Finite Group Databasebeta

← #13426582

Subgroups — #13426582

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×#2363C211yesmaximal, characteristic
2#11730957C211yesmaximal, characteristic, complemented
3#11729434C211yesmaximal, characteristic
4#11729434C211yesmaximal, characteristic
5C4×#6406C211yesmaximal, characteristic, complemented
6#11729360C211yesmaximal, characteristic, complemented
7#11729330C211yesmaximal, characteristic, complemented
8#11729329C211yesmaximal, characteristic, complemented
9#11727406C211yesmaximal, characteristic, complemented
10#11727394C211yesmaximal, characteristic, complemented
11#11727372C211yesmaximal, characteristic, complemented
12#11727356C211yesmaximal, characteristic, complemented
13#11727221C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C2×#6442—21no—
20C7×#390—21no—
21C7×#390C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#388—21no—
24C7×#386—21no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C22⋊C4—21no—
27C4×C7×C22⋊C4C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32#89984C2211yescharacteristic
33#89984C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89725—21no—
41#89707C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89431—21no—
48#89425—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
60C22×C7×D4—21no—
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×C4×C7C2311yescharacteristic
64C7×#121—22no—
65C7×#121—22no—
66C4×C7×D4—22no—
67C4×C7×D4—41no—
68C7×#117—22no—
69C7×#117—41no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4D411yescharacteristic
80C2×C7×C22⋊C4D411yescharacteristic
81C2×C42×C7—21no—
82C2×C42×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100#6442—44no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95D411yescharacteristic
104C7×#95D411yescharacteristic
105#6406—22no—
106#6406—22no—
107#6406—22no—
108#6406—41no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C22⋊(C7⋊C8)—22no—
115C22⋊(C7⋊C8)—41no—
116#6382—22no—
117#6382—22no—
118#6382—41no—
119#6381—22no—
120C4⋊(C7⋊C8)—22no—
121C4×C7⋊C8—41no—
122#2496—71no—
123#2363D711yescharacteristic
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×C7—21no—
138C24×C7C2×D411yescharacteristic
139C2×C7×D4—24no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—81no—
146C2×C7×D4—81no—
147C2×C7×D4#4112yes—
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×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—42no—
158C22×C4×C7—42no—
159C22×C4×C7—81no—
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—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—81no—
175C7×C4⋊C4#4112yes—
176C7×C4⋊C4#4112yes—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—22no—
179C7×C22⋊C4—22no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—42no—
182C7×C22⋊C4—42no—
183C7×C22⋊C4—81no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—42no—
189C42×C7—81no—
190C22⋊Dic7—42no—
191C22⋊Dic7—42no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—42no—
201C4⋊Dic7—42no—
202Dic7⋊C4—42no—
203Dic7⋊C4—42no—
204C4×Dic7—42no—
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—141no—
218C2×#121C2×D711yescharacteristic
219C2×C4×D4C2×D711yescharacteristic
220C2×#117C2×D711yescharacteristic
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—141no—
234#397—141no—
235#390—21no—
236#390C2×D711yescharacteristic
237#389—71no—
238#389—71no—
239#389C2×D711yescharacteristic
240#388—21no—
241#387—141no—
242#386—21no—
243#384—71no—
244#380—71no—
245#380—71no—
246C4×C4⋊C4—71no—
247C4×C4⋊C4C2×D711yescharacteristic
248C4×C22⋊C4—21no—
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—22no—
261C23×C7—21no—
262C23×C7—21no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7—41no—
274C23×C7C22×D411yescharacteristic
275C7×D4—22no—
276C7×D4—22no—
277C7×D4—41no—
278C7×D4—41no—
279C7×D4—84no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
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—41no—
305C2×C4×C7—41no—
306C2×C4×C7—84no—
307C2×C4×C7#13512yes—
308C2×C4×C7AGL(1,ℤ/8)12yes—
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7H(5,𝔽2)11yescharacteristic
314C2×C4×C72−1+411yescharacteristic
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—41no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C7⋊C8—22no—
328C7⋊C8—41no—
329C7⋊C8—41no—
330C7⋊C8—41no—
331C22×D4—21no—
332C22×D4C22×D711yescharacteristic
333C23×C4—21no—
334C23×C4C22×D711yescharacteristic
335C22×C8—71no—
336C22×C8—71no—
337C22×C8—71no—
338C22×C8—71no—
339#126—284no—
340#121—22no—
341#121—22no—
342C4×D4—22no—
343C4×D4—41no—
344#117—22no—
345#117—41no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—141no—
364C2×C22⋊C4—141no—
365C2×C22⋊C4C22⋊D711yescharacteristic
366C2×C22⋊C4C22⋊D711yescharacteristic
367C2×C42—21no—
368C2×C42—21no—
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—141no—
387#95—141no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95C22⋊D711yescharacteristic
392#95C22⋊D711yescharacteristic
393C22×C7—41no—
394C22×C7—41no—
395C22×C7—41no—
396C22×C7—42no—
397C22×C7—41no—
398C22×C7—42no—
399C22×C7—41no—
400C22×C7—42no—
401C22×C7—41no—
402C22×C7—41no—
403C22×C7—81no—
404C22×C7—81no—
405C22×C7—81no—
406C22×C7—81no—
407C22×C7#52511yescharacteristic
408C22×C7#54611yescharacteristic
409C22×C7#54711yescharacteristic
410C22×C7C2×#13511yescharacteristic
411C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
412C22×C7#57711yescharacteristic
413C22×C7#57711yescharacteristic
414C4×C7—22no—
415C4×C7—22no—
416C4×C7—41no—
417C4×C7—42no—
418C4×C7—41no—
419C4×C7—41no—
420C4×C7—42no—
421C4×C7—81no—
422C4×C7#52911yescharacteristic
423C4×C7#53911yescharacteristic
424C4×C7#55311yescharacteristic
425C4×C7#55611yescharacteristic
426Dic7—81no—
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430C24—21no—
431C24C2×C22⋊D711yescharacteristic
432C2×D4—24no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4#115712yes—
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—42no—
451C22×C4—42no—
452C22×C4—81no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—141no—
456C22×C4—141no—
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—81no—
480C4⋊C4—141no—
481C4⋊C4—141no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4#115712yes—
493C4⋊C4#115912yes—
494C22⋊C4—24no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—282no—
502C22⋊C4—282no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42—81no—
509C42—282no—
510C2×C7—81no—
511C2×C7—82no—
512C2×C7#246611yescharacteristic
513C2×C7#310611yescharacteristic
514C2×C7#312311yescharacteristic
515C2×C7#330411yescharacteristic
516C2×C7#330611yescharacteristic
517C2×C7#333711yescharacteristic
518C2×C7#335411yescharacteristic
519C23—22no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23C22×C22⋊D711yescharacteristic
534D4—22no—
535D4—22no—
536D4—41no—
537D4—41no—
538D4—84no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
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—84no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4#651212yes—
579C2×C4#651312yes—
580C2×C4C2×#115711yescharacteristic
581C2×C4C2×#115711yescharacteristic
582C2×C4C2×#115911yescharacteristic
583C2×C4#654811yescharacteristic
584C2×C4D7×#4111yescharacteristic
585C2×C4#655411yescharacteristic
586C8—142no—
587C8—281no—
588C8—281no—
589C8—281no—
590C7#2703911yescharacteristic, complemented, Sylow
591C22—41no—
592C22—41no—
593C22—42no—
594C22—41no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—42no—
600C22—41no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22#9048211yescharacteristic
606C22#9049711yescharacteristic
607C22C2×#651211yescharacteristic
608C22C2×#651311yescharacteristic
609C22#9051511yescharacteristic
610C22#9052711yescharacteristic
611C22#9052711yescharacteristic
612C4—22no—
613C4—22no—
614C4—42no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—81no—
620C4—561no—
621C4—561no—
622C4—561no—
623C4—561no—
624C4#9022911yescharacteristic
625C4#9023311yescharacteristic
626C4#9024911yescharacteristic
627C4#9025511yescharacteristic
628C2—81no—
629C2—82no—
630C2#1173981411yescharacteristic
631C2#1173997411yescharacteristic
632C2#1173999011yescharacteristic
633C2#1174094111yescharacteristic
634C2#1174101511yescharacteristic
635C2#1174105111yescharacteristic
636C2#1174109511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2363C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2363D7
hypocenterC7#27039
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
2. #11730957C2semidirect
5. C4×#6406C2semidirect
6. #11729360C2semidirect
7. #11729330C2semidirect
8. #11729329C2semidirect
9. #11727406C2semidirect
10. #11727394C2semidirect
11. #11727372C2semidirect
12. #11727356C2semidirect
13. #11727221C2semidirect
14. #11727215C2semidirect
15. #11727205C2semidirect
590. C7#27039semidirect

Maximal subgroups

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

subgroupindex
C7×#23632
#117309572
#117294342
C4×#64062
#117293602
#117293302
#117293292
#117274062
#117273942
#117273722
#117273562
#117272212
#117272152
#117272052
#270397