Finite Group Databasebeta

← #13416807

Subgroups — #13416807

Order 1792.

864 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2349C211yesmaximal, characteristic
2C2×C4×#1142C211yesmaximal, characteristic
3C2×#89779C211yesmaximal, characteristic
4#11731119C211yesmaximal, characteristic, complemented
5#11729366C211yesmaximal, characteristic
6#11729366C211yesmaximal, characteristic
7#11729361C211yesmaximal, characteristic
8#11729360C211yesmaximal, characteristic
9#11727389C211yesmaximal, characteristic, complemented
10#11727385C211yesmaximal, characteristic, complemented
11#11727373C211yesmaximal, characteristic, complemented
12#11727359C211yesmaximal, characteristic, complemented
13#11727315C211yesmaximal, characteristic, complemented
14#11727295C211yesmaximal, characteristic, complemented
15#11727236C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C2×#6432C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#385—21no—
25C7×#385—21no—
26C7×#382C2211yescharacteristic
27C7×#380—21no—
28C4×C7×C4⋊C4C2211yescharacteristic
29C2×C7×#95—21no—
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C4×#1142—24no—
34#89779—24no—
35#89753—21no—
36#89751—21no—
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89701C2211yescharacteristic
46#89701C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89429—21no—
52#89418—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26372—71nomaximal, Sylow
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C22×Dic14—21no—
67C22×Dic14C2311yescharacteristic
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C4×C7×D4—41no—
73C4×C7×D4—41no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×#1142—22no—
88C2×#1142—22no—
89C2×#1142Q814yescomplemented
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×C4×Dic7—21no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6432—22no—
103#6432—41no—
104C4×Dic14—22no—
105C4×Dic14—41no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95D411yescharacteristic
111C7×#95D411yescharacteristic
112#6406—22no—
113#6406—22no—
114#6406D414yes—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121#6384—22no—
122#6384—22no—
123#6384—41no—
124#6382—22no—
125#6382—41no—
126#6381—22no—
127#6381—22no—
128#6381—22no—
129C4⋊(C7⋊C8)—22no—
130C4×C7⋊C8—22no—
131C2×#474—71no—
132C2×C4×SD16—71no—
133#2451—71no—
134#2349D711yescharacteristic
135#2097—71no—
136#2085—71no—
137#2068—71no—
138#2058—71no—
139#1942—71no—
140#1916—71no—
141#1877—71no—
142#1877—71no—
143#1864—71no—
144#1863—71no—
145#1862—71no—
146C24×C7—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4C2×D412yes—
151C2×C7×D4C2×Q812yes—
152C2×C7×D4#4112yes—
153C22×C4×C7—21no—
154C22×C4×C7—24no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—42no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—81no—
162C22×C4×C7—81no—
163C22×C4×C7—81no—
164C22×C4×C7—81no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C2×Dic14—21no—
177C2×Dic14—21no—
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—42no—
182C2×Dic14—42no—
183C2×Dic14—81no—
184C2×Dic14C2×Q812yes—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—81no—
193C7×C4⋊C4—81no—
194C7×C4⋊C4#4112yes—
195C7×C22⋊C4—24no—
196C7×C22⋊C4—42no—
197C7×C22⋊C4—42no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—21no—
204#1142—28no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—42no—
211C4⋊Dic7C2×D412yes—
212Dic7⋊C4—42no—
213Dic7⋊C4—81no—
214Dic7⋊C4—81no—
215C4×Dic7—42no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8C2×D412yes—
227C2×C7⋊C8C2×Q812yes—
228C22×SD16—71no—
229C2×C4⋊Q8—71no—
230C2×C4×Q8—71no—
231C2×C4×D4C2×D711yescharacteristic
232C2×C4×D4C2×D711yescharacteristic
233C2×C4×D4C2×D711yescharacteristic
234#474—144no—
235C4×SD16—144no—
236C2×#107—71no—
237C2×#107—71no—
238C2×#107—71no—
239C2×#106—71no—
240C2×#106—71no—
241C2×C4⋊C8—71no—
242C2×#103—71no—
243C2×#103—71no—
244C2×#103—71no—
245C2×#102—71no—
246C2×#102—71no—
247C2×#102—71no—
248C2×C4×C8—71no—
249#398—141no—
250#395—141no—
251#391—71no—
252#391—71no—
253#391—141no—
254#389—71no—
255#389—71no—
256#389C2×D711yescharacteristic
257#389C2×D711yescharacteristic
258#387—141no—
259#385—21no—
260#385—21no—
261#382C2×D711yescharacteristic
262#380—21no—
263C4×C4⋊C4C2×D711yescharacteristic
264C2×#95—21no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#336—71no—
270#336—71no—
271#336—71no—
272#336—71no—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7—22no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7C22×D411yescharacteristic
281C7×D4—22no—
282C7×D4—22no—
283C7×D4—22no—
284C7×D4C22⋊Q814yes—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—22no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—84no—
311C2×C4×C7—84no—
312C2×C4×C7—84no—
313C2×C4×C7#13512yes—
314C2×C4×C7#13712yes—
315C2×C4×C7C22×D411yescharacteristic
316C2×C4×C7C22×Q811yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×C4×C7C2×#4111yescharacteristic
319C2×C4×C7H(5,𝔽2)11yescharacteristic
320C2×C4×C72−1+411yescharacteristic
321C2×Dic7—22no—
322C2×Dic7—22no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333Dic14—22no—
334Dic14—22no—
335Dic14—41no—
336Dic14—41no—
337Dic14—84no—
338C7⋊C8—22no—
339C7⋊C8—22no—
340C7⋊C8—41no—
341C7⋊C8—41no—
342C22×Q8—71no—
343C22×Q8—141no—
344C22×D4C22×D711yescharacteristic
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—21no—
348C2×SD16—74no—
349C2×SD16—142no—
350C2×SD16—142no—
351C22×C8—71no—
352C22×C8—71no—
353C22×C8—71no—
354C22×C8—71no—
355C4⋊Q8—142no—
356C4⋊Q8—281no—
357C4×Q8—142no—
358C4×Q8—281no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—41no—
364C4×D4—41no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4C22⋊D711yescharacteristic
375C2×C4⋊C4C22⋊D711yescharacteristic
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C42—141no—
383C2×C42C22×D711yescharacteristic
384C2×C42C22×D711yescharacteristic
385C2×C42C22×D711yescharacteristic
386#107—142no—
387#107—142no—
388#107—142no—
389#106—142no—
390#106—281no—
391C4⋊C8—142no—
392#103—142no—
393#103—142no—
394#103—281no—
395#102—74no—
396#102—142no—
397#102—142no—
398C4×C8—142no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—141no—
404#95—141no—
405#95—141no—
406#95—141no—
407#95—141no—
408#95—141no—
409#95C22⋊D711yescharacteristic
410#95C22⋊D711yescharacteristic
411C22×C7—24no—
412C22×C7—42no—
413C22×C7—42no—
414C22×C7—42no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7—42no—
418C22×C7#52511yescharacteristic
419C22×C7D4211yescharacteristic
420C22×C7D4×Q811yescharacteristic
421C22×C7C2×#13511yescharacteristic
422C22×C7C2×#13711yescharacteristic
423C22×C7#57611yescharacteristic
424C22×C7#57711yescharacteristic
425C4×C7—22no—
426C4×C7—22no—
427C4×C7—41no—
428C4×C7—41no—
429C4×C7—42no—
430C4×C7—41no—
431C4×C7—41no—
432C4×C7—81no—
433C4×C7—81no—
434C4×C7C2×C22⋊Q811yescharacteristic
435C4×C7#53911yescharacteristic
436C4×C7#54711yescharacteristic
437C4×C7#55411yescharacteristic
438Dic7—42no—
439Dic7—42no—
440Dic7—81no—
441Dic7—81no—
442C24—21no—
443C2×Q8—72no—
444C2×Q8—141no—
445C2×Q8—141no—
446C2×Q8—141no—
447C2×Q8—141no—
448C2×Q8—141no—
449C2×Q8—282no—
450C2×Q8—282no—
451C2×Q8—561no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4#115712yes—
456C2×D4D4×D712yes—
457C2×D4Q8×D712yes—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—81no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—141no—
471C22×C4—141no—
472C22×C4—141no—
473C22×C4—141no—
474C22×C4C2×C22⋊D711yescharacteristic
475C22×C4C2×C22⋊D711yescharacteristic
476C22×C4C2×C22⋊D711yescharacteristic
477C22×C4C2×C22⋊D711yescharacteristic
478C22×C4C2×C22⋊D711yescharacteristic
479C22×C4C2×C22⋊D711yescharacteristic
480C22×C4C23×D711yescharacteristic
481SD16—148no—
482C2×C8—72no—
483C2×C8—72no—
484C2×C8—141no—
485C2×C8—141no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C2×C8—141no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—72no—
502C4⋊C4—81no—
503C4⋊C4—81no—
504C4⋊C4—141no—
505C4⋊C4—141no—
506C4⋊C4—141no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—282no—
510C4⋊C4—282no—
511C4⋊C4—561no—
512C4⋊C4—561no—
513C4⋊C4#115712yes—
514C22⋊C4—24no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—282no—
524C2×C7—44no—
525C2×C7#244911yescharacteristic
526C2×C7#310711yescharacteristic
527C2×C7#311811yescharacteristic
528C2×C7#326611yescharacteristic
529C2×C7#328011yescharacteristic
530C2×C7#338111yescharacteristic
531C2×C7#338511yescharacteristic
532C23—22no—
533C23—22no—
534C23—22no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C22×C22⋊D711yescharacteristic
540Q8—142no—
541Q8—142no—
542Q8—281no—
543Q8—281no—
544Q8—564no—
545D4—22no—
546D4—22no—
547D4—22no—
548D4#645914yes—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—84no—
577C2×C4—142no—
578C2×C4—142no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4—281no—
589C2×C4#651312yes—
590C2×C4#651412yes—
591C2×C4C2×#115711yescharacteristic
592C2×C4C2×#115711yescharacteristic
593C2×C4C2×D4×D711yescharacteristic
594C2×C4#654811yescharacteristic
595C2×C4C2×Q8×D711yescharacteristic
596C2×C4#655111yescharacteristic
597C8—142no—
598C8—142no—
599C8—281no—
600C8—281no—
601C7#2637211yescharacteristic, complemented, Sylow
602C22—24no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22#9048211yescharacteristic
610C22D4×C22⋊D711yescharacteristic
611C22Q8×C22⋊D711yescharacteristic
612C22C2×#651311yescharacteristic
613C22C2×#651411yescharacteristic
614C22#9052611yescharacteristic
615C22#9052711yescharacteristic
616C4—22no—
617C4—22no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—81no—
624C4—81no—
625C4—282no—
626C4—282no—
627C4—561no—
628C4—561no—
629C4#9019811yescharacteristic
630C4C2×#645911yescharacteristic
631C4#9020211yescharacteristic
632C4#9020611yescharacteristic
633C2—44no—
634C2#1173961811yescharacteristic
635C2#1173997911yescharacteristic
636C2#1173998611yescharacteristic
637C2#1174095011yescharacteristic
638C2#1174095111yescharacteristic
639C2#1174107411yescharacteristic
640C2#1174107611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2349C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2349D7
hypocenterC7#26372
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
4. #11731119C2semidirect
9. #11727389C2semidirect
10. #11727385C2semidirect
11. #11727373C2semidirect
12. #11727359C2semidirect
13. #11727315C2semidirect
14. #11727295C2semidirect
15. #11727236C2semidirect
89. C2×#1142Q8semidirect
601. C7#26372semidirect

Maximal subgroups

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

subgroupindex
C7×#23492
C2×C4×#11422
C2×#897792
#117311192
#117293662
#117293612
#117293602
#117273892
#117273852
#117273732
#117273592
#117273152
#117272952
#117272362
#263727