Finite Group Databasebeta

← #13358136

Subgroups — #13358136

Order 1792.

874 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
1C2×#89936C211yesmaximal, characteristic
2#11733922C211yesmaximal, characteristic, complemented
3#11733734C211yesmaximal, characteristic
4C7×#1968C211yesmaximal, characteristic
5#11729804C211yesmaximal, characteristic
6#11728315C211yesmaximal, characteristic, complemented
7#11728307C211yesmaximal, characteristic, complemented
8#11728010C211yesmaximal, characteristic, complemented
9#11727974C211yesmaximal, characteristic, complemented
10#11727942C211yesmaximal, characteristic
11#11727906C211yesmaximal, characteristic
12#11727524C211yesmaximal, characteristic, complemented
13C4×#6389C211yesmaximal, characteristic, complemented
14#11727428C211yesmaximal, characteristic, complemented
15#11727398C211yesmaximal, characteristic
16C22×C7×SD16C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C2×#6440—21no—
20C2×C7×#106C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×#102C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C7×#391C2211yescharacteristic
25C7×#390C2211yescharacteristic
26#90002C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29#89993C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32#89936C418yescomplemented
33C2×#6392C2211yescharacteristic
34C2×#6392C2211yescharacteristic
35C2×#6389C2211yescharacteristic
36C2×#6389C2211yescharacteristic
37C2×#6389C2211yescharacteristic
38C2×#6389C2211yescharacteristic
39C4⋊C4×Dic7C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41#89745C2211yescharacteristic
42#89745C2211yescharacteristic
43#89717—21no—
44#89716—21no—
45C22⋊C4×Dic7—21no—
46C42×Dic7C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4×C7⋊C8C2211yescharacteristic
50#89426—21no—
51#89425—21no—
52#89421—21no—
53#89420—21no—
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21288—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×Dic7—21no—
64C2×C7×SD16—22no—
65C2×C7×SD16—22no—
66C2×C7×SD16C2×C414yes—
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C42×C7C2311yescharacteristic
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77Q8×Dic7—22no—
78Q8×Dic7C2×C414yes—
79D4×Dic7—22no—
80D4×Dic7C2×C414yes—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7D411yescharacteristic
94C2×C4×Dic7D411yescharacteristic
95C2×C4×Dic7D411yescharacteristic
96C2×C4×Dic7D411yescharacteristic
97C2×C4×Dic7C2311yescharacteristic
98C2×C4×Dic7C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101#6440—44no—
102C7×#106—22no—
103C7×#103—22no—
104C7×#102—22no—
105C4×C7×C8—22no—
106C7×#95—21no—
107C7×#95—21no—
108#6409—22no—
109#6409C2×C414yes—
110#6406—22no—
111#6406C2×C414yes—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120#6392—22no—
121#6392C2×C414yes—
122#6389—22no—
123#6389—22no—
124#6389—22no—
125#6389C2×C414yes—
126C4×C7⋊C8—22no—
127C2×#440—71no—
128#2295—71no—
129#2293—71no—
130#1973—71no—
131#1968D711yescharacteristic
132#1943—71no—
133#1942—71no—
134#1936—71no—
135#1935—71no—
136#1934—71no—
137#1933—71no—
138#1776—71no—
139#1763—71no—
140#1762—71no—
141C4×#97—71no—
142C24×C7—21no—
143C2×C7×Q8—21no—
144C2×C7×Q8—21no—
145C2×C7×Q8—21no—
146C2×C7×Q8—21no—
147C2×C7×Q8—21no—
148C2×C7×Q8C22×C412yes—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4C22×C412yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—24no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—42no—
167C22×Dic7—42no—
168C22×Dic7C2×D411yescharacteristic
169C22×Dic7C2×D411yescharacteristic
170C22×Dic7C2×D411yescharacteristic
171C22×Dic7C2×D411yescharacteristic
172C7×SD16—28no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8C22×C412yes—
176C2×C7×C8#4112yes—
177C2×C7×C8#4112yes—
178C2×C7×C8#4112yes—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C22⋊C4—42no—
184C7×C22⋊C4—42no—
185C42×C7—21no—
186C42×C7—21no—
187C22⋊Dic7—24no—
188C22⋊Dic7—42no—
189C22⋊Dic7—42no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—24no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7C22×C412yes—
198C4⋊Dic7C22×C412yes—
199Dic7⋊C4—42no—
200Dic7⋊C4—42no—
201Dic7⋊C4—42no—
202Dic7⋊C4—42no—
203C4×Dic7—22no—
204C4×Dic7—22no—
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—24no—
208C4×Dic7—41no—
209C4×Dic7—41no—
210C4×Dic7—42no—
211C4×Dic7—42no—
212C4×Dic7C22×C412yes—
213C4×Dic7#4112yes—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8C22×C412yes—
219C2×C7⋊C8#4112yes—
220C22×SD16C2×D711yescharacteristic
221C2×C4×Q8—71no—
222C2×C4×D4—71no—
223C2×#117—141no—
224#440—78no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#106C2×D711yescharacteristic
228C2×#103—71no—
229C2×#103—71no—
230C2×#103C2×D711yescharacteristic
231C2×#102—71no—
232C2×#102—71no—
233C2×#102C2×D711yescharacteristic
234C2×#97—71no—
235C2×#97—71no—
236C2×#97—71no—
237C2×#97—71no—
238C2×C4×C8—71no—
239C2×C4×C8C2×D711yescharacteristic
240#391—71no—
241#391C2×D711yescharacteristic
242#390—71no—
243#390C2×D711yescharacteristic
244#389—71no—
245#389—71no—
246#389—71no—
247#389—71no—
248#388—141no—
249#388—141no—
250#387—141no—
251#387—141no—
252C4×C4⋊C4—71no—
253C4×C4⋊C4—71no—
254C4×C4⋊C4—141no—
255C4×C22⋊C4—141no—
256#376—141no—
257C43—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264C23×C7—22no—
265C23×C7—41no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7C22×D411yescharacteristic
272C7×Q8—22no—
273C7×Q8—22no—
274C7×Q8—41no—
275C7×Q8—41no—
276C7×D4—22no—
277C7×D4—22no—
278C7×D4—41no—
279C7×D4—41no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7AGL(1,ℤ/8)12yes—
294C2×C4×C7#13712yes—
295C2×C4×C7C23×C411yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—22no—
306C2×Dic7—22no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—48no—
314C2×Dic7—41no—
315C2×Dic7—41no—
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×Dic7C4×D412yes—
325C2×Dic7C4×D412yes—
326C7×C8—22no—
327C7×C8#11714yescomplemented
328C7⋊C8—22no—
329C7⋊C8—22no—
330C22×Q8C22×D711yescharacteristic
331C22×D4C22×D711yescharacteristic
332C23×C4—141no—
333C2×SD16—22no—
334C2×SD16—22no—
335C2×SD16C4×D714yes—
336C22×C8—71no—
337C22×C8—71no—
338C22×C8C22×D711yescharacteristic
339C22×C8C22×D711yescharacteristic
340C4×Q8—74no—
341C4×Q8—142no—
342C4×D4—74no—
343C4×D4—142no—
344#117—284no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—141no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—141no—
359C2×C22⋊C4—141no—
360C2×C22⋊C4—141no—
361C2×C42—71no—
362C2×C42—71no—
363C2×C42—71no—
364C2×C42—71no—
365C2×C42—71no—
366C2×C42—71no—
367C2×C42—141no—
368C2×C42—141no—
369C2×C42—141no—
370C2×C42C22×D711yescharacteristic
371#107—74no—
372#107—142no—
373#106—22no—
374#103—22no—
375#103—74no—
376#103—142no—
377#102—22no—
378#102—74no—
379#102—142no—
380#97—74no—
381#97—142no—
382#97—142no—
383#97—142no—
384C4×C8—22no—
385C4×C8—142no—
386#95—21no—
387#95—21no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396C22×C7—42no—
397C22×C7—42no—
398C22×C7—42no—
399C22×C7—81no—
400C22×C7—81no—
401C22×C7—81no—
402C22×C7—81no—
403C22×C7C2×C4×D411yescharacteristic
404C22×C7#53211yescharacteristic
405C22×C7#53211yescharacteristic
406C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
407C22×C7C2×#13711yescharacteristic
408C22×C7#57511yescharacteristic
409C22×C7#57511yescharacteristic
410C4×C7—22no—
411C4×C7—22no—
412C4×C7—42no—
413C4×C7—81no—
414C4×C7—81no—
415C4×C7C2×#11711yescharacteristic
416C4×C7C4×#4111yescharacteristic
417C4×C7#52911yescharacteristic
418C4×C7#52911yescharacteristic
419Dic7—22no—
420Dic7—22no—
421Dic7—41no—
422Dic7—41no—
423Dic7—42no—
424Dic7—42no—
425Dic7—81no—
426Dic7—81no—
427C24—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8C2×C4×D712yes—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4C2×C4×D712yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—144no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—282no—
453C22×C4—282no—
454C22×C4D4×D711yescharacteristic
455C22×C4D4×D711yescharacteristic
456C22×C4C23×D711yescharacteristic
457SD16—28no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—72no—
461C2×C8—72no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8C2×C4×D712yes—
467C2×C8#115912yes—
468C2×C8#115912yes—
469C2×C8#115912yes—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—72no—
475C4⋊C4—72no—
476C4⋊C4—144no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—282no—
484C4⋊C4—282no—
485C4⋊C4—282no—
486C4⋊C4—282no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—144no—
490C22⋊C4—282no—
491C22⋊C4—282no—
492C42—21no—
493C42—21no—
494C42—72no—
495C42—72no—
496C42—144no—
497C42—142no—
498C42—142no—
499C42—141no—
500C42—141no—
501C42—281no—
502C42—281no—
503C42—282no—
504C42—282no—
505C2×C7—82no—
506C2×C7#229311yescharacteristic
507C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
508C2×C7C4×#13711yescharacteristic
509C2×C7#318011yescharacteristic
510C2×C7#318111yescharacteristic
511C2×C7#318211yescharacteristic
512C2×C7#318311yescharacteristic
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C2×D4×D711yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525D4—22no—
526D4—22no—
527D4—41no—
528D4—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—72no—
543C2×C4—72no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—288no—
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—281no—
567C2×C4#647712yes—
568C2×C4#647812yes—
569C2×C4C22×C4×D711yescharacteristic
570C2×C4C2×#115911yescharacteristic
571C2×C4C2×#115911yescharacteristic
572C2×C4C2×#115911yescharacteristic
573C2×C4D7×#4111yescharacteristic
574C2×C4D7×#4111yescharacteristic
575C8—22no—
576C8—142no—
577C8—142no—
578C8#644014yescomplemented
579C7#2128811yescharacteristic, complemented, Sylow
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C4×D4×D711yescharacteristic
588C22#9028711yescharacteristic
589C22#9040711yescharacteristic
590C22C2×#647711yescharacteristic
591C22C2×#647811yescharacteristic
592C22#9045911yescharacteristic
593C22#9045911yescharacteristic
594C4—22no—
595C4—22no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—142no—
600C4—142no—
601C4—282no—
602C4—282no—
603C4—281no—
604C4—281no—
605C4—561no—
606C4—561no—
607C4C2×#644011yescharacteristic
608C4C4×#115911yescharacteristic
609C4#9029111yescharacteristic
610C4#9037111yescharacteristic
611C2—82no—
612C2#1173373411yescharacteristic
613C2C4×#647711yescharacteristic
614C2C4×#647811yescharacteristic
615C2#1173808311yescharacteristic
616C2#1173808411yescharacteristic
617C2#1173919911yescharacteristic
618C2#1173925711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1968C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#1968D7
hypocenterC7#21288
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
2. #11733922C2semidirect
6. #11728315C2semidirect
7. #11728307C2semidirect
8. #11728010C2semidirect
9. #11727974C2semidirect
12. #11727524C2semidirect
13. C4×#6389C2semidirect
14. #11727428C2semidirect
32. #89936C4semidirect
327. C7×C8#117semidirect
578. C8#6440semidirect
579. C7#21288semidirect

Maximal subgroups

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

subgroupindex
C2×#899362
#117339222
#117337342
C7×#19682
#117298042
#117283152
#117283072
#117280102
#117279742
#117279422
#117279062
#117275242
C4×#63892
#117274282
#117273982
#212887