Finite Group Databasebeta

← #13416666

Subgroups — #13416666

Order 1792.

860 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2363C211yesmaximal, characteristic
2C2×#89781C211yesmaximal, characteristic, complemented
3C2×#89767C211yesmaximal, characteristic, complemented
4#11730940C211yesmaximal, characteristic, complemented
5#11729436C211yesmaximal, characteristic
6#11729431C211yesmaximal, characteristic
7#11729360C211yesmaximal, characteristic, complemented
8#11729330C211yesmaximal, characteristic, complemented
9#11727408C211yesmaximal, characteristic, complemented
10#11727406C211yesmaximal, characteristic, complemented
11#11727378C211yesmaximal, characteristic, complemented
12#11727360C211yesmaximal, characteristic, complemented
13#11727222C211yesmaximal, characteristic, complemented
14#11727212C211yesmaximal, characteristic, complemented
15#11727204C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C2×#6433C2211yescharacteristic
22C7×#390—21no—
23C7×#390C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C7×#388—21no—
26C7×#386—21no—
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C22⋊C4—21no—
29C4×C7×C22⋊C4C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34#89984C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36#89781—24no—
37#89767—24no—
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89718—21no—
45#89714—21no—
46#89708C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51#89432—21no—
52#89429—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26363—71nomaximal, Sylow
62C22×C7×D4—21no—
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×C4×C7C2311yescharacteristic
66C23×Dic7—21no—
67C7×#121—22no—
68C7×#121Q814yes—
69C4×C7×D4—22no—
70C4×C7×D4—41no—
71C7×#117—22no—
72C7×#117—41no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4D411yescharacteristic
83C2×C7×C22⋊C4D411yescharacteristic
84C2×C42×C7—21no—
85C2×C42×C7—21no—
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88C2×C22⋊Dic7—21no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—41no—
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×C4⋊Dic7C2311yescharacteristic
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×#1136—22no—
101C2×#1136—22no—
102C2×#1136—22no—
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107#6433—22no—
108#6433—41no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95D411yescharacteristic
112C7×#95D411yescharacteristic
113#6406—22no—
114#6406—22no—
115#6406—22no—
116#6406—41no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C22⋊(C7⋊C8)—22no—
123#6382—22no—
124#6382—22no—
125#6382—22no—
126#6381—22no—
127#6381—41no—
128#6381D414yescomplemented
129C4⋊(C7⋊C8)—22no—
130C4⋊(C7⋊C8)—22no—
131C2×#478—71no—
132C2×#428—71no—
133#2442—71no—
134#2363D711yescharacteristic
135#2092—71no—
136#2088—71no—
137#2065—71no—
138#2059—71no—
139#1949—71no—
140#1918—71no—
141#1895—71no—
142#1874—71no—
143#1862—71no—
144#1854—71no—
145#1851—71no—
146C24×C7—21no—
147C24×C7C2×D411yescharacteristic
148C2×C7×D4—24no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—81no—
155C2×C7×D4—81no—
156C2×C7×D4C2×Q812yes—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—21no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—21no—
163C22×C4×C7—21no—
164C22×C4×C7—42no—
165C22×C4×C7—42no—
166C22×C4×C7—81no—
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×Q812yes—
173C22×C4×C7#4112yes—
174C22×C4×C7C2411yescharacteristic
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—42no—
180C22×Dic7—42no—
181C22×Dic7—81no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—81no—
187C7×C4⋊C4C2×D412yes—
188C7×C4⋊C4C2×Q812yes—
189C7×C22⋊C4—24no—
190C7×C22⋊C4—41no—
191C7×C22⋊C4—41no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—42no—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—81no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199C42×C7—21no—
200C42×C7—42no—
201C42×C7—81no—
202C22⋊Dic7—42no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
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—42no—
214Dic7⋊C4—81no—
215Dic7⋊C4—81no—
216#1136—28no—
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⋊C8—21no—
227C2×C7⋊C8C2×D412yes—
228C2×C7⋊C8#4112yes—
229C22×M16—71no—
230C2×#125—71no—
231C2×#121C2×D711yescharacteristic
232C2×C4×D4C2×D711yescharacteristic
233C2×#117C2×D711yescharacteristic
234C22×C4⋊C4—71no—
235#478—144no—
236#428—144no—
237C2×#107—71no—
238C2×#107—71no—
239C2×#107—71no—
240C2×#106—71no—
241C2×#106—71no—
242C2×#106—71no—
243C2×C4⋊C8—71no—
244C2×C4⋊C8—71no—
245C2×#102—71no—
246C2×#102—71no—
247C2×#102—71no—
248C2×#102—71no—
249C2×C22⋊C8—71no—
250#400—141no—
251#398—141no—
252#390—21no—
253#390C2×D711yescharacteristic
254#389—71no—
255#389—71no—
256#389C2×D711yescharacteristic
257#388—21no—
258#386—21no—
259#385—141no—
260#382—71no—
261#380—71no—
262#380—141no—
263C4×C4⋊C4C2×D711yescharacteristic
264C4×C22⋊C4—21no—
265C4×C22⋊C4C2×D711yescharacteristic
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#336—71no—
273#336—71no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—21no—
278C23×C7—21no—
279C23×C7—21no—
280C23×C7—22no—
281C23×C7—41no—
282C23×C7—41no—
283C23×C7—41no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7C22⋊Q812yes—
288C23×C7C22×D411yescharacteristic
289C7×D4—22no—
290C7×D4—22no—
291C7×D4—41no—
292C7×D4—41no—
293C7×D4—84no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—22no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—21no—
311C2×C4×C7—21no—
312C2×C4×C7—21no—
313C2×C4×C7—21no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—41no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—41no—
321C2×C4×C7—84no—
322C2×C4×C7C22⋊Q812yes—
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C22×Q811yescharacteristic
325C2×C4×C7C2×#4111yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7H(5,𝔽2)11yescharacteristic
328C2×C4×C72−1+411yescharacteristic
329C2×Dic7—22no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—84no—
342C7⋊C8—22no—
343C7⋊C8—22no—
344C7⋊C8—41no—
345C7⋊C8—41no—
346C22×D4—21no—
347C22×D4C22×D711yescharacteristic
348C23×C4—21no—
349C23×C4—141no—
350C23×C4C22×D711yescharacteristic
351C2×M16—142no—
352C2×M16—142no—
353C2×M16—142no—
354C22×C8—71no—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8—71no—
358#125—142no—
359#125—281no—
360#121—22no—
361#121Dic1414yes—
362C4×D4—22no—
363C4×D4—41no—
364#117—22no—
365#117—41no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—71no—
368C2×C4⋊C4—71no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—142no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—141no—
377C2×C4⋊C4—281no—
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C4⋊C4C22×D711yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—141no—
387C2×C22⋊C4C22⋊D711yescharacteristic
388C2×C22⋊C4C22⋊D711yescharacteristic
389C2×C42—21no—
390C2×C42—21no—
391C2×C42C22×D711yescharacteristic
392C2×C42C22×D711yescharacteristic
393#107—74no—
394#107—142no—
395#107—281no—
396#106—142no—
397#106—142no—
398#106—142no—
399C4⋊C8—142no—
400C4⋊C8—142no—
401#102—142no—
402#102—142no—
403#102—142no—
404#102—281no—
405C22⋊C8—142no—
406#95—21no—
407#95—21no—
408#95—141no—
409#95—141no—
410#95—141no—
411#95—141no—
412#95—141no—
413#95C22⋊D711yescharacteristic
414#95C22⋊D711yescharacteristic
415C22×C7—22no—
416C22×C7—22no—
417C22×C7—22no—
418C22×C7—41no—
419C22×C7—41no—
420C22×C7—41no—
421C22×C7—41no—
422C22×C7—42no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—81no—
426C22×C7—81no—
427C22×C7—81no—
428C22×C7—81no—
429C22×C7C2×C22⋊Q811yescharacteristic
430C22×C7D4211yescharacteristic
431C22×C7#54711yescharacteristic
432C22×C7#57511yescharacteristic
433C22×C7#57511yescharacteristic
434C22×C7#57611yescharacteristic
435C22×C7#57711yescharacteristic
436C4×C7—22no—
437C4×C7—42no—
438C4×C7—41no—
439C4×C7—41no—
440C4×C7—41no—
441C4×C7—41no—
442C4×C7—42no—
443C4×C7—81no—
444C4×C7#52511yescharacteristic
445C4×C7#53911yescharacteristic
446C4×C7D4×Q811yescharacteristic
447C4×C7#55411yescharacteristic
448Dic7—42no—
449Dic7—81no—
450Dic7—81no—
451Dic7—81no—
452C24—21no—
453C24C2×C22⋊D711yescharacteristic
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—81no—
461C2×D4—81no—
462C2×D4C2×Dic1412yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4—141no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—282no—
478C22×C4—282no—
479C22×C4—561no—
480C22×C4C2×Dic1412yes—
481C22×C4#115712yes—
482C22×C4C2×C22⋊D711yescharacteristic
483C22×C4C2×C22⋊D711yescharacteristic
484C22×C4C2×C22⋊D711yescharacteristic
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C2×C22⋊D711yescharacteristic
487C22×C4C23×D711yescharacteristic
488M16—148no—
489C2×C8—72no—
490C2×C8—72no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—72no—
506C4⋊C4—81no—
507C4⋊C4—141no—
508C4⋊C4—141no—
509C4⋊C4—141no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—282no—
515C4⋊C4—282no—
516C4⋊C4—282no—
517C4⋊C4—561no—
518C4⋊C4—561no—
519C4⋊C4C2×Dic1412yes—
520C4⋊C4D4×D712yes—
521C22⋊C4—24no—
522C22⋊C4—41no—
523C22⋊C4—41no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—282no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—42no—
534C42—81no—
535C2×C7—42no—
536C2×C7—82no—
537C2×C7#244911yescharacteristic
538C2×C7#308011yescharacteristic
539C2×C7#308111yescharacteristic
540C2×C7#327111yescharacteristic
541C2×C7#327311yescharacteristic
542C2×C7#334811yescharacteristic
543C2×C7#335511yescharacteristic
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—22no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23#648712yes—
558C23C22×C22⋊D711yescharacteristic
559D4—22no—
560D4—22no—
561D4—41no—
562D4—41no—
563D4—84no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—142no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—564no—
605C2×C4#648712yes—
606C2×C4C22×Dic1411yescharacteristic
607C2×C4C2×#115711yescharacteristic
608C2×C4C2×D4×D711yescharacteristic
609C2×C4#654811yescharacteristic
610C2×C4D7×#4111yescharacteristic
611C2×C4#655411yescharacteristic
612C8—142no—
613C8—142no—
614C8—281no—
615C8—281no—
616C7#2636311yescharacteristic, complemented, Sylow
617C22—22no—
618C22—22no—
619C22—22no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C2×#648711yescharacteristic
632C22D4×C22⋊D711yescharacteristic
633C22#9051311yescharacteristic
634C22#9051311yescharacteristic
635C22#9051511yescharacteristic
636C22#9052611yescharacteristic
637C22#9052711yescharacteristic
638C4—22no—
639C4—41no—
640C4—42no—
641C4—42no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—81no—
646C4—282no—
647C4—561no—
648C4—561no—
649C4—561no—
650C4D4×Dic1411yescharacteristic
651C4#9023011yescharacteristic
652C4#9023811yescharacteristic
653C4#9025511yescharacteristic
654C2—42no—
655C2—82no—
656C2#1173980511yescharacteristic
657C2#1173995411yescharacteristic
658C2#1173998411yescharacteristic
659C2#1174094911yescharacteristic
660C2#1174102811yescharacteristic
661C2#1174105811yescharacteristic
662C2#1174109411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2363C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2363D7
hypocenterC7#26363
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
2. C2×#89781C2semidirect
3. C2×#89767C2semidirect
4. #11730940C2semidirect
7. #11729360C2semidirect
8. #11729330C2semidirect
9. #11727408C2semidirect
10. #11727406C2semidirect
11. #11727378C2semidirect
12. #11727360C2semidirect
13. #11727222C2semidirect
14. #11727212C2semidirect
15. #11727204C2semidirect
128. #6381D4semidirect
616. C7#26363semidirect

Maximal subgroups

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

subgroupindex
C7×#23632
C2×#897812
C2×#897672
#117309402
#117294362
#117294312
#117293602
#117293302
#117274082
#117274062
#117273782
#117273602
#117272222
#117272122
#117272042
#263637