Finite Group Databasebeta

← #13413908

Subgroups — #13413908

Order 1792.

792 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2351C211yesmaximal, characteristic
2C2×#89803C211yesmaximal, characteristic
3C2×C4×#1144C211yesmaximal, characteristic
4#11731117C211yesmaximal, characteristic
5#11729386C211yesmaximal, characteristic
6#11729386C211yesmaximal, characteristic
7#11729386C211yesmaximal, characteristic
8#11729383C211yesmaximal, characteristic
9#11727387C211yesmaximal, characteristic
10#11727386C211yesmaximal, characteristic
11#11727376C211yesmaximal, characteristic
12#11727368C211yesmaximal, characteristic
13#11727315C211yesmaximal, characteristic
14#11727314C211yesmaximal, characteristic
15#11727236C211yesmaximal, characteristic
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C22×#1144C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C7×#389—21no—
24C7×#389—21no—
25C7×#389C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#382—21no—
28C7×#382C2211yescharacteristic
29C4×C7×C4⋊C4—21no—
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33#89803—24no—
34C4×#1144—24no—
35#89756—21no—
36#89753—21no—
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6384C2211yescharacteristic
42C2×#6384C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89701C2211yescharacteristic
46#89701C2211yescharacteristic
47#89701C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89432—21no—
52#89427—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26167—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×Dic14C2311yescharacteristic
64C22×Dic14C2311yescharacteristic
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×Q8—22no—
68C4×C7×Q8—22no—
69C4×C7×Q8—41no—
70C4×C7×Q8—41no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
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×C42×C7—21no—
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×#1144—22no—
88C2×#1144—22no—
89C2×#1144—22no—
90C2×#1144—22no—
91C2×#1144—22no—
92C2×#1144D414yes—
93C2×C4⋊Dic7—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7C2311yescharacteristic
96C2×C4⋊Dic7C2311yescharacteristic
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104#6432—22no—
105#6432—41no—
106C4×Dic14—22no—
107C4×Dic14—41no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114#6409—22no—
115#6409—22no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123#6384—22no—
124#6384—22no—
125#6384—22no—
126#6384—22no—
127#6384—22no—
128#6384—41no—
129#6382—41no—
130#6381—22no—
131C4⋊(C7⋊C8)—22no—
132C4×C7⋊C8—22no—
133C2×#462—71no—
134C2×C4×Q16—71no—
135#2446—71no—
136#2351D711yescharacteristic
137#2097—71no—
138#2086—71no—
139#2074—71no—
140#2043—71no—
141#1933—71no—
142#1916—71no—
143#1878—71no—
144#1878—71no—
145#1878—71no—
146#1865—71no—
147#1865—71no—
148C2×C7×Q8—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8C2×D412yes—
152C2×C7×Q8#4112yes—
153C2×C7×Q8#4112yes—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14—21no—
176C2×Dic14—21no—
177C2×Dic14—21no—
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14C2×D412yes—
181C7×C4⋊C4—24no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4—22no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—42no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4—42no—
192C7×C4⋊C4—42no—
193C7×C4⋊C4#4112yes—
194C42×C7—24no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—42no—
202C42×C7—42no—
203#1144—28no—
204#1144—44no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—42no—
210C4⋊Dic7—42no—
211Dic7⋊C4—42no—
212Dic7⋊C4—42no—
213C4×Dic7—42no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
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⋊C8C2×D412yes—
226C22×Q16—71no—
227C22×Q16—71no—
228C2×C4⋊Q8—71no—
229C2×C4×Q8—71no—
230C2×C4×Q8C2×D711yescharacteristic
231C2×C4×Q8C2×D711yescharacteristic
232C2×C4×Q8C2×D711yescharacteristic
233#462—144no—
234C4×Q16—144no—
235C2×#107—71no—
236C2×#106—71no—
237C2×C4⋊C8—71no—
238C2×#103—71no—
239C2×#103—71no—
240C2×#103—71no—
241C2×#103—71no—
242C2×#103—71no—
243C2×#103—71no—
244C2×#103—71no—
245C2×#103—71no—
246C2×C4×C8—71no—
247#400—141no—
248#398—141no—
249#391—71no—
250#391—71no—
251#391—71no—
252#389—21no—
253#389—21no—
254#389—71no—
255#389C2×D711yescharacteristic
256#389C2×D711yescharacteristic
257#387—141no—
258#387—141no—
259#382—21no—
260#382C2×D711yescharacteristic
261C4×C4⋊C4—21no—
262C4×C4⋊C4C2×D711yescharacteristic
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#336—71no—
268#336—71no—
269#336—71no—
270#336—71no—
271C23×C7C22×D411yescharacteristic
272C7×Q8—22no—
273C7×Q8—22no—
274C7×Q8—22no—
275C7×Q8#12314yes—
276C2×C4×C7—22no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7#13512yes—
305C2×C4×C7#13712yes—
306C2×C4×C7C22×D411yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7H(5,𝔽2)11yescharacteristic
311C2×C4×C72−1+411yescharacteristic
312C2×Dic7—22no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324Dic14—22no—
325Dic14—22no—
326Dic14—41no—
327Dic14—41no—
328Dic14—41no—
329Dic14—41no—
330Dic14—41no—
331Dic14—41no—
332C7⋊C8—22no—
333C7⋊C8—41no—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C22×Q8—71no—
337C22×Q8—71no—
338C22×Q8C22×D711yescharacteristic
339C2×Q16—74no—
340C2×Q16—142no—
341C2×Q16—142no—
342C2×Q16—142no—
343C2×Q16—142no—
344C2×Q16—142no—
345C22×C8—71no—
346C22×C8—71no—
347C22×C8—71no—
348C22×C8—71no—
349C4⋊Q8—142no—
350C4⋊Q8—281no—
351C4×Q8—22no—
352C4×Q8—22no—
353C4×Q8—22no—
354C4×Q8—22no—
355C4×Q8—41no—
356C4×Q8—41no—
357C4×Q8—142no—
358C4×Q8—281no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—71no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4C22⋊D711yescharacteristic
371C2×C4⋊C4C22⋊D711yescharacteristic
372C2×C4⋊C4C22×D711yescharacteristic
373C2×C4⋊C4C22×D711yescharacteristic
374C2×C4⋊C4C22×D711yescharacteristic
375C2×C42—21no—
376C2×C42—21no—
377C2×C42—21no—
378C2×C42—141no—
379C2×C42C22×D711yescharacteristic
380C2×C42C22×D711yescharacteristic
381C2×C42C22×D711yescharacteristic
382#107—142no—
383#106—281no—
384C4⋊C8—142no—
385#103—142no—
386#103—142no—
387#103—142no—
388#103—142no—
389#103—142no—
390#103—142no—
391#103—142no—
392#103—281no—
393C4×C8—142no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—141no—
399#95—141no—
400#95—141no—
401#95—141no—
402#95—141no—
403#95—141no—
404#95—141no—
405#95C22⋊D711yescharacteristic
406#95C22⋊D711yescharacteristic
407C22×C7#52511yescharacteristic
408C22×C7D4211yescharacteristic
409C22×C7#54811yescharacteristic
410C22×C7C2×#13511yescharacteristic
411C22×C7C2×#13711yescharacteristic
412C22×C7#57811yescharacteristic
413C22×C7#57811yescharacteristic
414C4×C7—24no—
415C4×C7—22no—
416C4×C7—22no—
417C4×C7—41no—
418C4×C7—42no—
419C4×C7—41no—
420C4×C7—41no—
421C4×C7—41no—
422C4×C7—81no—
423C4×C7—81no—
424C4×C7C2×#12311yescharacteristic
425C4×C7#53911yescharacteristic
426C4×C7#54711yescharacteristic
427C4×C7#55311yescharacteristic
428Dic7—42no—
429Dic7—81no—
430Dic7—81no—
431Dic7—81no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—72no—
436C2×Q8—141no—
437C2×Q8—141no—
438C2×Q8—141no—
439C2×Q8—141no—
440C2×Q8—141no—
441C2×Q8—141no—
442C2×Q8—141no—
443C2×Q8—141no—
444C2×Q8—141no—
445C2×Q8—141no—
446C2×Q8—141no—
447C2×Q8#115712yes—
448C2×Q8D4×D712yes—
449C2×Q8#116112yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C2×C22⋊D711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C2×C22⋊D711yescharacteristic
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C23×D711yescharacteristic
465Q16—148no—
466Q16—284no—
467C2×C8—72no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—141no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—282no—
496C4⋊C4—282no—
497C4⋊C4—282no—
498C4⋊C4—282no—
499C4⋊C4#115712yes—
500C42—24no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—42no—
509C42—282no—
510C2×C7#244311yescharacteristic
511C2×C7#310711yescharacteristic
512C2×C7#312711yescharacteristic
513C2×C7#327011yescharacteristic
514C2×C7#328511yescharacteristic
515C2×C7#336411yescharacteristic
516C2×C7#337411yescharacteristic
517C23C22×C22⋊D711yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—22no—
521Q8—142no—
522Q8—142no—
523Q8—281no—
524Q8—281no—
525Q8—281no—
526Q8—281no—
527Q8—281no—
528Q8—281no—
529Q8#645714yes—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—142no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4#651312yes—
571C2×C4#651412yes—
572C2×C4C2×#115711yescharacteristic
573C2×C4C2×#115711yescharacteristic
574C2×C4C2×D4×D711yescharacteristic
575C2×C4#654811yescharacteristic
576C2×C4C2×#116111yescharacteristic
577C2×C4#655111yescharacteristic
578C8—142no—
579C8—281no—
580C8—281no—
581C8—281no—
582C7#2616711yescharacteristic, complemented, Sylow
583C22#9048211yescharacteristic
584C22D4×C22⋊D711yescharacteristic
585C22#9050811yescharacteristic
586C22C2×#651311yescharacteristic
587C22C2×#651411yescharacteristic
588C22#9052911yescharacteristic
589C22#9052911yescharacteristic
590C4—22no—
591C4—22no—
592C4—24no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—282no—
601C4—561no—
602C4—561no—
603C4—561no—
604C4C2×#645711yescharacteristic
605C4#9019811yescharacteristic
606C4#9020511yescharacteristic
607C4#9020611yescharacteristic
608C2#1173964011yescharacteristic
609C2#1173997911yescharacteristic
610C2#1174000011yescharacteristic
611C2#1174099111yescharacteristic
612C2#1174099411yescharacteristic
613C2#1174113411yescharacteristic
614C2#1174114311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2351C2
socle, prime radicalC23×C7C22×D4
hypercenter#2351D7
hypocenterC7#26167
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
582. C7#26167semidirect

Maximal subgroups

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

subgroupindex
C7×#23512
C2×#898032
C2×C4×#11442
#117311172
#117293862
#117293832
#117273872
#117273862
#117273762
#117273682
#117273152
#117273142
#117272362
#261677