Finite Group Databasebeta

← #13362378

Subgroups — #13362378

Order 1792.

1062 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2326C211yesmaximal, characteristic
2C2×#89832C211yesmaximal, characteristic
3C2×#89802C211yesmaximal, characteristic, complemented
4C2×C4×#1144C211yesmaximal, characteristic, complemented
5C2×C4×#1142C211yesmaximal, characteristic
6C2×#89790C211yesmaximal, characteristic
7#11730833C211yesmaximal, characteristic, complemented
8#11729479C211yesmaximal, characteristic
9#11727432C211yesmaximal, characteristic, complemented
10#11727431C211yesmaximal, characteristic, complemented
11#11727403C211yesmaximal, characteristic
12#11727400C211yesmaximal, characteristic
13#11727333C211yesmaximal, characteristic, complemented
14#11727332C211yesmaximal, characteristic, complemented
15#11727188C211yesmaximal, characteristic, complemented
16C2×C7×#124C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C22×#1144C2211yescharacteristic
20C22×#1144C2211yescharacteristic
21C22×#1142C2211yescharacteristic
22C22×#1142C2211yescharacteristic
23C2×#6432C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C2×C4×Dic14C2211yescharacteristic
26C7×#391C2211yescharacteristic
27C7×#390C2211yescharacteristic
28C7×#388—22no—
29C7×#385—22no—
30C7×#384C2211yescharacteristic
31C7×#376C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36#89832C418yescomplemented
37#89802—24no—
38C4×#1144—24no—
39#89790—24no—
40C4×#1142—24no—
41C2×#6384C2211yescharacteristic
42C2×#6384C2211yescharacteristic
43C2×#6384C2211yescharacteristic
44C2×#6384C2211yescharacteristic
45C2×#6382C2211yescharacteristic
46C2×#6382C2211yescharacteristic
47C2×#6381C2211yescharacteristic
48C2×#6381C2211yescharacteristic
49#89708C2211yescharacteristic
50#89701C2211yescharacteristic
51#89701C2211yescharacteristic
52C2×#6377C2211yescharacteristic
53C2×#6377C2211yescharacteristic
54C2×#6377C2211yescharacteristic
55C2×C4×C7⋊C8C2211yescharacteristic
56C2×C4×C7⋊C8C2211yescharacteristic
57C2×C4×C7⋊C8C2211yescharacteristic
58#89425—22no—
59#89425—22no—
60#21703—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×Dic14C2311yescharacteristic
65C22×Dic14C2311yescharacteristic
66C7×#124—22no—
67C7×#124C2×C414yes—
68C4×C7×Q8—22no—
69C4×C7×Q8—22no—
70C4×C7×D4—22no—
71C4×C7×D4—22no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—22no—
79C2×C42×C7—21no—
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×#1144—22no—
84C2×#1144—22no—
85C2×#1144—22no—
86C2×#1144—22no—
87C2×#1144C2×C414yescomplemented
88C2×#1144C2×C414yescomplemented
89C2×#1142—22no—
90C2×#1142—22no—
91C2×#1142—22no—
92C2×#1142—22no—
93C2×#1142C2×C414yes—
94C2×#1142C2×C414yes—
95C2×C4⋊Dic7—22no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106#6432—22no—
107#6432C2×C414yescomplemented
108C4×Dic14—22no—
109C4×Dic14—22no—
110C4×Dic14—22no—
111C4×Dic14—22no—
112C7×#95—22no—
113C7×#95D412yes—
114C7×#95D412yes—
115#6409—22no—
116#6409—22no—
117#6406—22no—
118#6406—22no—
119C7⋊#95—22no—
120C7⋊#95—22no—
121C7⋊#95—22no—
122#6384—22no—
123#6384—22no—
124#6384—22no—
125#6384—22no—
126#6382—22no—
127#6382—22no—
128#6381—22no—
129#6381—22no—
130#6377—22no—
131#6377C2×C414yescomplemented
132#6377D414yes—
133C4×C7⋊C8—22no—
134C4×C7⋊C8—22no—
135C4×C7⋊C8D414yes—
136C2×#497—71no—
137C2×#441—71no—
138C2×#440—71no—
139C2×C4×Q16—71no—
140C2×C4×SD16—71no—
141#2327—71no—
142#2326D711yescharacteristic
143#1973—71no—
144#1971—71no—
145#1969—71no—
146#1968—71no—
147#1958—71no—
148#1956—71no—
149#1847—71no—
150#1726—71no—
151C24×C7—21no—
152C2×C7×Q8—21no—
153C2×C7×Q8—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8C22×C412yes—
158C2×C7×D4—21no—
159C2×C7×D4—21no—
160C2×C7×D4—21no—
161C2×C7×D4—21no—
162C2×C7×D4—21no—
163C2×C7×D4C22×C412yes—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—42no—
168C22×C4×C7—82no—
169C22×C4×C7C2×D412yes—
170C22×C4×C7C2×D412yes—
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2411yescharacteristic
174C22×Dic7—21no—
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—21no—
178C2×Dic14—21no—
179C2×Dic14—21no—
180C2×Dic14—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14C22×C412yes—
189C2×Dic14C22×C412yes—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—21no—
194C7×C4⋊C4—42no—
195C7×C4⋊C4—82no—
196C7×C22⋊C4—28no—
197C7×C22⋊C4—42no—
198C7×C22⋊C4—82no—
199C42×C7—21no—
200C42×C7—21no—
201C42×C7—42no—
202C42×C7C22×C412yes—
203C42×C7C2×D412yes—
204C42×C7C2×D412yes—
205C42×C7#4112yes—
206#1144—28no—
207#1144—28no—
208#1142—28no—
209#1142—28no—
210C4⋊Dic7—28no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215Dic7⋊C4—42no—
216Dic7⋊C4—42no—
217C4×Dic7—42no—
218C4×Dic7—42no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8C22×C412yes—
224C2×C7⋊C8C22×C412yes—
225C2×C7⋊C8C2×D412yes—
226C2×C7⋊C8C2×D412yes—
227C2×C7⋊C8C2×D412yes—
228C2×C7⋊C8C2×D412yes—
229C2×C7⋊C8#4112yes—
230C2×C7⋊C8#4112yes—
231C22×Q16—71no—
232C22×Q16—71no—
233C22×SD16—71no—
234C22×SD16—71no—
235C2×C4⋊Q8—71no—
236C2×#124C2×D711yescharacteristic
237C2×C4×Q8—71no—
238C2×C4×Q8—71no—
239C2×C4×Q8C2×D711yescharacteristic
240C2×C4×D4C2×D711yescharacteristic
241#497—78no—
242#441—144no—
243#440—144no—
244C4×Q16—144no—
245C4×SD16—144no—
246C2×#107—71no—
247C2×#107—71no—
248C2×#106—71no—
249C2×#106—71no—
250C2×#103—71no—
251C2×#103—71no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#102—71no—
257C2×#102—71no—
258C2×#97—71no—
259C2×#97—71no—
260C2×#97—71no—
261C2×C4×C8—71no—
262C2×C4×C8—71no—
263C2×C4×C8—71no—
264#391—71no—
265#391—71no—
266#391C2×D711yescharacteristic
267#390C2×D711yescharacteristic
268#388—22no—
269#387—142no—
270#387—142no—
271#385—22no—
272#384C2×D711yescharacteristic
273#382—71no—
274#376C2×D711yescharacteristic
275C23×C7—22no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—42no—
279C23×C7—42no—
280C23×C7C22×D411yescharacteristic
281C7×Q8—22no—
282C7×Q8—22no—
283C7×Q8—41no—
284C7×Q8—41no—
285C7×D4—22no—
286C7×D4—22no—
287C7×D4—41no—
288C7×D4—41no—
289C2×C4×C7—22no—
290C2×C4×C7—22no—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—22no—
294C2×C4×C7—22no—
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—42no—
302C2×C4×C7—42no—
303C2×C4×C7—84no—
304C2×C4×C7C4×D414yes—
305C2×C4×C7#12712yes—
306C2×C4×C7#12712yes—
307C2×C4×C7C23×C411yescharacteristic
308C2×C4×C7C22×D411yescharacteristic
309C2×C4×C7C22×D411yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—42no—
322C2×Dic7—42no—
323Dic14—22no—
324Dic14—22no—
325Dic14—22no—
326Dic14—22no—
327Dic14—41no—
328Dic14—41no—
329Dic14—41no—
330Dic14—41no—
331C7⋊C8—22no—
332C7⋊C8—22no—
333C7⋊C8C4×D414yes—
334C7⋊C8C4×D414yes—
335C22×Q8—71no—
336C22×Q8—71no—
337C22×Q8C22×D711yescharacteristic
338C22×D4C22×D711yescharacteristic
339C23×C4—21no—
340C2×Q16—74no—
341C2×Q16—74no—
342C2×Q16—142no—
343C2×Q16—142no—
344C2×Q16—142no—
345C2×Q16—142no—
346C2×SD16—74no—
347C2×SD16—74no—
348C2×SD16—142no—
349C2×SD16—142no—
350C2×SD16—142no—
351C2×SD16—142no—
352C22×C8—71no—
353C22×C8—71no—
354C22×C8—71no—
355C22×C8—71no—
356C4⋊Q8—74no—
357C4⋊Q8—142no—
358#124—22no—
359#124C4×D714yes—
360C4×Q8—22no—
361C4×Q8—22no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×D4—22no—
367C4×D4—22no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—142no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4C22×D711yescharacteristic
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C22⋊C4—22no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—22no—
380C2×C42—21no—
381C2×C42—141no—
382C2×C42—141no—
383C2×C42C22×D711yescharacteristic
384C2×C42C22×D711yescharacteristic
385C2×C42C22×D711yescharacteristic
386#107—142no—
387#107—142no—
388#106—142no—
389#106—142no—
390#103—142no—
391#103—142no—
392#103—142no—
393#103—142no—
394#103—142no—
395#103—142no—
396#102—142no—
397#102—142no—
398#97—74no—
399#97—74no—
400#97—142no—
401C4×C8—74no—
402C4×C8—142no—
403C4×C8—142no—
404#95—22no—
405#95—142no—
406#95—142no—
407#95—142no—
408#95C22⋊D712yes—
409#95C22⋊D712yes—
410C22×C7—42no—
411C22×C7—44no—
412C22×C7—81no—
413C22×C7—81no—
414C22×C7—82no—
415C22×C7C2×C4×D411yescharacteristic
416C22×C7C2×#12711yescharacteristic
417C22×C7#53211yescharacteristic
418C22×C7#57712yes—
419C22×C7#57812yes—
420C4×C7—22no—
421C4×C7—22no—
422C4×C7—24no—
423C4×C7—42no—
424C4×C7—42no—
425C4×C7—81no—
426C4×C7—81no—
427C4×C7C2×C4×D411yescharacteristic
428C4×C7C2×C4×D411yescharacteristic
429C4×C7#53211yescharacteristic
430C4×C7#53211yescharacteristic
431Dic7—42no—
432Dic7—42no—
433Dic7—81no—
434Dic7—81no—
435C24—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—72no—
442C2×Q8—72no—
443C2×Q8—141no—
444C2×Q8—141no—
445C2×Q8—141no—
446C2×Q8—141no—
447C2×Q8—141no—
448C2×Q8—141no—
449C2×Q8—141no—
450C2×Q8—141no—
451C2×Q8—141no—
452C2×Q8—141no—
453C2×Q8C2×C4×D712yes—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4C2×C4×D712yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—42no—
464C22×C4—82no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4C2×C22⋊D712yes—
470C22×C4C2×C22⋊D712yes—
471C22×C4C2×C22⋊D711yescharacteristic
472C22×C4C2×C22⋊D711yescharacteristic
473C22×C4C23×D711yescharacteristic
474Q16—148no—
475Q16—148no—
476SD16—148no—
477SD16—148no—
478C2×C8—72no—
479C2×C8—72no—
480C2×C8—72no—
481C2×C8—72no—
482C2×C8—72no—
483C2×C8—72no—
484C2×C8—72no—
485C2×C8—72no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—141no—
489C2×C8—141no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—82no—
496C4⋊C4—148no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—141no—
500C4⋊C4—141no—
501C4⋊C4—282no—
502C4⋊C4—282no—
503C22⋊C4—28no—
504C22⋊C4—42no—
505C22⋊C4—82no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42—282no—
510C42—282no—
511C42C2×C4×D712yes—
512C42#115712yes—
513C42D4×D712yes—
514C42D4×D712yes—
515C2×C7—82no—
516C2×C7C4×#12711yescharacteristic
517C2×C7#294712yes—
518C2×C7#315411yescharacteristic
519C2×C7#315611yescharacteristic
520C2×C7#317812yes—
521C23—22no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23C22×C22⋊D711yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—41no—
530Q8—41no—
531Q8—142no—
532Q8—142no—
533Q8—142no—
534Q8—142no—
535Q8—281no—
536Q8—281no—
537Q8—281no—
538Q8—281no—
539D4—22no—
540D4—22no—
541D4—41no—
542D4—41no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—84no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4C4×C22⋊D714yes—
569C2×C4#650312yes—
570C2×C4#650312yes—
571C2×C4C22×C4×D711yescharacteristic
572C2×C4C2×#115711yescharacteristic
573C2×C4C2×D4×D711yescharacteristic
574C2×C4C2×D4×D711yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4D7×#4111yescharacteristic
577C8—74no—
578C8—74no—
579C8—142no—
580C8—142no—
581C7#2170311yescharacteristic, complemented, Sylow
582C22—44no—
583C22—42no—
584C22—81no—
585C22—81no—
586C22—82no—
587C22C2×C4×C22⋊D711yescharacteristic
588C22C2×#650311yescharacteristic
589C22#9052311yescharacteristic
590C22#9052712yes—
591C22#9052912yes—
592C4—24no—
593C4—22no—
594C4—22no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—282no—
600C4—282no—
601C4—561no—
602C4—561no—
603C4C4×D4×D711yescharacteristic
604C4C4×D4×D711yescharacteristic
605C4#9023911yescharacteristic
606C4#9023911yescharacteristic
607C2—82no—
608C2C4×#650311yescharacteristic
609C2#1174001612yes—
610C2#1174082611yescharacteristic
611C2#1174084811yescharacteristic
612C2#1174090312yes—
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2326C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#2326D7
hypocenterC7#21703
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
3. C2×#89802C2semidirect
4. C2×C4×#1144C2semidirect
7. #11730833C2semidirect
9. #11727432C2semidirect
10. #11727431C2semidirect
13. #11727333C2semidirect
14. #11727332C2semidirect
15. #11727188C2semidirect
36. #89832C4semidirect
87. C2×#1144C2×C4semidirect
88. C2×#1144C2×C4semidirect
107. #6432C2×C4semidirect
131. #6377C2×C4semidirect
581. C7#21703semidirect

Maximal subgroups

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

subgroupindex
C7×#23262
C2×#898322
C2×#898022
C2×C4×#11442
C2×C4×#11422
C2×#897902
#117308332
#117294792
#117274322
#117274312
#117274032
#117274002
#117273332
#117273322
#117271882
#217037