Finite Group Databasebeta

← #13357353

Subgroups — #13357353

Order 1792.

882 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89665C211yesmaximal, characteristic
2#11736653C211yesmaximal, characteristic
3D7×#428C211yesmaximal, characteristic
4C2×#89653C211yesmaximal, characteristic
5#11735993C211yesmaximal, characteristic
6C7×#1952C211yesmaximal, characteristic, complemented
7#11728454C211yesmaximal, characteristic, complemented
8#11728423C211yesmaximal, characteristic, complemented
9#11728422C211yesmaximal, characteristic, complemented
10#11728379C211yesmaximal, characteristic, complemented
11#11727647C211yesmaximal, characteristic, complemented
12#11726743C211yesmaximal, characteristic
13D7×#343C211yesmaximal, characteristic
14#11726739C211yesmaximal, characteristic, complemented
15#11726738C211yesmaximal, characteristic, complemented
16C2×D7×M16C2211yescharacteristic
17C2×D7×M16C2211yescharacteristic
18C2×D7×M16C2211yescharacteristic
19C22×#1131C2211yescharacteristic
20D7×#117C2211yescharacteristic
21C2×#6455C2211yescharacteristic
22C2×D7×C4⋊C4C2211yescharacteristic
23C2×C7×#108C2211yescharacteristic
24C7×#428C2211yescharacteristic
25C7×#428C2211yescharacteristic
26C2×C7×#107C2211yescharacteristic
27C7×#405C2211yescharacteristic
28C2×#6396C2211yescharacteristic
29C2×#6396C2211yescharacteristic
30#89894C2211yescharacteristic
31#89893C2211yescharacteristic
32#89893C2211yescharacteristic
33C2×#6393C2211yescharacteristic
34C2×#6391C2211yescharacteristic
35#89873C2211yescharacteristic
36#89769C2211yescharacteristic
37#89767C2211yescharacteristic
38#89767C2211yescharacteristic
39#89736C2211yescharacteristic
40#89736C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43#89665C418yescomplemented
44#89653—21no—
45#89653—21no—
46#89653—21no—
47#89653—21no—
48#89653—21no—
49#89653—21no—
50#89652—21no—
51D7×#107—21no—
52#89633—21no—
53#89633—21no—
54#89632—21no—
55D7×#106—21no—
56#89480—21no—
57#89479—21no—
58#89461—21no—
59C4×#1131—21no—
60C7×#343C2211yescharacteristic
61C7×#343C2211yescharacteristic
62#89349C2211yescharacteristic
63#89349C2211yescharacteristic
64#89349C2211yescharacteristic
65#89349C2211yescharacteristic
66#89325C2211yescharacteristic
67#89325C2211yescharacteristic
68#21182—71nomaximal, Sylow
69C22×C4×D7—21no—
70C22×C4×D7C2311yescharacteristic
71C2×C7×M16C2311yescharacteristic
72C2×C7×M16C2311yescharacteristic
73C2×C7×M16C2311yescharacteristic
74C22×C7×C8C2311yescharacteristic
75C7×#117C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×(C2×D7)⋊C4—21no—
79#6489C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×C4×Dic7—21no—
84C2×#1136C2311yescharacteristic
85C2×#1136C2311yescharacteristic
86C2×#1136C2311yescharacteristic
87C22×C7⋊C8C2311yescharacteristic
88D7×M16—22no—
89D7×M16—22no—
90D7×M16—22no—
91D7×M16C2×C414yes—
92D7×M16C2×C414yes—
93D7×M16Q814yes—
94C2×#1131—21no—
95C2×#1131—21no—
96C2×#1131—21no—
97C2×#1131C2×C414yes—
98C2×#1131D411yescharacteristic
99C2×#1131D411yescharacteristic
100C2×#1131D411yescharacteristic
101C2×#1131D411yescharacteristic
102C2×#1131Q814yes—
103C2×C8×D7—21no—
104C2×C8×D7—21no—
105C2×C8×D7—21no—
106#6455—21no—
107#6455—21no—
108#6455—21no—
109#6455—21no—
110#6455D411yescharacteristic
111#6455D411yescharacteristic
112D7×C4⋊C4—21no—
113D7×C4⋊C4—21no—
114D7×C4⋊C4—21no—
115D7×C4⋊C4—21no—
116D7×C4⋊C4D411yescharacteristic
117D7×C4⋊C4D411yescharacteristic
118D7×C22⋊C4—21no—
119#6440—21no—
120#6435—21no—
121C42×D7—21no—
122C7×#108C2×C414yescomplemented
123C7×#107—21no—
124C7×#107—21no—
125C7×#107—21no—
126C7×#107—21no—
127C7×#106—21no—
128C7×#106—21no—
129C7×#97—21no—
130C4×C7×C8—21no—
131#6396C2×C414yescomplemented
132#6396C2×C414yescomplemented
133#6393C2×C414yescomplemented
134#6392—21no—
135#6392—21no—
136#6391—21no—
137#6391—21no—
138#6391—21no—
139#6391—21no—
140#6389—21no—
141#6389—21no—
142C8×Dic7—21no—
143C8×Dic7—21no—
144#6382—21no—
145#6382—21no—
146#6382—21no—
147#6382—21no—
148#6382—21no—
149#6382—21no—
150#6381—21no—
151#6381—21no—
152#6381—21no—
153#6381—21no—
154#6381—21no—
155#6381—21no—
156#6377—21no—
157C4×C7⋊C8—21no—
158C2×#430—71no—
159#2911—71no—
160C2×#428—71no—
161C2×#428—71no—
162#2873—71no—
163#1952—71no—
164#1952—71no—
165#1952—71no—
166#1952D711yescharacteristic, complemented
167#1951—71no—
168#1951—71no—
169#1951—71no—
170#1951—71no—
171#1742—71no—
172C2×#343—71no—
173C23×D7#4111yescharacteristic
174C22×C4×C7—21no—
175C22×C4×C7—21no—
176C22×C4×C7C2411yescharacteristic
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7#4111yescharacteristic
180C2×C4×D7—21no—
181C2×C4×D7—21no—
182C2×C4×D7—41no—
183C2×C4×D7—41no—
184C2×C4×D7—82no—
185C2×C4×D7C22×C412yes—
186C2×C4×D7C2×D411yescharacteristic
187C2×C4×D7C2×D411yescharacteristic
188C2×C4×D7C2×D411yescharacteristic
189C2×C4×D7C2×D411yescharacteristic
190C2×C4×D7C2×Q812yes—
191C2×C4×D7C2×Q812yes—
192C2×C4×D7#4112yes—
193C7×M16—21no—
194C7×M16—21no—
195C7×M16—21no—
196C7×M16C22×C412yes—
197C7×M16C22×C412yes—
198C7×M16C2×Q812yes—
199C2×C7×C8—21no—
200C2×C7×C8—21no—
201C2×C7×C8—21no—
202C2×C7×C8C22×C412yes—
203C2×C7×C8C2×D411yescharacteristic
204C2×C7×C8C2×D411yescharacteristic
205C2×C7×C8C2×Q812yes—
206C7×C4⋊C4—21no—
207C7×C4⋊C4—21no—
208C7×C4⋊C4—21no—
209C7×C4⋊C4—21no—
210C7×C4⋊C4C2×D411yescharacteristic
211C7×C4⋊C4C2×D411yescharacteristic
212C7×C22⋊C4—21no—
213C42×C7—21no—
214C22⋊Dic7—21no—
215(C2×D7)⋊C4—21no—
216(C2×D7)⋊C4—21no—
217(C2×D7)⋊C4—41no—
218(C2×D7)⋊C4—41no—
219C4⋊Dic7—21no—
220C4⋊Dic7—21no—
221C4⋊Dic7—21no—
222C4⋊Dic7—21no—
223C4⋊Dic7C2×D411yescharacteristic
224C4⋊Dic7C2×D411yescharacteristic
225Dic7⋊C4—21no—
226Dic7⋊C4—21no—
227Dic7⋊C4—41no—
228Dic7⋊C4—41no—
229C4×Dic7—21no—
230C4×Dic7—21no—
231C4×Dic7—21no—
232C4×Dic7—41no—
233C4×Dic7—41no—
234#1136—21no—
235#1136—21no—
236#1136—21no—
237#1136C22×C412yes—
238#1136C22×C412yes—
239#1136C2×Q812yes—
240C2×C7⋊C8—21no—
241C2×C7⋊C8—21no—
242C2×C7⋊C8—21no—
243C2×C7⋊C8C22×C412yes—
244C2×C7⋊C8C2×D411yescharacteristic
245C2×C7⋊C8C2×D411yescharacteristic
246C2×C7⋊C8C2×Q812yes—
247#1131—24no—
248#1131—24no—
249#1131—24no—
250#1131C4⋊C414yes—
251#1131C4⋊C414yes—
252#1131C4⋊C414yes—
253#1131C4⋊C414yes—
254C8×D7—24no—
255C8×D7—24no—
256C8×D7—24no—
257C22×M16—71no—
258C22×M16—71no—
259C22×M16—71no—
260C22×M16—71no—
261C2×#117—71no—
262C2×#117—71no—
263C22×C4⋊C4—71no—
264#430—78no—
265C2×#108—71no—
266C2×#108—71no—
267C2×#108—71no—
268C2×#108C2×D711yescharacteristic
269#428—71no—
270#428—71no—
271#428—71no—
272#428—71no—
273#428—71no—
274#428—71no—
275#428—141no—
276#428—141no—
277#428—141no—
278#428—141no—
279#428—141no—
280#428—141no—
281#428—141no—
282#428—141no—
283#428C2×D711yescharacteristic
284#428C2×D711yescharacteristic
285#427—141no—
286#427—141no—
287C2×#107—71no—
288C2×#107—141no—
289C2×#107C2×D711yescharacteristic
290C2×#106—71no—
291C2×#106—71no—
292C2×#106—141no—
293#405—71no—
294#405—71no—
295#405—71no—
296#405—141no—
297#405—141no—
298#405C2×D711yescharacteristic
299C4×M16—141no—
300C4×M16—141no—
301#343—71no—
302#343—71no—
303#343—71no—
304#343—71no—
305#343—71no—
306#343—71no—
307#343C2×D711yescharacteristic
308#343C2×D711yescharacteristic
309C23×C7C2×#4111yescharacteristic
310C22×D7—41no—
311C22×D7—41no—
312C22×D7C4×Q812yes—
313C22×D7#12111yescharacteristic
314C22×D7#12111yescharacteristic
315C22×D7#12512yes—
316C2×C4×C7—21no—
317C2×C4×C7—21no—
318C2×C4×C7—41no—
319C2×C4×C7—41no—
320C2×C4×C7—81no—
321C2×C4×C7—81no—
322C2×C4×C7C23×C411yescharacteristic
323C2×C4×C7C22×D411yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×Q811yescharacteristic
326C2×C4×C7C22×Q811yescharacteristic
327C2×C4×C7C2×#4111yescharacteristic
328C2×Dic7—21no—
329C2×Dic7—21no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—81no—
333C2×Dic7—81no—
334C2×Dic7C4×Q812yes—
335C2×Dic7#12111yescharacteristic
336C2×Dic7#12111yescharacteristic
337C2×Dic7#12512yes—
338C4×D7—81no—
339C4×D7—81no—
340C4×D7—82no—
341C4×D7—82no—
342C4×D7C2×C4⋊C412yes—
343C4×D7C2×C4⋊C412yes—
344C4×D7C4×D412yes—
345C4×D7C4×D412yes—
346C4×D7C22⋊Q812yes—
347C4×D7C22⋊Q812yes—
348C4×D7C4⋊Q812yes—
349C4×D7C4⋊Q812yes—
350C7×C8—22no—
351C7×C8—22no—
352C7×C8—22no—
353C7×C8C2×C4⋊C412yes—
354C7×C8C2×C4⋊C412yes—
355C7⋊C8—22no—
356C7⋊C8—22no—
357C7⋊C8—22no—
358C7⋊C8C2×C4⋊C412yes—
359C7⋊C8C2×C4⋊C412yes—
360C23×C4—71no—
361C23×C4—141no—
362C2×M16—74no—
363C2×M16—71no—
364C2×M16—71no—
365C2×M16—71no—
366C2×M16—71no—
367C2×M16—74no—
368C2×M16—71no—
369C2×M16—71no—
370C2×M16—71no—
371C2×M16—74no—
372C2×M16—74no—
373C2×M16—74no—
374C2×M16—141no—
375C2×M16—141no—
376C2×M16—141no—
377C2×M16—142no—
378C2×M16—142no—
379C2×M16—142no—
380C2×M16C22×D711yescharacteristic
381C2×M16C22×D711yescharacteristic
382C2×M16C22×D711yescharacteristic
383C22×C8—71no—
384C22×C8—141no—
385C22×C8—141no—
386C22×C8—141no—
387C22×C8C22×D711yescharacteristic
388#117—71no—
389#117—71no—
390#117—71no—
391#117—141no—
392#117—141no—
393#117—141no—
394#117—141no—
395#117—141no—
396#117—141no—
397#117C22×D711yescharacteristic
398C2×C4⋊C4—71no—
399C2×C4⋊C4—71no—
400C2×C4⋊C4—71no—
401C2×C4⋊C4—71no—
402C2×C4⋊C4—141no—
403C2×C4⋊C4—141no—
404C2×C4⋊C4—141no—
405C2×C4⋊C4—141no—
406C2×C4⋊C4—141no—
407C2×C4⋊C4C22×D711yescharacteristic
408C2×C4⋊C4C22×D711yescharacteristic
409C2×C22⋊C4—141no—
410C2×C22⋊C4—141no—
411C2×C42—141no—
412C2×C42—141no—
413#108—74no—
414#108—74no—
415#108—74no—
416#108C4×D714yescomplemented
417#107—21no—
418#107—21no—
419#107—21no—
420#107—21no—
421#107—141no—
422#107—141no—
423#107—141no—
424#107—141no—
425#107—141no—
426#107—141no—
427#107—141no—
428#107—141no—
429#106—21no—
430#106—21no—
431#106—141no—
432#106—141no—
433#106—141no—
434#106—141no—
435#106—141no—
436#106—141no—
437#106—141no—
438#106—141no—
439#106—141no—
440#106—141no—
441#97—21no—
442#97—141no—
443#97—141no—
444#97—141no—
445C4×C8—21no—
446C4×C8—141no—
447C4×C8—141no—
448C4×C8—141no—
449C22×C7—21no—
450C22×C7—21no—
451C22×C7C2×C4×Q811yescharacteristic
452C22×C7C2×#12111yescharacteristic
453C22×C7C2×#12511yescharacteristic
454C2×D7—21no—
455C2×D7—21no—
456C2×D7—21no—
457C2×D7—21no—
458C2×D7—42no—
459C2×D7—42no—
460C2×D7#38912yes—
461C2×D7#38912yes—
462C4×C7—41no—
463C4×C7—41no—
464C4×C7—81no—
465C4×C7—81no—
466C4×C7C22×C4⋊C411yescharacteristic
467C4×C7C2×C4×D411yescharacteristic
468C4×C7C2×C22⋊Q811yescharacteristic
469C4×C7C2×C4⋊Q811yescharacteristic
470Dic7—41no—
471Dic7—41no—
472Dic7—81no—
473Dic7—81no—
474Dic7#38912yes—
475Dic7#38912yes—
476C24—71no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—71no—
480C22×C4—72no—
481C22×C4—71no—
482C22×C4—71no—
483C22×C4—71no—
484C22×C4—71no—
485C22×C4—72no—
486C22×C4—72no—
487C22×C4—72no—
488C22×C4—141no—
489C22×C4—141no—
490C22×C4—141no—
491C22×C4—141no—
492C22×C4—281no—
493C22×C4—281no—
494C22×C4—562no—
495C22×C4C23×D711yescharacteristic
496M16—21no—
497M16—21no—
498M16—21no—
499M16—74no—
500M16—74no—
501M16—74no—
502M16—74no—
503M16—72no—
504M16—72no—
505M16—72no—
506M16—144no—
507M16—144no—
508M16—144no—
509M16—141no—
510M16—141no—
511M16—141no—
512M16C2×C4×D712yes—
513M16C2×C4×D712yes—
514M16Q8×D712yes—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—72no—
519C2×C8—71no—
520C2×C8—71no—
521C2×C8—72no—
522C2×C8—144no—
523C2×C8—144no—
524C2×C8—144no—
525C2×C8—141no—
526C2×C8—141no—
527C2×C8—141no—
528C2×C8C2×C4×D712yes—
529C2×C8D4×D711yescharacteristic
530C2×C8D4×D711yescharacteristic
531C2×C8Q8×D712yes—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—71no—
537C4⋊C4—71no—
538C4⋊C4—141no—
539C4⋊C4—141no—
540C4⋊C4—141no—
541C4⋊C4—141no—
542C4⋊C4—141no—
543C4⋊C4—141no—
544C4⋊C4—281no—
545C4⋊C4—281no—
546C4⋊C4D4×D711yescharacteristic
547C4⋊C4D4×D711yescharacteristic
548C22⋊C4—21no—
549C22⋊C4—141no—
550C22⋊C4—141no—
551C22⋊C4—141no—
552C22⋊C4—281no—
553C22⋊C4—281no—
554C42—21no—
555C42—141no—
556C42—141no—
557C42—141no—
558C42—281no—
559C42—281no—
560C2×C7—21no—
561C2×C7—21no—
562C2×C7C2×#38911yescharacteristic
563C2×C7#306811yescharacteristic
564C2×C7#306911yescharacteristic
565D7—22no—
566D7—22no—
567C23—71no—
568C23—71no—
569C23—72no—
570C23—72no—
571C23—281no—
572C23—281no—
573C23D7×#4111yescharacteristic
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—72no—
579C2×C4—72no—
580C2×C4—71no—
581C2×C4—71no—
582C2×C4—72no—
583C2×C4—72no—
584C2×C4—72no—
585C2×C4—72no—
586C2×C4—72no—
587C2×C4—72no—
588C2×C4—72no—
589C2×C4—72no—
590C2×C4—81no—
591C2×C4—81no—
592C2×C4—141no—
593C2×C4—141no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—561no—
597C2×C4—561no—
598C2×C4—562no—
599C2×C4—562no—
600C2×C4—561no—
601C2×C4—561no—
602C2×C4C22×C4×D711yescharacteristic
603C2×C4C2×D4×D711yescharacteristic
604C2×C4C2×D4×D711yescharacteristic
605C2×C4C2×Q8×D711yescharacteristic
606C2×C4C2×Q8×D711yescharacteristic
607C2×C4D7×#4111yescharacteristic
608C8—22no—
609C8—22no—
610C8—22no—
611C8—72no—
612C8—72no—
613C8—142no—
614C8—142no—
615C8—142no—
616C8D7×C4⋊C412yes—
617C8D7×C4⋊C412yes—
618C7#2118211yescharacteristic, complemented, Sylow
619C22—21no—
620C22—21no—
621C22—72no—
622C22—72no—
623C22—141no—
624C22—141no—
625C22—141no—
626C22—141no—
627C22—282no—
628C22—282no—
629C22C4×Q8×D711yescharacteristic
630C22D7×#12111yescharacteristic
631C22D7×#12511yescharacteristic
632C4—41no—
633C4—41no—
634C4—72no—
635C4—72no—
636C4—81no—
637C4—81no—
638C4—281no—
639C4—281no—
640C4—561no—
641C4—561no—
642C4C2×D7×C4⋊C411yescharacteristic
643C4C4×D4×D711yescharacteristic
644C4D7×C22⋊Q811yescharacteristic
645C4D7×C4⋊Q811yescharacteristic
646C2—21no—
647C2—21no—
648C2—142no—
649C2—142no—
650C2D7×#38911yescharacteristic
651C2#1173838311yescharacteristic
652C2#1173838911yescharacteristic
subgroupG/H
centerC22D7×#121
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1952C2
socleC22×C7C2×#121
prime radicalC23×D7#41
cosocleC22×C4×C7C24
hypercenter#1952D7
hypocenterC7#21182
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
6. C7×#1952C2semidirect
7. #11728454C2semidirect
8. #11728423C2semidirect
9. #11728422C2semidirect
10. #11728379C2semidirect
11. #11727647C2semidirect
14. #11726739C2semidirect
15. #11726738C2semidirect
43. #89665C4semidirect
122. C7×#108C2×C4semidirect
131. #6396C2×C4semidirect
132. #6396C2×C4semidirect
133. #6393C2×C4semidirect
166. #1952D7semidirect
416. #108C4×D7semidirect
618. C7#21182semidirect

Maximal subgroups

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

subgroupindex
C2×#896652
#117366532
D7×#4282
C2×#896532
#117359932
C7×#19522
#117284542
#117284232
#117284222
#117283792
#117276472
#117267432
D7×#3432
#117267392
#117267382
#211827