Finite Group Databasebeta

← #72246658

Subgroups — #72246658

Order 4032.

788 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 114 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#72246652C212yesmaximal, complemented
2C3×#10620516C212yesmaximal, complemented
3C4×C3⋊D3×Dic7C211yesmaximal, characteristic, complemented
4#72245744C211yesmaximal, characteristic
5C7×#63818C211yesmaximal, characteristic, complemented
6D3×#6399—32nomaximal
7C2×C4×C7×C3⋊D3C2211yescharacteristic
8C3×#66728C2212yes—
9#11759644—21no—
10#11759637—21no—
11#11759635C411yescharacteristic, complemented
12C4×(C3⋊D3)⋊D7—21no—
13#11759631—21no—
14#11759630C411yescharacteristic, complemented
15#11759628C412yescomplemented
16C2×C3⋊D3×Dic7—21no—
17C4×C3⋊Dic21C2211yescharacteristic, complemented
18C3×C7×#208C2212yescomplemented
19C32×C4×Dic7C2211yescharacteristic, complemented
20C7×#3712—21no—
21C7×#3710—21no—
22Dic7×C3⋊Dic3—21no—
23D3×#1157—32no—
24D3×C7×M16—32no—
25C4×D3×Dic7—32no—
26C4×D3×Dic7—61no—
27#10620664—32no—
28C3×#6399—32no—
29#10620516D312yescomplemented
30#10620511—32no—
31#10590393—71nomaximal, Hall
32C22×C7×C3⋊D3D411yescharacteristic
33C2×(C3⋊D3)⋊D7—21no—
34C2×C32×C4×C7C2311yescharacteristic
35C2×C3⋊Dic21—21no—
36C2×C7×C3⋊Dic3D411yescharacteristic
37C4×C7×C3⋊D3C2×C411yescharacteristic
38C4×C7×C3⋊D3C2×C411yescharacteristic
39C4×C7×C3⋊D3D411yescharacteristic
40C4×C7×C3⋊D3D411yescharacteristic
41C3×#4268—22no—
42C3×D84C2×C412yescomplemented
43C3×C4×D21—22no—
44C3×Dic42C2×C412yescomplemented
45C2×C32×Dic7—21no—
46#92696—21no—
47#92695—21no—
48#92694—22no—
49#92693—21no—
50D21⋊Dic3—22no—
51C3⋊D3×Dic7—24no—
52C3×C7×C3⋊C8—22no—
53C2×#6399—91noHall
54#66728C2×D312yes—
55C2×C4×D3×C7—32no—
56C2×C4×D3×C7—61no—
57C3×#1157—32no—
58D3×C22⋊D7—62no—
59C2×D3×Dic7—62no—
60C2×D3×Dic7—61no—
61C2×D3×Dic7—61no—
62#66683—62no—
63D3×D28—32no—
64C4×D3×D7—62no—
65#66675—62no—
66#66673—32no—
67D3×Dic14—32no—
68#66669—32no—
69C3×C7×M16—32no—
70C4×Dic21—32no—
71C4×Dic21—61no—
72C7×#208C2×D312yescomplemented
73C7×ΓL(1,𝔽25)—62no—
74D3×C7×C8—62no—
75C3×C4×Dic7—21no—
76C3×C4×Dic7C2×D312yescomplemented
77Dic3×Dic7—62no—
78Dic3×Dic7—61no—
79Dic3×Dic7—61no—
80#64306—71no—
81C42×C3⋊D3—71no—
82#63818D711yescharacteristic, complemented
83C3×#805—72no—
84#63568—72no—
85C22×C32×C7C2×D411yescharacteristic
86C2×C7×C3⋊D3—22no—
87C2×C7×C3⋊D3C22⋊C411yescharacteristic
88C2×C7×C3⋊D3C22⋊C411yescharacteristic
89C2×C3×D21—22no—
90(C3⋊D3)⋊D7—42no—
91C32×C4×C7C22×C411yescharacteristic
92C32×C4×C7C2×D411yescharacteristic
93C3⋊Dic21—22no—
94C7×C3⋊Dic3C22⋊C411yescharacteristic
95C7×C3⋊Dic3C22⋊C411yescharacteristic
96C3×Dic21—22no—
97C32×Dic7—22no—
98C2×#1157—91no—
99C2×C7×M16—91no—
100C2×C4×Dic7—91no—
101#6399—32no—
102#6399—92no—
103D3×#104—212no—
104C22×D3×C7—32no—
105C22×D3×C7—61no—
106C2×D3×D7—62no—
107C2×C3×C4×C7—21no—
108C2×C3×C4×C7C22×D312yes—
109#4268—22no—
110C2×Dic21—61no—
111C2×Dic21—61no—
112C2×Dic21—62no—
113D84C4×D312yescomplemented
114C4×D21—22no—
115Dic42C4×D312yescomplemented
116C2×C7×Dic3—32no—
117C2×C7×Dic3—61no—
118C4×D3×C7—32no—
119C4×D3×C7—32no—
120C4×D3×C7—32no—
121C4×D3×C7—32no—
122C4×D3×C7—61no—
123C4×D3×C7—61no—
124C4×D3×C7—61no—
125C4×D3×C7—61no—
126C3×C22⋊D7—62no—
127C2×C3×Dic7—21no—
128C2×C3×Dic7—21no—
129C2×C3×Dic7—22no—
130C3×D28—32no—
131C3×C4×D7—62no—
132C3×Dic14—32no—
133#4248—62no—
134#4247—62no—
135#4246—62no—
136#4245—62no—
137D21⋊C4—62no—
138D3×Dic7—68no—
139D3×Dic7—64no—
140D3×Dic7—64no—
141D3×Dic7—62no—
142Dic3×D7—62no—
143C3×C7×C8—62no—
144C7×C3⋊C8—22no—
145C2×C4×C3⋊D3—141no—
146C2×C4×C3⋊D3C2×D711yescharacteristic
147C3×#235—72no—
148#3811—141no—
149#3805—141no—
150#3802—71no—
151C4×D32—141no—
152#3798—141no—
153#3797—71no—
154#3796—72no—
155C32×C42—71no—
156C4×C3⋊Dic3—71no—
157C4×C3⋊Dic3—141no—
158C3×#208C2×D712yescomplemented
159#3712—21no—
160#3710—21no—
161C2×C32×C7—21no—
162C2×C32×C7C2×C22⋊C411yescharacteristic
163C7×C3⋊D3—21no—
164C7×C3⋊D3#10412yescomplemented
165C3×D21—42no—
166C22×C4×C7—91no—
167C2×C22⋊D7—181no—
168C22×Dic7—181no—
169#1157—32no—
170#1157—92no—
171#1157—181no—
172#1157—181no—
173C2×D28—91no—
174C2×C4×D7—181no—
175C2×Dic14—91no—
176C7×M16—32no—
177C7×M16—181no—
178C2×C7×C8—181no—
179C4×Dic7—91no—
180C4×Dic7—181no—
181C4×Dic7D3211yescharacteristic, complemented
182D3×#41—212no—
183D3×M16—32no—
184C42×D3—212no—
185C42×D3—421no—
186C3×#104—212no—
187#837—212no—
188#825—212no—
189#805—72no—
190C22×C3×C7—21no—
191C22×C3×C7D3×D412yes—
192C2×D21—22no—
193C2×D3×C7—32no—
194C2×D3×C7—32no—
195C2×D3×C7—61no—
196C2×D3×C7—61no—
197C2×D3×C7—64no—
198C2×D3×C7—62no—
199C2×D3×C7—62no—
200C2×C3×D7—62no—
201D3×D7—124no—
202C3×C4×C7—21no—
203C3×C4×C7—21no—
204C3×C4×C7C2×C4×D312yes—
205C3×C4×C7D3×D412yes—
206Dic21—22no—
207Dic21—62no—
208Dic21—62no—
209Dic21—64no—
210C3×Dic7—22no—
211C3×Dic7—22no—
212C3×Dic7—24no—
213C3×Dic7—62no—
214C7×Dic3—32no—
215C7×Dic3—32no—
216C7×Dic3—61no—
217C7×Dic3—61no—
218C22×C3⋊D3D2811yescharacteristic
219C2×D32—141no—
220C2×C32×C4—141no—
221C2×C32×C4C22×D711yescharacteristic
222C2×C3⋊Dic3—141no—
223C2×C3⋊Dic3D2811yescharacteristic
224C4×C3⋊D3—144no—
225C4×C3⋊D3C4×D711yescharacteristic
226C4×C3⋊D3C4×D711yescharacteristic
227C4×C3⋊D3C22⋊D711yescharacteristic
228C4×C3⋊D3C22⋊D711yescharacteristic
229C3×#67—142no—
230C3×D12—72no—
231C3×C4×D3—142no—
232C3×Dic6—72no—
233#627—141no—
234#626—142no—
235#625—141no—
236#624—141no—
237D3×Dic3—142no—
238C3×C3⋊C8—22no—
239C2×#104—631noSylow
240C32×C7C2×#10411yescharacteristic, complemented, Hall
241C23×C7—91no—
242C22×D7—181no—
243C2×C4×C7—91no—
244C2×C4×C7—91no—
245C2×C4×C7—91no—
246C2×C4×C7—91no—
247C2×C4×C7—91no—
248C2×C4×C7C2×D3211yescharacteristic
249C22⋊D7—62no—
250C22⋊D7—182no—
251C2×Dic7—21no—
252C2×Dic7—184no—
253C2×Dic7—181no—
254C2×Dic7—181no—
255D28—32no—
256D28—181no—
257C4×D7—62no—
258C4×D7—182no—
259Dic14—32no—
260Dic14—181no—
261C7×C8—62no—
262C3×#41—212no—
263#239—212no—
264D3×Q8—212no—
265#237—212no—
266#237—422no—
267D3×D4—212no—
268D3×D4—422no—
269#235—72no—
270#235—422no—
271C2×C4×D3—32no—
272C2×C4×D3—61no—
273C2×C4×D3—422no—
274C2×C4×D3—422no—
275C2×C4×D3—421no—
276C2×C4×D3—421no—
277C3×M16—32no—
278C3×C42—72no—
279C3×C42—141no—
280C4×Dic3—212no—
281C4×Dic3—421no—
282C4×Dic3—422no—
283C4×Dic3—421no—
284C4×Dic3—421no—
285#208D3×D712yescomplemented
286ΓL(1,𝔽25)—62no—
287D3×C8—62no—
288C2×C3×C7—21no—
289C2×C3×C7—21no—
290C2×C3×C7—22no—
291C2×C3×C7—21no—
292C2×C3×C7D3×C22⋊C412yes—
293D21—42no—
294C3×D7—122no—
295D3×C7—34no—
296D3×C7—62no—
297D3×C7—61no—
298D3×C7—61no—
299D3×C7—62no—
300C22×C32C2×D2811yescharacteristic
301C2×C3⋊D3—22no—
302C2×C3⋊D3(C2×D7)⋊C411yescharacteristic
303C2×C3⋊D3(C2×D7)⋊C411yescharacteristic
304C2×C3×D3—142no—
305D32—282no—
306C32×C4—142no—
307C32×C4C2×C4×D711yescharacteristic
308C32×C4C2×C22⋊D711yescharacteristic
309C3⋊Dic3—142no—
310C3⋊Dic3(C2×D7)⋊C411yescharacteristic
311C3⋊Dic3(C2×D7)⋊C411yescharacteristic
312C3×Dic3—142no—
313C2×#41—631no—
314C2×M16—91no—
315C2×C42—631no—
316#104—212no—
317#104—632no—
318C22×C7—91no—
319C22×C7—91no—
320C22×C7—182no—
321C22×C7#381111yescharacteristic
322C2×D7—62no—
323C2×D7—362no—
324C4×C7—91no—
325C4×C7—91no—
326C4×C7C4×D3211yescharacteristic
327C4×C7#380211yescharacteristic
328Dic7—22no—
329Dic7—62no—
330Dic7—182no—
331C22×D3—32no—
332C22×D3—61no—
333C22×D3—422no—
334C3×Q8—212no—
335C3×D4—212no—
336C3×D4—422no—
337C2×C3×C4—21no—
338C2×C3×C4—141no—
339C2×C3×C4—141no—
340C2×C3×C4—142no—
341C2×C3×C4—422no—
342C2×C3×C4C2×D3×D712yes—
343#67—142no—
344#67—422no—
345#67—422no—
346C2×Dic3—32no—
347C2×Dic3—61no—
348C2×Dic3—422no—
349C2×Dic3—421no—
350C2×Dic3—421no—
351C2×Dic3—422no—
352D12—72no—
353D12—422no—
354C4×D3—32no—
355C4×D3—32no—
356C4×D3—32no—
357C4×D3—32no—
358C4×D3—61no—
359C4×D3—61no—
360C4×D3—61no—
361C4×D3—61no—
362C4×D3—142no—
363C4×D3—422no—
364C4×D3—422no—
365C4×D3—428no—
366C4×D3—424no—
367C4×D3—424no—
368Dic6—72no—
369Dic6—422no—
370C3×C8—62no—
371C3⋊C8—22no—
372C3×C7—21no—
373C3×C7D3×#10412yescomplemented
374C2×C32—21no—
375C2×C32C2×(C2×D7)⋊C411yescharacteristic
376C3⋊D3—21no—
377C3⋊D3#639912yescomplemented
378C3×D3—282no—
379#41—212no—
380#41—632no—
381#41—1261no—
382#41—1261no—
383C2×Q8—631no—
384C2×D4—631no—
385C2×D4—1261no—
386C22×C4—91no—
387C22×C4—1261no—
388C22×C4—1261no—
389M16—32no—
390M16—181no—
391C2×C8—181no—
392C42—71no—
393C42—631no—
394C42—1261no—
395C2×C7—21no—
396C2×C7—92no—
397C2×C7—181no—
398C2×C7#6392111yescharacteristic
399D7—122no—
400C22×C3—21no—
401C22×C3—422no—
402C22×C3D3×D2812yes—
403C2×D3—32no—
404C2×D3—32no—
405C2×D3—61no—
406C2×D3—61no—
407C2×D3—64no—
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—142no—
411C2×D3—844no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—142no—
415C3×C4—142no—
416C3×C4—144no—
417C3×C4—422no—
418C3×C4C4×D3×D712yes—
419C3×C4D3×C22⋊D712yes—
420Dic3—32no—
421Dic3—32no—
422Dic3—61no—
423Dic3—61no—
424Dic3—142no—
425Dic3—422no—
426Dic3—422no—
427Dic3—424no—
428C32C2×#639911yescharacteristic, complemented, Sylow
429C23—91no—
430C23—1261no—
431Q8—212no—
432Q8—1261no—
433D4—212no—
434D4—422no—
435D4—1261no—
436D4—1262no—
437C2×C4—91no—
438C2×C4—91no—
439C2×C4—91no—
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—141no—
443C2×C4—422no—
444C2×C4—1262no—
445C2×C4—1264no—
446C2×C4—1261no—
447C2×C4—1261no—
448C2×C4D32×D711yescharacteristic
449C8—62no—
450C7#1059039311yescharacteristic, complemented, Sylow
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—22no—
454C2×C3—21no—
455C2×C3—842no—
456C2×C3D3×(C2×D7)⋊C412yes—
457D3—34no—
458D3—62no—
459D3—61no—
460D3—61no—
461D3—62no—
462D3—282no—
463C22—91no—
464C22—91no—
465C22—182no—
466C22—422no—
467C22—2522no—
468C22#1175942811yescharacteristic
469C4—91no—
470C4—91no—
471C4—142no—
472C4—422no—
473C4—1262no—
474C4#1175941911yescharacteristic
475C4#1175943111yescharacteristic
476C3—21no—
477C3D3×#639912yescomplemented
478C2—21no—
479C2—92no—
480C2—181no—
481C2—842no—
482C2#7224665411yescharacteristic
subgroupG/H
centerC4#11759431
commutatorC32×C4×C7C22×C4
Frattini, hypercenterC2×C4D32×D7
Fitting, cosocleC2×C32×C4×C7C23
socleC2×C32×C7C2×C22⋊C4
prime radical#72245744C2
hypocenterC32×C7C2×#104
LCSG ▷ C32×C4×C7 ▷ C2×C32×C7 ▷ C32×C7
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #72246652C2semidirect
2. C3×#10620516C2semidirect
3. C4×C3⋊D3×Dic7C2semidirect
5. C7×#63818C2semidirect
11. #11759635C4semidirect
14. #11759630C4semidirect
15. #11759628C4semidirect
17. C4×C3⋊Dic21C22semidirect
18. C3×C7×#208C22semidirect
19. C32×C4×Dic7C22semidirect
29. #10620516D3semidirect
42. C3×D84C2×C4semidirect
44. C3×Dic42C2×C4semidirect
72. C7×#208C2×D3semidirect
76. C3×C4×Dic7C2×D3semidirect
82. #63818D7semidirect
113. D84C4×D3semidirect
115. Dic42C4×D3semidirect
158. C3×#208C2×D7semidirect
164. C7×C3⋊D3#104semidirect
181. C4×Dic7D32semidirect
240. C32×C7C2×#104semidirect
285. #208D3×D7semidirect
373. C3×C7D3×#104semidirect
377. C3⋊D3#6399semidirect
428. C32C2×#6399semidirect
450. C7#10590393semidirect
477. C3D3×#6399semidirect

Maximal subgroups

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

subgroupindex
#722466522
C3×#106205162
C4×C3⋊D3×Dic72
#722457442
C7×#638182
D3×#63993
#105903937