Finite Group Databasebeta

← #13426507

Subgroups — #13426507

Order 1792.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2360C211yesmaximal, characteristic
2#11730957C211yesmaximal, characteristic
3#11729448C211yesmaximal, characteristic
4#11729448C211yesmaximal, characteristic
5C4×#6409C211yesmaximal, characteristic, complemented
6#11729383C211yesmaximal, characteristic, complemented
7#11729310C211yesmaximal, characteristic
8#11729308C211yesmaximal, characteristic
9#11727436C211yesmaximal, characteristic, complemented
10#11727424C211yesmaximal, characteristic, complemented
11#11727372C211yesmaximal, characteristic, complemented
12#11727356C211yesmaximal, characteristic, complemented
13#11727221C211yesmaximal, characteristic, complemented
14#11727213C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×C4×C7×Q8C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C2×#6442—21no—
20C7×#391C2211yescharacteristic
21C7×#389—21no—
22C7×#389C2211yescharacteristic
23C7×#387—21no—
24C7×#385—21no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C22⋊C4—21no—
27C4×C7×C22⋊C4C2211yescharacteristic
28C2×#6409C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32#89984C2211yescharacteristic
33#89984C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×C22⋊(C7⋊C8)C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
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#27034—71nomaximal, Sylow
60C22×C7×Q8C2311yescharacteristic
61C23×C4×C7—21no—
62C23×C4×C7C2311yescharacteristic
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8—22no—
65C4×C7×Q8—22no—
66C4×C7×Q8—41no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—41no—
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4D411yescharacteristic
78C2×C7×C22⋊C4D411yescharacteristic
79C2×C42×C7—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C22⋊Dic7—21no—
84C2×C22⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—21no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98#6442—44no—
99C7×#95—21no—
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—
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—22no—
119#6382—41no—
120C4⋊(C7⋊C8)—22no—
121C4×C7⋊C8—41no—
122#2496—71no—
123#2360D711yescharacteristic
124#2072—71no—
125#2066—71no—
126#2035—71no—
127#2035—71no—
128#1948—71no—
129#1941—71no—
130#1923—71no—
131#1920—71no—
132#1919—71no—
133#1865—71no—
134#1853—71no—
135#1824—71no—
136C4×#103—71no—
137C24×C7C2×D411yescharacteristic
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8#4112yes—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
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—42no—
155C22×C4×C7—42no—
156C22×C4×C7—81no—
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—24no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—42no—
173C7×C4⋊C4—42no—
174C7×C4⋊C4—81no—
175C7×C4⋊C4#4112yes—
176C7×C4⋊C4#4112yes—
177C7×C22⋊C4—24no—
178C7×C22⋊C4—41no—
179C7×C22⋊C4SD1614yes—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—42no—
185C42×C7—81no—
186C22⋊Dic7—42no—
187C22⋊Dic7—42no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—42no—
197C4⋊Dic7—42no—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200C4×Dic7—42no—
201C2×C7⋊C8—21no—
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#4112yes—
213C2×#126—141no—
214C2×C22⋊Q8C2×D711yescharacteristic
215C2×C4×Q8C2×D711yescharacteristic
216C22×C4⋊C4C2×D711yescharacteristic
217C2×#106—71no—
218C2×#106—71no—
219C2×#106—71no—
220C2×#106—71no—
221C2×C4⋊C8—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×#103—71no—
225C2×#103—71no—
226C2×C22⋊C8—71no—
227C2×C22⋊C8—71no—
228C2×C4×C8—71no—
229#400—141no—
230#397—141no—
231#391C2×D711yescharacteristic
232#389—21no—
233#389—71no—
234#389—71no—
235#389C2×D711yescharacteristic
236#387—21no—
237#387—141no—
238#385—21no—
239#384—71no—
240#380—71no—
241#380—71no—
242C4×C4⋊C4—71no—
243C4×C4⋊C4C2×D711yescharacteristic
244C4×C22⋊C4—21no—
245C4×C22⋊C4C2×D711yescharacteristic
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#336—71no—
255#336—71no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—21no—
262C23×C7C2×SD1612yes—
263C23×C7C22×D411yescharacteristic
264C7×Q8—22no—
265C7×Q8—22no—
266C7×Q8—41no—
267C7×Q8—41no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—22no—
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—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
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—84no—
298C2×C4×C7C2×SD1612yes—
299C2×C4×C7C2×SD1612yes—
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7C2×#4111yescharacteristic
302C2×C4×C7C2×#4111yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7H(5,𝔽2)11yescharacteristic
305C2×C4×C72−1+411yescharacteristic
306C2×Dic7—41no—
307C2×Dic7—41no—
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—
318C7⋊C8—22no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C7⋊C8—41no—
322C22×Q8C22×D711yescharacteristic
323C23×C4—21no—
324C23×C4C22×D711yescharacteristic
325C22×C8—71no—
326C22×C8—71no—
327C22×C8—71no—
328C22×C8—71no—
329#126—284no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C4×Q8—22no—
333C4×Q8—41no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—41no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
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×C22⋊C4—21no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4C22⋊D711yescharacteristic
355C2×C22⋊C4C22⋊D711yescharacteristic
356C2×C42—21no—
357C2×C42—21no—
358C2×C42—141no—
359C2×C42C22×D711yescharacteristic
360C2×C42C22×D711yescharacteristic
361#106—142no—
362#106—142no—
363#106—142no—
364#106—281no—
365C4⋊C8—142no—
366#103—142no—
367#103—142no—
368#103—142no—
369#103—281no—
370C22⋊C8—142no—
371C22⋊C8—281no—
372C4×C8—281no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—21no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95C22⋊D711yescharacteristic
383#95C22⋊D711yescharacteristic
384C22×C7—22no—
385C22×C7—22no—
386C22×C7—22no—
387C22×C7—41no—
388C22×C7—41no—
389C22×C7—41no—
390C22×C7—41no—
391C22×C7#52511yescharacteristic
392C22×C7#54611yescharacteristic
393C22×C7#54711yescharacteristic
394C22×C7C22×SD1611yescharacteristic
395C22×C7#57511yescharacteristic
396C22×C7#57711yescharacteristic
397C22×C7#57811yescharacteristic
398C4×C7—22no—
399C4×C7—22no—
400C4×C7—41no—
401C4×C7—42no—
402C4×C7—41no—
403C4×C7—42no—
404C4×C7—41no—
405C4×C7—42no—
406C4×C7—81no—
407C4×C7#52911yescharacteristic
408C4×C7#53911yescharacteristic
409C4×C7#55311yescharacteristic
410C4×C7#55611yescharacteristic
411Dic7—81no—
412Dic7—81no—
413Dic7—81no—
414Dic7—81no—
415C24C2×C22⋊D711yescharacteristic
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8#115712yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
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×C4C23×D711yescharacteristic
445C2×C8—72no—
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—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—282no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—282no—
477C4⋊C4#115712yes—
478C4⋊C4#115912yes—
479C22⋊C4—24no—
480C22⋊C4—41no—
481C22⋊C4—282no—
482C22⋊C4—282no—
483C22⋊C4#114314yes—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—42no—
489C42—81no—
490C42—282no—
491C2×C7—42no—
492C2×C7#246611yescharacteristic
493C2×C7#310311yescharacteristic
494C2×C7#312011yescharacteristic
495C2×C7#329811yescharacteristic
496C2×C7#331111yescharacteristic
497C2×C7#333511yescharacteristic
498C2×C7#335211yescharacteristic
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23C2×#114312yes—
506C23C22×C22⋊D711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
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×C4C2×#114312yes—
554C2×C4C2×#114312yes—
555C2×C4C2×#115711yescharacteristic
556C2×C4C2×#115711yescharacteristic
557C2×C4C2×#115911yescharacteristic
558C2×C4#654811yescharacteristic
559C2×C4D7×#4111yescharacteristic
560C2×C4#655411yescharacteristic
561C8—142no—
562C8—281no—
563C8—281no—
564C8—281no—
565C7#2703411yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#9048211yescharacteristic
574C22#9049711yescharacteristic
575C22C22×#114311yescharacteristic
576C22#9051311yescharacteristic
577C22#9051511yescharacteristic
578C22#9052711yescharacteristic
579C22#9052911yescharacteristic
580C4—22no—
581C4—22no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—81no—
589C4—561no—
590C4—561no—
591C4—561no—
592C4—561no—
593C4#9022911yescharacteristic
594C4#9023311yescharacteristic
595C4#9024911yescharacteristic
596C4#9025511yescharacteristic
597C2—42no—
598C2#1173981411yescharacteristic
599C2#1173988711yescharacteristic
600C2#1174002011yescharacteristic
601C2#1174094211yescharacteristic
602C2#1174099511yescharacteristic
603C2#1174104211yescharacteristic
604C2#1174106711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2360C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2360D7
hypocenterC7#27034
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
5. C4×#6409C2semidirect
6. #11729383C2semidirect
9. #11727436C2semidirect
10. #11727424C2semidirect
11. #11727372C2semidirect
12. #11727356C2semidirect
13. #11727221C2semidirect
14. #11727213C2semidirect
565. C7#27034semidirect

Maximal subgroups

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

subgroupindex
C7×#23602
#117309572
#117294482
C4×#64092
#117293832
#117293102
#117293082
#117274362
#117274242
#117273722
#117273562
#117272212
#117272132
#117272052
#270347