Finite Group Databasebeta

← #13416849

Subgroups — #13416849

Order 1792.

820 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2454C211yesmaximal, characteristic
2C2×#89835C211yesmaximal, characteristic, complemented
3C2×C4×#1142C211yesmaximal, characteristic
4#11731082C211yesmaximal, characteristic, complemented
5#11729493C211yesmaximal, characteristic, complemented
6#11729360C211yesmaximal, characteristic
7#11727414C211yesmaximal, characteristic
8#11727414C211yesmaximal, characteristic
9#11727408C211yesmaximal, characteristic
10#11727406C211yesmaximal, characteristic
11#11727386C211yesmaximal, characteristic, complemented
12#11727385C211yesmaximal, characteristic, complemented
13#11727372C211yesmaximal, characteristic, complemented
14#11727363C211yesmaximal, characteristic, complemented
15#11727236C211yesmaximal, characteristic, complemented
16C2×C7×#125C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C22×#1142C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#399—21no—
23C7×#396—21no—
24C7×#390C2211yescharacteristic
25C7×#390C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C7×#385—21no—
29C7×#380—21no—
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34#89835—24no—
35C4×#1142—24no—
36#89744—21no—
37#89744—21no—
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6382C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46#89708C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89422—21no—
52#89421—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26375—71nomaximal, Sylow
62C22×C7×D4C2311yescharacteristic
63C23×C4×C7—21no—
64C22×Dic14C2311yescharacteristic
65C7×#125—22no—
66C7×#125—41no—
67C4×C7×D4—22no—
68C4×C7×D4—41no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×#1142—22no—
87C2×#1142—22no—
88C2×#1142Q814yes—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C4×Dic14—22no—
105C4×Dic14—22no—
106C4×Dic14—22no—
107C4×Dic14—22no—
108C4×Dic14—41no—
109C4×Dic14—41no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114#6406—22no—
115#6406—22no—
116#6406—41no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121#6384—22no—
122#6384—22no—
123#6384D414yes—
124#6382—22no—
125#6382—22no—
126#6382—22no—
127#6381—22no—
128#6381—41no—
129C4⋊(C7⋊C8)—22no—
130C4×C7⋊C8—22no—
131C2×#479—71no—
132C2×C4×SD16—71no—
133#2454D711yescharacteristic
134#2351—71no—
135#2092—71no—
136#2085—71no—
137#2065—71no—
138#2059—71no—
139#1941—71no—
140#1916—71no—
141#1880—71no—
142#1880—71no—
143#1865—71no—
144#1863—71no—
145#1862—71no—
146C24×C7—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4C2×Q812yes—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—81no—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2411yescharacteristic
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C22×Dic7—21no—
170C2×Dic14—21no—
171C2×Dic14—21no—
172C2×Dic14—21no—
173C2×Dic14C2×D412yes—
174C2×Dic14C2×Q812yes—
175C2×Dic14#4112yes—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—41no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4—81no—
186C7×C4⋊C4—81no—
187C7×C4⋊C4C2×D412yes—
188C7×C22⋊C4—42no—
189C7×C22⋊C4—42no—
190C7×C22⋊C4—81no—
191C7×C22⋊C4—81no—
192C7×C22⋊C4—81no—
193C42×C7—21no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7—21no—
199#1142—28no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7#4112yes—
206Dic7⋊C4—24no—
207Dic7⋊C4—42no—
208Dic7⋊C4—42no—
209Dic7⋊C4—42no—
210Dic7⋊C4—42no—
211C4×Dic7—24no—
212C4×Dic7—42no—
213C4×Dic7—42no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8C2×D412yes—
225C2×C7⋊C8C2×Q812yes—
226C22×SD16—71no—
227C2×#125C2×D711yescharacteristic
228C2×C4×Q8—71no—
229C2×C4×Q8—71no—
230C2×C4×Q8—71no—
231C2×C4×D4C2×D711yescharacteristic
232#479—144no—
233C4×SD16—144no—
234C2×#107—71no—
235C2×#107—71no—
236C2×#106—71no—
237C2×#106—71no—
238C2×#106—71no—
239C2×C4⋊C8—71no—
240C2×#103—71no—
241C2×#103—71no—
242C2×#103—71no—
243C2×#102—71no—
244C2×#102—71no—
245C2×#102—71no—
246C2×C4×C8—71no—
247#399—21no—
248#396—21no—
249#390C2×D711yescharacteristic
250#390C2×D711yescharacteristic
251#389—71no—
252#389—71no—
253#389—141no—
254#389—141no—
255#389C2×D711yescharacteristic
256#389C2×D711yescharacteristic
257#385—21no—
258#382—71no—
259#382—141no—
260#380—21no—
261C4×C4⋊C4—141no—
262C4×C4⋊C4C2×D711yescharacteristic
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#336—71no—
268#336—71no—
269#336—71no—
270#336—71no—
271C23×C7—22no—
272C23×C7—41no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7C22×D411yescharacteristic
279C7×D4—22no—
280C7×D4—22no—
281C7×D4—41no—
282C7×D4—41no—
283C2×C4×C7—21no—
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—22no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—41no—
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×C7—41no—
309C2×C4×C7—84no—
310C2×C4×C7#13512yes—
311C2×C4×C7C22×D411yescharacteristic
312C2×C4×C7C22×Q811yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7H(5,𝔽2)11yescharacteristic
316C2×C4×C72−1+411yescharacteristic
317C2×Dic7—22no—
318C2×Dic7—22no—
319C2×Dic7—22no—
320C2×Dic7—22no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—41no—
329Dic14—22no—
330Dic14—22no—
331Dic14—22no—
332Dic14C22⋊Q814yes—
333C7⋊C8—22no—
334C7⋊C8—22no—
335C7⋊C8—41no—
336C7⋊C8—41no—
337C22×Q8—71no—
338C22×D4C22×D711yescharacteristic
339C23×C4—21no—
340C2×SD16—74no—
341C2×SD16—142no—
342C2×SD16—142no—
343C22×C8—71no—
344C22×C8—71no—
345C22×C8—71no—
346C22×C8—71no—
347#125—22no—
348#125—41no—
349C4×Q8—142no—
350C4×Q8—142no—
351C4×Q8—142no—
352C4×Q8—142no—
353C4×Q8—281no—
354C4×Q8—281no—
355C4×D4—22no—
356C4×D4—41no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—141no—
365C2×C4⋊C4—141no—
366C2×C4⋊C4—141no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4C22⋊D711yescharacteristic
370C2×C4⋊C4C22⋊D711yescharacteristic
371C2×C4⋊C4C22×D711yescharacteristic
372C2×C4⋊C4C22×D711yescharacteristic
373C2×C4⋊C4C22×D711yescharacteristic
374C2×C22⋊C4—21no—
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C42—141no—
380C2×C42—141no—
381C2×C42—141no—
382C2×C42C22×D711yescharacteristic
383C2×C42C22×D711yescharacteristic
384C2×C42C22×D711yescharacteristic
385#107—142no—
386#107—281no—
387#106—142no—
388#106—142no—
389#106—142no—
390C4⋊C8—142no—
391#103—74no—
392#103—142no—
393#103—142no—
394#102—142no—
395#102—142no—
396#102—281no—
397C4×C8—142no—
398#95—21no—
399#95—21no—
400#95—141no—
401#95—141no—
402#95—141no—
403#95—141no—
404#95C22⋊D711yescharacteristic
405#95C22⋊D711yescharacteristic
406C22×C7—42no—
407C22×C7—42no—
408C22×C7—42no—
409C22×C7—81no—
410C22×C7—81no—
411C22×C7—81no—
412C22×C7—81no—
413C22×C7#52511yescharacteristic
414C22×C7D4211yescharacteristic
415C22×C7D4×Q811yescharacteristic
416C22×C7C2×#13511yescharacteristic
417C22×C7#57511yescharacteristic
418C22×C7#57711yescharacteristic
419C22×C7#57711yescharacteristic
420C4×C7—22no—
421C4×C7—41no—
422C4×C7—41no—
423C4×C7—41no—
424C4×C7—41no—
425C4×C7—41no—
426C4×C7—42no—
427C4×C7—81no—
428C4×C7—81no—
429C4×C7C2×C22⋊Q811yescharacteristic
430C4×C7#53911yescharacteristic
431C4×C7#54711yescharacteristic
432C4×C7#55411yescharacteristic
433Dic7—24no—
434Dic7—42no—
435Dic7—81no—
436Dic7—81no—
437C24—21no—
438C2×Q8—72no—
439C2×Q8—72no—
440C2×Q8—72no—
441C2×Q8—141no—
442C2×Q8—141no—
443C2×Q8—141no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4Q8×D712yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—81no—
456C22×C4—141no—
457C22×C4—141no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C2×C22⋊D711yescharacteristic
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C2×C22⋊D711yescharacteristic
465C22×C4C2×C22⋊D711yescharacteristic
466C22×C4C23×D711yescharacteristic
467SD16—148no—
468C2×C8—72no—
469C2×C8—72no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—141no—
479C2×C8—141no—
480C4⋊C4—22no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—72no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—144no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—282no—
499C4⋊C4—282no—
500C4⋊C4—282no—
501C4⋊C4—282no—
502C4⋊C4D4×D712yes—
503C22⋊C4—42no—
504C22⋊C4—42no—
505C22⋊C4—81no—
506C22⋊C4—81no—
507C22⋊C4—81no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—144no—
515C42—282no—
516C42—282no—
517C2×C7—82no—
518C2×C7#244911yescharacteristic
519C2×C7#310211yescharacteristic
520C2×C7#312611yescharacteristic
521C2×C7#327111yescharacteristic
522C2×C7#328211yescharacteristic
523C2×C7#338111yescharacteristic
524C2×C7#338811yescharacteristic
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C22×C22⋊D711yescharacteristic
533Q8—74no—
534Q8—142no—
535Q8—142no—
536Q8—142no—
537D4—22no—
538D4—22no—
539D4—41no—
540D4—41no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4#651312yes—
581C2×C4C2×D4×D711yescharacteristic
582C2×C4#654811yescharacteristic
583C2×C4C2×Q8×D711yescharacteristic
584C2×C4#655111yescharacteristic
585C2×C4D7×#4111yescharacteristic
586C2×C4D7×#4111yescharacteristic
587C8—142no—
588C8—142no—
589C8—281no—
590C8—281no—
591C7#2637511yescharacteristic, complemented, Sylow
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22D4×C22⋊D711yescharacteristic
600C22Q8×C22⋊D711yescharacteristic
601C22C2×#651311yescharacteristic
602C22#9051311yescharacteristic
603C22#9051911yescharacteristic
604C22#9052711yescharacteristic
605C22#9052711yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—81no—
614C4—81no—
615C4—144no—
616C4—282no—
617C4—561no—
618C4—561no—
619C4D7×C22⋊Q811yescharacteristic
620C4#9032211yescharacteristic
621C4#9032711yescharacteristic
622C4#9032911yescharacteristic
623C2—82no—
624C2#1174026211yescharacteristic
625C2#1174058511yescharacteristic
626C2#1174088411yescharacteristic
627C2#1174095411yescharacteristic
628C2#1174102811yescharacteristic
629C2#1174107411yescharacteristic
630C2#1174107711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2454C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2454D7
hypocenterC7#26375
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2454

Complements

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

NH ≅ G/Ntype
2. C2×#89835C2semidirect
4. #11731082C2semidirect
5. #11729493C2semidirect
11. #11727386C2semidirect
12. #11727385C2semidirect
13. #11727372C2semidirect
14. #11727363C2semidirect
15. #11727236C2semidirect
591. C7#26375semidirect

Maximal subgroups

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

subgroupindex
C7×#24542
C2×#898352
C2×C4×#11422
#117310822
#117294932
#117293602
#117274142
#117274082
#117274062
#117273862
#117273852
#117273722
#117273632
#117272362
#263757