Finite Group Databasebeta

← #13405688

Subgroups — #13405688

Order 1792.

774 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×#2580C211yesmaximal, characteristic
2#11739928C211yesmaximal, characteristic, complemented
3C2×Q8×C7⋊C8C211yesmaximal, characteristic, complemented
4C2×#89781C211yesmaximal, characteristic
5C4×#6406C211yesmaximal, characteristic
6#11727407C211yesmaximal, characteristic
7#11727406C211yesmaximal, characteristic
8#11727398C211yesmaximal, characteristic
9#11727398C211yesmaximal, characteristic
10#11727377C211yesmaximal, characteristic, complemented
11#11727373C211yesmaximal, characteristic, complemented
12#11727372C211yesmaximal, characteristic, complemented
13#11727364C211yesmaximal, characteristic, complemented
14#11727308C211yesmaximal, characteristic, complemented
15#11727294C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×#6433C2211yescharacteristic
19C7×#399—21no—
20C7×#393—21no—
21C7×#390C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C7×#385—21no—
26C7×#385—21no—
27C7×#382C2211yescharacteristic
28#90001—21no—
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33Q8×C7⋊C8—24no—
34#89781—24no—
35#89752—21no—
36#89744—21no—
37#89744—21no—
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43#89700C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47C2×C4⋊(C7⋊C8)C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51C2×C4×C7⋊C8C2211yescharacteristic
52C2×C4×C7⋊C8C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#25473—71nomaximal, Sylow
62C22×C7×Q8D411yescharacteristic
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C4×C7×Q8—22no—
66C4×C7×Q8—22no—
67C4×C7×D4—22no—
68C4×C7×D4—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4D411yescharacteristic
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⋊C4—21no—
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100#6433—22no—
101#6433—22no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106#6406—22no—
107#6406—22no—
108#6406—22no—
109#6406Q814yescomplemented
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115#6382—22no—
116#6382—22no—
117#6381—22no—
118#6381—22no—
119C4⋊(C7⋊C8)—22no—
120C4⋊(C7⋊C8)—22no—
121C4⋊(C7⋊C8)D414yes—
122C4×C7⋊C8—22no—
123C4×C7⋊C8—22no—
124C4×C7⋊C8—22no—
125C2×#478—71no—
126C2×C8×Q8—71no—
127#2584—71no—
128#2580D711yescharacteristic
129#2088—71no—
130#2087—71no—
131#2067—71no—
132#2065—71no—
133#1991—71no—
134#1949—71no—
135#1942—71no—
136#1941—71no—
137#1934—71no—
138#1934—71no—
139C4×#102—71no—
140C24×C7—21no—
141C2×C7×Q8—22no—
142C2×C7×Q8—22no—
143C2×C7×Q8—22no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4—21no—
149C2×C7×D4C2×Q812yes—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—42no—
154C22×C4×C7—81no—
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—22no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—81no—
177C7×C4⋊C4—81no—
178C7×C22⋊C4—42no—
179C7×C22⋊C4—42no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—81no—
182C7×C22⋊C4—81no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7C2×D412yes—
188C42×C7#4112yes—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7C2×Q812yes—
196C4⋊Dic7#4112yes—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200Dic7⋊C4—42no—
201Dic7⋊C4—81no—
202Dic7⋊C4—81no—
203C4×Dic7—81no—
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⋊C8C2×D412yes—
214C2×C7⋊C8C2×D412yes—
215C2×C7⋊C8C2×Q812yes—
216C2×#125—71no—
217C2×C4×Q8C2×D711yescharacteristic
218C2×C4×D4C2×D711yescharacteristic
219#478—144no—
220C8×Q8—144no—
221C2×#107—71no—
222C2×#107—71no—
223C2×#106—71no—
224C2×#106—71no—
225C2×C4⋊C8—71no—
226C2×C4⋊C8—71no—
227C2×C4⋊C8—71no—
228C2×#102—71no—
229C2×#102—71no—
230C2×#102—71no—
231C2×#102—71no—
232C2×C4×C8—71no—
233C2×C4×C8—71no—
234C2×C4×C8—71no—
235#399—21no—
236#398—141no—
237#395—141no—
238#393—21no—
239#390C2×D711yescharacteristic
240#390C2×D711yescharacteristic
241#389—71no—
242#389—71no—
243#389—71no—
244#389—71no—
245#389—141no—
246#389—141no—
247#389C2×D711yescharacteristic
248#389C2×D711yescharacteristic
249#385—21no—
250#385—21no—
251#382C2×D711yescharacteristic
252C4×C4⋊C4—71no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261C23×C7—22no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7—41no—
267C23×C7—41no—
268C23×C7C22×D411yescharacteristic
269C7×Q8—28no—
270C7×D4—22no—
271C7×D4—22no—
272C7×D4—41no—
273C7×D4—41no—
274C2×C4×C7—21no—
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—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×C7#12112yes—
299C2×C4×C7#12112yes—
300C2×C4×C7#13512yes—
301C2×C4×C7#13512yes—
302C2×C4×C7C22×D411yescharacteristic
303C2×C4×C7C22×Q811yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C72−1+411yescharacteristic
307C2×C4×C72−1+411yescharacteristic
308C2×Dic7—22no—
309C2×Dic7—22no—
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—
319C2×Dic7—41no—
320C7⋊C8—22no—
321C7⋊C8—22no—
322C7⋊C8—22no—
323C7⋊C8—41no—
324C22×Q8C22⋊D711yescharacteristic
325C22×D4C22×D711yescharacteristic
326C23×C4—21no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8—71no—
330C22×C8—71no—
331#125—142no—
332#125—142no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×D4—22no—
336C4×D4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4C22⋊D711yescharacteristic
350C2×C4⋊C4C22⋊D711yescharacteristic
351C2×C4⋊C4C22⋊D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C4⋊C4C22×D711yescharacteristic
354C2×C4⋊C4C22×D711yescharacteristic
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C42—141no—
361C2×C42C22×D711yescharacteristic
362C2×C42C22×D711yescharacteristic
363C2×C42C22×D711yescharacteristic
364#107—142no—
365#107—142no—
366#106—142no—
367#106—142no—
368C4⋊C8—74no—
369C4⋊C8—142no—
370C4⋊C8—142no—
371#102—74no—
372#102—142no—
373#102—142no—
374#102—142no—
375C4×C8—142no—
376C4×C8—142no—
377C4×C8—142no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—81no—
391C22×C7—81no—
392C22×C7—81no—
393C22×C7—81no—
394C22×C7C2×#12111yescharacteristic
395C22×C7#54811yescharacteristic
396C22×C7D4×Q811yescharacteristic
397C22×C7C2×#13511yescharacteristic
398C22×C7C2×#13511yescharacteristic
399C22×C7#57611yescharacteristic
400C22×C7#57711yescharacteristic
401C4×C7—22no—
402C4×C7—22no—
403C4×C7—22no—
404C4×C7—22no—
405C4×C7—41no—
406C4×C7—41no—
407C4×C7—81no—
408C4×C7—81no—
409C4×C7—81no—
410C4×C7#53311yescharacteristic
411C4×C7#54111yescharacteristic
412C4×C7#54711yescharacteristic
413C4×C7D4×Q811yescharacteristic
414Dic7—42no—
415Dic7—42no—
416Dic7—81no—
417Dic7—81no—
418C24—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4Q8×D712yes—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4C2×C22⋊D711yescharacteristic
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—72no—
447C2×C8—72no—
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—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—72no—
467C4⋊C4—72no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—282no—
477C4⋊C4—282no—
478C4⋊C4—282no—
479C4⋊C4—282no—
480C4⋊C4—561no—
481C4⋊C4—561no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—561no—
492C42D4×D712yes—
493C42#115912yes—
494C2×C7—82no—
495C2×C7#242911yescharacteristic
496C2×C7#305011yescharacteristic
497C2×C7#306211yescharacteristic
498C2×C7#336111yescharacteristic
499C2×C7#336511yescharacteristic
500C2×C7#338011yescharacteristic
501C2×C7#338111yescharacteristic
502C23—22no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23C22×C22⋊D711yescharacteristic
510Q8—28no—
511D4—22no—
512D4—22no—
513D4—41no—
514D4—41no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
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—84no—
539C2×C4—142no—
540C2×C4—142no—
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#650212yes—
552C2×C4#650212yes—
553C2×C4#651312yes—
554C2×C4#651312yes—
555C2×C4C2×D4×D711yescharacteristic
556C2×C4C2×#115911yescharacteristic
557C2×C4C2×Q8×D711yescharacteristic
558C2×C4#655111yescharacteristic
559C2×C4D7×#4111yescharacteristic
560C2×C4#655411yescharacteristic
561C8—142no—
562C8—142no—
563C8—142no—
564C8—281no—
565C7#2547311yescharacteristic, complemented, Sylow
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#650211yescharacteristic
574C22Q8×C22⋊D711yescharacteristic
575C22C2×#651311yescharacteristic
576C22C2×#651311yescharacteristic
577C22#9052211yescharacteristic
578C22#9052611yescharacteristic
579C22#9052711yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—282no—
590C4—282no—
591C4—561no—
592C4—561no—
593C4#9031311yescharacteristic
594C4#9031611yescharacteristic
595C4#9032211yescharacteristic
596C4#9032311yescharacteristic
597C2—82no—
598C2#1174020311yescharacteristic
599C2#1174036711yescharacteristic
600C2#1174082411yescharacteristic
601C2#1174107311yescharacteristic
602C2#1174107411yescharacteristic
603C2#1174115811yescharacteristic
604C2#1174116211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2580C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2580D7
hypocenterC7#25473
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. #11739928C2semidirect
3. C2×Q8×C7⋊C8C2semidirect
10. #11727377C2semidirect
11. #11727373C2semidirect
12. #11727372C2semidirect
13. #11727364C2semidirect
14. #11727308C2semidirect
15. #11727294C2semidirect
109. #6406Q8semidirect
565. C7#25473semidirect

Maximal subgroups

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

subgroupindex
C7×#25802
#117399282
C2×Q8×C7⋊C82
C2×#897812
C4×#64062
#117274072
#117274062
#117273982
#117273772
#117273732
#117273722
#117273642
#117273082
#117272942
#254737