Finite Group Databasebeta

← #13362383

Subgroups — #13362383

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
1C2×C7×#497C211yesmaximal, characteristic
2C2×#89956C211yesmaximal, characteristic, complemented
3C2×Q16×Dic7C211yesmaximal, characteristic, complemented
4C2×#89936C211yesmaximal, characteristic
5C2×SD16×Dic7C211yesmaximal, characteristic
6#11740672C211yesmaximal, characteristic, complemented
7#11740398C211yesmaximal, characteristic
8#11729821C211yesmaximal, characteristic, complemented
9#11729820C211yesmaximal, characteristic, complemented
10#11729804C211yesmaximal, characteristic
11#11729803C211yesmaximal, characteristic
12#11729539C211yesmaximal, characteristic, complemented
13#11729479C211yesmaximal, characteristic
14#11727570C211yesmaximal, characteristic, complemented
15#11727558C211yesmaximal, characteristic, complemented
16C22×C7×Q16C2211yescharacteristic
17C22×C7×Q16C2211yescharacteristic
18C22×C7×SD16C2211yescharacteristic
19C22×C7×SD16C2211yescharacteristic
20C2×C7×C4⋊Q8C2211yescharacteristic
21C2×C7×#124C2211yescharacteristic
22C2×Q8×Dic7C2211yescharacteristic
23C2×Q8×Dic7C2211yescharacteristic
24C2×Q8×Dic7C2211yescharacteristic
25C2×D4×Dic7C2211yescharacteristic
26C7×#497C418yescomplemented
27C2×C7×#97C2211yescharacteristic
28#90002C2211yescharacteristic
29#90002C2211yescharacteristic
30#90002C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33C2×#6409C2211yescharacteristic
34C2×#6409C2211yescharacteristic
35C2×#6409C2211yescharacteristic
36C2×#6409C2211yescharacteristic
37#89993C2211yescharacteristic
38C2×#6406C2211yescharacteristic
39C2×#6406C2211yescharacteristic
40#89956—24no—
41Q16×Dic7—24no—
42#89936—24no—
43SD16×Dic7—24no—
44C2×#6392C2211yescharacteristic
45C2×#6392C2211yescharacteristic
46C2×#6391C2211yescharacteristic
47C2×#6391C2211yescharacteristic
48C2×#6389C2211yescharacteristic
49C2×#6389C2211yescharacteristic
50C2×C8×Dic7C2211yescharacteristic
51C2×C8×Dic7C2211yescharacteristic
52#89753—22no—
53#89753—22no—
54#89723—22no—
55#89722—22no—
56#89708C2211yescharacteristic
57#89707C2211yescharacteristic
58#89704C2211yescharacteristic
59C2×C4×C7⋊C8C2211yescharacteristic
60#21703—71nomaximal, Sylow
61C22×C7×Q8C2311yescharacteristic
62C22×C7×Q8C2311yescharacteristic
63C22×C7×Q8C2311yescharacteristic
64C22×C7×D4C2311yescharacteristic
65C23×Dic7—21no—
66C2×C7×Q16—22no—
67C2×C7×Q16—22no—
68C2×C7×Q16—22no—
69C2×C7×Q16—22no—
70C2×C7×Q16C2×C414yescomplemented
71C2×C7×Q16C2×C414yescomplemented
72C2×C7×SD16—22no—
73C2×C7×SD16—22no—
74C2×C7×SD16—22no—
75C2×C7×SD16—22no—
76C2×C7×SD16C2×C414yes—
77C2×C7×SD16C2×C414yes—
78C22×C7×C8C2311yescharacteristic
79C22×C7×C8C2311yescharacteristic
80C7×C4⋊Q8—22no—
81C7×C4⋊Q8C2×C414yescomplemented
82C7×#124—22no—
83C7×#124C2×C414yes—
84C2×C7×C4⋊C4—22no—
85C2×C7×C22⋊C4—22no—
86C2×C42×C7C2311yescharacteristic
87C2×C22⋊Dic7—21no—
88C2×C22⋊Dic7—22no—
89Q8×Dic7—22no—
90Q8×Dic7—22no—
91Q8×Dic7—22no—
92Q8×Dic7—22no—
93Q8×Dic7—22no—
94Q8×Dic7—22no—
95D4×Dic7—22no—
96D4×Dic7—22no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7—21no—
100C2×C4⋊Dic7—22no—
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4⋊Dic7C2311yescharacteristic
103C2×C4⋊Dic7C2311yescharacteristic
104C2×C4⋊Dic7C2311yescharacteristic
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7—21no—
108C2×C4×Dic7C2311yescharacteristic
109C2×C4×Dic7C2311yescharacteristic
110C22×C7⋊C8C2311yescharacteristic
111C22×C7⋊C8C2311yescharacteristic
112C7×#97C2×C414yescomplemented
113#6409—22no—
114#6409—22no—
115#6409—22no—
116#6409—22no—
117#6409—22no—
118#6409—22no—
119#6406—22no—
120#6406—22no—
121C7⋊#95—22no—
122C7⋊#95—22no—
123C7⋊#95—22no—
124C7⋊#95—22no—
125C7⋊#95D412yes—
126C7⋊#95D412yes—
127#6392—22no—
128#6392—22no—
129#6391—22no—
130#6391—22no—
131#6389—22no—
132#6389D414yes—
133C8×Dic7—22no—
134C8×Dic7D414yes—
135C4×C7⋊C8—22no—
136C2×#497D711yescharacteristic
137C2×#441—71no—
138C2×#440—71no—
139C2×C4×Q16—71no—
140C2×C4×SD16—71no—
141#2327—71no—
142#2326—71no—
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×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8—21no—
164C2×C7×Q8—21no—
165C2×C7×Q8—21no—
166C2×C7×Q8—21no—
167C2×C7×Q8C22×C412yes—
168C2×C7×Q8C22×C412yes—
169C2×C7×Q8C22×C412yes—
170C2×C7×D4—21no—
171C2×C7×D4—21no—
172C2×C7×D4—21no—
173C2×C7×D4—21no—
174C2×C7×D4—21no—
175C2×C7×D4C22×C412yes—
176C22×C4×C7—21no—
177C22×C4×C7—21no—
178C22×C4×C7—21no—
179C22×C4×C7C2×D412yes—
180C22×C4×C7C2411yescharacteristic
181C22×Dic7—21no—
182C22×Dic7—21no—
183C22×Dic7—21no—
184C22×Dic7—21no—
185C22×Dic7—42no—
186C22×Dic7—82no—
187C22×Dic7C2×D412yes—
188C22×Dic7C2×D411yescharacteristic
189C22×Dic7C2×D411yescharacteristic
190C7×Q16—28no—
191C7×Q16—28no—
192C7×SD16—28no—
193C7×SD16—28no—
194C2×C7×C8C22×C412yes—
195C2×C7×C8C22×C412yes—
196C2×C7×C8C2×D412yes—
197C2×C7×C8C2×D412yes—
198C2×C7×C8#4112yes—
199C2×C7×C8#4112yes—
200C7×C4⋊C4—28no—
201C7×C22⋊C4—28no—
202C42×C7C22×C412yes—
203C42×C7#4112yes—
204C22⋊Dic7—42no—
205C22⋊Dic7—82no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—42no—
215C4⋊Dic7—42no—
216C4⋊Dic7—42no—
217C4⋊Dic7—82no—
218C4×Dic7—21no—
219C4×Dic7—21no—
220C4×Dic7—42no—
221C4×Dic7—42no—
222C4×Dic7—42no—
223C4×Dic7C2×D412yes—
224C4×Dic7C2×D412yes—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8C2×D412yes—
230C2×C7⋊C8C2×D412yes—
231C22×Q16C2×D711yescharacteristic
232C22×Q16C2×D711yescharacteristic
233C22×SD16C2×D711yescharacteristic
234C22×SD16C2×D711yescharacteristic
235C2×C4⋊Q8C2×D711yescharacteristic
236C2×#124C2×D711yescharacteristic
237C2×C4×Q8—71no—
238C2×C4×Q8—71no—
239C2×C4×Q8—71no—
240C2×C4×D4—71no—
241#497Dic718yescomplemented
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×#97C2×D711yescharacteristic
261C2×C4×C8—71no—
262C2×C4×C8—71no—
263C2×C4×C8—71no—
264#391—71no—
265#391—71no—
266#391—71no—
267#390—71no—
268#388—142no—
269#387—142no—
270#387—142no—
271#385—142no—
272#384—71no—
273#382—71no—
274#376—71no—
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—22no—
284C7×Q8—22no—
285C7×Q8—22no—
286C7×Q8—22no—
287C7×Q8—41no—
288C7×Q8—41no—
289C7×Q8—41no—
290C7×Q8—41no—
291C7×Q8—41no—
292C7×Q8—41no—
293C7×D4—22no—
294C7×D4—22no—
295C7×D4—41no—
296C7×D4—41no—
297C2×C4×C7—22no—
298C2×C4×C7—22no—
299C2×C4×C7—22no—
300C2×C4×C7—22no—
301C2×C4×C7—22no—
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×C7C4×D414yes—
309C2×C4×C7C23×C411yescharacteristic
310C2×C4×C7C22×D411yescharacteristic
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×Dic7—22no—
316C2×Dic7—22no—
317C2×Dic7—22no—
318C2×Dic7—22no—
319C2×Dic7—22no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—42no—
325C2×Dic7—42no—
326C2×Dic7—42no—
327C2×Dic7—42no—
328C2×Dic7—84no—
329C2×Dic7#12712yes—
330C2×Dic7#12712yes—
331C7×C8C4×D414yes—
332C7×C8C4×D414yes—
333C7⋊C8—22no—
334C7⋊C8—22no—
335C22×Q8C22×D711yescharacteristic
336C22×Q8C22×D711yescharacteristic
337C22×Q8C22×D711yescharacteristic
338C22×D4C22×D711yescharacteristic
339C23×C4—141no—
340C2×Q16—22no—
341C2×Q16—22no—
342C2×Q16—22no—
343C2×Q16—22no—
344C2×Q16C2×Dic714yescomplemented
345C2×Q16C2×Dic714yescomplemented
346C2×SD16—22no—
347C2×SD16—22no—
348C2×SD16—22no—
349C2×SD16—22no—
350C2×SD16C2×Dic714yes—
351C2×SD16C2×Dic714yes—
352C22×C8—71no—
353C22×C8—71no—
354C22×C8C22×D711yescharacteristic
355C22×C8C22×D711yescharacteristic
356C4⋊Q8—22no—
357C4⋊Q8C2×Dic714yescomplemented
358#124—22no—
359#124C2×Dic714yes—
360C4×Q8—142no—
361C4×Q8—142no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×Q8—142no—
366C4×D4—142no—
367C4×D4—142no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4—142no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—141no—
379C2×C22⋊C4—142no—
380C2×C42—71no—
381C2×C42—71no—
382C2×C42—141no—
383C2×C42—141no—
384C2×C42—141no—
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—142no—
400#97C2×Dic714yescomplemented
401C4×C8—74no—
402C4×C8—142no—
403C4×C8—142no—
404#95—72no—
405#95—72no—
406#95—142no—
407#95—142no—
408#95—142no—
409#95—142no—
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—24no—
421C4×C7—42no—
422C4×C7—42no—
423C4×C7—42no—
424C4×C7C2×C4×D411yescharacteristic
425C4×C7C2×C4×D411yescharacteristic
426C4×C7#53211yescharacteristic
427C4×C7#53211yescharacteristic
428Dic7—22no—
429Dic7—22no—
430Dic7—42no—
431Dic7—81no—
432Dic7—81no—
433Dic7—81no—
434Dic7—81no—
435C24—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8C22×Dic712yes—
452C2×Q8C22×Dic712yes—
453C2×Q8C22×Dic712yes—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4C22×Dic712yes—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—71no—
464C22×C4—71no—
465C22×C4—72no—
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—141no—
470C22×C4—282no—
471C22×C4—562no—
472C22×C4D4×D712yes—
473C22×C4C23×D711yescharacteristic
474Q16—28no—
475Q16—28no—
476SD16—28no—
477SD16—28no—
478C2×C8—72no—
479C2×C8—72no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8D4×D712yes—
485C2×C8D4×D712yes—
486C2×C8#115912yes—
487C2×C8#115912yes—
488C2×C8C22×Dic712yes—
489C2×C8C22×Dic712yes—
490C4⋊C4—28no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—282no—
500C4⋊C4—282no—
501C4⋊C4—282no—
502C4⋊C4—562no—
503C22⋊C4—28no—
504C22⋊C4—282no—
505C22⋊C4—562no—
506C42—72no—
507C42—72no—
508C42—141no—
509C42—141no—
510C42—282no—
511C42—282no—
512C42—282no—
513C42#115912yes—
514C42C22×Dic712yes—
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—
526C23C2×D4×D711yescharacteristic
527Q8—22no—
528Q8—22no—
529Q8—22no—
530Q8—22no—
531Q8—22no—
532Q8—22no—
533Q8—41no—
534Q8—41no—
535Q8—41no—
536Q8—41no—
537Q8—41no—
538Q8—41no—
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—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—72no—
555C2×C4—72no—
556C2×C4—142no—
557C2×C4—142no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—282no—
566C2×C4—282no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—564no—
570C2×C4D4×Dic714yes—
571C2×C4C2×D4×D711yescharacteristic
572C2×C4C2×D4×D711yescharacteristic
573C2×C4C2×#115911yescharacteristic
574C2×C4C2×#115911yescharacteristic
575C2×C4C2×#115911yescharacteristic
576C2×C4C23×Dic711yescharacteristic
577C8—142no—
578C8—142no—
579C8D4×Dic714yes—
580C8D4×Dic714yes—
581C7#2170311yescharacteristic, complemented, Sylow
582C22—42no—
583C22—44no—
584C22—82no—
585C22—81no—
586C22—81no—
587C22D7×#12711yescharacteristic
588C22#9040711yescharacteristic
589C22#9046912yes—
590C22#9047012yes—
591C22C2×D4×Dic711yescharacteristic
592C4—24no—
593C4—42no—
594C4—42no—
595C4—42no—
596C4—142no—
597C4—142no—
598C4—282no—
599C4—561no—
600C4—561no—
601C4—561no—
602C4—561no—
603C4#9040711yescharacteristic
604C4#9040711yescharacteristic
605C4C2×D4×Dic711yescharacteristic
606C4C2×D4×Dic711yescharacteristic
607C2—82no—
608C2#1173930712yes—
609C2#1173931711yescharacteristic
610C2#1173931811yescharacteristic
611C2Dic7×#12711yescharacteristic
612C2#1174141012yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#497C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#497D7
hypocenterC7#21703
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
2. C2×#89956C2semidirect
3. C2×Q16×Dic7C2semidirect
6. #11740672C2semidirect
8. #11729821C2semidirect
9. #11729820C2semidirect
12. #11729539C2semidirect
14. #11727570C2semidirect
15. #11727558C2semidirect
26. C7×#497C4semidirect
70. C2×C7×Q16C2×C4semidirect
71. C2×C7×Q16C2×C4semidirect
81. C7×C4⋊Q8C2×C4semidirect
112. C7×#97C2×C4semidirect
241. #497Dic7semidirect
344. C2×Q16C2×Dic7semidirect
345. C2×Q16C2×Dic7semidirect
357. C4⋊Q8C2×Dic7semidirect
400. #97C2×Dic7semidirect
581. C7#21703semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4972
C2×#899562
C2×Q16×Dic72
C2×#899362
C2×SD16×Dic72
#117406722
#117403982
#117298212
#117298202
#117298042
#117298032
#117295392
#117294792
#117275702
#117275582
#217037