Finite Group Databasebeta

← #13426447

Subgroups — #13426447

Order 1792.

812 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2361C211yesmaximal, characteristic
2#11730957C211yesmaximal, characteristic, complemented
3#11729433C211yesmaximal, characteristic
4#11729433C211yesmaximal, characteristic
5C4×#6406C211yesmaximal, characteristic, complemented
6#11729360C211yesmaximal, characteristic, complemented
7#11729309C211yesmaximal, characteristic, complemented
8#11729307C211yesmaximal, characteristic, complemented
9#11727407C211yesmaximal, characteristic, complemented
10#11727394C211yesmaximal, characteristic, complemented
11#11727373C211yesmaximal, characteristic, complemented
12#11727354C211yesmaximal, characteristic, complemented
13#11727220C211yesmaximal, characteristic, complemented
14#11727212C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×C4×C7×D4—21no—
18C2×C4×C7×D4C2211yescharacteristic
19C22×C7×C4⋊C4C2211yescharacteristic
20C2×#6442—21no—
21C7×#390C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#388—21no—
24C7×#386—21no—
25C7×#385—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
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×#6381C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
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#27030—71nomaximal, Sylow
60C22×C7×D4—21no—
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×C4×C7—21no—
64C23×C4×C7C2311yescharacteristic
65C7×#121—22no—
66C7×#121—22no—
67C4×C7×D4—22no—
68C4×C7×D4—41no—
69C4×C7×D4—44no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—41no—
74C2×C7×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
82C2×C7×C22⋊C4D411yescharacteristic
83C2×C42×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×C4×Dic7—21no—
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101#6442—44no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95D411yescharacteristic
105C7×#95D411yescharacteristic
106#6406—22no—
107#6406—22no—
108#6406—22no—
109#6406—41no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C22⋊(C7⋊C8)—22no—
116C22⋊(C7⋊C8)—41no—
117#6381—22no—
118#6381—22no—
119#6381—22no—
120#6381—41no—
121C4⋊(C7⋊C8)—22no—
122C4×C7⋊C8—41no—
123#2496—71no—
124#2361D711yescharacteristic
125#2071—71no—
126#2067—71no—
127#2033—71no—
128#2033—71no—
129#1947—71no—
130#1942—71no—
131#1923—71no—
132#1918—71no—
133#1918—71no—
134#1862—71no—
135#1852—71no—
136#1823—71no—
137C4×#102—71no—
138C24×C7—21no—
139C24×C7C2×D411yescharacteristic
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—42no—
147C2×C7×D4—42no—
148C2×C7×D4#4112yes—
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—21no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—81no—
163C22×C4×C7—81no—
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2411yescharacteristic
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—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#4112yes—
181C7×C4⋊C4#4112yes—
182C7×C22⋊C4—24no—
183C7×C22⋊C4—41no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—81no—
187C7×C22⋊C4D814yes—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—21no—
192C42×C7—42no—
193C22⋊Dic7—42no—
194C22⋊Dic7—42no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—42no—
204C4⋊Dic7—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207C4×Dic7—42no—
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#4112yes—
220C2×#126—141no—
221C2×#121C2×D711yescharacteristic
222C2×C4×D4—21no—
223C2×C4×D4C2×D711yescharacteristic
224C22×C4⋊C4C2×D711yescharacteristic
225C2×#107—71no—
226C2×#107—71no—
227C2×#107—71no—
228C2×#107—71no—
229C2×C4⋊C8—71no—
230C2×#102—71no—
231C2×#102—71no—
232C2×#102—71no—
233C2×#102—71no—
234C2×C22⋊C8—71no—
235C2×C22⋊C8—71no—
236C2×C4×C8—71no—
237#400—141no—
238#397—141no—
239#390C2×D711yescharacteristic
240#389—71no—
241#389—71no—
242#389C2×D711yescharacteristic
243#388—21no—
244#387—141no—
245#386—21no—
246#385—21no—
247#384—71no—
248#380—71no—
249#380—71no—
250C4×C4⋊C4—71no—
251C4×C4⋊C4C2×D711yescharacteristic
252C4×C22⋊C4C2×D711yescharacteristic
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#336—71no—
262#336—71no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—22no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7—41no—
274C23×C7—41no—
275C23×C7—41no—
276C23×C7C2×D812yes—
277C23×C7C22×D411yescharacteristic
278C7×D4—22no—
279C7×D4—22no—
280C7×D4—48no—
281C7×D4—41no—
282C7×D4—41no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—22no—
289C2×C4×C7—21no—
290C2×C4×C7—22no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
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—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—84no—
310C2×C4×C7—84no—
311C2×C4×C7C2×D812yes—
312C2×C4×C7C2×D812yes—
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7C2×#4111yescharacteristic
315C2×C4×C7C2×#4111yescharacteristic
316C2×C4×C7C2×#4111yescharacteristic
317C2×C4×C7H(5,𝔽2)11yescharacteristic
318C2×C4×C72−1+411yescharacteristic
319C2×Dic7—41no—
320C2×Dic7—41no—
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—
331C7⋊C8—22no—
332C7⋊C8—41no—
333C7⋊C8—41no—
334C7⋊C8—41no—
335C22×D4—21no—
336C22×D4C22×D711yescharacteristic
337C23×C4—21no—
338C23×C4—21no—
339C23×C4C22×D711yescharacteristic
340C22×C8—71no—
341C22×C8—71no—
342C22×C8—71no—
343C22×C8—71no—
344#126—284no—
345#121—22no—
346#121—22no—
347C4×D4—22no—
348C4×D4—41no—
349C4×D4—44no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—41no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
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—141no—
370C2×C22⋊C4—141no—
371C2×C22⋊C4C22⋊D711yescharacteristic
372C2×C22⋊C4C22⋊D711yescharacteristic
373C2×C42—21no—
374C2×C42—141no—
375C2×C42C22×D711yescharacteristic
376C2×C42C22×D711yescharacteristic
377#107—142no—
378#107—142no—
379#107—142no—
380#107—281no—
381C4⋊C8—142no—
382#102—142no—
383#102—142no—
384#102—142no—
385#102—281no—
386C22⋊C8—142no—
387C22⋊C8—281no—
388C4×C8—281no—
389#95—21no—
390#95—21no—
391#95—141no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95C22⋊D711yescharacteristic
397#95C22⋊D711yescharacteristic
398C22×C7—22no—
399C22×C7—22no—
400C22×C7—22no—
401C22×C7—41no—
402C22×C7—41no—
403C22×C7—41no—
404C22×C7—41no—
405C22×C7—42no—
406C22×C7—42no—
407C22×C7—42no—
408C22×C7—81no—
409C22×C7—81no—
410C22×C7—81no—
411C22×C7—81no—
412C22×C7#52511yescharacteristic
413C22×C7#54611yescharacteristic
414C22×C7#54711yescharacteristic
415C22×C7C22×D811yescharacteristic
416C22×C7#57511yescharacteristic
417C22×C7#57711yescharacteristic
418C22×C7#57811yescharacteristic
419C4×C7—22no—
420C4×C7—22no—
421C4×C7—41no—
422C4×C7—42no—
423C4×C7—41no—
424C4×C7—42no—
425C4×C7—41no—
426C4×C7—81no—
427C4×C7#52911yescharacteristic
428C4×C7#53911yescharacteristic
429C4×C7#55311yescharacteristic
430C4×C7#55611yescharacteristic
431Dic7—81no—
432Dic7—81no—
433Dic7—81no—
434Dic7—81no—
435C24—21no—
436C24C2×C22⋊D711yescharacteristic
437C2×D4—24no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4#115712yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—141no—
462C22×C4—141no—
463C22×C4—141no—
464C22×C4—141no—
465C22×C4C2×C22⋊D711yescharacteristic
466C22×C4C2×C22⋊D711yescharacteristic
467C22×C4C2×C22⋊D711yescharacteristic
468C22×C4C2×C22⋊D711yescharacteristic
469C22×C4C2×C22⋊D711yescharacteristic
470C22×C4C23×D711yescharacteristic
471C2×C8—72no—
472C2×C8—141no—
473C2×C8—141no—
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—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—282no—
498C4⋊C4—282no—
499C4⋊C4—282no—
500C4⋊C4—282no—
501C4⋊C4#115712yes—
502C4⋊C4#115912yes—
503C22⋊C4—24no—
504C22⋊C4—41no—
505C22⋊C4—42no—
506C22⋊C4—42no—
507C22⋊C4—81no—
508C22⋊C4—282no—
509C22⋊C4—282no—
510C22⋊C4#114114yes—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—282no—
517C2×C7—42no—
518C2×C7—82no—
519C2×C7#246611yescharacteristic
520C2×C7#310011yescharacteristic
521C2×C7#312011yescharacteristic
522C2×C7#329411yescharacteristic
523C2×C7#331311yescharacteristic
524C2×C7#333411yescharacteristic
525C2×C7#335111yescharacteristic
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—22no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23C2×#114112yes—
540C23C22×C22⋊D711yescharacteristic
541D4—22no—
542D4—22no—
543D4—48no—
544D4—41no—
545D4—41no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—84no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
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×C4C2×#114112yes—
587C2×C4C2×#114112yes—
588C2×C4C2×#115711yescharacteristic
589C2×C4C2×#115711yescharacteristic
590C2×C4C2×#115911yescharacteristic
591C2×C4#654811yescharacteristic
592C2×C4D7×#4111yescharacteristic
593C2×C4#655411yescharacteristic
594C8—142no—
595C8—281no—
596C8—281no—
597C8—281no—
598C7#2703011yescharacteristic, complemented, Sylow
599C22—22no—
600C22—22no—
601C22—22no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22#9048211yescharacteristic
614C22C22×#114111yescharacteristic
615C22#9049711yescharacteristic
616C22#9051311yescharacteristic
617C22#9051511yescharacteristic
618C22#9052711yescharacteristic
619C22#9052911yescharacteristic
620C4—22no—
621C4—22no—
622C4—41no—
623C4—42no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—81no—
628C4—561no—
629C4—561no—
630C4—561no—
631C4—561no—
632C4#9022911yescharacteristic
633C4#9023311yescharacteristic
634C4#9024911yescharacteristic
635C4#9025511yescharacteristic
636C2—42no—
637C2—82no—
638C2#1173975311yescharacteristic
639C2#1173981411yescharacteristic
640C2#1174002011yescharacteristic
641C2#1174092711yescharacteristic
642C2#1174096111yescharacteristic
643C2#1174098211yescharacteristic
644C2#1174105711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2361C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2361D7
hypocenterC7#27030
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
2. #11730957C2semidirect
5. C4×#6406C2semidirect
6. #11729360C2semidirect
7. #11729309C2semidirect
8. #11729307C2semidirect
9. #11727407C2semidirect
10. #11727394C2semidirect
11. #11727373C2semidirect
12. #11727354C2semidirect
13. #11727220C2semidirect
14. #11727212C2semidirect
15. #11727205C2semidirect
598. C7#27030semidirect

Maximal subgroups

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

subgroupindex
C7×#23612
#117309572
#117294332
C4×#64062
#117293602
#117293092
#117293072
#117274072
#117273942
#117273732
#117273542
#117272202
#117272122
#117272052
#270307