Finite Group Databasebeta

← #13416661

Subgroups — #13416661

Order 1792.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740192C211yesmaximal, characteristic
2#11739694C211yesmaximal, characteristic
3C2×#89767C211yesmaximal, characteristic
4C2×#89568C211yesmaximal, characteristic, complemented
5C7×#1873C211yesmaximal, characteristic
6#11729451C211yesmaximal, characteristic
7#11728015C211yesmaximal, characteristic, complemented
8#11728003C211yesmaximal, characteristic, complemented
9#11727989C211yesmaximal, characteristic, complemented
10#11727705C211yesmaximal, characteristic
11#11727681C211yesmaximal, characteristic
12#11726583C211yesmaximal, characteristic, complemented
13#11726571C211yesmaximal, characteristic, complemented
14#11726561C211yesmaximal, characteristic, complemented
15#11726549C211yesmaximal, characteristic, complemented
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C2×#6453C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C7×#103C2211yescharacteristic
24C2×C7×C22⋊C8C2211yescharacteristic
25#90002C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28#89984C2211yescharacteristic
29C4×C22⋊Dic7C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6392C2211yescharacteristic
32C2×Dic7⋊C8C2211yescharacteristic
33C2×Dic7⋊C8C2211yescharacteristic
34#89767—24no—
35#89757—21no—
36#89757C2211yescharacteristic
37#89757C2211yescharacteristic
38#89753—21no—
39#89748C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41#89745C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89722—21no—
47#89721—21no—
48#89720—21no—
49C22⋊C4×Dic7—21no—
50#89705—21no—
51#89700—21no—
52#89568—24no—
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89345C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#26362—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C23×Dic7—21no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×C22⋊Q8—22no—
68C7×C22⋊Q8Q814yes—
69C7×#117—22no—
70C7×#117—41no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C42×C7—21no—
78C2×C22⋊Dic7—21no—
79C2×C22⋊Dic7D411yescharacteristic
80C2×C22⋊Dic7D411yescharacteristic
81Q8×Dic7—22no—
82Q8×Dic7—41no—
83C2×C4⋊Dic7—22no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—41no—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C2×C4×Dic7C2311yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C2×#1136—22no—
103C2×#1136—22no—
104C2×#1136—22no—
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107#6453—22no—
108#6453—41no—
109C7×#103—22no—
110C7×#103—41no—
111C7×C22⋊C8—22no—
112#6409—22no—
113#6409—22no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95D411yescharacteristic
122C7⋊#95D411yescharacteristic
123#6392—22no—
124#6392—41no—
125Dic7⋊C8—22no—
126Dic7⋊C8—22no—
127#6382—22no—
128#6382D414yes—
129#6381—22no—
130#6381—22no—
131C2×#479—71no—
132C2×#428—71no—
133#2441—71no—
134#2360—71no—
135#2095—71no—
136#2086—71no—
137#2066—71no—
138#2059—71no—
139#1947—71no—
140#1919—71no—
141#1894—71no—
142#1873D711yescharacteristic
143#1863—71no—
144#1852—71no—
145#1851—71no—
146C24×C7C2×D411yescharacteristic
147C2×C7×Q8—21no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8C2×Q812yes—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×Q812yes—
163C22×C4×C7#4112yes—
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—42no—
170C22×Dic7—42no—
171C22×Dic7—81no—
172C22×Dic7C2×D411yescharacteristic
173C22×Dic7C2×D411yescharacteristic
174C22×Dic7C2×D411yescharacteristic
175C22×Dic7C2×D411yescharacteristic
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C7×C4⋊C4—24no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4C2×D412yes—
188C7×C4⋊C4C2×Q812yes—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—42no—
191C42×C7—42no—
192C22⋊Dic7—22no—
193C22⋊Dic7—41no—
194C22⋊Dic7—81no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—42no—
203C4⋊Dic7—42no—
204C4⋊Dic7—42no—
205C4⋊Dic7—42no—
206C4⋊Dic7—81no—
207C4⋊Dic7C2×D412yes—
208Dic7⋊C4—42no—
209Dic7⋊C4—42no—
210Dic7⋊C4—81no—
211Dic7⋊C4—81no—
212C4×Dic7—21no—
213C4×Dic7—21no—
214C4×Dic7—21no—
215C4×Dic7—21no—
216C4×Dic7—42no—
217C4×Dic7—81no—
218#1136—28no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8C2×D412yes—
224C2×C7⋊C8#4112yes—
225C22×M16—71no—
226C2×#125—71no—
227C2×C22⋊Q8C2×D711yescharacteristic
228C2×C4×Q8—71no—
229C2×#117C2×D711yescharacteristic
230C22×C4⋊C4—71no—
231#479—144no—
232#428—144no—
233C2×#107—71no—
234C2×#107—71no—
235C2×#107—71no—
236C2×#107—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×C4⋊C8—71no—
240C2×C4⋊C8—71no—
241C2×#103—71no—
242C2×#103—71no—
243C2×#103C2×D711yescharacteristic
244C2×#103C2×D711yescharacteristic
245C2×C22⋊C8C2×D711yescharacteristic
246#399—141no—
247#396—141no—
248#391—71no—
249#389—71no—
250#389—71no—
251#389—71no—
252#389—141no—
253#389—141no—
254#387—141no—
255#387—141no—
256#385—141no—
257#384—71no—
258#380—71no—
259C4×C4⋊C4—71no—
260C4×C22⋊C4—71no—
261C4×C22⋊C4—141no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340C2×D711yescharacteristic
267#340C2×D711yescharacteristic
268#336—71no—
269#336—71no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7C22⋊Q812yes—
277C23×C7C22×D411yescharacteristic
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—41no—
281C7×Q8—41no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7C22⋊Q812yes—
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C22×Q811yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7H(5,𝔽2)11yescharacteristic
306C2×C4×C72−1+411yescharacteristic
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—22no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—21no—
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—84no—
331C2×Dic7#13712yes—
332C7×C8—41no—
333C7×C8—41no—
334C7⋊C8—22no—
335C7⋊C8—22no—
336C22×Q8C22×D711yescharacteristic
337C23×C4—141no—
338C23×C4C22×D711yescharacteristic
339C2×M16—142no—
340C2×M16—142no—
341C2×M16—142no—
342C22×C8—71no—
343C22×C8—71no—
344C22×C8C22×D711yescharacteristic
345C22×C8C22×D711yescharacteristic
346#125—142no—
347#125—281no—
348C22⋊Q8—22no—
349C22⋊Q8Dic1414yes—
350C4×Q8—142no—
351C4×Q8—281no—
352#117—22no—
353#117—41no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—142no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—281no—
370C2×C4⋊C4C22×D711yescharacteristic
371C2×C4⋊C4C22×D711yescharacteristic
372C2×C4⋊C4C22×D711yescharacteristic
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—71no—
376C2×C22⋊C4—71no—
377C2×C22⋊C4—141no—
378C2×C42—21no—
379C2×C42—71no—
380C2×C42—71no—
381C2×C42—141no—
382C2×C42—141no—
383#107—142no—
384#107—142no—
385#107—142no—
386#107—281no—
387#106—74no—
388#106—142no—
389C4⋊C8—142no—
390C4⋊C8—142no—
391#103—22no—
392#103—41no—
393#103—142no—
394#103—142no—
395C22⋊C8—22no—
396#95—71no—
397#95—71no—
398#95—141no—
399#95—141no—
400#95—141no—
401#95—141no—
402#95—141no—
403#95—141no—
404#95—141no—
405C22×C7—22no—
406C22×C7—22no—
407C22×C7—22no—
408C22×C7—41no—
409C22×C7—41no—
410C22×C7—41no—
411C22×C7—41no—
412C22×C7C2×C22⋊Q811yescharacteristic
413C22×C7D4211yescharacteristic
414C22×C7#54711yescharacteristic
415C22×C7C2×#13711yescharacteristic
416C22×C7#57511yescharacteristic
417C22×C7#57611yescharacteristic
418C22×C7#57811yescharacteristic
419C4×C7—22no—
420C4×C7—42no—
421C4×C7—42no—
422C4×C7—42no—
423C4×C7—81no—
424C4×C7#52511yescharacteristic
425C4×C7#53911yescharacteristic
426C4×C7D4×Q811yescharacteristic
427C4×C7#55411yescharacteristic
428Dic7—22no—
429Dic7—41no—
430Dic7—41no—
431Dic7—42no—
432Dic7—41no—
433Dic7—81no—
434Dic7—81no—
435Dic7—81no—
436C24D4×D711yescharacteristic
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8C2×Dic1412yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—71no—
454C22×C4—71no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—141no—
459C22×C4—282no—
460C22×C4—282no—
461C22×C4—561no—
462C22×C4C2×Dic1412yes—
463C22×C4D4×D711yescharacteristic
464C22×C4#115912yes—
465C22×C4C23×D711yescharacteristic
466M16—148no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—72no—
474C2×C8—72no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—141no—
479C4⋊C4—24no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—72no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—282no—
496C4⋊C4—282no—
497C4⋊C4—282no—
498C4⋊C4—561no—
499C4⋊C4—561no—
500C4⋊C4—561no—
501C4⋊C4C2×Dic1412yes—
502C4⋊C4D4×D712yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—142no—
506C22⋊C4—281no—
507C22⋊C4—561no—
508C42—42no—
509C42—141no—
510C42—141no—
511C42—141no—
512C42—141no—
513C42—282no—
514C42—561no—
515C2×C7—42no—
516C2×C7#244911yescharacteristic
517C2×C7#307811yescharacteristic
518C2×C7#308311yescharacteristic
519C2×C7#326811yescharacteristic
520C2×C7#327611yescharacteristic
521C2×C7#334611yescharacteristic
522C2×C7#335211yescharacteristic
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23#644112yes—
530C23C2×D4×D711yescharacteristic
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—72no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4—141no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—141no—
564C2×C4—142no—
565C2×C4—141no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—564no—
578C2×C4#644112yes—
579C2×C4C22×Dic1411yescharacteristic
580C2×C4C2×D4×D711yescharacteristic
581C2×C4C2×#115911yescharacteristic
582C2×C4#654811yescharacteristic
583C2×C4D7×#4111yescharacteristic
584C2×C4#655411yescharacteristic
585C8—41no—
586C8—41no—
587C8—142no—
588C8—142no—
589C7#2636211yescharacteristic, complemented, Sylow
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#644111yescharacteristic
598C22#9028111yescharacteristic
599C22#9034611yescharacteristic
600C22#9046011yescharacteristic
601C22#9046211yescharacteristic
602C22D7×#13711yescharacteristic
603C22#9046811yescharacteristic
604C4—22no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—81no—
609C4—142no—
610C4—281no—
611C4—281no—
612C4—282no—
613C4—281no—
614C4—561no—
615C4—561no—
616C4—561no—
617C4D4×Dic1411yescharacteristic
618C4#9023011yescharacteristic
619C4#9029311yescharacteristic
620C4#9029711yescharacteristic
621C2—42no—
622C2#1173215411yescharacteristic
623C2#1173626811yescharacteristic
624C2#1173639411yescharacteristic
625C2#1173771111yescharacteristic
626C2#1173771911yescharacteristic
627C2#1173895611yescharacteristic
628C2#1173898011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1873C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1873D7
hypocenterC7#26362
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
4. C2×#89568C2semidirect
7. #11728015C2semidirect
8. #11728003C2semidirect
9. #11727989C2semidirect
12. #11726583C2semidirect
13. #11726571C2semidirect
14. #11726561C2semidirect
15. #11726549C2semidirect
589. C7#26362semidirect

Maximal subgroups

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

subgroupindex
#117401922
#117396942
C2×#897672
C2×#895682
C7×#18732
#117294512
#117280152
#117280032
#117279892
#117277052
#117276812
#117265832
#117265712
#117265612
#117265492
#263627