Finite Group Databasebeta

← #13422436

Subgroups — #13422436

Order 1792.

732 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2367C211yesmaximal, characteristic
2#11731127C211yesmaximal, characteristic, complemented
3#11729476C211yesmaximal, characteristic
4#11729476C211yesmaximal, characteristic
5#11729382C211yesmaximal, characteristic, complemented
6#11729380C211yesmaximal, characteristic, complemented
7C4×#6406C211yesmaximal, characteristic
8#11729360C211yesmaximal, characteristic
9#11727428C211yesmaximal, characteristic, complemented
10#11727420C211yesmaximal, characteristic, complemented
11#11727406C211yesmaximal, characteristic
12#11727394C211yesmaximal, characteristic
13#11727236C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727214C211yesmaximal, characteristic, complemented
16C2×C7×#124C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6452—21no—
20C7×#391C2211yescharacteristic
21C7×#390C2211yescharacteristic
22C7×#388—21no—
23C7×#388—21no—
24C7×#385—21no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C4×C7×C22⋊C4—21no—
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36#89755—21no—
37C2×#6382C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89707C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41#89700C2211yescharacteristic
42#89700C2211yescharacteristic
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#89319C2211yescharacteristic
56#89319C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26761—71nomaximal, Sylow
60C22×C7×Q8C2311yescharacteristic
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C7×#124—22no—
64C7×#124—22no—
65C4×C7×Q8—22no—
66C4×C7×Q8—41no—
67C4×C7×D4—22no—
68C4×C7×D4—41no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D411yescharacteristic
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×C42×C7—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×C4×Dic7—21no—
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99#6452—44no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95D411yescharacteristic
104C7×#95D411yescharacteristic
105#6409—22no—
106#6409—22no—
107#6409—22no—
108#6409—41no—
109#6406—22no—
110#6406—22no—
111#6406—22no—
112#6406—41no—
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#2506—71no—
123#2367D711yescharacteristic
124#2071—71no—
125#2065—71no—
126#2042—71no—
127#2042—71no—
128#1933—71no—
129#1927—71no—
130#1921—71no—
131#1920—71no—
132#1916—71no—
133#1863—71no—
134#1862—71no—
135#1803—71no—
136C4×#102—71no—
137C24×C7—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8#4112yes—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4#4112yes—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7—81no—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2411yescharacteristic
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C22×Dic7—21no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—42no—
173C7×C4⋊C4#4112yes—
174C7×C22⋊C4—24no—
175C7×C22⋊C4—24no—
176C7×C22⋊C4—42no—
177C7×C22⋊C4—42no—
178C7×C22⋊C4—81no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—42no—
186C42×C7—81no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—42no—
195C4⋊Dic7—42no—
196C4⋊Dic7#4112yes—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200Dic7⋊C4—42no—
201C4×Dic7—42no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
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×#125—141no—
215C2×#124C2×D711yescharacteristic
216C2×C4×Q8C2×D711yescharacteristic
217C2×C4×D4C2×D711yescharacteristic
218C2×#107—71no—
219C2×#106—71no—
220C2×C4⋊C8—71no—
221C2×#103—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×#103—71no—
225C2×#102—71no—
226C2×#102—71no—
227C2×#102—71no—
228C2×#102—71no—
229C2×C4×C8—71no—
230#400—141no—
231#398—141no—
232#391C2×D711yescharacteristic
233#390C2×D711yescharacteristic
234#389—71no—
235#389—71no—
236#389—71no—
237#389—71no—
238#388—21no—
239#388—21no—
240#387—141no—
241#385—21no—
242#384—71no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4C2×D711yescharacteristic
245C4×C4⋊C4C2×D711yescharacteristic
246C4×C22⋊C4—21no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#336—71no—
254#336—71no—
255#336—71no—
256#336—71no—
257C23×C7—22no—
258C23×C7—41no—
259C23×C7—41no—
260C23×C7—41no—
261C23×C7—41no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7C22×D411yescharacteristic
265C7×Q8—22no—
266C7×Q8—22no—
267C7×Q8—41no—
268C7×Q8—41no—
269C7×D4—22no—
270C7×D4—22no—
271C7×D4—41no—
272C7×D4—41no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—22no—
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—21no—
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—84no—
299C2×C4×C7#13512yes—
300C2×C4×C7AGL(1,ℤ/8)12yes—
301C2×C4×C7C2×#4111yescharacteristic
302C2×C4×C7C2×#4111yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7H(5,𝔽2)11yescharacteristic
306C2×C4×C72−1+411yescharacteristic
307C2×Dic7—22no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C7⋊C8—41no—
322C7⋊C8—41no—
323C22×Q8C22×D711yescharacteristic
324C22×D4C22×D711yescharacteristic
325C23×C4—21no—
326C22×C8—71no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8—71no—
330#125—284no—
331#124—22no—
332#124—22no—
333C4×Q8—22no—
334C4×Q8—41no—
335C4×D4—22no—
336C4×D4—41no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4C22⋊D711yescharacteristic
349C2×C4⋊C4C22⋊D711yescharacteristic
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—141no—
360C2×C42C22×D711yescharacteristic
361C2×C42C22×D711yescharacteristic
362C2×C42C22×D711yescharacteristic
363#107—142no—
364#106—142no—
365C4⋊C8—281no—
366#103—142no—
367#103—142no—
368#103—142no—
369#103—281no—
370#102—142no—
371#102—142no—
372#102—142no—
373#102—281no—
374C4×C8—281no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95C22⋊D711yescharacteristic
384#95C22⋊D711yescharacteristic
385C22×C7—42no—
386C22×C7—42no—
387C22×C7—42no—
388C22×C7—81no—
389C22×C7—81no—
390C22×C7—81no—
391C22×C7—81no—
392C22×C7#52511yescharacteristic
393C22×C7#54611yescharacteristic
394C22×C7#54811yescharacteristic
395C22×C7C2×#13511yescharacteristic
396C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
397C22×C7#57711yescharacteristic
398C22×C7#57811yescharacteristic
399C4×C7—22no—
400C4×C7—22no—
401C4×C7—41no—
402C4×C7—41no—
403C4×C7—42no—
404C4×C7—41no—
405C4×C7—41no—
406C4×C7—42no—
407C4×C7—81no—
408C4×C7#52911yescharacteristic
409C4×C7#53911yescharacteristic
410C4×C7#55211yescharacteristic
411C4×C7#55311yescharacteristic
412Dic7—42no—
413Dic7—81no—
414Dic7—81no—
415Dic7—81no—
416C24—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8#115712yes—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4#115712yes—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—81no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C2×C22⋊D711yescharacteristic
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C23×D711yescharacteristic
446C2×C8—141no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—72no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—282no—
473C4⋊C4—282no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—282no—
477C4⋊C4—282no—
478C4⋊C4#115912yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—81no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42—81no—
492C42—282no—
493C2×C7—82no—
494C2×C7#246011yescharacteristic
495C2×C7#310611yescharacteristic
496C2×C7#312011yescharacteristic
497C2×C7#330011yescharacteristic
498C2×C7#330611yescharacteristic
499C2×C7#336511yescharacteristic
500C2×C7#336711yescharacteristic
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C22×C22⋊D711yescharacteristic
509Q8—22no—
510Q8—22no—
511Q8—41no—
512Q8—41no—
513D4—22no—
514D4—22no—
515D4—41no—
516D4—41no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—142no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4#651212yes—
556C2×C4#651312yes—
557C2×C4C2×#115711yescharacteristic
558C2×C4C2×#115711yescharacteristic
559C2×C4C2×#115911yescharacteristic
560C2×C4#654811yescharacteristic
561C2×C4D7×#4111yescharacteristic
562C2×C4#655411yescharacteristic
563C8—281no—
564C8—281no—
565C8—281no—
566C8—281no—
567C7#2676111yescharacteristic, complemented, Sylow
568C22—42no—
569C22—42no—
570C22—42no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22#9048211yescharacteristic
576C22#9049711yescharacteristic
577C22C2×#651211yescharacteristic
578C22C2×#651311yescharacteristic
579C22#9052211yescharacteristic
580C22#9052711yescharacteristic
581C22#9052911yescharacteristic
582C4—22no—
583C4—22no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—42no—
590C4—81no—
591C4—282no—
592C4—561no—
593C4—561no—
594C4—561no—
595C4#9022911yescharacteristic
596C4#9023111yescharacteristic
597C4#9023211yescharacteristic
598C4#9025511yescharacteristic
599C2—82no—
600C2#1173981111yescharacteristic
601C2#1173997411yescharacteristic
602C2#1174002011yescharacteristic
603C2#1174094111yescharacteristic
604C2#1174097911yescharacteristic
605C2#1174111311yescharacteristic
606C2#1174116211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2367C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2367D7
hypocenterC7#26761
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2367

Complements

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

NH ≅ G/Ntype
2. #11731127C2semidirect
5. #11729382C2semidirect
6. #11729380C2semidirect
9. #11727428C2semidirect
10. #11727420C2semidirect
13. #11727236C2semidirect
14. #11727215C2semidirect
15. #11727214C2semidirect
567. C7#26761semidirect

Maximal subgroups

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

subgroupindex
C7×#23672
#117311272
#117294762
#117293822
#117293802
C4×#64062
#117293602
#117274282
#117274202
#117274062
#117273942
#117272362
#117272152
#117272142
#267617