Finite Group Databasebeta

← #13357192

Subgroups — #13357192

Order 1792.

982 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89653C211yesmaximal, characteristic
2C2×#89633C211yesmaximal, characteristic
3C2×#89480C211yesmaximal, characteristic
4#11733640C211yesmaximal, characteristic
5#11733413C211yesmaximal, characteristic
6C7×#1944C211yesmaximal, characteristic, complemented
7#11728365C211yesmaximal, characteristic, complemented
8#11728314C211yesmaximal, characteristic, complemented
9#11727573C211yesmaximal, characteristic, complemented
10#11727373C211yesmaximal, characteristic, complemented
11#11727372C211yesmaximal, characteristic, complemented
12#11726620C211yesmaximal, characteristic
13D7×#340C211yesmaximal, characteristic
14#11726615C211yesmaximal, characteristic, complemented
15#11726614C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C22×C8×D7C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4C2211yescharacteristic
21C2×D7×C4⋊C4C2211yescharacteristic
22C2×#6435C2211yescharacteristic
23C2×C7×#107C2211yescharacteristic
24C2×C7×#106C2211yescharacteristic
25C2×C7×#97C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6391C2211yescharacteristic
30C2×#6389C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89761—21no—
33#89759—21no—
34#89757C2211yescharacteristic
35#89757C2211yescharacteristic
36#89748—21no—
37#89745C2211yescharacteristic
38#89745C2211yescharacteristic
39#89744—21no—
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89653—24no—
48#89633C418yescomplemented
49#89480—24no—
50#89440—21no—
51#89439—21no—
52#89422—21no—
53#89419—21no—
54C7×#340C2211yescharacteristic
55C7×#340C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21160—71nomaximal, Sylow
63C22×C4×D7—21no—
64C22×C4×D7—21no—
65C22×C4×D7C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C42×C7C2311yescharacteristic
75C2×(C2×D7)⋊C4—21no—
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4D411yescharacteristic
78C2×(C2×D7)⋊C4D411yescharacteristic
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4D411yescharacteristic
92C2×Dic7⋊C4D411yescharacteristic
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C2×#1131C2×C414yes—
99C2×#1131D414yes—
100C2×#1131Q814yes—
101C2×C8×D7—22no—
102C2×C8×D7—22no—
103C2×C8×D7Q814yes—
104#6455—22no—
105#6455—22no—
106#6455—22no—
107#6455C2×C414yes—
108D7×C4⋊C4—22no—
109D7×C4⋊C4—22no—
110D7×C4⋊C4—22no—
111D7×C4⋊C4C2×C414yes—
112#6435—22no—
113#6435—22no—
114C7×#107—22no—
115C7×#106C2×C414yescomplemented
116C7×#97—22no—
117C7×#95—21no—
118C7×#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124C7⋊#95—21no—
125#6392C2×C414yescomplemented
126#6391—22no—
127#6389—22no—
128C8×Dic7—22no—
129#6382—22no—
130#6382C2×C414yescomplemented
131#6381—22no—
132#6381C2×C414yescomplemented
133C4×C7⋊C8—22no—
134C2×#428—71no—
135C2×#428—71no—
136C2×#405—71no—
137#2317—71no—
138#2316—71no—
139#1944—71no—
140#1944D711yescharacteristic, complemented
141#1943—71no—
142#1943—71no—
143#1942—71no—
144#1942—71no—
145#1941—71no—
146#1941—71no—
147#1735—71no—
148C2×#340—71no—
149C23×D7C2×D411yescharacteristic
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C2×C4×D7—24no—
165C2×C4×D7—42no—
166C2×C4×D7—81no—
167C2×C4×D7—81no—
168C2×C4×D7—81no—
169C2×C4×D7—81no—
170C2×C4×D7C22×C412yes—
171C2×C4×D7C2×D412yes—
172C2×C4×D7C2×Q812yes—
173C2×C4×D7C2×Q812yes—
174C2×C4×D7#4112yes—
175C2×C4×D7#4112yes—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8C22×C412yes—
179C2×C7×C8C2×D412yes—
180C2×C7×C8C2×Q812yes—
181C2×C7×C8C2×Q812yes—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—81no—
189C7×C4⋊C4—81no—
190C7×C4⋊C4C22×C412yes—
191C7×C4⋊C4C22×C412yes—
192C42×C7—21no—
193C42×C7—21no—
194(C2×D7)⋊C4—24no—
195(C2×D7)⋊C4—22no—
196(C2×D7)⋊C4—22no—
197(C2×D7)⋊C4—42no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—81no—
205C4⋊Dic7—81no—
206C4⋊Dic7C22×C412yes—
207C4⋊Dic7C22×C412yes—
208Dic7⋊C4—24no—
209Dic7⋊C4—22no—
210Dic7⋊C4—22no—
211Dic7⋊C4—42no—
212Dic7⋊C4—81no—
213Dic7⋊C4—81no—
214Dic7⋊C4—81no—
215Dic7⋊C4—81no—
216C4×Dic7—21no—
217C4×Dic7—21no—
218C4×Dic7—24no—
219C4×Dic7—42no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8C22×C412yes—
223C2×C7⋊C8C2×D412yes—
224C2×C7⋊C8C2×Q812yes—
225C2×C7⋊C8C2×Q812yes—
226#1131C4⋊C4116yescomplemented
227C8×D7—28no—
228C22×M16—71no—
229C23×C8—71no—
230C2×#117—71no—
231C2×#117—71no—
232C2×#117—71no—
233C22×C4⋊C4—71no—
234C22×C4⋊C4—71no—
235#428—78no—
236#428—144no—
237C2×#107—71no—
238C2×#107—71no—
239C2×#107—71no—
240C2×#107C2×D711yescharacteristic
241C2×#106—71no—
242C2×#106—71no—
243C2×#106—71no—
244C2×#106C2×D711yescharacteristic
245#405—144no—
246C2×#97—71no—
247C2×#97C2×D711yescharacteristic
248C2×C4×C8—71no—
249C2×C4×C8—71no—
250#389—71no—
251#389—71no—
252#389—71no—
253#389—71no—
254#389—71no—
255#389—71no—
256#389—141no—
257#389—141no—
258#389C2×D711yescharacteristic
259#389C2×D711yescharacteristic
260#385—141no—
261#385—141no—
262#384—141no—
263#382—141no—
264#380—141no—
265#380—141no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#340—71no—
271#340—71no—
272#340C2×D711yescharacteristic
273#340C2×D711yescharacteristic
274C23×C7C22×D411yescharacteristic
275C22×D7—21no—
276C22×D7—21no—
277C22×D7—21no—
278C22×D7—21no—
279C22×D7C4×D412yes—
280C22×D7#12112yes—
281C22×D7C4⋊Q812yes—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
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×C7C23×C411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C2×C4×C7C22×Q811yescharacteristic
303C2×C4×C7C22×Q811yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
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×Dic7#12112yes—
326C2×Dic7C4⋊Q812yes—
327C4×D7—84no—
328C4×D7—84no—
329C4×D7C2×C4⋊C414yes—
330C4×D7C4×Q814yes—
331C4×D7C22⋊Q814yes—
332C4×D7#12514yes—
333C7×C8—22no—
334C7×C8C2×C4⋊C414yes—
335C7⋊C8—22no—
336C7⋊C8C2×C4⋊C414yes—
337C23×C4—71no—
338C23×C4—141no—
339C23×C4—141no—
340C2×M16—74no—
341C2×M16—74no—
342C2×M16—74no—
343C22×C8—71no—
344C22×C8—71no—
345C22×C8—74no—
346C22×C8—142no—
347C22×C8—142no—
348C22×C8C22×D711yescharacteristic
349C22×C8C22×D711yescharacteristic
350#117—74no—
351#117—142no—
352#117—142no—
353#117—142no—
354#117—142no—
355#117—142no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—74no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—71no—
365C2×C4⋊C4—142no—
366C2×C4⋊C4—142no—
367C2×C4⋊C4—142no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—141no—
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C22⋊C4—71no—
381C2×C22⋊C4—71no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—141no—
384C2×C42—71no—
385C2×C42—141no—
386C2×C42—141no—
387C2×C42C22×D711yescharacteristic
388#107—22no—
389#107—74no—
390#107—74no—
391#107—142no—
392#106—74no—
393#106—142no—
394#106—142no—
395#106C4×D714yescomplemented
396#97—22no—
397#97—142no—
398C4×C8—142no—
399C4×C8—142no—
400#95—21no—
401#95—21no—
402#95—141no—
403#95—141no—
404#95—141no—
405#95—141no—
406#95—141no—
407#95—141no—
408C22×C7C2×C4×D411yescharacteristic
409C22×C7C2×#12111yescharacteristic
410C22×C7C2×C4⋊Q811yescharacteristic
411C22×C7#57611yescharacteristic
412C22×C7#57711yescharacteristic
413C22×C7#57711yescharacteristic
414C22×C7#57811yescharacteristic
415C2×D7—22no—
416C2×D7—22no—
417C2×D7—22no—
418C2×D7—22no—
419C2×D7—22no—
420C2×D7—22no—
421C2×D7#38914yes—
422C4×C7—42no—
423C4×C7—42no—
424C4×C7—41no—
425C4×C7—41no—
426C4×C7—81no—
427C4×C7—81no—
428C4×C7C22×C4⋊C411yescharacteristic
429C4×C7C2×C4×Q811yescharacteristic
430C4×C7C2×C22⋊Q811yescharacteristic
431C4×C7C2×#12511yescharacteristic
432Dic7—42no—
433Dic7—42no—
434Dic7—41no—
435Dic7—41no—
436Dic7—81no—
437Dic7—81no—
438Dic7#38914yes—
439C24—71no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—71no—
445C22×C4—72no—
446C22×C4—72no—
447C22×C4—72no—
448C22×C4—72no—
449C22×C4—72no—
450C22×C4—72no—
451C22×C4—71no—
452C22×C4—71no—
453C22×C4—144no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—282no—
459C22×C4—561no—
460C22×C4—561no—
461C22×C4—561no—
462C22×C4—561no—
463C22×C4D4×D711yescharacteristic
464C22×C4D4×D711yescharacteristic
465C22×C4C23×D711yescharacteristic
466M16—716no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—72no—
470C2×C8—72no—
471C2×C8—72no—
472C2×C8—72no—
473C2×C8—148no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8C2×C4×D712yes—
477C2×C8D4×D712yes—
478C2×C8Q8×D712yes—
479C2×C8Q8×D712yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—72no—
487C4⋊C4—72no—
488C4⋊C4—81no—
489C4⋊C4—81no—
490C4⋊C4—144no—
491C4⋊C4—142no—
492C4⋊C4—142no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—282no—
500C4⋊C4—561no—
501C4⋊C4—561no—
502C4⋊C4—561no—
503C4⋊C4—561no—
504C4⋊C4—561no—
505C4⋊C4—561no—
506C4⋊C4C2×C4×D712yes—
507C4⋊C4C2×C4×D712yes—
508C22⋊C4—142no—
509C22⋊C4—142no—
510C22⋊C4—144no—
511C22⋊C4—282no—
512C42—21no—
513C42—21no—
514C42—144no—
515C42—141no—
516C42—141no—
517C42—282no—
518C2×C7C2×#38911yescharacteristic
519C2×C7#294611yescharacteristic
520C2×C7#294711yescharacteristic
521C2×C7#306211yescharacteristic
522C2×C7#306311yescharacteristic
523C2×C7#316311yescharacteristic
524C2×C7#316411yescharacteristic
525D7—24no—
526C23—72no—
527C23—72no—
528C23—72no—
529C23—141no—
530C23—141no—
531C23—141no—
532C23—141no—
533C23C2×D4×D711yescharacteristic
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
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—72no—
553C2×C4—72no—
554C2×C4—72no—
555C2×C4—74no—
556C2×C4—74no—
557C2×C4—74no—
558C2×C4—74no—
559C2×C4—141no—
560C2×C4—141no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—141no—
566C2×C4—141no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—564no—
578C2×C4—564no—
579C2×C4C22×C4×D711yescharacteristic
580C2×C4C2×D4×D711yescharacteristic
581C2×C4C2×Q8×D711yescharacteristic
582C2×C4C2×Q8×D711yescharacteristic
583C2×C4D7×#4111yescharacteristic
584C2×C4D7×#4111yescharacteristic
585C8—22no—
586C8—74no—
587C8—142no—
588C8D7×C4⋊C414yes—
589C7#2116011yescharacteristic, complemented, Sylow
590C22—74no—
591C22—142no—
592C22—142no—
593C22—142no—
594C22—142no—
595C22—142no—
596C22—142no—
597C22C4×D4×D711yescharacteristic
598C22D7×#12111yescharacteristic
599C22D7×C4⋊Q811yescharacteristic
600C22#9046511yescharacteristic
601C22#9046611yescharacteristic
602C22#9046911yescharacteristic
603C22#9047011yescharacteristic
604C4—42no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—74no—
609C4—81no—
610C4—81no—
611C4—282no—
612C4—282no—
613C4—281no—
614C4—281no—
615C4—561no—
616C4—561no—
617C4C2×D7×C4⋊C411yescharacteristic
618C4C4×Q8×D711yescharacteristic
619C4D7×C22⋊Q811yescharacteristic
620C4D7×#12511yescharacteristic
621C2—144no—
622C2D7×#38911yescharacteristic
623C2#1173722011yescharacteristic
624C2#1173722111yescharacteristic
625C2#1173831211yescharacteristic
626C2#1173835211yescharacteristic
627C2#1173941911yescharacteristic
628C2#1173942011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1944C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1944D7
hypocenterC7#21160
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1944

Complements

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

NH ≅ G/Ntype
6. C7×#1944C2semidirect
7. #11728365C2semidirect
8. #11728314C2semidirect
9. #11727573C2semidirect
10. #11727373C2semidirect
11. #11727372C2semidirect
14. #11726615C2semidirect
15. #11726614C2semidirect
48. #89633C4semidirect
115. C7×#106C2×C4semidirect
125. #6392C2×C4semidirect
130. #6382C2×C4semidirect
132. #6381C2×C4semidirect
140. #1944D7semidirect
226. #1131C4⋊C4semidirect
395. #106C4×D7semidirect
589. C7#21160semidirect

Maximal subgroups

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

subgroupindex
C2×#896532
C2×#896332
C2×#894802
#117336402
#117334132
C7×#19442
#117283652
#117283142
#117275732
#117273732
#117273722
#117266202
D7×#3402
#117266152
#117266142
#211607