Finite Group Databasebeta

← #13413938

Subgroups — #13413938

Order 1792.

826 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#89800C211yesmaximal, characteristic
3C2×C4×#1144C211yesmaximal, characteristic, complemented
4#11731118C211yesmaximal, characteristic
5#11729387C211yesmaximal, characteristic
6#11729387C211yesmaximal, characteristic
7#11729386C211yesmaximal, characteristic, complemented
8#11729382C211yesmaximal, characteristic, complemented
9#11727385C211yesmaximal, characteristic, complemented
10#11727382C211yesmaximal, characteristic
11#11727376C211yesmaximal, characteristic, complemented
12#11727369C211yesmaximal, characteristic
13#11727343C211yesmaximal, characteristic, complemented
14#11727340C211yesmaximal, characteristic
15#11727236C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×Q8C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C22×#1143C2211yescharacteristic
21C2×#6438C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C7×#389—21no—
24C7×#389—21no—
25C7×#389C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#384—21no—
28C7×#384C2211yescharacteristic
29C4×C7×C4⋊C4—21no—
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C4×#1144—24no—
34#89800—24no—
35#89764—21no—
36#89762—21no—
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6383C2211yescharacteristic
41C2×#6383C2211yescharacteristic
42C2×#6383C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89711C2211yescharacteristic
46#89711C2211yescharacteristic
47#89701C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89447—21no—
52#89441—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26169—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C22×D28C2311yescharacteristic
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⋊C4—21no—
77C2×C7×C4⋊C4—21no—
78C2×C7×C4⋊C4D411yescharacteristic
79C2×C7×C4⋊C4D411yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C4⋊C4C2311yescharacteristic
82C2×C7×C4⋊C4C2311yescharacteristic
83C2×C42×C7—21no—
84C2×C42×C7—21no—
85C2×C42×C7—21no—
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×C42×C7C2311yescharacteristic
89C2×#1144—22no—
90C2×#1144—22no—
91C2×#1144D414yes—
92C2×#1143—22no—
93C2×#1143—22no—
94C2×#1143—22no—
95C2×(C2×D7)⋊C4—21no—
96C2×(C2×D7)⋊C4—21no—
97C2×(C2×D7)⋊C4—21no—
98C2×(C2×D7)⋊C4—21no—
99C2×(C2×D7)⋊C4—21no—
100C2×(C2×D7)⋊C4—21no—
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4⋊Dic7C2311yescharacteristic
103C2×Dic7⋊C4—21no—
104C2×Dic7⋊C4—21no—
105C2×C4×Dic7—21no—
106C22×C7⋊C8C2311yescharacteristic
107C22×C7⋊C8C2311yescharacteristic
108C22×C7⋊C8C2311yescharacteristic
109C22×C7⋊C8C2311yescharacteristic
110#6438—22no—
111#6438—41no—
112C4×Dic14—22no—
113C4×Dic14—41no—
114C7×#95—21no—
115C7×#95—21no—
116C7×#95D411yescharacteristic
117C7×#95D411yescharacteristic
118#6409—22no—
119#6409—22no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123#6384—22no—
124#6384—22no—
125#6384—22no—
126#6383—22no—
127#6383—22no—
128#6383—41no—
129#6381—22no—
130#6381—41no—
131C4⋊(C7⋊C8)—22no—
132C4×C7⋊C8—22no—
133C2×#464—71no—
134C2×C4×Q16—71no—
135#2448—71no—
136Q8×C4⋊C4D711yescharacteristic, complemented
137#2093—71no—
138#2086—71no—
139#2072—71no—
140#2042—71no—
141#1934—71no—
142#1916—71no—
143#1880—71no—
144#1880—71no—
145#1878—71no—
146#1863—71no—
147#1863—71no—
148C23×D7—21no—
149C2×C7×Q8—21no—
150C2×C7×Q8—21no—
151C2×C7×Q8—21no—
152C2×C7×Q8C2×D412yes—
153C2×C7×Q8#4112yes—
154C2×C7×Q8#4112yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2411yescharacteristic
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C2×D28—21no—
170C2×D28—21no—
171C2×D28—21no—
172C2×D28—21no—
173C2×D28—21no—
174C2×D28—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—81no—
193C7×C4⋊C4—81no—
194C7×C4⋊C4—81no—
195C7×C4⋊C4#4112yes—
196C42×C7—24no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—42no—
204C42×C7—42no—
205#1144—28no—
206#1143—44no—
207(C2×D7)⋊C4—42no—
208(C2×D7)⋊C4—42no—
209(C2×D7)⋊C4—42no—
210(C2×D7)⋊C4—42no—
211(C2×D7)⋊C4—81no—
212(C2×D7)⋊C4—81no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—21no—
217Dic7⋊C4—42no—
218Dic7⋊C4—81no—
219C4×Dic7—42no—
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⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8C2×D412yes—
232C22×Q16—71no—
233C22×SD16—71no—
234C2×#124—71no—
235C2×C4×Q8—71no—
236C2×C4×Q8C2×D711yescharacteristic
237C2×C4×Q8C2×D711yescharacteristic
238C2×C4×Q8C2×D711yescharacteristic
239#464—144no—
240C4×Q16—144no—
241C2×#107—71no—
242C2×#107—71no—
243C2×C4⋊C8—71no—
244C2×#103—71no—
245C2×#103—71no—
246C2×#103—71no—
247C2×#103—71no—
248C2×#103—71no—
249C2×#102—71no—
250C2×#102—71no—
251C2×#102—71no—
252C2×C4×C8—71no—
253#399—141no—
254#397—141no—
255#391—71no—
256#390—71no—
257#390—71no—
258#389—21no—
259#389—21no—
260#389—71no—
261#389C2×D711yescharacteristic
262#389C2×D711yescharacteristic
263#388—141no—
264#388—141no—
265#384—21no—
266#384C2×D711yescharacteristic
267C4×C4⋊C4—21no—
268C4×C4⋊C4C2×D711yescharacteristic
269#340—71no—
270#340—71no—
271#340—71no—
272#340—71no—
273#336—71no—
274#336—71no—
275#336—71no—
276#336—71no—
277C23×C7C22×D411yescharacteristic
278C22×D7—41no—
279C22×D7—41no—
280C22×D7—41no—
281C22×D7—41no—
282C22×D7—41no—
283C22×D7—41no—
284C22×D7—41no—
285C7×Q8—22no—
286C7×Q8—22no—
287C7×Q8—22no—
288C7×Q8#12314yes—
289C2×C4×C7—21no—
290C2×C4×C7—22no—
291C2×C4×C7—21no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—22no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7#13512yes—
318C2×C4×C7#13712yes—
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7H(5,𝔽2)11yescharacteristic
324C2×C4×C72−1+411yescharacteristic
325C2×Dic7—22no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334D28—41no—
335D28—41no—
336D28—41no—
337D28—41no—
338Dic14—22no—
339Dic14—22no—
340Dic14—41no—
341Dic14—41no—
342C7⋊C8—22no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C7⋊C8—41no—
346C22×Q8—71no—
347C22×Q8C22×D711yescharacteristic
348C22×D4—71no—
349C2×Q16—74no—
350C2×Q16—142no—
351C2×Q16—142no—
352C2×SD16—142no—
353C2×SD16—142no—
354C2×SD16—142no—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8—71no—
358C22×C8—71no—
359#124—142no—
360#124—281no—
361C4×Q8—22no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—22no—
365C4×Q8—41no—
366C4×Q8—41no—
367C4×Q8—142no—
368C4×Q8—281no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4C22⋊D711yescharacteristic
381C2×C4⋊C4C22⋊D711yescharacteristic
382C2×C4⋊C4C22×D711yescharacteristic
383C2×C4⋊C4C22×D711yescharacteristic
384C2×C4⋊C4C22×D711yescharacteristic
385C2×C22⋊C4—141no—
386C2×C22⋊C4—141no—
387C2×C22⋊C4—141no—
388C2×C22⋊C4—141no—
389C2×C22⋊C4—141no—
390C2×C22⋊C4—141no—
391C2×C42—21no—
392C2×C42—21no—
393C2×C42—21no—
394C2×C42—141no—
395C2×C42C22×D711yescharacteristic
396C2×C42C22×D711yescharacteristic
397C2×C42C22×D711yescharacteristic
398#107—142no—
399#107—281no—
400C4⋊C8—142no—
401#103—142no—
402#103—142no—
403#103—142no—
404#103—142no—
405#103—142no—
406#102—142no—
407#102—142no—
408#102—281no—
409C4×C8—142no—
410#95—21no—
411#95—21no—
412#95—141no—
413#95—141no—
414#95—141no—
415#95C22⋊D711yescharacteristic
416#95C22⋊D711yescharacteristic
417C22×C7#52511yescharacteristic
418C22×C7D4211yescharacteristic
419C22×C7#54811yescharacteristic
420C22×C7C2×#13511yescharacteristic
421C22×C7C2×#13711yescharacteristic
422C22×C7#57611yescharacteristic
423C22×C7#57811yescharacteristic
424C2×D7—81no—
425C2×D7—81no—
426C2×D7—81no—
427C2×D7—81no—
428C2×D7—81no—
429C2×D7—81no—
430C2×D7—81no—
431C4×C7—22no—
432C4×C7—24no—
433C4×C7—22no—
434C4×C7—41no—
435C4×C7—42no—
436C4×C7—41no—
437C4×C7—41no—
438C4×C7—41no—
439C4×C7—81no—
440C4×C7—81no—
441C4×C7C2×#12311yescharacteristic
442C4×C7#53911yescharacteristic
443C4×C7#54711yescharacteristic
444C4×C7#55311yescharacteristic
445Dic7—42no—
446Dic7—81no—
447Dic7—81no—
448C24—141no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—72no—
453C2×Q8—141no—
454C2×Q8—141no—
455C2×Q8—141no—
456C2×Q8—141no—
457C2×Q8—141no—
458C2×Q8#115712yes—
459C2×Q8D4×D712yes—
460C2×Q8#116112yes—
461C2×D4—141no—
462C2×D4—141no—
463C2×D4—141no—
464C2×D4—141no—
465C2×D4—141no—
466C2×D4—141no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
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
481Q16—148no—
482SD16—284no—
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—
494C2×C8—141no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—81no—
507C4⋊C4—81no—
508C4⋊C4—81no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—282no—
514C4⋊C4—561no—
515C4⋊C4#115712yes—
516C22⋊C4—282no—
517C22⋊C4—282no—
518C22⋊C4—282no—
519C22⋊C4—282no—
520C22⋊C4—561no—
521C22⋊C4—561no—
522C42—24no—
523C42—21no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—42no—
530C42—42no—
531C42—282no—
532C2×C7#244311yescharacteristic
533C2×C7#310811yescharacteristic
534C2×C7#312411yescharacteristic
535C2×C7#327011yescharacteristic
536C2×C7#328711yescharacteristic
537C2×C7#336411yescharacteristic
538C2×C7#337511yescharacteristic
539D7—161no—
540C23—281no—
541C23—281no—
542C23—281no—
543C23—281no—
544C23—281no—
545C23—281no—
546C23—281no—
547C23C22×C22⋊D711yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—22no—
551Q8—142no—
552Q8—142no—
553Q8—281no—
554Q8—281no—
555Q8#645714yes—
556D4—281no—
557D4—281no—
558D4—281no—
559D4—281no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—142no—
589C2×C4—281no—
590C2×C4—281no—
591C2×C4—281no—
592C2×C4—281no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4#650512yes—
598C2×C4#651312yes—
599C2×C4C2×#115711yescharacteristic
600C2×C4C2×#115711yescharacteristic
601C2×C4C2×D4×D711yescharacteristic
602C2×C4#654811yescharacteristic
603C2×C4C2×#116111yescharacteristic
604C2×C4#655111yescharacteristic
605C8—142no—
606C8—281no—
607C8—281no—
608C8—281no—
609C7#2616911yescharacteristic, complemented, Sylow
610C22—561no—
611C22—561no—
612C22—561no—
613C22—561no—
614C22—561no—
615C22—561no—
616C22—561no—
617C22#9048211yescharacteristic
618C22D4×C22⋊D711yescharacteristic
619C22C2×#650511yescharacteristic
620C22#9050811yescharacteristic
621C22C2×#651311yescharacteristic
622C22#9052611yescharacteristic
623C22#9052911yescharacteristic
624C4—22no—
625C4—22no—
626C4—24no—
627C4—41no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—282no—
635C4—561no—
636C4—561no—
637C4C2×#645711yescharacteristic
638C4#9019811yescharacteristic
639C4#9020511yescharacteristic
640C4#9020611yescharacteristic
641C2—1121no—
642C2#1173964011yescharacteristic
643C2#1173988811yescharacteristic
644C2#1174001411yescharacteristic
645C2#1174101011yescharacteristic
646C2#1174103011yescharacteristic
647C2#1174113411yescharacteristic
648C2#1174115311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×Q8×C4⋊C4C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenterQ8×C4⋊C4D7
hypocenterC7#26169
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C7×Q8×C4⋊C4C2semidirect
3. C2×C4×#1144C2semidirect
7. #11729386C2semidirect
8. #11729382C2semidirect
9. #11727385C2semidirect
11. #11727376C2semidirect
13. #11727343C2semidirect
15. #11727236C2semidirect
136. Q8×C4⋊C4D7semidirect
609. C7#26169semidirect

Maximal subgroups

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

subgroupindex
C7×Q8×C4⋊C42
C2×#898002
C2×C4×#11442
#117311182
#117293872
#117293862
#117293822
#117273852
#117273822
#117273762
#117273692
#117273432
#117273402
#117272362
#261697