Finite Group Databasebeta

← #13422346

Subgroups — #13422346

Order 1792.

804 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2366C211yesmaximal, characteristic
2#11731133C211yesmaximal, characteristic, complemented
3#11729516C211yesmaximal, characteristic
4#11729516C211yesmaximal, characteristic
5C4×#6406C211yesmaximal, characteristic, complemented
6#11729360C211yesmaximal, characteristic, complemented
7#11729360C211yesmaximal, characteristic, complemented
8#11729359C211yesmaximal, characteristic, complemented
9#11727406C211yesmaximal, characteristic, complemented
10#11727398C211yesmaximal, characteristic, complemented
11#11727395C211yesmaximal, characteristic, complemented
12#11727391C211yesmaximal, characteristic, complemented
13#11727236C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727213C211yesmaximal, characteristic, complemented
16C2×C7×#127C2211yescharacteristic
17C2×C4×C7×D4—21no—
18C2×C4×C7×D4C2211yescharacteristic
19C2×C4×C7×D4C2211yescharacteristic
20C7×#390—21no—
21C7×#390C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#386—21no—
24C7×#386—21no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35#89754—21no—
36#89744—21no—
37C2×#6382C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89708C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41#89700C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89430—21no—
48#89424—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89319C2211yescharacteristic
56#89319C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26755—71nomaximal, Sylow
60C22×C7×D4—21no—
61C22×C7×D4—21no—
62C22×C7×D4C2311yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C7×#127—22no—
67C7×#127—22no—
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C4×C7×D4—41no—
71C4×C7×D4—41no—
72C4×C7×D4—44no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C22⋊C4—21no—
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×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C7×#95—21no—
101C7×#95D411yescharacteristic
102C7×#95D411yescharacteristic
103#6406—22no—
104#6406—22no—
105#6406—22no—
106#6406—22no—
107#6406—22no—
108#6406—22no—
109#6406—41no—
110#6406—41no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118#6382—22no—
119#6381—22no—
120C4⋊(C7⋊C8)—41no—
121C4×C7⋊C8—41no—
122#2509—71no—
123#2366D711yescharacteristic
124#2073—71no—
125#2065—71no—
126#2041—71no—
127#2041—71no—
128#1934—71no—
129#1926—71no—
130#1921—71no—
131#1919—71no—
132#1916—71no—
133#1862—71no—
134#1862—71no—
135#1804—71no—
136C4×#102—71no—
137C24×C7—21no—
138C24×C7—21no—
139C2×C7×D4—24no—
140C2×C7×D4—24no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—42no—
152C2×C7×D4—42no—
153C2×C7×D4—81no—
154C2×C7×D4—81no—
155C2×C7×D4#4112yes—
156C2×C7×D4#4112yes—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7—81no—
164C22×C4×C7—81no—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4#4112yes—
182C7×C22⋊C4—42no—
183C7×C22⋊C4—42no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—81no—
187C7×C22⋊C4—81no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—21no—
193C42×C7—21no—
194C42×C7—42no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7#4112yes—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—81no—
206Dic7⋊C4—81no—
207C4×Dic7—81no—
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—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×#127C2×D711yescharacteristic
221C2×C4×D4—21no—
222C2×C4×D4C2×D711yescharacteristic
223C2×C4×D4C2×D711yescharacteristic
224C2×#107—71no—
225C2×#106—71no—
226C2×C4⋊C8—71no—
227C2×#102—71no—
228C2×#102—71no—
229C2×#102—71no—
230C2×#102—71no—
231C2×#102—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×#102—71no—
235C2×C4×C8—71no—
236#400—141no—
237#400—141no—
238#390—21no—
239#390C2×D711yescharacteristic
240#390C2×D711yescharacteristic
241#389—71no—
242#389—71no—
243#389—71no—
244#389—71no—
245#389—141no—
246#387—141no—
247#386—21no—
248#386—21no—
249#382—71no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4C2×D711yescharacteristic
252C4×C4⋊C4C2×D711yescharacteristic
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#336—71no—
260#336—71no—
261#336—71no—
262#336—71no—
263C23×C7—22no—
264C23×C7—22no—
265C23×C7—41no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7C22×D411yescharacteristic
278C7×D4—22no—
279C7×D4—22no—
280C7×D4—22no—
281C7×D4—22no—
282C7×D4—48no—
283C7×D4—41no—
284C7×D4—41no—
285C7×D4—41no—
286C7×D4—41no—
287C7×D4—84no—
288C2×C4×C7—21no—
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—21no—
296C2×C4×C7—22no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—84no—
311C2×C4×C7—84no—
312C2×C4×C7#13512yes—
313C2×C4×C7AGL(1,ℤ/8)12yes—
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7C2×#4111yescharacteristic
316C2×C4×C7C2×#4111yescharacteristic
317C2×C4×C7C2×#4111yescharacteristic
318C2×C4×C7H(5,𝔽2)11yescharacteristic
319C2×C4×C72−1+411yescharacteristic
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—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C7⋊C8—41no—
333C7⋊C8—41no—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C22×D4—21no—
337C22×D4—21no—
338C22×D4C22×D711yescharacteristic
339C22×D4C22×D711yescharacteristic
340C23×C4—21no—
341C23×C4—21no—
342C22×C8—71no—
343C22×C8—71no—
344C22×C8—71no—
345C22×C8—71no—
346#127—22no—
347#127—22no—
348C4×D4—22no—
349C4×D4—22no—
350C4×D4—41no—
351C4×D4—41no—
352C4×D4—44no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C22⋊D711yescharacteristic
362C2×C4⋊C4C22⋊D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C42—21no—
372C2×C42—141no—
373C2×C42C22×D711yescharacteristic
374C2×C42C22×D711yescharacteristic
375C2×C42C22×D711yescharacteristic
376#107—142no—
377#106—142no—
378C4⋊C8—281no—
379#102—142no—
380#102—142no—
381#102—142no—
382#102—142no—
383#102—142no—
384#102—142no—
385#102—281no—
386#102—281no—
387C4×C8—281no—
388#95—21no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95C22⋊D711yescharacteristic
397#95C22⋊D711yescharacteristic
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—42no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—81no—
405C22×C7—81no—
406C22×C7—81no—
407C22×C7—81no—
408C22×C7—81no—
409C22×C7—81no—
410C22×C7—81no—
411C22×C7—81no—
412C22×C7#52511yescharacteristic
413C22×C7#54611yescharacteristic
414C22×C7#54811yescharacteristic
415C22×C7C2×#13511yescharacteristic
416C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
417C22×C7#57611yescharacteristic
418C22×C7#57711yescharacteristic
419C4×C7—22no—
420C4×C7—22no—
421C4×C7—41no—
422C4×C7—41no—
423C4×C7—41no—
424C4×C7—41no—
425C4×C7—42no—
426C4×C7—81no—
427C4×C7#52911yescharacteristic
428C4×C7#53911yescharacteristic
429C4×C7#55211yescharacteristic
430C4×C7#55311yescharacteristic
431Dic7—42no—
432Dic7—81no—
433Dic7—81no—
434Dic7—81no—
435C24—21no—
436C24—21no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—81no—
452C2×D4—81no—
453C2×D4#115712yes—
454C2×D4#115712yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—141no—
464C22×C4—141no—
465C22×C4—141no—
466C22×C4—141no—
467C22×C4C2×C22⋊D711yescharacteristic
468C22×C4C2×C22⋊D711yescharacteristic
469C22×C4C2×C22⋊D711yescharacteristic
470C22×C4C2×C22⋊D711yescharacteristic
471C22×C4C2×C22⋊D711yescharacteristic
472C22×C4C2×C22⋊D711yescharacteristic
473C22×C4C23×D711yescharacteristic
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—141no—
479C2×C8—141no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—141no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—72no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—282no—
500C4⋊C4—282no—
501C4⋊C4—561no—
502C4⋊C4—561no—
503C4⋊C4#115912yes—
504C22⋊C4—42no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—42no—
508C22⋊C4—81no—
509C22⋊C4—81no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—42no—
517C42—561no—
518C2×C7—82no—
519C2×C7—82no—
520C2×C7#246011yescharacteristic
521C2×C7#310511yescharacteristic
522C2×C7#311811yescharacteristic
523C2×C7#330411yescharacteristic
524C2×C7#331811yescharacteristic
525C2×C7#336111yescharacteristic
526C2×C7#336811yescharacteristic
527C23—22no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23C22×C22⋊D711yescharacteristic
542D4—22no—
543D4—22no—
544D4—22no—
545D4—22no—
546D4—48no—
547D4—41no—
548D4—41no—
549D4—41no—
550D4—41no—
551D4—84no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—84no—
575C2×C4—84no—
576C2×C4—142no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—281no—
582C2×C4—281no—
583C2×C4—281no—
584C2×C4—281no—
585C2×C4—281no—
586C2×C4—281no—
587C2×C4—281no—
588C2×C4#649512yes—
589C2×C4#651312yes—
590C2×C4C2×#115711yescharacteristic
591C2×C4C2×#115711yescharacteristic
592C2×C4C2×#115911yescharacteristic
593C2×C4#654811yescharacteristic
594C2×C4D7×#4111yescharacteristic
595C2×C4#655411yescharacteristic
596C8—281no—
597C8—281no—
598C8—281no—
599C8—281no—
600C7#2675511yescharacteristic, complemented, Sylow
601C22—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#9048211yescharacteristic
616C22C2×#649511yescharacteristic
617C22#9049711yescharacteristic
618C22C2×#651311yescharacteristic
619C22#9052211yescharacteristic
620C22#9052611yescharacteristic
621C22#9052711yescharacteristic
622C4—22no—
623C4—22no—
624C4—41no—
625C4—42no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—282no—
631C4—561no—
632C4—561no—
633C4—561no—
634C4#9022911yescharacteristic
635C4#9023111yescharacteristic
636C4#9023211yescharacteristic
637C4#9025511yescharacteristic
638C2—82no—
639C2—82no—
640C2#1173975411yescharacteristic
641C2#1173981111yescharacteristic
642C2#1173998611yescharacteristic
643C2#1174093011yescharacteristic
644C2#1174101511yescharacteristic
645C2#1174103211yescharacteristic
646C2#1174115811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2366C2
socleC23×C7C22×D4
prime radicalC2×C7×#127C22
hypercenter#2366D7
hypocenterC7#26755
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2366

Complements

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

NH ≅ G/Ntype
2. #11731133C2semidirect
5. C4×#6406C2semidirect
6. #11729360C2semidirect
7. #11729360C2semidirect
8. #11729359C2semidirect
9. #11727406C2semidirect
10. #11727398C2semidirect
11. #11727395C2semidirect
12. #11727391C2semidirect
13. #11727236C2semidirect
14. #11727215C2semidirect
15. #11727213C2semidirect
600. C7#26755semidirect

Maximal subgroups

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

subgroupindex
C7×#23662
#117311332
#117295162
C4×#64062
#117293602
#117293592
#117274062
#117273982
#117273952
#117273912
#117272362
#117272152
#117272132
#267557