Finite Group Databasebeta

← #13412597

Subgroups — #13412597

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
1C2×#89633C211yesmaximal, characteristic
2C2×#89454C211yesmaximal, characteristic, complemented
3#11733668C211yesmaximal, characteristic
4#11733443C211yesmaximal, characteristic
5C7×#1941C211yesmaximal, characteristic, complemented
6#11728332C211yesmaximal, characteristic
7#11728315C211yesmaximal, characteristic, complemented
8#11728310C211yesmaximal, characteristic, complemented
9#11727535C211yesmaximal, characteristic
10#11727359C211yesmaximal, characteristic, complemented
11#11727358C211yesmaximal, characteristic, complemented
12#11726631C211yesmaximal, characteristic
13#11726628C211yesmaximal, characteristic
14#11726591C211yesmaximal, characteristic, complemented
15#11726588C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C2×#6459—21no—
18C2×#6455C2211yescharacteristic
19C2×D7×C4⋊C4C2211yescharacteristic
20C2×#6432C2211yescharacteristic
21C2×#6432C2211yescharacteristic
22C2×C7×#106C2211yescharacteristic
23C2×C7×#106C2211yescharacteristic
24C2×C4×C7×C8C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C2×(C2×D7)⋊C8C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6392C2211yescharacteristic
32C2×#6389C2211yescharacteristic
33#89766—21no—
34#89759C2211yescharacteristic
35#89759C2211yescharacteristic
36#89753—21no—
37#89751—21no—
38#89749C2211yescharacteristic
39#89748C2211yescharacteristic
40#89745C2211yescharacteristic
41#89745C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C4×(C2×D7)⋊C4C2211yescharacteristic
47#89633—24no—
48#89454—24no—
49#89445—21no—
50#89440—21no—
51#89429—21no—
52#89418—21no—
53C7×#340C2211yescharacteristic
54C7×#340C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89319C2211yescharacteristic
61#26074—71nomaximal, Sylow
62C22×C4×D7—21no—
63C22×C4×D7C2311yescharacteristic
64C22×Dic14—21no—
65C22×Dic14—21no—
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)⋊C4—21no—
78C2×(C2×D7)⋊C4D411yescharacteristic
79C2×(C2×D7)⋊C4D411yescharacteristic
80C2×C4⋊Dic7—21no—
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×Dic7⋊C4—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C2×#1131—22no—
96C2×#1131—22no—
97C2×#1131Q814yescomplemented
98#6459—44no—
99#6455—22no—
100#6455—41no—
101D7×C4⋊C4—22no—
102D7×C4⋊C4—41no—
103#6432—22no—
104#6432—22no—
105#6432—41no—
106#6432—41no—
107C7×#106—22no—
108C7×#106—22no—
109C4×C7×C8—22no—
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(C2×D7)⋊C8Q814yescomplemented
121#6392—22no—
122#6392—22no—
123#6392—22no—
124#6392D414yes—
125#6389—22no—
126#6382—22no—
127#6382—41no—
128#6381—22no—
129#6381—41no—
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#1941D711yescharacteristic, complemented
140#1854—71no—
141#1852—71no—
142#1847—71no—
143#1809—71no—
144#1753—71no—
145C23×D7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C2×C4×D7—21no—
161C2×C4×D7—21no—
162C2×C4×D7—21no—
163C2×C4×D7—21no—
164C2×C4×D7—42no—
165C2×C4×D7—42no—
166C2×C4×D7—81no—
167C2×C4×D7C2×Q812yes—
168C2×C4×D7C2×Q812yes—
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×C8C2×D412yes—
178C2×C7×C8C2×Q812yes—
179C2×C7×C8C2×Q812yes—
180C2×C7×C8#4112yes—
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⋊C4—21no—
189C7×C4⋊C4—42no—
190C7×C4⋊C4—42no—
191C42×C7—21no—
192C42×C7—21no—
193(C2×D7)⋊C4—22no—
194(C2×D7)⋊C4—22no—
195(C2×D7)⋊C4—42no—
196(C2×D7)⋊C4—42no—
197(C2×D7)⋊C4—81no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—42no—
205C4⋊Dic7—42no—
206C4⋊Dic7C2×D412yes—
207C4⋊Dic7C2×D412yes—
208Dic7⋊C4—42no—
209Dic7⋊C4—42no—
210Dic7⋊C4—42no—
211Dic7⋊C4—81no—
212Dic7⋊C4—81no—
213C4×Dic7—21no—
214C4×Dic7—21no—
215C4×Dic7—42no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8C2×Q812yes—
221C2×C7⋊C8C2×Q812yes—
222#1131—28no—
223C22×M16—71no—
224C2×C4⋊Q8—71no—
225C2×C4⋊Q8—71no—
226C2×C22⋊Q8—141no—
227C2×#117—71no—
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×#107—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×#106C2×D711yescharacteristic
240C2×#106C2×D711yescharacteristic
241C2×C22⋊C8—71no—
242C2×#97—71no—
243C2×C4×C8C2×D711yescharacteristic
244#398—141no—
245#396—141no—
246#395—141no—
247#393—141no—
248#391—141no—
249#389—71no—
250#389—71no—
251#389C2×D711yescharacteristic
252#389C2×D711yescharacteristic
253#387—141no—
254#385—141no—
255#384—71no—
256#382—71no—
257#380—71no—
258#380—71no—
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×C7C22×D411yescharacteristic
269C22×D7—21no—
270C22×D7—21no—
271C22×D7—21no—
272C22×D7—21no—
273C22×D7—21no—
274C22×D7—21no—
275C22×D7C4⋊Q812yes—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
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×C7AGL(1,ℤ/8)12yes—
293C2×C4×C7#13712yes—
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C22×Q811yescharacteristic
296C2×C4×C7C22×Q811yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7H(5,𝔽2)11yescharacteristic
299C2×C4×C72−1+411yescharacteristic
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—22no—
303C2×Dic7—22no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—41no—
311C2×Dic7—41no—
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×Dic7C4⋊Q812yes—
321C4×D7—22no—
322C4×D7—22no—
323C4×D7—22no—
324C4×D7—22no—
325C4×D7—84no—
326Dic14—84no—
327Dic14—84no—
328C7×C8—22no—
329C7×C8C22⋊Q814yescomplemented
330C7⋊C8—22no—
331C7⋊C8—22no—
332C22×Q8—141no—
333C22×Q8—141no—
334C23×C4—71no—
335C23×C4—141no—
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—142no—
349#117—281no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—142no—
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—71no—
370C2×C22⋊C4—71no—
371C2×C22⋊C4—141no—
372C2×C22⋊C4—141no—
373C2×C22⋊C4—141no—
374C2×C42—71no—
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377#107—74no—
378#107—142no—
379#107—142no—
380#107—142no—
381#107—142no—
382#107—281no—
383#106—22no—
384#106—22no—
385#106—142no—
386#106—281no—
387C22⋊C8—74no—
388#97—142no—
389C4×C8—22no—
390#95—21no—
391#95—21no—
392#95—71no—
393#95—71no—
394#95—141no—
395#95—141no—
396#95—141no—
397#95—141no—
398#95—141no—
399#95—141no—
400C22×C7C2×C4⋊Q811yescharacteristic
401C22×C7D4211yescharacteristic
402C22×C7#54711yescharacteristic
403C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
404C22×C7C2×#13711yescharacteristic
405C22×C7#57611yescharacteristic
406C22×C7#57811yescharacteristic
407C2×D7—22no—
408C2×D7—22no—
409C2×D7—22no—
410C2×D7—41no—
411C2×D7—41no—
412C2×D7—41no—
413C2×D7—41no—
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—81no—
417C4×C7—81no—
418C4×C7—81no—
419C4×C7—81no—
420C4×C7C2×C22⋊Q811yescharacteristic
421C4×C7#54411yescharacteristic
422C4×C7D4×Q811yescharacteristic
423C4×C7#55411yescharacteristic
424Dic7—22no—
425Dic7—41no—
426Dic7—41no—
427Dic7—42no—
428Dic7—42no—
429Dic7—81no—
430Dic7—81no—
431C24—71no—
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—71no—
443C22×C4—72no—
444C22×C4—72no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—282no—
456C22×C4—282no—
457C22×C4—561no—
458C22×C4D4×D711yescharacteristic
459C22×C4D4×D711yescharacteristic
460C22×C4C23×D711yescharacteristic
461M16—148no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—72no—
465C2×C8—72no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8#115712yes—
471C2×C8D4×D712yes—
472C2×C8Q8×D712yes—
473C2×C8Q8×D712yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—72no—
485C4⋊C4—72no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—282no—
493C4⋊C4—282no—
494C4⋊C4—282no—
495C4⋊C4—282no—
496C4⋊C4—282no—
497C4⋊C4—561no—
498C4⋊C4—561no—
499C22⋊C4—142no—
500C22⋊C4—142no—
501C22⋊C4—282no—
502C22⋊C4—282no—
503C22⋊C4—561no—
504C42—21no—
505C42—21no—
506C42—141no—
507C42—141no—
508C42—282no—
509C2×C7#244011yescharacteristic
510C2×C7#316411yescharacteristic
511C2×C7#316511yescharacteristic
512C2×C7#326411yescharacteristic
513C2×C7#327011yescharacteristic
514C2×C7#335311yescharacteristic
515C2×C7#335711yescharacteristic
516D7—42no—
517C23—72no—
518C23—141no—
519C23—141no—
520C23—141no—
521C23—141no—
522C23—141no—
523C23—141no—
524C23C2×D4×D711yescharacteristic
525Q8—564no—
526Q8—564no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
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—72no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—142no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—141no—
551C2×C4—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—142no—
557C2×C4—142no—
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—564no—
569C2×C4#647712yes—
570C2×C4#647812yes—
571C2×C4C2×#115711yescharacteristic
572C2×C4C2×D4×D711yescharacteristic
573C2×C4C2×Q8×D711yescharacteristic
574C2×C4C2×Q8×D711yescharacteristic
575C2×C4#655311yescharacteristic
576C2×C4#655411yescharacteristic
577C8—22no—
578C8—142no—
579C8—142no—
580C8#645914yescomplemented
581C7#2607411yescharacteristic, complemented, Sylow
582C22—142no—
583C22—142no—
584C22—142no—
585C22—281no—
586C22—281no—
587C22—281no—
588C22—281no—
589C22#9024211yescharacteristic
590C22#9030011yescharacteristic
591C22D7×C4⋊Q811yescharacteristic
592C22C2×#647711yescharacteristic
593C22C2×#647811yescharacteristic
594C22#9046011yescharacteristic
595C22#9046211yescharacteristic
596C4—22no—
597C4—22no—
598C4—81no—
599C4—81no—
600C4—81no—
601C4—81no—
602C4—142no—
603C4—282no—
604C4—282no—
605C4—281no—
606C4—281no—
607C4—561no—
608C4—561no—
609C4C2×#645911yescharacteristic
610C4#9027011yescharacteristic
611C4#9032311yescharacteristic
612C4#9038011yescharacteristic
613C2—282no—
614C2#1173344311yescharacteristic
615C2#1173713711yescharacteristic
616C2#1173713811yescharacteristic
617C2#1173811911yescharacteristic
618C2#1173813211yescharacteristic
619C2#1173933811yescharacteristic
620C2#1173936311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1941C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenter#1941D7
hypocenterC7#26074
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
2. C2×#89454C2semidirect
5. C7×#1941C2semidirect
7. #11728315C2semidirect
8. #11728310C2semidirect
10. #11727359C2semidirect
11. #11727358C2semidirect
14. #11726591C2semidirect
15. #11726588C2semidirect
97. C2×#1131Q8semidirect
120. (C2×D7)⋊C8Q8semidirect
139. #1941D7semidirect
329. C7×C8C22⋊Q8semidirect
580. C8#6459semidirect
581. C7#26074semidirect

Maximal subgroups

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

subgroupindex
C2×#896332
C2×#894542
#117336682
#117334432
C7×#19412
#117283322
#117283152
#117283102
#117275352
#117273592
#117273582
#117266312
#117266282
#117265912
#117265882
#260747