Finite Group Databasebeta

← #11732331

Subgroups — #11732331

Order 896.

1056 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#1157C211yesmaximal, characteristic, complemented
2C2×#6500C211yesmaximal, characteristic, complemented
3C2×#6445C211yesmaximal, characteristic, complemented
4C2×D7×C22⋊C4C211yesmaximal, characteristic, complemented
5C7×#386C211yesmaximal, characteristic, complemented
6#89995C211yesmaximal, characteristic, complemented
7#89985C211yesmaximal, characteristic, complemented
8#89732C211yesmaximal, characteristic, complemented
9#89728C211yesmaximal, characteristic, complemented
10#89728C211yesmaximal, characteristic, complemented
11#89719C211yesmaximal, characteristic, complemented
12#89445C211yesmaximal, characteristic, complemented
13#89437C211yesmaximal, characteristic, complemented
14#89437C211yesmaximal, characteristic, complemented
15#89436C211yesmaximal, characteristic, complemented
16C24×D7C2211yescharacteristic
17C22×C7×D4C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic, complemented
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C22×C22⋊D7C2211yescharacteristic
22C22×C22⋊D7C2211yescharacteristic, complemented
23C2×#1157—24no—
24C2×#1157—24no—
25C2×#1157—24no—
26C22×D28C2211yescharacteristic
27C22×D28C2211yescharacteristic
28C22×C4×D7C2211yescharacteristic
29C22×C4×D7C2211yescharacteristic
30C22×C4×D7C2211yescharacteristic, complemented
31C22×Dic14C2211yescharacteristic, complemented
32C2×C7×C22⋊C4C2211yescharacteristic
33C2×C7×C22⋊C4C2211yescharacteristic
34C2×C7×C22⋊C4C2211yescharacteristic, complemented
35C2×C22⋊Dic7C2211yescharacteristic
36C2×C22⋊Dic7C2211yescharacteristic
37C2×C22⋊Dic7C2211yescharacteristic, complemented
38#6500—24no—
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×(C2×D7)⋊C4C2211yescharacteristic
42C2×(C2×D7)⋊C4C2211yescharacteristic
43C2×(C2×D7)⋊C4C2211yescharacteristic
44C2×(C2×D7)⋊C4C2211yescharacteristic
45C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
46C2×(C2×D7)⋊C4C2211yescharacteristic, complemented
47#6445—24no—
48D7×C22⋊C4—24no—
49C7×#95C2211yescharacteristic, complemented
50C7×#95C2211yescharacteristic, complemented
51C7⋊#95C2211yescharacteristic, complemented
52C7⋊#95C2211yescharacteristic, complemented
53C7⋊#95C2211yescharacteristic, complemented
54C7⋊#95C2211yescharacteristic, complemented
55C7⋊#95C2211yescharacteristic, complemented
56C7⋊#95C2211yescharacteristic, complemented
57#2404—71nomaximal, Sylow
58C24×C7C2311yescharacteristic
59C24×C7C2311yescharacteristic
60C23×D7—22no—
61C23×D7—22no—
62C23×D7—22no—
63C23×D7—22no—
64C23×D7—22no—
65C23×D7—41no—
66C23×D7—41no—
67C23×D7C2311yescharacteristic
68C23×D7C2311yescharacteristic
69C23×D7C2311yescharacteristic
70C2×C7×D4—22no—
71C2×C7×D4—22no—
72C2×C7×D4—41no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—22no—
76C22×C4×C7C2311yescharacteristic
77C22×C4×C7C2311yescharacteristic
78C22×C4×C7C2311yescharacteristic
79C22×C4×C7C2311yescharacteristic
80C22×C4×C7C2311yescharacteristic
81C2×C22⋊D7—22no—
82C2×C22⋊D7—22no—
83C2×C22⋊D7—22no—
84C2×C22⋊D7—22no—
85C2×C22⋊D7—22no—
86C2×C22⋊D7—22no—
87C2×C22⋊D7—22no—
88C2×C22⋊D7—22no—
89C2×C22⋊D7—22no—
90C2×C22⋊D7—41no—
91C2×C22⋊D7—41no—
92C2×C22⋊D7D414yescomplemented
93C22×Dic7C2311yescharacteristic
94C22×Dic7C2311yescharacteristic
95C22×Dic7C2311yescharacteristic
96C22×Dic7C2311yescharacteristic
97C22×Dic7C2311yescharacteristic
98#1157—232no—
99C2×D28—22no—
100C2×D28—22no—
101C2×D28—22no—
102C2×D28—22no—
103C2×D28—41no—
104C2×D28D414yescomplemented
105C2×C4×D7—22no—
106C2×C4×D7—22no—
107C2×C4×D7—22no—
108C2×C4×D7—22no—
109C2×C4×D7—22no—
110C2×C4×D7—22no—
111C2×C4×D7—41no—
112C2×C4×D7—41no—
113C2×C4×D7D414yescomplemented
114C2×Dic14—22no—
115C2×Dic14—22no—
116C2×Dic14—22no—
117C7×C22⋊C4—22no—
118C7×C22⋊C4—22no—
119C7×C22⋊C4—41no—
120C22⋊Dic7—22no—
121C22⋊Dic7—22no—
122C22⋊Dic7—41no—
123(C2×D7)⋊C4—22no—
124(C2×D7)⋊C4—22no—
125(C2×D7)⋊C4—22no—
126(C2×D7)⋊C4—22no—
127(C2×D7)⋊C4—22no—
128(C2×D7)⋊C4—22no—
129(C2×D7)⋊C4—41no—
130(C2×D7)⋊C4—41no—
131C22×#41—71no—
132C2×#120—71no—
133C2×#120—71no—
134C22×C22⋊C4—71no—
135#393—71no—
136#393—71no—
137#392—71no—
138#392—71no—
139#386—71no—
140#386—71no—
141#386—71no—
142#386—71no—
143#386—71no—
144#386D711yescharacteristic, complemented
145#381—71no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—21no—
157C23×C7—21no—
158C23×C7C2×D412yes—
159C23×C7#4112yescomplemented
160C23×C7C2411yescharacteristic
161C22×D7—21no—
162C22×D7—21no—
163C22×D7—21no—
164C22×D7—21no—
165C22×D7—28no—
166C22×D7—21no—
167C22×D7—21no—
168C22×D7—21no—
169C22×D7—21no—
170C22×D7—21no—
171C22×D7—21no—
172C22×D7—21no—
173C22×D7—21no—
174C22×D7—21no—
175C22×D7—21no—
176C22×D7—21no—
177C22×D7—21no—
178C22×D7—44no—
179C22×D7—44no—
180C22×D7—44no—
181C22×D7—44no—
182C22×D7—44no—
183C22×D7—44no—
184C22×D7C2×D412yes—
185C22×D7C2×D412yes—
186C22×D7C2×D412yes—
187C22×D7C2×D412yes—
188C22×D7#4112yescomplemented
189C7×D4—44no—
190C2×C4×C7—28no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—21no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—21no—
202C2×C4×C7—21no—
203C2×C4×C7—21no—
204C2×C4×C7C2×D412yes—
205C2×C4×C7C2×D412yes—
206C22⋊D7—28no—
207C22⋊D7—28no—
208C22⋊D7—44no—
209C22⋊D7—44no—
210C2×Dic7—21no—
211C2×Dic7—21no—
212C2×Dic7—21no—
213C2×Dic7—21no—
214C2×Dic7—21no—
215C2×Dic7—21no—
216C2×Dic7—21no—
217C2×Dic7—21no—
218C2×Dic7—21no—
219C2×Dic7—21no—
220C2×Dic7—21no—
221C2×Dic7—21no—
222C2×Dic7—21no—
223C2×Dic7C2×D412yes—
224C2×Dic7C2×D412yes—
225D28—28no—
226D28—44no—
227C4×D7—28no—
228C4×D7—28no—
229C4×D7—44no—
230Dic14—28no—
231C25—71no—
232C2×#41—144no—
233C2×#41—144no—
234C2×#41—144no—
235C22×Q8—71no—
236C22×D4—71no—
237C22×D4—71no—
238C22×D4—71no—
239C22×D4—71no—
240C22×D4—71no—
241C22×D4—71no—
242C22×D4C2×D711yescharacteristic
243C23×C4—71no—
244C23×C4—71no—
245C23×C4—71no—
246C23×C4C2×D711yescharacteristic, complemented
247#120—144no—
248#120—144no—
249C2×C22⋊C4—71no—
250C2×C22⋊C4—71no—
251C2×C22⋊C4—71no—
252C2×C22⋊C4—71no—
253C2×C22⋊C4—71no—
254C2×C22⋊C4—71no—
255C2×C22⋊C4—71no—
256C2×C22⋊C4—71no—
257C2×C22⋊C4—71no—
258C2×C22⋊C4—71no—
259C2×C22⋊C4—71no—
260C2×C22⋊C4—144no—
261C2×C22⋊C4C2×D711yescharacteristic
262C2×C22⋊C4C2×D711yescharacteristic
263C2×C22⋊C4C2×D711yescharacteristic, complemented
264#95—71no—
265#95—71no—
266#95—71no—
267#95—71no—
268#95—71no—
269#95—71no—
270#95C2×D711yescharacteristic, complemented
271#95C2×D711yescharacteristic, complemented
272C22×C7—22no—
273C22×C7—22no—
274C22×C7—22no—
275C22×C7—22no—
276C22×C7—22no—
277C22×C7—22no—
278C22×C7—41no—
279C22×C7—41no—
280C22×C7—41no—
281C22×C7—41no—
282C22×C7—41no—
283C22×C7—41no—
284C22×C7—41no—
285C22×C7—41no—
286C22×C7C22×D411yescharacteristic
287C22×C7C22×D411yescharacteristic
288C22×C7C22×D411yescharacteristic
289C22×C7C2×#4111yescharacteristic
290C22×C7C2×#4111yescharacteristic
291C22×C7H(5,𝔽2)11yescharacteristic
292C22×C7H(5,𝔽2)11yescharacteristic
293C2×D7—22no—
294C2×D7—22no—
295C2×D7—22no—
296C2×D7—22no—
297C2×D7—22no—
298C2×D7—22no—
299C2×D7—22no—
300C2×D7—22no—
301C2×D7—22no—
302C2×D7—22no—
303C2×D7—22no—
304C2×D7—22no—
305C2×D7—416no—
306C2×D7—41no—
307C2×D7—41no—
308C2×D7—41no—
309C2×D7—41no—
310C2×D7—41no—
311C2×D7—41no—
312C2×D7—41no—
313C2×D7—41no—
314C2×D7#12014yes—
315C4×C7—22no—
316C4×C7—22no—
317C4×C7—41no—
318C4×C7—41no—
319C4×C7—41no—
320Dic7—22no—
321Dic7—22no—
322Dic7—41no—
323Dic7—41no—
324Dic7—41no—
325C24—71no—
326C24—71no—
327C24—71no—
328C24—142no—
329C24—142no—
330C24—142no—
331C24—142no—
332C24—142no—
333C24—281no—
334C24—281no—
335C24C22×D711yescharacteristic
336C24C22×D711yescharacteristic
337#41—1432no—
338C2×Q8—142no—
339C2×Q8—142no—
340C2×Q8—142no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—41no—
344C2×D4—74no—
345C2×D4—74no—
346C2×D4—142no—
347C2×D4—142no—
348C2×D4—142no—
349C2×D4—142no—
350C2×D4—142no—
351C2×D4—142no—
352C2×D4—142no—
353C2×D4—142no—
354C2×D4—142no—
355C2×D4—142no—
356C2×D4—142no—
357C2×D4—142no—
358C2×D4—142no—
359C2×D4—281no—
360C2×D4—281no—
361C2×D4—281no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—71no—
366C22×C4—71no—
367C22×C4—74no—
368C22×C4—71no—
369C22×C4—71no—
370C22×C4—71no—
371C22×C4—142no—
372C22×C4—142no—
373C22×C4—142no—
374C22×C4—142no—
375C22×C4—142no—
376C22×C4—142no—
377C22×C4—281no—
378C22×C4—281no—
379C22×C4C22×D711yescharacteristic
380C22×C4C22×D711yescharacteristic
381C22×C4C22×D711yescharacteristic
382C22×C4C22×D711yescharacteristic
383C22×C4C22×D711yescharacteristic
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—41no—
387C22⋊C4—142no—
388C22⋊C4—142no—
389C22⋊C4—142no—
390C22⋊C4—142no—
391C22⋊C4—142no—
392C22⋊C4—142no—
393C22⋊C4—142no—
394C22⋊C4—142no—
395C22⋊C4—281no—
396C22⋊C4—281no—
397C22⋊C4—281no—
398C2×C7—42no—
399C2×C7—42no—
400C2×C7C2×#12011yescharacteristic
401C2×C7#52511yescharacteristic
402C2×C7#53511yescharacteristic
403C2×C7#54611yescharacteristic
404C2×C7#54611yescharacteristic
405C2×C7#54611yescharacteristic
406C2×C7#54611yescharacteristic
407D7—24no—
408D7—42no—
409D7—42no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—72no—
423C23—72no—
424C23—72no—
425C23—72no—
426C23—72no—
427C23—148no—
428C23—141no—
429C23—141no—
430C23—141no—
431C23—141no—
432C23—141no—
433C23—141no—
434C23—141no—
435C23—141no—
436C23—141no—
437C23—141no—
438C23—141no—
439C23—141no—
440C23—141no—
441C23—141no—
442C23—141no—
443C23—141no—
444C23—284no—
445C23—284no—
446C23—284no—
447C23—284no—
448C23—284no—
449C23—284no—
450C23#115712yescomplemented
451C23D4×D712yes—
452C23C23×D711yescharacteristic
453Q8—148no—
454D4—44no—
455D4—148no—
456D4—148no—
457D4—148no—
458D4—284no—
459D4—284no—
460D4—284no—
461C2×C4—28no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—72no—
476C2×C4—72no—
477C2×C4—148no—
478C2×C4—148no—
479C2×C4—141no—
480C2×C4—141no—
481C2×C4—141no—
482C2×C4—141no—
483C2×C4—141no—
484C2×C4—141no—
485C2×C4—141no—
486C2×C4—141no—
487C2×C4—141no—
488C2×C4—141no—
489C2×C4—141no—
490C2×C4—141no—
491C2×C4—141no—
492C2×C4—284no—
493C2×C4D4×D712yes—
494C2×C4D4×D712yes—
495C7#240411yescharacteristic, complemented, Sylow
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—41no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—74no—
511C22—142no—
512C22—142no—
513C22—142no—
514C22—142no—
515C22—142no—
516C22—142no—
517C22—142no—
518C22—142no—
519C22—142no—
520C22—142no—
521C22—142no—
522C22—142no—
523C22—2816no—
524C22—281no—
525C22—281no—
526C22—281no—
527C22—281no—
528C22—281no—
529C22—281no—
530C22—281no—
531C22—281no—
532C22C2×#115711yescharacteristic
533C22C2×D4×D711yescharacteristic
534C22C2×D4×D711yescharacteristic
535C22C2×D4×D711yescharacteristic
536C22#654811yescharacteristic
537C22D7×#4111yescharacteristic
538C22#655311yescharacteristic
539C4—22no—
540C4—22no—
541C4—41no—
542C4—41no—
543C4—41no—
544C4—142no—
545C4—142no—
546C4—281no—
547C4—281no—
548C4—281no—
549C2—42no—
550C2—42no—
551C2—144no—
552C2—282no—
553C2—282no—
554C2#9018111yescharacteristic
555C2#9023811yescharacteristic
556C2#9024111yescharacteristic
557C2D7×#12011yescharacteristic
558C2#9029811yescharacteristic
559C2#9030311yescharacteristic
560C2#9034511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#386C2
hypercenter#386D7
hypocenterC7#2404
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C22×#1157C2semidirect
2. C2×#6500C2semidirect
3. C2×#6445C2semidirect
4. C2×D7×C22⋊C4C2semidirect
5. C7×#386C2semidirect
6. #89995C2semidirect
7. #89985C2semidirect
8. #89732C2semidirect
9. #89728C2semidirect
10. #89728C2semidirect
11. #89719C2semidirect
12. #89445C2semidirect
13. #89437C2semidirect
14. #89437C2semidirect
15. #89436C2semidirect
18. C23×C4×C7C22semidirect
22. C22×C22⋊D7C22semidirect
30. C22×C4×D7C22semidirect
31. C22×Dic14C22semidirect
34. C2×C7×C22⋊C4C22semidirect
37. C2×C22⋊Dic7C22semidirect
45. C2×(C2×D7)⋊C4C22semidirect
46. C2×(C2×D7)⋊C4C22semidirect
49. C7×#95C22semidirect
50. C7×#95C22semidirect
51. C7⋊#95C22semidirect
52. C7⋊#95C22semidirect
53. C7⋊#95C22semidirect
54. C7⋊#95C22semidirect
55. C7⋊#95C22semidirect
56. C7⋊#95C22semidirect
92. C2×C22⋊D7D4semidirect
104. C2×D28D4semidirect
113. C2×C4×D7D4semidirect
144. #386D7semidirect
159. C23×C7#41semidirect
188. C22×D7#41semidirect
246. C23×C4C2×D7semidirect
263. C2×C22⋊C4C2×D7semidirect
270. #95C2×D7semidirect
271. #95C2×D7semidirect
450. C23#1157semidirect
495. C7#2404semidirect

Maximal subgroups

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

subgroupindex
C22×#11572
C2×#65002
C2×#64452
C2×D7×C22⋊C42
C7×#3862
#899952
#899852
#897322
#897282
#897192
#894452
#894372
#894362
#24047