Finite Group Databasebeta

← #13426492

Subgroups — #13426492

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×#2361C211yesmaximal, characteristic
2#11730955C211yesmaximal, characteristic, complemented
3#11729434C211yesmaximal, characteristic
4#11729433C211yesmaximal, characteristic
5C4×#6406C211yesmaximal, characteristic, complemented
6#11729360C211yesmaximal, characteristic, complemented
7#11729309C211yesmaximal, characteristic, complemented
8#11729308C211yesmaximal, characteristic, complemented
9#11727407C211yesmaximal, characteristic, complemented
10#11727395C211yesmaximal, characteristic, complemented
11#11727373C211yesmaximal, characteristic, complemented
12#11727355C211yesmaximal, characteristic, complemented
13#11727222C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic, complemented
16C2×C7×#121C2211yescharacteristic
17C2×C4×C7×D4—21no—
18C2×C4×C7×D4C2211yescharacteristic
19C22×C7×C4⋊C4C2211yescharacteristic
20C7×#390C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#388—21no—
23C7×#386—21no—
24C7×#385—21no—
25C4×C7×C4⋊C4C2211yescharacteristic
26C4×C7×C22⋊C4C2211yescharacteristic
27C2×#6406C2211yescharacteristic
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31#89984C2211yescharacteristic
32#89984C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2211yescharacteristic
34C2×C22⋊(C7⋊C8)C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39#89724—21no—
40#89717—21no—
41#89708C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47C7⋊#401—21no—
48#89424—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#27033—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⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×C4×Dic7—21no—
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C7×#95—21no—
100C7×#95—21no—
101C7×#95D411yescharacteristic
102C7×#95D411yescharacteristic
103#6406—22no—
104#6406—22no—
105#6406—22no—
106#6406—41no—
107C7⋊#95—21no—
108C7⋊#95—21no—
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#6381—22no—
118#6381—22no—
119#6381—41no—
120C4⋊(C7⋊C8)—22no—
121C4×C7⋊C8—41no—
122#2494—71no—
123#2361D711yescharacteristic
124#2073—71no—
125#2067—71no—
126#2034—71no—
127#2033—71no—
128#1949—71no—
129#1942—71no—
130#1923—71no—
131#1921—71no—
132#1919—71no—
133#1862—71no—
134#1852—71no—
135#1824—71no—
136C4×#102—71no—
137C24×C7—21no—
138C24×C7C2×D411yescharacteristic
139C2×C7×D4—24no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—42no—
146C2×C7×D4—42no—
147C2×C7×D4#4112yes—
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—21no—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—42no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—81no—
162C22×C4×C7—81no—
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2411yescharacteristic
169C22×Dic7—21no—
170C22×Dic7—21no—
171C22×Dic7—21no—
172C22×Dic7—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4#4112yes—
180C7×C4⋊C4#4112yes—
181C7×C22⋊C4—24no—
182C7×C22⋊C4—22no—
183C7×C22⋊C4—22no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—81no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—42no—
192C22⋊Dic7—42no—
193C22⋊Dic7—81no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202Dic7⋊C4—42no—
203Dic7⋊C4—81no—
204C4×Dic7—81no—
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×#121C2×D711yescharacteristic
218C2×C4×D4—21no—
219C2×C4×D4C2×D711yescharacteristic
220C22×C4⋊C4C2×D711yescharacteristic
221C2×#107—71no—
222C2×#107—71no—
223C2×#107—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#401—141no—
234#397—141no—
235#390C2×D711yescharacteristic
236#389—71no—
237#389—71no—
238#389C2×D711yescharacteristic
239#388—21no—
240#388—141no—
241#387—141no—
242#386—21no—
243#385—21no—
244#382—71no—
245#380—71no—
246#380—71no—
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—22no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7#13512yes—
274C23×C7C22×D411yescharacteristic
275C7×D4—22no—
276C7×D4—22no—
277C7×D4—48no—
278C7×D4—41no—
279C7×D4—41no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—22no—
284C2×C4×C7—21no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
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—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×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—84no—
308C2×C4×C7—84no—
309C2×C4×C7AGL(1,ℤ/8)12yes—
310C2×C4×C7C2×#4111yescharacteristic
311C2×C4×C7C2×#4111yescharacteristic
312C2×C4×C7C2×#4111yescharacteristic
313C2×C4×C7C2×#4111yescharacteristic
314C2×C4×C7H(5,𝔽2)11yescharacteristic
315C2×C4×C72−1+411yescharacteristic
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
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—
328C7⋊C8—22no—
329C7⋊C8—41no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C22×D4—21no—
333C22×D4C22×D711yescharacteristic
334C23×C4—21no—
335C23×C4—21no—
336C23×C4C22×D711yescharacteristic
337C22×C8—71no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—71no—
341#121—22no—
342#121—22no—
343C4×D4—22no—
344C4×D4—41no—
345C4×D4—44no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—41no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
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#107—142no—
373#107—281no—
374#106—142no—
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—141no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
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—42no—
402C22×C7—42no—
403C22×C7—42no—
404C22×C7—81no—
405C22×C7—81no—
406C22×C7—81no—
407C22×C7—81no—
408C22×C7#52511yescharacteristic
409C22×C7#54611yescharacteristic
410C22×C7#54711yescharacteristic
411C22×C7C2×#13511yescharacteristic
412C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
413C22×C7#57611yescharacteristic
414C22×C7#57811yescharacteristic
415C4×C7—22no—
416C4×C7—41no—
417C4×C7—41no—
418C4×C7—42no—
419C4×C7—41no—
420C4×C7—41no—
421C4×C7—42no—
422C4×C7—81no—
423C4×C7#52911yescharacteristic
424C4×C7#53911yescharacteristic
425C4×C7#55311yescharacteristic
426C4×C7#55611yescharacteristic
427Dic7—81no—
428Dic7—81no—
429Dic7—81no—
430Dic7—81no—
431C24—21no—
432C24C2×C22⋊D711yescharacteristic
433C2×D4—24no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—42no—
440C2×D4—42no—
441C2×D4#115712yes—
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—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—81no—
456C22×C4—81no—
457C22×C4—141no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4—141no—
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—42no—
484C4⋊C4—42no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—141no—
489C4⋊C4—141no—
490C4⋊C4—141no—
491C4⋊C4—141no—
492C4⋊C4—141no—
493C4⋊C4—282no—
494C4⋊C4—561no—
495C4⋊C4#115712yes—
496C4⋊C4#115912yes—
497C22⋊C4—24no—
498C22⋊C4—22no—
499C22⋊C4—22no—
500C22⋊C4—42no—
501C22⋊C4—42no—
502C22⋊C4—81no—
503C22⋊C4—282no—
504C22⋊C4—561no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—561no—
511C2×C7—42no—
512C2×C7—82no—
513C2×C7#246611yescharacteristic
514C2×C7#310511yescharacteristic
515C2×C7#312411yescharacteristic
516C2×C7#330011yescharacteristic
517C2×C7#331811yescharacteristic
518C2×C7#333911yescharacteristic
519C2×C7#335311yescharacteristic
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—22no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23#651312yes—
534C23C22×C22⋊D711yescharacteristic
535D4—22no—
536D4—22no—
537D4—48no—
538D4—41no—
539D4—41no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
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—84no—
568C2×C4—84no—
569C2×C4—281no—
570C2×C4—281no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4#649512yes—
582C2×C4C2×#115711yescharacteristic
583C2×C4C2×#115711yescharacteristic
584C2×C4C2×#115911yescharacteristic
585C2×C4#654811yescharacteristic
586C2×C4D7×#4111yescharacteristic
587C2×C4#655411yescharacteristic
588C8—142no—
589C8—281no—
590C8—281no—
591C8—281no—
592C7#2703311yescharacteristic, complemented, Sylow
593C22—22no—
594C22—22no—
595C22—22no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22#9048211yescharacteristic
608C22C2×#649511yescharacteristic
609C22#9049711yescharacteristic
610C22C2×#651311yescharacteristic
611C22#9051511yescharacteristic
612C22#9052611yescharacteristic
613C22#9052911yescharacteristic
614C4—22no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—42no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—561no—
623C4—561no—
624C4—561no—
625C4—561no—
626C4#9022911yescharacteristic
627C4#9023311yescharacteristic
628C4#9024911yescharacteristic
629C4#9025511yescharacteristic
630C2—42no—
631C2—82no—
632C2#1173975411yescharacteristic
633C2#1173981411yescharacteristic
634C2#1174001411yescharacteristic
635C2#1174093011yescharacteristic
636C2#1174096511yescharacteristic
637C2#1174097911yescharacteristic
638C2#1174105211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2361C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2361D7
hypocenterC7#27033
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. #11730955C2semidirect
5. C4×#6406C2semidirect
6. #11729360C2semidirect
7. #11729309C2semidirect
8. #11729308C2semidirect
9. #11727407C2semidirect
10. #11727395C2semidirect
11. #11727373C2semidirect
12. #11727355C2semidirect
13. #11727222C2semidirect
14. #11727215C2semidirect
15. #11727205C2semidirect
592. C7#27033semidirect

Maximal subgroups

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

subgroupindex
C7×#23612
#117309552
#117294342
#117294332
C4×#64062
#117293602
#117293092
#117293082
#117274072
#117273952
#117273732
#117273552
#117272222
#117272152
#117272052
#270337