Finite Group Databasebeta

← #13426464

Subgroups — #13426464

Order 1792.

784 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2495C211yesmaximal, characteristic, complemented
2#11730912C211yesmaximal, characteristic
3#11727383C211yesmaximal, characteristic, complemented
4#11727380C211yesmaximal, characteristic, complemented
5#11727372C211yesmaximal, characteristic, complemented
6#11727355C211yesmaximal, characteristic, complemented
7#11727311C211yesmaximal, characteristic, complemented
8#11727294C211yesmaximal, characteristic, complemented
9#11727278C211yesmaximal, characteristic
10#11727277C211yesmaximal, characteristic
11#11727259C211yesmaximal, characteristic, complemented
12#11727256C211yesmaximal, characteristic, complemented
13C4×#6383C211yesmaximal, characteristic, complemented
14#11727212C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic, complemented
16C2×#6493C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C2×(C22⋊D7)⋊C4—21no—
19C2×C4×D28C2211yescharacteristic
20C7×#400—21no—
21C7×#397—21no—
22C7×#389C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#388—21no—
25C7×#383—21no—
26C7×#382C2211yescharacteristic
27C7×#380C2211yescharacteristic
28C7×#380C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C4×C7×C22⋊C4C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6383C2211yescharacteristic
34C2×#6383C2211yescharacteristic
35C2×#6383C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89730—21no—
42#89711C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89442—21no—
48#89440—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#27031—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C22×C22⋊D7—21no—
62C23×Dic7—21no—
63C22×D28C2311yescharacteristic
64C22×C4×D7—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4D411yescharacteristic
74C2×C7×C22⋊C4D411yescharacteristic
75C2×C42×C7—21no—
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78#6493—22no—
79#6493—22no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×C4⋊Dic7—22no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7—41no—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×Dic7⋊C4—21no—
91C2×C4×Dic7—21no—
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96(C22⋊D7)⋊C4—44no—
97C4×D28—22no—
98C4×D28—41no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95D411yescharacteristic
107C7×#95D411yescharacteristic
108C7⋊#95—21no—
109C7⋊#95—21no—
110C22⋊(C7⋊C8)—22no—
111C22⋊(C7⋊C8)—41no—
112#6383—22no—
113#6383—22no—
114#6383—22no—
115#6383—41no—
116#6382—22no—
117#6382—22no—
118#6382—41no—
119#6381—22no—
120C4⋊(C7⋊C8)—22no—
121C4×C7⋊C8—41no—
122#2495D711yescharacteristic, complemented
123#2361—71no—
124#2073—71no—
125#2065—71no—
126#2034—71no—
127#2033—71no—
128#1949—71no—
129#1941—71no—
130#1923—71no—
131#1919—71no—
132#1918—71no—
133#1864—71no—
134#1853—71no—
135#1824—71no—
136C4×#102—71no—
137C24×C7C2×D411yescharacteristic
138C23×D7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C2×C22⋊D7—24no—
156C2×C22⋊D7—42no—
157C2×C22⋊D7—42no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—42no—
162C22×Dic7—42no—
163C22×Dic7—81no—
164C2×D28—21no—
165C2×D28—21no—
166C2×D28—21no—
167C2×D28—21no—
168C2×D28—21no—
169C2×D28#4112yes—
170C2×C4×D7—42no—
171C2×C4×D7—42no—
172C2×C4×D7—81no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—81no—
182C7×C4⋊C4—81no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—22no—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—81no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—42no—
192(C2×D7)⋊C4—24no—
193(C2×D7)⋊C4—42no—
194(C2×D7)⋊C4—42no—
195(C2×D7)⋊C4—81no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—42no—
201C4⋊Dic7#4112yes—
202C4⋊Dic7#4112yes—
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×#121—71no—
218C2×C4×D4—71no—
219C2×C4×D4—141no—
220C22×C4⋊C4—71no—
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—21no—
234#397—21no—
235#390—71no—
236#389—71no—
237#389C2×D711yescharacteristic
238#389C2×D711yescharacteristic
239#388—21no—
240#388—141no—
241#386—141no—
242#385—141no—
243#383—21no—
244#382C2×D711yescharacteristic
245#380C2×D711yescharacteristic
246#380C2×D711yescharacteristic
247C4×C4⋊C4—71no—
248C4×C4⋊C4C2×D711yescharacteristic
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—21no—
261C23×C7—21no—
262C23×C7—21no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7#13512yes—
267C23×C7C22×D411yescharacteristic
268C22×D7—22no—
269C22×D7—41no—
270C22×D7—41no—
271C22×D7—41no—
272C22×D7—41no—
273C22×D7—41no—
274C22×D7—41no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
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—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
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—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×C7AGL(1,ℤ/8)12yes—
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7H(5,𝔽2)11yescharacteristic
311C2×C4×C72−1+411yescharacteristic
312C22⋊D7—48no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—84no—
323D28—22no—
324D28—22no—
325D28—41no—
326D28—41no—
327C4×D7—84no—
328C7⋊C8—22no—
329C7⋊C8—41no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C22×D4—71no—
333C22×D4—141no—
334C23×C4—141no—
335C23×C4—141no—
336C23×C4C22×D711yescharacteristic
337C22×C8—71no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—71no—
341#121—142no—
342#121—142no—
343C4×D4—142no—
344C4×D4—281no—
345C4×D4—284no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—142no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—281no—
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—141no—
362C2×C22⋊C4—141no—
363C2×C22⋊C4—141no—
364C2×C22⋊C4—141no—
365C2×C22⋊C4C22⋊D711yescharacteristic
366C2×C22⋊C4C22⋊D711yescharacteristic
367C2×C42—21no—
368C2×C42—141no—
369C2×C42C22×D711yescharacteristic
370C2×C42C22×D711yescharacteristic
371#107—142no—
372#106—142no—
373#106—142no—
374#106—281no—
375C4⋊C8—142no—
376#102—142no—
377#102—142no—
378#102—142no—
379#102—281no—
380C22⋊C8—142no—
381C22⋊C8—281no—
382C4×C8—281no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—141no—
391#95—141no—
392#95C22⋊D711yescharacteristic
393#95C22⋊D711yescharacteristic
394C22×C7—22no—
395C22×C7—22no—
396C22×C7—22no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7—41no—
400C22×C7—41no—
401C22×C7#52511yescharacteristic
402C22×C7#54611yescharacteristic
403C22×C7#54711yescharacteristic
404C22×C7C2×#13511yescharacteristic
405C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
406C22×C7#57711yescharacteristic
407C22×C7#57711yescharacteristic
408C2×D7—42no—
409C2×D7—42no—
410C2×D7—42no—
411C2×D7—81no—
412C2×D7—81no—
413C2×D7—81no—
414C2×D7—81no—
415C4×C7—22no—
416C4×C7—41no—
417C4×C7—41no—
418C4×C7—41no—
419C4×C7—41no—
420C4×C7—81no—
421C4×C7—81no—
422C4×C7—81no—
423C4×C7—81no—
424C4×C7#52911yescharacteristic
425C4×C7#53911yescharacteristic
426C4×C7#55311yescharacteristic
427C4×C7#55611yescharacteristic
428Dic7—42no—
429Dic7—42no—
430Dic7—81no—
431C24—141no—
432C24C2×C22⋊D711yescharacteristic
433C2×D4—72no—
434C2×D4—144no—
435C2×D4—141no—
436C2×D4—141no—
437C2×D4—141no—
438C2×D4—141no—
439C2×D4—141no—
440C2×D4—282no—
441C2×D4—282no—
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—21no—
451C22×C4—21no—
452C22×C4—141no—
453C22×C4—141no—
454C22×C4—141no—
455C22×C4—282no—
456C22×C4—282no—
457C22×C4—282no—
458C22×C4—282no—
459C22×C4—561no—
460C22×C4—561no—
461C22×C4C2×C22⋊D711yescharacteristic
462C22×C4C2×C22⋊D711yescharacteristic
463C22×C4C2×C22⋊D711yescharacteristic
464C22×C4C2×C22⋊D711yescharacteristic
465C22×C4C2×C22⋊D711yescharacteristic
466C22×C4C23×D711yescharacteristic
467C2×C8—72no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8—141no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—141no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—72no—
488C4⋊C4—72no—
489C4⋊C4—81no—
490C4⋊C4—81no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—141no—
494C4⋊C4—141no—
495C4⋊C4—282no—
496C4⋊C4—282no—
497C22⋊C4—22no—
498C22⋊C4—22no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—144no—
502C22⋊C4—282no—
503C22⋊C4—282no—
504C22⋊C4—561no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—282no—
511C2×C7—42no—
512C2×C7#246611yescharacteristic
513C2×C7#310511yescharacteristic
514C2×C7#312511yescharacteristic
515C2×C7#330111yescharacteristic
516C2×C7#331011yescharacteristic
517C2×C7#333711yescharacteristic
518C2×C7#335411yescharacteristic
519D7—82no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—142no—
527C23—281no—
528C23—281no—
529C23—281no—
530C23—281no—
531C23—281no—
532C23—281no—
533C23#651312yes—
534C23C22×C22⋊D711yescharacteristic
535D4—142no—
536D4—142no—
537D4—288no—
538D4—281no—
539D4—281no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
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—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—564no—
580C2×C4—564no—
581C2×C4#651212yes—
582C2×C4D7×#4111yescharacteristic
583C2×C4D7×#4111yescharacteristic
584C2×C4D7×#4111yescharacteristic
585C2×C4D7×#4111yescharacteristic
586C2×C4#655311yescharacteristic
587C2×C4#655411yescharacteristic
588C8—142no—
589C8—281no—
590C8—281no—
591C8—281no—
592C7#2703111yescharacteristic, complemented, Sylow
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—282no—
601C22—282no—
602C22—282no—
603C22—561no—
604C22—561no—
605C22—561no—
606C22—561no—
607C22C2×#651211yescharacteristic
608C22C2×#651311yescharacteristic
609C22#9051511yescharacteristic
610C22#9051911yescharacteristic
611C22#9052011yescharacteristic
612C22#9052711yescharacteristic
613C22#9052811yescharacteristic
614C4—22no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—282no—
624C4—282no—
625C4—561no—
626C4#9039311yescharacteristic
627C4#9039711yescharacteristic
628C4#9039811yescharacteristic
629C4#9040011yescharacteristic
630C2—42no—
631C2—562no—
632C2#1174055011yescharacteristic
633C2#1174058611yescharacteristic
634C2#1174090711yescharacteristic
635C2#1174103611yescharacteristic
636C2#1174104411yescharacteristic
637C2#1174106311yescharacteristic
638C2#1174109511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2495C2
socleC23×C7C22×D4
prime radicalC2×#6493C22
hypercenter#2495D7
hypocenterC7#27031
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2495

Complements

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

NH ≅ G/Ntype
1. C7×#2495C2semidirect
3. #11727383C2semidirect
4. #11727380C2semidirect
5. #11727372C2semidirect
6. #11727355C2semidirect
7. #11727311C2semidirect
8. #11727294C2semidirect
11. #11727259C2semidirect
12. #11727256C2semidirect
13. C4×#6383C2semidirect
14. #11727212C2semidirect
15. #11727205C2semidirect
122. #2495D7semidirect
592. C7#27031semidirect

Maximal subgroups

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

subgroupindex
C7×#24952
#117309122
#117273832
#117273802
#117273722
#117273552
#117273112
#117272942
#117272782
#117272772
#117272592
#117272562
C4×#63832
#117272122
#117272052
#270317