Finite Group Databasebeta

← #13412600

Subgroups — #13412600

Order 1792.

826 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739693C211yesmaximal, characteristic
2C2×#89767C211yesmaximal, characteristic
3C2×#89646C211yesmaximal, characteristic, complemented
4#11731344C211yesmaximal, characteristic
5C7×#1854C211yesmaximal, characteristic
6#11728373C211yesmaximal, characteristic, complemented
7#11728319C211yesmaximal, characteristic, complemented
8#11728310C211yesmaximal, characteristic, complemented
9#11727703C211yesmaximal, characteristic
10#11727681C211yesmaximal, characteristic
11#11727267C211yesmaximal, characteristic
12#11726614C211yesmaximal, characteristic, complemented
13#11726612C211yesmaximal, characteristic, complemented
14#11726591C211yesmaximal, characteristic, complemented
15#11726580C211yesmaximal, characteristic, complemented
16C2×C7×#117C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×#6451C2211yescharacteristic
20C2×#6451C2211yescharacteristic
21C2×#6441—21no—
22C2×C7×#107C2211yescharacteristic
23C2×C7×#106C2211yescharacteristic
24C2×C7×C22⋊C8C2211yescharacteristic
25C7×#380C2211yescharacteristic
26#89984C2211yescharacteristic
27C4×C22⋊Dic7C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6389C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89767—24no—
33#89757C2211yescharacteristic
34#89757C2211yescharacteristic
35#89749C2211yescharacteristic
36#89748C2211yescharacteristic
37#89745C2211yescharacteristic
38#89745C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89720—21no—
46#89719—21no—
47#89718—21no—
48#89705—21no—
49#89701—21no—
50#89646—24no—
51#89429—21no—
52#89428—21no—
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89345C2211yescharacteristic
58#89345C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#26074—71nomaximal, Sylow
62C23×C4×C7C2311yescharacteristic
63C23×Dic7—21no—
64C22×Dic14—21no—
65C22×Dic14—21no—
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C7×#117—22no—
69C7×#117—41no—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C42×C7—21no—
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7D411yescharacteristic
79C2×C22⋊Dic7D411yescharacteristic
80C2×C4⋊Dic7—22no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—41no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7C2311yescharacteristic
96C2×C4×Dic7C2311yescharacteristic
97C2×#1136—22no—
98C2×#1136—22no—
99C2×#1136Q814yescomplemented
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6451—22no—
103#6451—22no—
104#6451—41no—
105#6451—41no—
106#6441—44no—
107C7×#107—22no—
108C7×#106—41no—
109C7×C22⋊C8Q814yescomplemented
110C7×#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95D411yescharacteristic
119C7⋊#95D411yescharacteristic
120#6392—22no—
121#6392—41no—
122#6389—22no—
123C8×Dic7—22no—
124#6382—22no—
125#6382—22no—
126#6382—22no—
127#6381—22no—
128#6381—22no—
129#6381D414yes—
130C2×#500—71no—
131C2×#428—71no—
132#2440—71no—
133#2439—71no—
134#2085—71no—
135#2083—71no—
136#2058—71no—
137#2057—71no—
138#1943—71no—
139#1941—71no—
140#1854D711yescharacteristic
141#1852—71no—
142#1847—71no—
143#1809—71no—
144#1753—71no—
145C24×C7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×Q812yes—
156C22×C4×C7C2×Q812yes—
157C22×C4×C7C2411yescharacteristic
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—42no—
163C22×Dic7—42no—
164C22×Dic7—81no—
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C22×Dic7C2×D411yescharacteristic
169C2×Dic14—42no—
170C2×Dic14—42no—
171C2×Dic14—42no—
172C2×Dic14—42no—
173C2×Dic14—81no—
174C2×Dic14—81no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8C2×Q812yes—
180C2×C7×C8C2×Q812yes—
181C7×C4⋊C4—21no—
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⋊C4C2×D412yes—
189C7×C22⋊C4—42no—
190C7×C22⋊C4—42no—
191C42×C7—42no—
192C22⋊Dic7—22no—
193C22⋊Dic7—22no—
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—42no—
204C4⋊Dic7—81no—
205C4⋊Dic7C2×D412yes—
206Dic7⋊C4—42no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209Dic7⋊C4—42no—
210Dic7⋊C4—42no—
211Dic7⋊C4—81no—
212C4×Dic7—21no—
213C4×Dic7—21no—
214C4×Dic7—21no—
215C4×Dic7—21no—
216#1136—28no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8C2×D412yes—
220C2×C7⋊C8C2×Q812yes—
221C2×C7⋊C8C2×Q812yes—
222C2×C7⋊C8#4112yes—
223C22×M16—71no—
224C2×C4⋊Q8—71no—
225C2×C4⋊Q8—71no—
226C2×C22⋊Q8—141no—
227C2×#117C2×D711yescharacteristic
228C22×C4⋊C4—71no—
229#500—144no—
230#428—144no—
231C2×#107—71no—
232C2×#107—71no—
233C2×#107—71no—
234C2×#107—71no—
235C2×#107—71no—
236C2×#107C2×D711yescharacteristic
237C2×#106—71no—
238C2×#106—71no—
239C2×#106—71no—
240C2×#106C2×D711yescharacteristic
241C2×C22⋊C8C2×D711yescharacteristic
242C2×#97—71no—
243C2×C4×C8—71no—
244#398—141no—
245#396—141no—
246#395—141no—
247#393—141no—
248#391—141no—
249#389—71no—
250#389—71no—
251#389—71no—
252#389—71no—
253#387—141no—
254#385—141no—
255#384—71no—
256#382—71no—
257#380—71no—
258#380C2×D711yescharacteristic
259C4×C22⋊C4—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340C2×D711yescharacteristic
265#340C2×D711yescharacteristic
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
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
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×C7—41no—
294C2×C4×C7C4⋊Q812yes—
295C2×C4×C7C22×D411yescharacteristic
296C2×C4×C7C22×Q811yescharacteristic
297C2×C4×C7C22×Q811yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7H(5,𝔽2)11yescharacteristic
300C2×C4×C72−1+411yescharacteristic
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—22no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—41no—
313C2×Dic7—41no—
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—84no—
324C2×Dic7AGL(1,ℤ/8)12yes—
325C2×Dic7#13712yes—
326Dic14—84no—
327Dic14—84no—
328C7×C8—22no—
329C7×C8—22no—
330C7⋊C8—22no—
331C7⋊C8C22⋊Q814yescomplemented
332C22×Q8—141no—
333C22×Q8—141no—
334C23×C4—141no—
335C23×C4C22×D711yescharacteristic
336C2×M16—74no—
337C2×M16—142no—
338C2×M16—142no—
339C22×C8—71no—
340C22×C8—71no—
341C22×C8C22×D711yescharacteristic
342C22×C8C22×D711yescharacteristic
343C4⋊Q8—142no—
344C4⋊Q8—142no—
345C4⋊Q8—281no—
346C4⋊Q8—281no—
347C22⋊Q8—284no—
348#117—22no—
349#117—41no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—142no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4—281no—
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C4⋊C4C22×D711yescharacteristic
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—71no—
372C2×C22⋊C4—71no—
373C2×C22⋊C4—141no—
374C2×C42—21no—
375C2×C42—71no—
376C2×C42—71no—
377#107—22no—
378#107—74no—
379#107—142no—
380#107—142no—
381#107—142no—
382#107—281no—
383#106—41no—
384#106—142no—
385#106—142no—
386#106—142no—
387C22⋊C8Dic1414yescomplemented
388#97—142no—
389C4×C8—142no—
390#95—21no—
391#95—71no—
392#95—71no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398#95—141no—
399#95—141no—
400C22×C7—22no—
401C22×C7—22no—
402C22×C7—22no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—41no—
406C22×C7—41no—
407C22×C7C2×C4⋊Q811yescharacteristic
408C22×C7D4211yescharacteristic
409C22×C7#54711yescharacteristic
410C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
411C22×C7C2×#13711yescharacteristic
412C22×C7#57611yescharacteristic
413C22×C7#57811yescharacteristic
414C4×C7—22no—
415C4×C7—42no—
416C4×C7—81no—
417C4×C7—81no—
418C4×C7—81no—
419C4×C7C2×C22⋊Q811yescharacteristic
420C4×C7#54411yescharacteristic
421C4×C7D4×Q811yescharacteristic
422C4×C7#55411yescharacteristic
423Dic7—22no—
424Dic7—22no—
425Dic7—41no—
426Dic7—41no—
427Dic7—42no—
428Dic7—81no—
429Dic7—81no—
430Dic7—81no—
431C24D4×D711yescharacteristic
432C2×Q8—282no—
433C2×Q8—282no—
434C2×Q8—282no—
435C2×Q8—282no—
436C2×Q8—561no—
437C2×Q8—561no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—282no—
455C22×C4—282no—
456C22×C4—561no—
457C22×C4C2×Dic1412yes—
458C22×C4D4×D711yescharacteristic
459C22×C4Q8×D712yes—
460C22×C4C23×D711yescharacteristic
461M16—148no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—72no—
467C2×C8—72no—
468C2×C8—72no—
469C2×C8—72no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8C2×Dic1412yes—
473C2×C8C2×Dic1412yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—72no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—282no—
496C4⋊C4—561no—
497C4⋊C4—561no—
498C4⋊C4D4×D712yes—
499C22⋊C4—42no—
500C22⋊C4—42no—
501C22⋊C4—142no—
502C22⋊C4—142no—
503C22⋊C4—561no—
504C42—42no—
505C42—141no—
506C42—141no—
507C42—141no—
508C42—141no—
509C2×C7—42no—
510C2×C7#244011yescharacteristic
511C2×C7#316411yescharacteristic
512C2×C7#316511yescharacteristic
513C2×C7#326411yescharacteristic
514C2×C7#327011yescharacteristic
515C2×C7#335311yescharacteristic
516C2×C7#335711yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23#645112yes—
524C23C2×D4×D711yescharacteristic
525Q8—564no—
526Q8—564no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
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—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—72no—
546C2×C4—72no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—142no—
556C2×C4—141no—
557C2×C4—141no—
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—281no—
568C2×C4—281no—
569C2×C4—564no—
570C2×C4#645112yes—
571C2×C4C22×Dic1411yescharacteristic
572C2×C4C2×D4×D711yescharacteristic
573C2×C4#654811yescharacteristic
574C2×C4C2×Q8×D711yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4#655411yescharacteristic
577C8—22no—
578C8—22no—
579C8—74no—
580C8—142no—
581C7#2607411yescharacteristic, complemented, Sylow
582C22—22no—
583C22—22no—
584C22—22no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22C2×#645111yescharacteristic
590C22#9028111yescharacteristic
591C22#9034611yescharacteristic
592C22#9046011yescharacteristic
593C22#9046211yescharacteristic
594C22D7×AGL(1,ℤ/8)11yescharacteristic
595C22D7×#13711yescharacteristic
596C4—22no—
597C4—42no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—142no—
602C4—142no—
603C4—281no—
604C4—281no—
605C4—282no—
606C4—561no—
607C4—561no—
608C4—561no—
609C4D4×Dic1411yescharacteristic
610C4#9023011yescharacteristic
611C4#9031411yescharacteristic
612C4D7×C22⋊Q811yescharacteristic
613C2—42no—
614C2#1173186111yescharacteristic
615C2#1173661811yescharacteristic
616C2#1173664811yescharacteristic
617C2#1173756511yescharacteristic
618C2#1173768711yescharacteristic
619C2#1173881211yescharacteristic
620C2#1173895811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1854C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1854D7
hypocenterC7#26074
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
3. C2×#89646C2semidirect
6. #11728373C2semidirect
7. #11728319C2semidirect
8. #11728310C2semidirect
12. #11726614C2semidirect
13. #11726612C2semidirect
14. #11726591C2semidirect
15. #11726580C2semidirect
99. C2×#1136Q8semidirect
109. C7×C22⋊C8Q8semidirect
331. C7⋊C8C22⋊Q8semidirect
387. C22⋊C8Dic14semidirect
581. C7#26074semidirect

Maximal subgroups

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

subgroupindex
#117396932
C2×#897672
C2×#896462
#117313442
C7×#18542
#117283732
#117283192
#117283102
#117277032
#117276812
#117272672
#117266142
#117266122
#117265912
#117265802
#260747