)—422no—
293ΓL(1,𝔽25)—422no—
294D3×C8—422no—
295D3×C8—422no—
296C2×C3×C7—21no—
297C2×C3×C7—21no—
298C2×C3×C7—21no—
299C2×C3×C7—21no—
300C2×C3×C7—21no—
301C2×C3×C7—42no—
302C2×C3×C7—42no—
303C2×C3×C7—42no—
304C2×C3×C7—42no—
305C2×C3×C7D3×C22⋊C411yescharacteristic
306C2×C3×C7D3×C22⋊C411yescharacteristic
307D21—32no—
308D21—32no—
309D21—62no—
310D21—61no—
311D21—61no—
312D21—61no—
313D21—61no—
314D21—122no—
315D21—122no—
316D21—122no—
317D21—122no—
318C22×C32—22no—
319C22×C32—42no—
320C22×C32C2×C4×D711yescharacteristic
321C2×C3⋊D3—71no—
322C2×C3⋊D3—71no—
323C2×C3⋊D3—142no—
324C2×C3⋊D3—142no—
325C2×C3⋊D3—284no—
326C2×C3⋊D3—284no—
327C2×C3⋊D3—282no—
328C32×C4D4×D712yes—
329C3⋊Dic3—72no—
330C22×D4—631no—
331C2×M16—632no—
332#100—31no—
333#100—211no—
334#100—632no—
335C22×C7—22no—
336C22×C7—42no—
337C22×C7C2×#62411yescharacteristic
338C2×D7—91no—
339C2×D7—91no—
340C2×D7—182no—
341C2×D7—182no—
342C2×D7—364no—
343C2×D7—364no—
344C2×D7—362no—
345C4×C7#381112yes—
346Dic7—92no—
347C23×C3—22no—
348C23×C3D21⋊C412yes—
349C23×C3C4×D2112yes—
350C22×D3—211no—
351C22×D3—211no—
352C22×D3—212no—
353C22×D3—212no—
354C22×D3—421no—
355C22×D3—422no—
356C22×D3—422no—
357C22×D3—422no—
358C22×D3—422no—
359C22×D3—422no—
360C22×D3—422no—
361C22×D3—422no—
362C22×D3—422no—
363C22×D3—422no—
364C22×D3—844no—
365C22×D3—844no—
366C22×D3—844no—
367C22×D3—844no—
368C3×D4—22no—
369C3×D4—22no—
370C3×D4—22no—
371C3×D4—22no—
372C2×C3×C4—21no—
373C2×C3×C4C2×D3×D711yescharacteristic
374C2×C3×C4C22×D2111yescharacteristic
375#67—424no—
376#67—424no—
377#67—424no—
378#67—424no—
379C2×Dic3—211no—
380C2×Dic3—211no—
381C2×Dic3—421no—
382D12—422no—
383D12—422no—
384D12—422no—
385D12—422no—
386C4×D3—212no—
387C4×D3—212no—
388C4×D3—212no—
389C4×D3—212no—
390C4×D3—422no—
391C4×D3—422no—
392C3×C8—62no—
393C3×C8—422no—
394C3⋊C8—22no—
395C3⋊C8—142no—
396C3×C7—21no—
397C3×C7D3×#10011yescharacteristic, complemented
398C3×C7D3×#10011yescharacteristic, complemented
399C2×C32—21no—
400C2×C32—42no—
401C2×C32D7×C22⋊C411yescharacteristic
402C3⋊D3—72no—
403C3⋊D3—141no—
404C3⋊D3—282no—
405C24—632no—
406C2×D4—632no—
407C2×D4—631no—
408C2×D4—1262no—
409C2×D4—1262no—
410C2×D4D3×D2111yescharacteristic
411C22×C4—631no—
412M16—32no—
413M16—212no—
414M16—1262no—
415C2×C8—1262no—
416C2×C7—21no—
417C2×C7—42no—
418C2×C7#6397411yescharacteristic
419D7—92no—
420D7—181no—
421D7—362no—
422C22×C3—22no—
423C22×C3—22no—
424C22×C3—22no—
425C22×C3—22no—
426C22×C3—21no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—42no—
430C22×C3—42no—
431C22×C3C2×D21⋊C411yescharacteristic
432C22×C3C2×C4×D2111yescharacteristic
433C2×D3—211no—
434C2×D3—211no—
435C2×D3—211no—
436C2×D3—211no—
437C2×D3—421no—
438C2×D3—421no—
439C2×D3—422no—
440C2×D3—422no—
441C2×D3—422no—
442C2×D3—422no—
443C2×D3—422no—
444C2×D3—422no—
445C2×D3—422no—
446C2×D3—422no—
447C2×D3—844no—
448C2×D3—844no—
449C2×D3—844no—
450C2×D3—844no—
451C2×D3—844no—
452C2×D3—844no—
453C2×D3—844no—
454C2×D3—844no—
455C2×D3—842no—
456C2×D3—842no—
457C2×D3—842no—
458C2×D3—842no—
459C3×C4—22no—
460C3×C4#6669412yes—
461C3×C4D4×D2112yes—
462Dic3—212no—
463Dic3—212no—
464Dic3—422no—
465C32D7×#10011yescharacteristic, complemented, Sylow
466C23—631no—
467C23—632no—
468C23—1262no—
469C23—1262no—
470C23—2524no—
471C23#9268712yes—
472D4—22no—
473D4—1264no—
474D4—1262no—
475C2×C4—631no—
476C2×C4—632no—
477C2×C4—632no—
478C2×C4C2×D3×D2111yescharacteristic
479C8—62no—
480C8—422no—
481C7#1059150311yescharacteristic, complemented, Sylow
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—42no—
488C2×C3—42no—
489C2×C3—42no—
490C2×C3—42no—
491C2×C3#1062112311yescharacteristic
492C2×C3C22⋊C4×D2111yescharacteristic
493D3—212no—
494D3—212no—
495D3—422no—
496D3—421no—
497D3—421no—
498D3—421no—
499D3—421no—
500D3—842no—
501D3—842no—
502D3—842no—
503D3—842no—
504C22—22no—
505C22—42no—
506C22—631no—
507C22—631no—
508C22—1262no—
509C22—1262no—
510C22—2522no—
511C22—2524no—
512C22—2524no—
513C22C2×#9268711yescharacteristic
514C4—632no—
515C4#1175962612yes—
516C3—21no—
517C3#1305449311yescharacteristic, complemented
518C3#100×D2111yescharacteristic, complemented
519C2—21no—
520C2—42no—
521C2—632no—
522C2—1261no—
523C2—2522no—
524C2#7224681611yescharacteristic
subgroupG/H
centerC2#72246816
commutatorC22×C32×C7C22×C4
FrattiniC2×C4C2×D3×D21
FittingC2×C32×C7×D4C22
socleC2×C32×C7C2×C22⋊C4
prime radicalC2×D4×C3⋊D21C2
cosocleC2×C32×C4×C7C23
hypercenterC2×D4D3×D21
hypocenterC32×C7C2×#100
LCSG ▷ C22×C32×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#10620672C2semidirect
2. #72246807C2semidirect
3. C3×C7×#833C2semidirect
4. #72245722C2semidirect
15. C3×#66627C22semidirect
16. C3×C7×#208C22semidirect
25. #10620672D3semidirect
27. C7×#833D3semidirect
71. #66627C2×D3semidirect
74. C7×#208C2×D3semidirect
81. C3×#833D7semidirect
159. C3×#208C2×D7semidirect
167. C3⋊D21#100semidirect
187. #833D21semidirect
249. C32×C7C2×#100semidirect
291. #208C2×D21semidirect
397. C3×C7D3×#100semidirect
398. C3×C7D3×#100semidirect
465. C32D7×#100semidirect
481. C7#10591503semidirect
517. C3#13054493semidirect
518. C3#100×D21semidirect

Maximal subgroups

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

subgroupindex
C3×#106206722
#722468072
C3×C7×#8332
#722457222
C2×D4×C3⋊D212
#100×D213
#130544933
#105915037