Finite Group Databasebeta

← #13323840

Subgroups — #13323840

Order 1792.

692 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737038C211yesmaximal, characteristic
2#11736983C211yesmaximal, characteristic, complemented
3C8×#1159C211yesmaximal, characteristic, complemented
4C7×#1581C211yesmaximal, characteristic, complemented
5#11728632C211yesmaximal, characteristic, complemented
6#11728618C211yesmaximal, characteristic, complemented
7#11728595C211yesmaximal, characteristic, complemented
8#11726329C211yesmaximal, characteristic, complemented
9#11726312C211yesmaximal, characteristic, complemented
10#11725928C211yesmaximal, characteristic, complemented
11#11725925C211yesmaximal, characteristic, complemented
12#11725905C211yesmaximal, characteristic, complemented
13#11725904C211yesmaximal, characteristic, complemented
14#11725839C211yesmaximal, characteristic, complemented
15#11725823C211yesmaximal, characteristic, complemented
16C2×#6470—21no—
17#90239C2211yescharacteristic
18C4×#1159C2211yescharacteristic
19C4×C7×D8C2211yescharacteristic
20C7×#436C2211yescharacteristic
21C7×C8×D4C2211yescharacteristic
22#89930C411yescharacteristic
23#89922C411yescharacteristic
24#89900—21no—
25C8×C22⋊D7—21no—
26C2×C8×Dic7—21no—
27C4×#1142C2211yescharacteristic
28C4×#1141C2211yescharacteristic
29D4×C7⋊C8C2211yescharacteristic
30D4×C7⋊C8C2211yescharacteristic
31#89657C411yescharacteristic
32#89647C411yescharacteristic
33#89609C2211yescharacteristic
34#89606C2211yescharacteristic
35#89605C2211yescharacteristic
36#89602C2211yescharacteristic
37#89553C411yescharacteristic
38#89551C411yescharacteristic
39#89534C411yescharacteristic
40#89533C411yescharacteristic
41#89499—21no—
42(C22⋊D7)⋊C8—21no—
43(C22⋊D7)⋊C8—21no—
44#89490—21no—
45C4×#1132C2211yescharacteristic
46#89460C2211yescharacteristic
47C4×#1131C2211yescharacteristic
48#89457C2211yescharacteristic
49C4×C8×D7C2211yescharacteristic
50C8×Dic14C2211yescharacteristic
51C7×#335C2211yescharacteristic
52C7×D4⋊C8C2211yescharacteristic
53C7×D4⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic
55D4⋊(C7⋊C8)C2211yescharacteristic
56D4⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89278C2211yescharacteristic
59#89275C2211yescharacteristic
60#89274C2211yescharacteristic
61#89254C2211yescharacteristic
62#89252C2211yescharacteristic
63#89251C2211yescharacteristic
64#89250C2211yescharacteristic
65#89250C2211yescharacteristic
66#89249C2211yescharacteristic
67#17917—71nomaximal, Sylow
68C2×#1159C2×C411yescharacteristic
69C2×#1157—21no—
70C2×C7×D8C2×C411yescharacteristic
71C2×C7×M16—21no—
72C22×C7×C8—21no—
73C4×C7×D4C2311yescharacteristic
74C4×C7×D4C2311yescharacteristic
75#6503C2×C411yescharacteristic
76#6501C2×C411yescharacteristic
77D4×Dic7C2×C411yescharacteristic
78C2×#1142C2×C411yescharacteristic
79C2×#1141C2×C411yescharacteristic
80C4×C22⋊D7—21no—
81C4×C22⋊D7—21no—
82C2×C4×Dic7—21no—
83C22×C7⋊C8—21no—
84C22×C7⋊C8—21no—
85#6470—44no—
86C2×#1132C2×C411yescharacteristic
87C2×#1131—21no—
88C2×#1131—21no—
89C2×#1131—21no—
90C2×C8×D7—21no—
91C2×C8×D7D411yescharacteristic
92C2×C8×D7D411yescharacteristic
93#6458C2×C411yescharacteristic
94#6455C2×C411yescharacteristic
95#6451C2×C411yescharacteristic
96#6450C2×C411yescharacteristic
97#6448—21no—
98#6447—21no—
99(C22⋊D7)⋊C4—21no—
100#6440—21no—
101C4×D28C2311yescharacteristic
102C42×D7C2311yescharacteristic
103C4×Dic14C2311yescharacteristic
104C7×#107C2×C411yescharacteristic
105C7×C4⋊C8C2311yescharacteristic
106C7×C4⋊C8C2311yescharacteristic
107C7×#102C2×C411yescharacteristic
108C7×#102C2×C411yescharacteristic
109C7×C22⋊C8—21no—
110C7×C22⋊C8—21no—
111C4×C7×C8C2311yescharacteristic
112C4×C7×C8C2311yescharacteristic
113C4×C7×C8C2311yescharacteristic
114#6406C2×C411yescharacteristic
115#6406C2×C411yescharacteristic
116C22⋊(C7⋊C8)—21no—
117C22⋊(C7⋊C8)—21no—
118#6395C2×C411yescharacteristic
119(C2×D7)⋊C8—21no—
120(C2×D7)⋊C8—21no—
121(C2×D7)⋊C8—21no—
122(C2×D7)⋊C8C2×C411yescharacteristic
123(C2×D7)⋊C8C2×C411yescharacteristic
124#6391C2×C411yescharacteristic
125#6390C2×C411yescharacteristic
126#6389—21no—
127Dic7⋊C8—21no—
128Dic7⋊C8—21no—
129Dic7⋊C8—21no—
130Dic7⋊C8C2×C411yescharacteristic
131Dic7⋊C8C2×C411yescharacteristic
132C8×Dic7—21no—
133C8×Dic7—21no—
134C8×Dic7—21no—
135C8×Dic7D411yescharacteristic
136C8×Dic7D411yescharacteristic
137#6384C2×C411yescharacteristic
138#6383C2×C411yescharacteristic
139#6382C2×C411yescharacteristic
140#6381C2×C411yescharacteristic
141C4⋊(C7⋊C8)C2311yescharacteristic
142C4⋊(C7⋊C8)C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145C4×C7⋊C8C2311yescharacteristic
146#3106—71no—
147C8×#41—71no—
148#2931—71no—
149#1583—71no—
150#1583—71no—
151#1581—71no—
152#1581D711yescharacteristic, complemented
153#1580—71no—
154#1580—71no—
155SD16⋊C8—71no—
156SD16⋊C8—71no—
157#1574—71no—
158#1489—71no—
159#1484—71no—
160#1451—71no—
161C2×C7×D4C22×C411yescharacteristic
162C2×C7×D4C22×C411yescharacteristic
163C22×C4×C7—21no—
164C22×C4×C7—21no—
165C2×C22⋊D7—21no—
166C2×C22⋊D7—21no—
167C2×C22⋊D7—21no—
168C2×C22⋊D7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171#1159—22no—
172#1159—22no—
173#1157—42no—
174#1157—42no—
175C2×D28—21no—
176C2×D28C22×C411yescharacteristic
177C2×C4×D7—21no—
178C2×C4×D7C22×C411yescharacteristic
179C2×C4×D7C2×D411yescharacteristic
180C2×C4×D7#4111yescharacteristic
181C2×Dic14—21no—
182C2×Dic14C22×C411yescharacteristic
183C7×D8—22no—
184C7×M16—42no—
185C2×C7×C8—21no—
186C2×C7×C8—21no—
187C2×C7×C8—21no—
188C2×C7×C8—42no—
189C2×C7×C8C22×C411yescharacteristic
190C2×C7×C8C22×C411yescharacteristic
191C2×C7×C8C2×D411yescharacteristic
192C2×C7×C8C2×D411yescharacteristic
193C7×C4⋊C4C22×C411yescharacteristic
194C7×C4⋊C4C22×C411yescharacteristic
195C7×C22⋊C4—21no—
196C7×C22⋊C4—21no—
197C42×C7C2411yescharacteristic
198C22⋊Dic7—21no—
199C22⋊Dic7—21no—
200#1142—22no—
201#1141—22no—
202(C2×D7)⋊C4—21no—
203(C2×D7)⋊C4—21no—
204(C2×D7)⋊C4—21no—
205C4⋊Dic7C22×C411yescharacteristic
206C4⋊Dic7C22×C411yescharacteristic
207Dic7⋊C4—21no—
208Dic7⋊C4—21no—
209Dic7⋊C4—21no—
210C4×Dic7—21no—
211C4×Dic7—21no—
212C4×Dic7—21no—
213C4×Dic7C22×C411yescharacteristic
214C4×Dic7C2×D411yescharacteristic
215C4×Dic7#4111yescharacteristic
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—42no—
220C2×C7⋊C8—42no—
221C2×C7⋊C8C22×C411yescharacteristic
222C2×C7⋊C8C22×C411yescharacteristic
223C2×C7⋊C8C2×D411yescharacteristic
224C2×C7⋊C8C2×D411yescharacteristic
225#1132—22no—
226#1131—42no—
227#1131—42no—
228#1131—42no—
229C8×D7—22no—
230C8×D7—41no—
231C8×D7—42no—
232C2×#131—141no—
233#532—71no—
234C4×#41—71no—
235#482—71no—
236#482—71no—
237#477—71no—
238#476—71no—
239#460—71no—
240#459—71no—
241#449—71no—
242#449—71no—
243C8×Q8—71no—
244C8×Q8—71no—
245C4×SD16—71no—
246C4×SD16—71no—
247C4×D8—71no—
248C4×D8C2×D711yescharacteristic
249#436—71no—
250#436—141no—
251#436C2×D711yescharacteristic
252C8×D4—71no—
253C8×D4—71no—
254C8×D4—71no—
255C8×D4—141no—
256C8×D4—141no—
257C8×D4—141no—
258C8×D4C2×D711yescharacteristic
259#431—71no—
260#431—71no—
261#431—141no—
262#423—71no—
263#408—141no—
264C4×M16—71no—
265C2×C4×C8—71no—
266C2×C4×C8—141no—
267#335—71no—
268#335C2×D711yescharacteristic
269#334—71no—
270#334—71no—
271Q8⋊C8—71no—
272Q8⋊C8—71no—
273D4⋊C8—71no—
274D4⋊C8—71no—
275D4⋊C8—71no—
276D4⋊C8—71no—
277D4⋊C8C2×D711yescharacteristic
278D4⋊C8C2×D711yescharacteristic
279#322—71no—
280#322—71no—
281#322—71no—
282#322C2×D711yescharacteristic
283C23×C7—21no—
284C23×C7—21no—
285C22×D7—21no—
286C22×D7C4×D411yescharacteristic
287C7×D4—21no—
288C7×D4—21no—
289C7×D4—21no—
290C7×D4#13112yescomplemented
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7C23×C411yescharacteristic
298C2×C4×C7C22×D411yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C22⋊D7—24no—
301C22⋊D7—42no—
302C22⋊D7—42no—
303C22⋊D7—42no—
304C2×Dic7—24no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—42no—
309C2×Dic7C4×D411yescharacteristic
310C2×Dic7C4×D411yescharacteristic
311C2×Dic7C4×D411yescharacteristic
312D28—21no—
313D28—21no—
314D28—42no—
315C4×D7—21no—
316C4×D7—21no—
317C4×D7—21no—
318C4×D7—21no—
319C4×D7—22no—
320C4×D7—41no—
321C4×D7—41no—
322Dic14—21no—
323Dic14—42no—
324Dic14#13112yescomplemented
325C7×C8—21no—
326C7×C8—21no—
327C7×C8—21no—
328C7×C8—41no—
329C7×C8—41no—
330C7×C8—41no—
331C7×C8AGL(1,ℤ/8)12yes—
332C7⋊C8—21no—
333C7⋊C8—21no—
334C7⋊C8—21no—
335C7⋊C8—21no—
336C7⋊C8—41no—
337C7⋊C8—41no—
338C7⋊C8—41no—
339C2×#41—71no—
340C2×#41—141no—
341C2×SD16—71no—
342C2×SD16—71no—
343C2×D8—71no—
344C2×D8C4×D711yescharacteristic
345#131—284no—
346C2×M16—21no—
347C2×M16—141no—
348C2×M16—141no—
349C2×M16—141no—
350C22×C8—21no—
351C22×C8—71no—
352C22×C8—71no—
353C22×C8—141no—
354C22×C8—141no—
355C22×C8—141no—
356C4⋊Q8—71no—
357#127—71no—
358#124—141no—
359#121—71no—
360#121—71no—
361#121—141no—
362C4×Q8—71no—
363C4×Q8—71no—
364C4×D4—71no—
365C4×D4—71no—
366C4×D4—141no—
367C4×D4—141no—
368C4×D4—141no—
369C4×D4C22×D711yescharacteristic
370C4×D4C22×D711yescharacteristic
371#117—71no—
372#117—141no—
373C2×C42—71no—
374C2×C42—141no—
375#107—71no—
376#107C4×D711yescharacteristic
377#106—71no—
378#106—71no—
379C4⋊C8—71no—
380C4⋊C8—71no—
381C4⋊C8—71no—
382C4⋊C8—71no—
383C4⋊C8—141no—
384C4⋊C8—141no—
385C4⋊C8—141no—
386C4⋊C8C22×D711yescharacteristic
387C4⋊C8C22×D711yescharacteristic
388#103—71no—
389#103—71no—
390#102—71no—
391#102—71no—
392#102—71no—
393#102—71no—
394#102C4×D711yescharacteristic
395#102C4×D711yescharacteristic
396C22⋊C8—21no—
397C22⋊C8—21no—
398C22⋊C8—71no—
399C22⋊C8—71no—
400C22⋊C8—141no—
401C22⋊C8—141no—
402C22⋊C8—141no—
403C22⋊C8—141no—
404C22⋊C8—141no—
405#97—141no—
406C4×C8—71no—
407C4×C8—71no—
408C4×C8—71no—
409C4×C8—71no—
410C4×C8—71no—
411C4×C8—141no—
412C4×C8—141no—
413C4×C8—141no—
414C4×C8C22×D711yescharacteristic
415C4×C8C22×D711yescharacteristic
416C4×C8C22×D711yescharacteristic
417C22×C7—22no—
418C22×C7—41no—
419C22×C7—41no—
420C22×C7—41no—
421C22×C7—41no—
422C22×C7—41no—
423C22×C7C2×C4×D411yescharacteristic
424C2×D7—21no—
425C2×D7—21no—
426C2×D7—21no—
427C2×D7—41no—
428C2×D7—41no—
429C2×D7—41no—
430C4×C7—21no—
431C4×C7—41no—
432C4×C7—41no—
433C4×C7C2×#13111yescharacteristic
434C4×C7C2×#13111yescharacteristic
435C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
436C4×C7#57511yescharacteristic
437Dic7—22no—
438Dic7—21no—
439Dic7—21no—
440Dic7—41no—
441Dic7—41no—
442Dic7#44412yes—
443#41—142no—
444#41—142no—
445#41—282no—
446#41—282no—
447C2×Q8—71no—
448C2×Q8—141no—
449C2×D4—71no—
450C2×D4—141no—
451C2×D4—141no—
452C2×D4—141no—
453C2×D4—141no—
454C2×D4—141no—
455C2×D4C2×C4×D711yescharacteristic
456C2×D4C2×C4×D711yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—71no—
460C22×C4—71no—
461C22×C4—71no—
462C22×C4—141no—
463C22×C4—141no—
464C22×C4—141no—
465SD16—142no—
466SD16—142no—
467D8—22no—
468D8—142no—
469M16—42no—
470M16—282no—
471M16—282no—
472M16—282no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—42no—
477C2×C8—71no—
478C2×C8—71no—
479C2×C8—71no—
480C2×C8—71no—
481C2×C8—142no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—281no—
486C2×C8—282no—
487C2×C8—282no—
488C2×C8—282no—
489C2×C8C2×C4×D711yescharacteristic
490C2×C8C2×C4×D711yescharacteristic
491C2×C8D4×D711yescharacteristic
492C2×C8D4×D711yescharacteristic
493C4⋊C4—71no—
494C4⋊C4—71no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4C2×C4×D711yescharacteristic
499C4⋊C4C2×C4×D711yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—141no—
503C22⋊C4—141no—
504C22⋊C4—141no—
505C22⋊C4—141no—
506C22⋊C4—141no—
507C42—71no—
508C42—71no—
509C42—71no—
510C42—141no—
511C42—141no—
512C42—141no—
513C42C23×D711yescharacteristic
514C2×C7—42no—
515C2×C7—81no—
516C2×C7#292911yescharacteristic
517C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
518C2×C7C2×#44411yescharacteristic
519D7—41no—
520D7—81no—
521C23—21no—
522C23—21no—
523C23—71no—
524C23—141no—
525Q8—72no—
526Q8—141no—
527Q8—282no—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—144no—
532D4—141no—
533D4—141no—
534D4—282no—
535D4—282no—
536D4—282no—
537D4—282no—
538D4#647012yescomplemented
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—71no—
546C2×C4—71no—
547C2×C4—71no—
548C2×C4—142no—
549C2×C4—144no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—282no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4C22×C4×D711yescharacteristic
561C2×C4C2×D4×D711yescharacteristic
562C2×C4D7×#4111yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—41no—
569C8—141no—
570C8—141no—
571C8—141no—
572C8—141no—
573C8—281no—
574C8—281no—
575C8—281no—
576C8#647412yes—
577C7#1791711yescharacteristic, complemented, Sylow
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—141no—
585C22—141no—
586C22—141no—
587C22—281no—
588C22—281no—
589C22—281no—
590C22C4×D4×D711yescharacteristic
591C4—21no—
592C4—41no—
593C4—41no—
594C4—72no—
595C4—141no—
596C4—142no—
597C4—141no—
598C4—281no—
599C4—281no—
600C4C2×#647011yescharacteristic
601C4D7×#13111yescharacteristic
602C4C2×#647411yescharacteristic
603C4#9045911yescharacteristic
604C2—42no—
605C2—81no—
606C2—281no—
607C2—561no—
608C2#1173692311yescharacteristic
609C2C4×#647411yescharacteristic
610C2D7×#44411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1581C2
socleC22×C7C2×C4×D4
prime radical#11737038C2
cosocleC42×C7C24
hypercenter#1581D7
hypocenterC7#17917
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1581

Complements

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

NH ≅ G/Ntype
2. #11736983C2semidirect
3. C8×#1159C2semidirect
4. C7×#1581C2semidirect
5. #11728632C2semidirect
6. #11728618C2semidirect
7. #11728595C2semidirect
8. #11726329C2semidirect
9. #11726312C2semidirect
10. #11725928C2semidirect
11. #11725925C2semidirect
12. #11725905C2semidirect
13. #11725904C2semidirect
14. #11725839C2semidirect
15. #11725823C2semidirect
152. #1581D7semidirect
290. C7×D4#131semidirect
324. Dic14#131semidirect
538. D4#6470semidirect
577. C7#17917semidirect

Maximal subgroups

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

subgroupindex
#117370382
#117369832
C8×#11592
C7×#15812
#117286322
#117286182
#117285952
#117263292
#117263122
#117259282
#117259252
#117259052
#117259042
#117258392
#117258232
#179177