Finite Group Databasebeta

← #13416726

Subgroups — #13416726

Order 1792.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2362C211yesmaximal, characteristic
2C2×#89794C211yesmaximal, characteristic, complemented
3C2×#89767C211yesmaximal, characteristic
4#11730940C211yesmaximal, characteristic
5#11729451C211yesmaximal, characteristic
6#11729446C211yesmaximal, characteristic
7#11729383C211yesmaximal, characteristic, complemented
8#11729330C211yesmaximal, characteristic
9#11727440C211yesmaximal, characteristic, complemented
10#11727437C211yesmaximal, characteristic, complemented
11#11727377C211yesmaximal, characteristic, complemented
12#11727360C211yesmaximal, characteristic, complemented
13#11727222C211yesmaximal, characteristic, complemented
14#11727213C211yesmaximal, characteristic, complemented
15#11727204C211yesmaximal, characteristic
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C22×C4⋊Dic7C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C2×#6433C2211yescharacteristic
22C7×#391C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#388—21no—
25C7×#388—21no—
26C7×#387—21no—
27C4×C7×C4⋊C4—21no—
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C22⋊C4C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34#89984C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36#89794—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#26367—71nomaximal, Sylow
62C22×C7×Q8C2311yescharacteristic
63C23×C4×C7C2311yescharacteristic
64C23×Dic7—21no—
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8Q814yes—
67C4×C7×Q8—22no—
68C4×C7×Q8—41no—
69C7×#117—22no—
70C7×#117—41no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
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⋊C4D411yescharacteristic
80C2×C7×C22⋊C4D411yescharacteristic
81C2×C42×C7—21no—
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C22⋊Dic7—21no—
87C2×C4⋊Dic7—22no—
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×#1136—22no—
99C2×#1136—22no—
100C2×#1136—22no—
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105#6433—22no—
106#6433—41no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95D411yescharacteristic
112C7×#95D411yescharacteristic
113#6409—22no—
114#6409—22no—
115#6409—22no—
116#6409—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—41no—
125#6382D414yes—
126#6381—22no—
127#6381—22no—
128#6381—22no—
129C4⋊(C7⋊C8)—22no—
130C4⋊(C7⋊C8)—22no—
131C2×#479—71no—
132C2×#428—71no—
133#2442—71no—
134#2362D711yescharacteristic
135#2097—71no—
136#2087—71no—
137#2068—71no—
138#2059—71no—
139#1949—71no—
140#1919—71no—
141#1894—71no—
142#1872—71no—
143#1865—71no—
144#1854—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×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×Q812yes—
167C22×C4×C7#4112yes—
168C22×C4×C7C2411yescharacteristic
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—42no—
174C22×Dic7—42no—
175C22×Dic7—81no—
176C7×C4⋊C4—24no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4C2×D412yes—
184C7×C4⋊C4C2×Q812yes—
185C7×C22⋊C4—24no—
186C7×C22⋊C4—41no—
187C7×C22⋊C4—41no—
188C7×C22⋊C4—42no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—42no—
194C42×C7—42no—
195C42×C7—42no—
196C22⋊Dic7—42no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—42no—
205C4⋊Dic7C2×D412yes—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—81no—
209Dic7⋊C4—81no—
210#1136—28no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
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⋊C8C2×D412yes—
222C2×C7⋊C8#4112yes—
223C22×M16—71no—
224C2×#125—71no—
225C2×C22⋊Q8C2×D711yescharacteristic
226C2×C4×Q8C2×D711yescharacteristic
227C2×#117C2×D711yescharacteristic
228C22×C4⋊C4—71no—
229#479—144no—
230#428—144no—
231C2×#107—71no—
232C2×#107—71no—
233C2×#107—71no—
234C2×#106—71no—
235C2×#106—71no—
236C2×#106—71no—
237C2×C4⋊C8—71no—
238C2×C4⋊C8—71no—
239C2×#103—71no—
240C2×#103—71no—
241C2×#103—71no—
242C2×#103—71no—
243C2×C22⋊C8—71no—
244#400—141no—
245#398—141no—
246#391C2×D711yescharacteristic
247#389—71no—
248#389—71no—
249#389C2×D711yescharacteristic
250#388—21no—
251#388—21no—
252#387—21no—
253#385—141no—
254#382—71no—
255#380—71no—
256#380—141no—
257C4×C4⋊C4—21no—
258C4×C4⋊C4C2×D711yescharacteristic
259C4×C22⋊C4C2×D711yescharacteristic
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#336—71no—
267#336—71no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7C22⋊Q812yes—
275C23×C7C22×D411yescharacteristic
276C7×Q8—22no—
277C7×Q8—22no—
278C7×Q8—41no—
279C7×Q8—41no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—22no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—22no—
291C2×C4×C7—21no—
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—41no—
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×C7C22⋊Q812yes—
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C22×Q811yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7H(5,𝔽2)11yescharacteristic
316C2×C4×C72−1+411yescharacteristic
317C2×Dic7—22no—
318C2×Dic7—41no—
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—84no—
330C7⋊C8—22no—
331C7⋊C8—22no—
332C7⋊C8—41no—
333C7⋊C8—41no—
334C22×Q8C22×D711yescharacteristic
335C23×C4—141no—
336C23×C4C22×D711yescharacteristic
337C2×M16—142no—
338C2×M16—142no—
339C2×M16—142no—
340C22×C8—71no—
341C22×C8—71no—
342C22×C8—71no—
343C22×C8—71no—
344#125—142no—
345#125—281no—
346C22⋊Q8—22no—
347C22⋊Q8Dic1414yes—
348C4×Q8—22no—
349C4×Q8—41no—
350#117—22no—
351#117—41no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
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—281no—
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—141no—
372C2×C22⋊C4C22⋊D711yescharacteristic
373C2×C22⋊C4C22⋊D711yescharacteristic
374C2×C42—21no—
375C2×C42—21no—
376C2×C42—21no—
377C2×C42C22×D711yescharacteristic
378C2×C42C22×D711yescharacteristic
379#107—142no—
380#107—142no—
381#107—142no—
382#106—74no—
383#106—142no—
384#106—281no—
385C4⋊C8—142no—
386C4⋊C8—142no—
387#103—142no—
388#103—142no—
389#103—142no—
390#103—281no—
391C22⋊C8—142no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—141no—
397#95—141no—
398#95—141no—
399#95—141no—
400#95—141no—
401#95C22⋊D711yescharacteristic
402#95C22⋊D711yescharacteristic
403C22×C7—22no—
404C22×C7—22no—
405C22×C7—22no—
406C22×C7—41no—
407C22×C7—41no—
408C22×C7—41no—
409C22×C7—41no—
410C22×C7C2×C22⋊Q811yescharacteristic
411C22×C7D4211yescharacteristic
412C22×C7#54711yescharacteristic
413C22×C7#57511yescharacteristic
414C22×C7#57511yescharacteristic
415C22×C7#57711yescharacteristic
416C22×C7#57811yescharacteristic
417C4×C7—22no—
418C4×C7—42no—
419C4×C7—41no—
420C4×C7—41no—
421C4×C7—42no—
422C4×C7—41no—
423C4×C7—42no—
424C4×C7—41no—
425C4×C7—81no—
426C4×C7#52511yescharacteristic
427C4×C7#53911yescharacteristic
428C4×C7D4×Q811yescharacteristic
429C4×C7#55411yescharacteristic
430Dic7—42no—
431Dic7—81no—
432Dic7—81no—
433Dic7—81no—
434C24C2×C22⋊D711yescharacteristic
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×Dic1412yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—282no—
454C22×C4—282no—
455C22×C4—561no—
456C22×C4C2×Dic1412yes—
457C22×C4#115712yes—
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C2×C22⋊D711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C2×C22⋊D711yescharacteristic
463C22×C4C23×D711yescharacteristic
464M16—148no—
465C2×C8—72no—
466C2×C8—72no—
467C2×C8—141no—
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—
477C4⋊C4—24no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—42no—
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—561no—
496C4⋊C4—561no—
497C4⋊C4C2×Dic1412yes—
498C4⋊C4D4×D712yes—
499C22⋊C4—24no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—42no—
503C22⋊C4—282no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42—42no—
510C42—42no—
511C2×C7—42no—
512C2×C7#244911yescharacteristic
513C2×C7#308011yescharacteristic
514C2×C7#308211yescharacteristic
515C2×C7#326911yescharacteristic
516C2×C7#327511yescharacteristic
517C2×C7#334911yescharacteristic
518C2×C7#335111yescharacteristic
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23#648712yes—
526C23C22×C22⋊D711yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
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—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—142no—
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—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—564no—
574C2×C4#648712yes—
575C2×C4C22×Dic1411yescharacteristic
576C2×C4C2×#115711yescharacteristic
577C2×C4C2×D4×D711yescharacteristic
578C2×C4#654811yescharacteristic
579C2×C4D7×#4111yescharacteristic
580C2×C4#655411yescharacteristic
581C8—142no—
582C8—142no—
583C8—281no—
584C8—281no—
585C7#2636711yescharacteristic, complemented, Sylow
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#648711yescharacteristic
594C22D4×C22⋊D711yescharacteristic
595C22#9051311yescharacteristic
596C22#9051311yescharacteristic
597C22#9051511yescharacteristic
598C22#9052811yescharacteristic
599C22#9052911yescharacteristic
600C4—22no—
601C4—41no—
602C4—42no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—41no—
608C4—81no—
609C4—282no—
610C4—561no—
611C4—561no—
612C4—561no—
613C4D4×Dic1411yescharacteristic
614C4#9023011yescharacteristic
615C4#9023811yescharacteristic
616C4#9025511yescharacteristic
617C2—42no—
618C2#1173980511yescharacteristic
619C2#1173995411yescharacteristic
620C2#1173999511yescharacteristic
621C2#1174097311yescharacteristic
622C2#1174098911yescharacteristic
623C2#1174104111yescharacteristic
624C2#1174110511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2362C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2362D7
hypocenterC7#26367
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
2. C2×#89794C2semidirect
7. #11729383C2semidirect
9. #11727440C2semidirect
10. #11727437C2semidirect
11. #11727377C2semidirect
12. #11727360C2semidirect
13. #11727222C2semidirect
14. #11727213C2semidirect
585. C7#26367semidirect

Maximal subgroups

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

subgroupindex
C7×#23622
C2×#897942
C2×#897672
#117309402
#117294512
#117294462
#117293832
#117293302
#117274402
#117274372
#117273772
#117273602
#117272222
#117272132
#117272042
#263677