Finite Group Databasebeta

← #13426584

Subgroups — #13426584

Order 1792.

762 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2496C211yesmaximal, characteristic, complemented
2#11730914C211yesmaximal, characteristic
3#11727383C211yesmaximal, characteristic, complemented
4#11727379C211yesmaximal, characteristic, complemented
5#11727372C211yesmaximal, characteristic, complemented
6#11727357C211yesmaximal, characteristic, complemented
7#11727339C211yesmaximal, characteristic, complemented
8#11727322C211yesmaximal, characteristic, complemented
9#11727278C211yesmaximal, characteristic
10#11727278C211yesmaximal, characteristic
11#11727260C211yesmaximal, characteristic, complemented
12#11727257C211yesmaximal, characteristic, complemented
13C4×#6383C211yesmaximal, characteristic, complemented
14#11727214C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic, complemented
16C2×C7×#126—21no—
17C2×#6493C2211yescharacteristic
18C2×#6489C2211yescharacteristic
19C2×C4×D28C2211yescharacteristic
20C7×#400—21no—
21C7×#397—21no—
22C7×#389C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#387—21no—
25C7×#384C2211yescharacteristic
26C7×#380C2211yescharacteristic
27C7×#380C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C22⋊C4C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×#6383C2211yescharacteristic
33C2×#6383C2211yescharacteristic
34C2×#6383C2211yescharacteristic
35C2×#6383C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89729—21no—
41#89728—21no—
42#89711C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89441—21no—
48#89438—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#27039—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C22×C22⋊D7—21no—
62C22×D28C2311yescharacteristic
63C22×C4×D7—21no—
64C7×#126—44no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4D411yescharacteristic
76C2×C7×C22⋊C4D411yescharacteristic
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—21no—
81#6493—22no—
82#6493—22no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4—21no—
86C2×(C2×D7)⋊C4—21no—
87#6489—22no—
88#6489—41no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C4×D28—22no—
100C4×D28—41no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95D411yescharacteristic
107C7×#95D411yescharacteristic
108C7⋊#95—21no—
109C7⋊#95—21no—
110C22⋊(C7⋊C8)—22no—
111C22⋊(C7⋊C8)—41no—
112#6383—22no—
113#6383—22no—
114#6383—22no—
115#6383—41no—
116#6382—22no—
117#6382—22no—
118#6382—41no—
119#6381—22no—
120C4⋊(C7⋊C8)—22no—
121C4×C7⋊C8—41no—
122#2496D711yescharacteristic, complemented
123#2363—71no—
124#2071—71no—
125#2065—71no—
126#2034—71no—
127#2034—71no—
128#1948—71no—
129#1941—71no—
130#1923—71no—
131#1921—71no—
132#1920—71no—
133#1862—71no—
134#1854—71no—
135#1825—71no—
136C4×#102—71no—
137C24×C7C2×D411yescharacteristic
138C23×D7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C2×C22⋊D7—24no—
156C2×C22⋊D7—81no—
157C2×C22⋊D7—81no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C2×D28—21no—
162C2×D28—21no—
163C2×D28—21no—
164C2×D28—21no—
165C2×D28—21no—
166C2×D28#4112yes—
167C2×C4×D7—42no—
168C2×C4×D7—42no—
169C2×C4×D7—81no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4—42no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—42no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—42no—
191C22⋊Dic7—42no—
192(C2×D7)⋊C4—24no—
193(C2×D7)⋊C4—42no—
194(C2×D7)⋊C4—42no—
195(C2×D7)⋊C4—81no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7#4112yes—
201C4⋊Dic7#4112yes—
202Dic7⋊C4—81no—
203C4×Dic7—42no—
204C4×Dic7—81no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8#4112yes—
217C2×#126—21no—
218C2×#121—71no—
219C2×C4×D4—71no—
220C2×#117—71no—
221C2×#107—71no—
222C2×#106—71no—
223C2×#106—71no—
224C2×#106—71no—
225C2×C4⋊C8—71no—
226C2×#102—71no—
227C2×#102—71no—
228C2×#102—71no—
229C2×#102—71no—
230C2×C22⋊C8—71no—
231C2×C22⋊C8—71no—
232C2×C4×C8—71no—
233#400—21no—
234#397—21no—
235#390—71no—
236#390—141no—
237#389—71no—
238#389C2×D711yescharacteristic
239#389C2×D711yescharacteristic
240#388—141no—
241#387—21no—
242#386—141no—
243#384C2×D711yescharacteristic
244#380C2×D711yescharacteristic
245#380C2×D711yescharacteristic
246C4×C4⋊C4—71no—
247C4×C4⋊C4C2×D711yescharacteristic
248C4×C22⋊C4—141no—
249C4×C22⋊C4C2×D711yescharacteristic
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#336—71no—
259#336—71no—
260C23×C7—21no—
261C23×C7—21no—
262C23×C7—21no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7C22×D411yescharacteristic
268C22×D7—22no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C22×D7—41no—
274C22×D7—41no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
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×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—41no—
303C2×C4×C7—41no—
304C2×C4×C7#13512yes—
305C2×C4×C7AGL(1,ℤ/8)12yes—
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7H(5,𝔽2)11yescharacteristic
311C2×C4×C72−1+411yescharacteristic
312C22⋊D7—84no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322D28—22no—
323D28—22no—
324D28—41no—
325D28—41no—
326C4×D7—84no—
327C7⋊C8—22no—
328C7⋊C8—41no—
329C7⋊C8—41no—
330C7⋊C8—41no—
331C22×D4—71no—
332C22×D4—141no—
333C23×C4—141no—
334C23×C4C22×D711yescharacteristic
335C22×C8—71no—
336C22×C8—71no—
337C22×C8—71no—
338C22×C8—71no—
339#126—44no—
340#121—142no—
341#121—142no—
342C4×D4—142no—
343C4×D4—281no—
344#117—142no—
345#117—281no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—141no—
361C2×C22⋊C4—141no—
362C2×C22⋊C4—141no—
363C2×C22⋊C4—141no—
364C2×C22⋊C4—141no—
365C2×C22⋊C4C22⋊D711yescharacteristic
366C2×C22⋊C4C22⋊D711yescharacteristic
367C2×C42—21no—
368C2×C42—141no—
369C2×C42—141no—
370C2×C42C22×D711yescharacteristic
371C2×C42C22×D711yescharacteristic
372#107—142no—
373#106—142no—
374#106—142no—
375#106—281no—
376C4⋊C8—142no—
377#102—142no—
378#102—142no—
379#102—142no—
380#102—281no—
381C22⋊C8—142no—
382C22⋊C8—281no—
383C4×C8—281no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—141no—
390#95—141no—
391#95C22⋊D711yescharacteristic
392#95C22⋊D711yescharacteristic
393C22×C7—41no—
394C22×C7—41no—
395C22×C7—41no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7—41no—
400C22×C7#52511yescharacteristic
401C22×C7#54611yescharacteristic
402C22×C7#54711yescharacteristic
403C22×C7C2×#13511yescharacteristic
404C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
405C22×C7#57711yescharacteristic
406C22×C7#57711yescharacteristic
407C2×D7—42no—
408C2×D7—42no—
409C2×D7—42no—
410C2×D7—81no—
411C2×D7—81no—
412C2×D7—81no—
413C2×D7—81no—
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—41no—
417C4×C7—41no—
418C4×C7—41no—
419C4×C7—81no—
420C4×C7—81no—
421C4×C7—81no—
422C4×C7—81no—
423C4×C7#52911yescharacteristic
424C4×C7#53911yescharacteristic
425C4×C7#55311yescharacteristic
426C4×C7#55611yescharacteristic
427Dic7—42no—
428Dic7—42no—
429Dic7—81no—
430C24—141no—
431C24C2×C22⋊D711yescharacteristic
432C2×D4—72no—
433C2×D4—144no—
434C2×D4—141no—
435C2×D4—141no—
436C2×D4—141no—
437C2×D4—141no—
438C2×D4—141no—
439C2×D4—561no—
440C2×D4—561no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—141no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—282no—
455C22×C4—282no—
456C22×C4—561no—
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C2×C22⋊D711yescharacteristic
459C22×C4C2×C22⋊D711yescharacteristic
460C22×C4C2×C22⋊D711yescharacteristic
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C23×D711yescharacteristic
463C2×C8—72no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—72no—
488C4⋊C4—72no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—561no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—42no—
497C22⋊C4—42no—
498C22⋊C4—144no—
499C22⋊C4—282no—
500C22⋊C4—282no—
501C22⋊C4—282no—
502C22⋊C4—561no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—282no—
509C42—561no—
510C2×C7—81no—
511C2×C7#246611yescharacteristic
512C2×C7#310611yescharacteristic
513C2×C7#312311yescharacteristic
514C2×C7#330411yescharacteristic
515C2×C7#330611yescharacteristic
516C2×C7#333711yescharacteristic
517C2×C7#335411yescharacteristic
518D7—82no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—142no—
527C23—281no—
528C23—281no—
529C23—281no—
530C23—281no—
531C23—281no—
532C23—281no—
533C23C22×C22⋊D711yescharacteristic
534D4—142no—
535D4—142no—
536D4—281no—
537D4—281no—
538D4—564no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
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—41no—
568C2×C4—142no—
569C2×C4—142no—
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#649512yes—
579C2×C4#651312yes—
580C2×C4D7×#4111yescharacteristic
581C2×C4D7×#4111yescharacteristic
582C2×C4D7×#4111yescharacteristic
583C2×C4D7×#4111yescharacteristic
584C2×C4#655311yescharacteristic
585C2×C4#655411yescharacteristic
586C8—142no—
587C8—281no—
588C8—281no—
589C8—281no—
590C7#2703911yescharacteristic, complemented, Sylow
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—282no—
599C22—282no—
600C22—282no—
601C22—561no—
602C22—561no—
603C22—561no—
604C22—561no—
605C22C2×#649511yescharacteristic
606C22C2×#651311yescharacteristic
607C22#9051511yescharacteristic
608C22#9051911yescharacteristic
609C22#9052011yescharacteristic
610C22#9052811yescharacteristic
611C22#9052811yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—282no—
622C4—282no—
623C4—561no—
624C4#9039311yescharacteristic
625C4#9039711yescharacteristic
626C4#9039811yescharacteristic
627C4#9040011yescharacteristic
628C2—81no—
629C2—562no—
630C2#1174055011yescharacteristic
631C2#1174074411yescharacteristic
632C2#1174089211yescharacteristic
633C2#1174096611yescharacteristic
634C2#1174097111yescharacteristic
635C2#1174103611yescharacteristic
636C2#1174111811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2496C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenter#2496D7
hypocenterC7#27039
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2496

Complements

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

NH ≅ G/Ntype
1. C7×#2496C2semidirect
3. #11727383C2semidirect
4. #11727379C2semidirect
5. #11727372C2semidirect
6. #11727357C2semidirect
7. #11727339C2semidirect
8. #11727322C2semidirect
11. #11727260C2semidirect
12. #11727257C2semidirect
13. C4×#6383C2semidirect
14. #11727214C2semidirect
15. #11727205C2semidirect
122. #2496D7semidirect
590. C7#27039semidirect

Maximal subgroups

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

subgroupindex
C7×#24962
#117309142
#117273832
#117273792
#117273722
#117273572
#117273392
#117273222
#117272782
#117272602
#117272572
C4×#63832
#117272142
#117272052
#270397