Finite Group Databasebeta

← #13412650

Subgroups — #13412650

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2440C211yesmaximal, characteristic
2C2×#89852C211yesmaximal, characteristic, complemented
3C2×#89736C211yesmaximal, characteristic
4#11730939C211yesmaximal, characteristic
5#11729537C211yesmaximal, characteristic, complemented
6#11729310C211yesmaximal, characteristic
7#11727374C211yesmaximal, characteristic, complemented
8#11727373C211yesmaximal, characteristic, complemented
9#11727362C211yesmaximal, characteristic, complemented
10#11727360C211yesmaximal, characteristic, complemented
11#11727330C211yesmaximal, characteristic, complemented
12#11727323C211yesmaximal, characteristic, complemented
13#11727267C211yesmaximal, characteristic
14#11727260C211yesmaximal, characteristic
15#11727202C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×C22⋊Q8—21no—
18C22×C7×C4⋊C4C2211yescharacteristic
19C2×#6489C2211yescharacteristic
20C22×#1136C2211yescharacteristic
21C2×#6433C2211yescharacteristic
22C7×#398—21no—
23C7×#395—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#385—21no—
27C7×#384C2211yescharacteristic
28C7×#380C2211yescharacteristic
29C4×C7×C22⋊C4C2211yescharacteristic
30#89984C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32#89852—24no—
33#89736—24no—
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89721—21no—
45#89720—21no—
46#89707C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×#6377C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89426—21no—
52#89419—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26078—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C23×C4×C7—21no—
64C23×C4×C7C2311yescharacteristic
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—41no—
67C7×C22⋊Q8—44no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—41no—
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4D411yescharacteristic
82C2×C7×C22⋊C4D411yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87#6489—22no—
88#6489—41no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100C2×#1136—22no—
101C2×#1136—22no—
102C2×#1136Q814yescomplemented
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×#95—21no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C22⋊(C7⋊C8)Q814yescomplemented
118#6382—22no—
119#6382—22no—
120#6382—22no—
121#6382—22no—
122#6382—41no—
123#6382—41no—
124#6381—22no—
125#6381—22no—
126#6381—22no—
127#6381D414yes—
128#6377—22no—
129C4×C7⋊C8—22no—
130C2×#501—71no—
131C2×#428—71no—
132#2441—71no—
133#2440D711yescharacteristic
134#2084—71no—
135#2083—71no—
136#2059—71no—
137#2058—71no—
138#1943—71no—
139#1942—71no—
140#1854—71no—
141#1853—71no—
142#1844—71no—
143#1809—71no—
144#1753—71no—
145C24×C7C2×D411yescharacteristic
146C2×C7×Q8—42no—
147C2×C7×Q8—42no—
148C2×C7×Q8—81no—
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—21no—
156C22×C4×C7—21no—
157C22×C4×C7—42no—
158C22×C4×C7—42no—
159C22×C4×C7—81no—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×Q812yes—
166C22×C4×C7C2×Q812yes—
167C22×C4×C7C2411yescharacteristic
168C22×Dic7—21no—
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—81no—
185C7×C4⋊C4C2×D412yes—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—22no—
188C7×C22⋊C4—42no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C22⋊Dic7—42no—
194C22⋊Dic7—81no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—42no—
203C4⋊Dic7—81no—
204C4⋊Dic7C2×D412yes—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—81no—
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⋊C8C2×D412yes—
218C2×C7⋊C8C2×Q812yes—
219C2×C7⋊C8C2×Q812yes—
220C2×C7⋊C8C2×Q812yes—
221C2×C7⋊C8C2×Q812yes—
222C2×C7⋊C8#4112yes—
223C22×M16—71no—
224C2×C4⋊Q8C2×D711yescharacteristic
225C2×#125—71no—
226C2×C22⋊Q8—21no—
227C2×#117—71no—
228C22×C4⋊C4C2×D711yescharacteristic
229#501—144no—
230#428—144no—
231C2×#107—71no—
232C2×#107—71no—
233C2×#107—71no—
234C2×#107—71no—
235C2×#106—71no—
236C2×#106—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#106—71no—
240C2×#106—71no—
241C2×C22⋊C8—71no—
242C2×#97—71no—
243C2×C4×C8—71no—
244#399—141no—
245#398—21no—
246#396—141no—
247#395—21no—
248#389—71no—
249#389—71no—
250#389—141no—
251#389C2×D711yescharacteristic
252#389C2×D711yescharacteristic
253#387—141no—
254#385—21no—
255#384—71no—
256#384C2×D711yescharacteristic
257#380—71no—
258#380C2×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×C7C4⋊Q812yes—
275C23×C7C22×D411yescharacteristic
276C7×Q8—84no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—22no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—41no—
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—84no—
306C2×C4×C7C4⋊Q812yes—
307C2×C4×C7AGL(1,ℤ/8)12yes—
308C2×C4×C7#13712yes—
309C2×C4×C7C22×D411yescharacteristic
310C2×C4×C7C22×Q811yescharacteristic
311C2×C4×C7C22×Q811yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7H(5,𝔽2)11yescharacteristic
314C2×C4×C72−1+411yescharacteristic
315C2×Dic7—22no—
316C2×Dic7—41no—
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—
326C2×Dic7—41no—
327C7⋊C8—22no—
328C7⋊C8—22no—
329C7⋊C8—22no—
330C7⋊C8C22⋊Q814yescomplemented
331C22×Q8—21no—
332C23×C4—21no—
333C23×C4C22×D711yescharacteristic
334C2×M16—74no—
335C2×M16—142no—
336C2×M16—142no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—71no—
341C4⋊Q8—22no—
342C4⋊Q8—41no—
343#125—142no—
344#125—281no—
345C22⋊Q8—44no—
346#117—142no—
347#117—281no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—41no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C4⋊C4C22×D711yescharacteristic
370C2×C22⋊C4—21no—
371C2×C22⋊C4—141no—
372C2×C22⋊C4—141no—
373C2×C22⋊C4C22⋊D711yescharacteristic
374C2×C22⋊C4C22⋊D711yescharacteristic
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377C2×C42C22×D711yescharacteristic
378#107—74no—
379#107—142no—
380#107—142no—
381#107—142no—
382#106—142no—
383#106—142no—
384#106—142no—
385#106—142no—
386#106—281no—
387#106—281no—
388C22⋊C8—74no—
389#97—142no—
390C4×C8—142no—
391#95—21no—
392#95—21no—
393#95—21no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95C22⋊D711yescharacteristic
398#95C22⋊D711yescharacteristic
399C22×C7—22no—
400C22×C7—22no—
401C22×C7—22no—
402C22×C7—41no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—41no—
406C22×C7C2×C4⋊Q811yescharacteristic
407C22×C7D4211yescharacteristic
408C22×C7#54711yescharacteristic
409C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
410C22×C7C2×#13711yescharacteristic
411C22×C7#57711yescharacteristic
412C22×C7#57711yescharacteristic
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—41no—
417C4×C7—42no—
418C4×C7—41no—
419C4×C7—81no—
420C4×C7—81no—
421C4×C7—81no—
422C4×C7C2×C22⋊Q811yescharacteristic
423C4×C7#54411yescharacteristic
424C4×C7D4×Q811yescharacteristic
425C4×C7#55411yescharacteristic
426Dic7—42no—
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430C24C2×C22⋊D711yescharacteristic
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—81no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—81no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4Q8×D712yes—
450C22×C4Q8×D712yes—
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457M16—148no—
458C2×C8—72no—
459C2×C8—72no—
460C2×C8—72no—
461C2×C8—72no—
462C2×C8—72no—
463C2×C8—72no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—72no—
483C4⋊C4—81no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—561no—
496C4⋊C4—561no—
497C4⋊C4D4×D712yes—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—42no—
501C22⋊C4—282no—
502C22⋊C4—561no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—282no—
508C2×C7—42no—
509C2×C7#244011yescharacteristic
510C2×C7#316311yescharacteristic
511C2×C7#316511yescharacteristic
512C2×C7#327211yescharacteristic
513C2×C7#327811yescharacteristic
514C2×C7#334411yescharacteristic
515C2×C7#334511yescharacteristic
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23#650712yes—
523C23C22×C22⋊D711yescharacteristic
524Q8—84no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—84no—
554C2×C4—142no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4#649512yes—
567C2×C4#650512yes—
568C2×C4#650712yes—
569C2×C4C2×D4×D711yescharacteristic
570C2×C4#654811yescharacteristic
571C2×C4C2×Q8×D711yescharacteristic
572C2×C4C2×Q8×D711yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4#655411yescharacteristic
575C8—74no—
576C8—142no—
577C8—142no—
578C8—142no—
579C7#2607811yescharacteristic, complemented, Sylow
580C22—22no—
581C22—22no—
582C22—22no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22C2×#649511yescharacteristic
588C22D4×C22⋊D711yescharacteristic
589C22C2×#650511yescharacteristic
590C22C2×#650711yescharacteristic
591C22#9051511yescharacteristic
592C22#9052711yescharacteristic
593C22#9052811yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—42no—
599C4—41no—
600C4—81no—
601C4—81no—
602C4—81no—
603C4—282no—
604C4—561no—
605C4—561no—
606C4—561no—
607C4#9031411yescharacteristic
608C4D7×C22⋊Q811yescharacteristic
609C4#9032311yescharacteristic
610C4#9032711yescharacteristic
611C2—42no—
612C2#1174020811yescharacteristic
613C2#1174065211yescharacteristic
614C2#1174091711yescharacteristic
615C2#1174093411yescharacteristic
616C2#1174096411yescharacteristic
617C2#1174101111yescharacteristic
618C2#1174106911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2440C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2440D7
hypocenterC7#26078
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
2. C2×#89852C2semidirect
5. #11729537C2semidirect
7. #11727374C2semidirect
8. #11727373C2semidirect
9. #11727362C2semidirect
10. #11727360C2semidirect
11. #11727330C2semidirect
12. #11727323C2semidirect
102. C2×#1136Q8semidirect
117. C22⋊(C7⋊C8)Q8semidirect
330. C7⋊C8C22⋊Q8semidirect
579. C7#26078semidirect

Maximal subgroups

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

subgroupindex
C7×#24402
C2×#898522
C2×#897362
#117309392
#117295372
#117293102
#117273742
#117273732
#117273622
#117273602
#117273302
#117273232
#117272672
#117272602
#117272022
#260787