Finite Group Databasebeta

← #13416728

Subgroups — #13416728

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×#2442C211yesmaximal, characteristic
2C2×#89835C211yesmaximal, characteristic, complemented
3C2×#89736C211yesmaximal, characteristic
4#11730913C211yesmaximal, characteristic
5#11729493C211yesmaximal, characteristic, complemented
6#11729313C211yesmaximal, characteristic
7#11727389C211yesmaximal, characteristic, complemented
8#11727386C211yesmaximal, characteristic, complemented
9#11727377C211yesmaximal, characteristic, complemented
10#11727355C211yesmaximal, characteristic, complemented
11#11727315C211yesmaximal, characteristic, complemented
12#11727294C211yesmaximal, characteristic, complemented
13#11727270C211yesmaximal, characteristic
14#11727260C211yesmaximal, characteristic
15#11727204C211yesmaximal, characteristic
16C2×C7×#125C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C2×#6489C2211yescharacteristic
19C2×#6487C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#400—21no—
23C7×#398—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#385—21no—
27C7×#382C2211yescharacteristic
28C7×#380—21no—
29C7×#380C2211yescharacteristic
30C4×C7×C22⋊C4C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32#89835—24no—
33C2×#6384C2211yescharacteristic
34C2×#6384C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36C2×#6384C2211yescharacteristic
37#89736—24no—
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89717—21no—
45#89716—21no—
46C4×C4⋊Dic7C2211yescharacteristic
47#89701C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51#89424—21no—
52#89421—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26367—71nomaximal, Sylow
62C23×C4×C7—21no—
63C23×C4×C7C2311yescharacteristic
64C22×Dic14C2311yescharacteristic
65C7×#125—22no—
66C7×#125—41no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—41no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4D411yescharacteristic
80C2×C7×C22⋊C4D411yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7—21no—
85#6489—22no—
86#6489—41no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90#6487—22no—
91#6487Q814yes—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C2×#1136—22no—
99C2×#1136—22no—
100C2×#1136—22no—
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C4×Dic14—22no—
106C4×Dic14—41no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C22⋊(C7⋊C8)—22no—
119#6384—22no—
120#6384—22no—
121#6384—22no—
122#6384—41no—
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#2442D711yescharacteristic
134#2362—71no—
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
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—42no—
156C22×C4×C7—42no—
157C22×C4×C7—81no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×Q812yes—
164C22×C4×C7#4112yes—
165C22×C4×C7C2411yescharacteristic
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14—21no—
174C2×Dic14—21no—
175C2×Dic14C2×Q812yes—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4—81no—
187C7×C4⋊C4—81no—
188C7×C4⋊C4C2×D412yes—
189C7×C22⋊C4—41no—
190C7×C22⋊C4—41no—
191C7×C22⋊C4—42no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—21no—
196C22⋊Dic7—24no—
197C22⋊Dic7—42no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7C2×D412yes—
203C4⋊Dic7C2×Q812yes—
204Dic7⋊C4—24no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207C4×Dic7—42no—
208C4×Dic7—42no—
209C4×Dic7—42no—
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×#125C2×D711yescharacteristic
225C2×C22⋊Q8—71no—
226C2×C4×Q8—71no—
227C2×#117—71no—
228C22×C4⋊C4C2×D711yescharacteristic
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—21no—
245#398—21no—
246#391—71no—
247#389—71no—
248#389C2×D711yescharacteristic
249#389C2×D711yescharacteristic
250#388—141no—
251#388—141no—
252#387—141no—
253#385—21no—
254#382C2×D711yescharacteristic
255#380—21no—
256#380C2×D711yescharacteristic
257C4×C4⋊C4—71no—
258C4×C4⋊C4—141no—
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
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
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—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—41no—
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—41no—
305C2×C4×C7—41no—
306C2×C4×C7—84no—
307C2×C4×C7C22⋊Q812yes—
308C2×C4×C7C22×D411yescharacteristic
309C2×C4×C7C22×Q811yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7H(5,𝔽2)11yescharacteristic
313C2×C4×C72−1+411yescharacteristic
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326Dic14—22no—
327Dic14—22no—
328Dic14—41no—
329Dic14—41no—
330C7⋊C8—22no—
331C7⋊C8—22no—
332C7⋊C8—41no—
333C7⋊C8—41no—
334C22×Q8—71no—
335C23×C4—21no—
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—22no—
345#125—41no—
346C22⋊Q8—74no—
347C22⋊Q8—142no—
348C4×Q8—142no—
349C4×Q8—281no—
350#117—142no—
351#117—281no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—41no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—141no—
371C2×C22⋊C4—141no—
372C2×C22⋊C4C22⋊D711yescharacteristic
373C2×C22⋊C4C22⋊D711yescharacteristic
374C2×C42—141no—
375C2×C42—141no—
376C2×C42—141no—
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—21no—
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—41no—
419C4×C7—41no—
420C4×C7—41no—
421C4×C7—41no—
422C4×C7—42no—
423C4×C7—81no—
424C4×C7—81no—
425C4×C7—81no—
426C4×C7#52511yescharacteristic
427C4×C7#53911yescharacteristic
428C4×C7D4×Q811yescharacteristic
429C4×C7#55411yescharacteristic
430Dic7—42no—
431Dic7—42no—
432Dic7—42no—
433Dic7—81no—
434C24C2×C22⋊D711yescharacteristic
435C2×Q8—72no—
436C2×Q8—141no—
437C2×Q8—141no—
438C2×Q8—141no—
439C2×Q8—141no—
440C2×Q8—141no—
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—42no—
450C22×C4—42no—
451C22×C4—81no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4Q8×D712yes—
457C22×C4#116112yes—
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—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—72no—
488C4⋊C4—72no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—144no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—282no—
497C4⋊C4—282no—
498C4⋊C4D4×D712yes—
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—42no—
502C22⋊C4—144no—
503C22⋊C4—282no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—282no—
509C42—282no—
510C42—282no—
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#650912yes—
526C23C22×C22⋊D711yescharacteristic
527Q8—142no—
528Q8—142no—
529Q8—281no—
530Q8—281no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
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—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—84no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
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—281no—
574C2×C4#650912yes—
575C2×C4C2×D4×D711yescharacteristic
576C2×C4#654811yescharacteristic
577C2×C4C2×Q8×D711yescharacteristic
578C2×C4C2×#116111yescharacteristic
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—
593C22D4×C22⋊D711yescharacteristic
594C22C2×#650911yescharacteristic
595C22#9051311yescharacteristic
596C22#9051311yescharacteristic
597C22#9051511yescharacteristic
598C22#9052711yescharacteristic
599C22#9052911yescharacteristic
600C4—22no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—282no—
610C4—282no—
611C4—282no—
612C4—561no—
613C4#9031811yescharacteristic
614C4#9032311yescharacteristic
615C4#9032711yescharacteristic
616C4#9033611yescharacteristic
617C2—42no—
618C2#1174020911yescharacteristic
619C2#1174046211yescharacteristic
620C2#1174092511yescharacteristic
621C2#1174095311yescharacteristic
622C2#1174098811yescharacteristic
623C2#1174104011yescharacteristic
624C2#1174105711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2442C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2442D7
hypocenterC7#26367
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
2. C2×#89835C2semidirect
5. #11729493C2semidirect
7. #11727389C2semidirect
8. #11727386C2semidirect
9. #11727377C2semidirect
10. #11727355C2semidirect
11. #11727315C2semidirect
12. #11727294C2semidirect
585. C7#26367semidirect

Maximal subgroups

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

subgroupindex
C7×#24422
C2×#898352
C2×#897362
#117309132
#117294932
#117293132
#117273892
#117273862
#117273772
#117273552
#117273152
#117272942
#117272702
#117272602
#117272042
#263677