Finite Group Databasebeta

← #13373233

Subgroups — #13373233

Order 1792.

866 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2354C211yesmaximal, characteristic
2C2×#89790C211yesmaximal, characteristic
3#11731082C211yesmaximal, characteristic, complemented
4#11729359C211yesmaximal, characteristic
5#11727414C211yesmaximal, characteristic
6#11727414C211yesmaximal, characteristic
7#11727399C211yesmaximal, characteristic
8#11727398C211yesmaximal, characteristic
9#11727342C211yesmaximal, characteristic, complemented
10#11727329C211yesmaximal, characteristic, complemented
11#11727232C211yesmaximal, characteristic, complemented
12#11727229C211yesmaximal, characteristic, complemented
13C4×#6384C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727212C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×C7×#117—21no—
18C22×#1142C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#390C2211yescharacteristic
24C7×#388—21no—
25C7×#388—21no—
26C7×#384C2211yescharacteristic
27C7×#380—21no—
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30C4×C7×C4⋊C4C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34#89790C418yescomplemented
35#89744—21no—
36#89744—21no—
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41#89708C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×#6377C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47C2×C4×C7⋊C8C2211yescharacteristic
48#89422—21no—
49#89421—21no—
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22781—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C22×Dic14C2311yescharacteristic
64C4×C7×D4—22no—
65C4×C7×D4C2×C414yes—
66C7×#117—44no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4D411yescharacteristic
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C22⋊C4—21no—
75C2×C7×C22⋊C4—21no—
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4—21no—
79C2×C42×C7—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×#1142—22no—
85C2×#1142—22no—
86C2×#1142C2×C414yes—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×C4×Dic7—21no—
96C2×C4×Dic7—21no—
97C2×C4×Dic7—21no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C4×Dic14—22no—
103C4×Dic14—22no—
104C4×Dic14—22no—
105C4×Dic14—41no—
106C4×Dic14—41no—
107C4×Dic14C2×C414yes—
108C7×#95—21no—
109C7×#95—21no—
110C7×#95D411yescharacteristic
111C7×#95D411yescharacteristic
112#6406—41no—
113#6406—41no—
114#6406C2×C414yes—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119#6384—22no—
120#6384—22no—
121#6384C2×C414yes—
122#6381C2×C414yes—
123C4⋊(C7⋊C8)—22no—
124#6377—41no—
125#6377C2×C414yes—
126C4×C7⋊C8—41no—
127C2×#440—71no—
128#2354D711yescharacteristic
129#2351—71no—
130#1936—71no—
131#1934—71no—
132#1921—71no—
133#1918—71no—
134#1880—71no—
135#1880—71no—
136#1836—71no—
137#1804—71no—
138#1803—71no—
139#1771—71no—
140#1763—71no—
141C4×#103—71no—
142C24×C7—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4C22×C412yes—
149C22×C4×C7—21no—
150C22×C4×C7—24no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—42no—
154C22×C4×C7—42no—
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14C22×C412yes—
170C2×Dic14#4112yes—
171C2×Dic14#4112yes—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—41no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4—81no—
181C7×C4⋊C4C22×C412yes—
182C7×C22⋊C4—24no—
183C7×C22⋊C4—42no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—81no—
186C7×C22⋊C4—81no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7—42no—
192C42×C7—42no—
193C42×C7C22×C412yes—
194C42×C7#4112yes—
195#1142—28no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7C22×C412yes—
202Dic7⋊C4—24no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207C4×Dic7—24no—
208C4×Dic7—42no—
209C4×Dic7—42no—
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—21no—
220C2×C7⋊C8C22×C412yes—
221C2×C7⋊C8C22×C412yes—
222C22×SD16—71no—
223C2×C4×Q8—71no—
224C2×C4×Q8—71no—
225C2×C4×Q8—71no—
226C2×C4×D4C2×D711yescharacteristic
227C2×#117—21no—
228#440—78no—
229C2×#107—71no—
230C2×C4⋊C8—71no—
231C2×#103—71no—
232C2×#103—71no—
233C2×#103—71no—
234C2×#102—71no—
235C2×#102—71no—
236C2×#102—71no—
237C2×#97—71no—
238C2×#97—71no—
239C2×C4×C8—71no—
240#390C2×D711yescharacteristic
241#390C2×D711yescharacteristic
242#389—71no—
243#389—71no—
244#389—141no—
245#389—141no—
246#388—21no—
247#388—21no—
248#384C2×D711yescharacteristic
249#382—71no—
250#382—141no—
251#380—21no—
252C4×C4⋊C4—141no—
253C4×C4⋊C4C2×D711yescharacteristic
254C4×C4⋊C4C2×D711yescharacteristic
255C4×C4⋊C4C2×D711yescharacteristic
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263#340—71no—
264#336—71no—
265#336—71no—
266C23×C7—22no—
267C23×C7—41no—
268C23×C7—41no—
269C23×C7—41no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7C22×D411yescharacteristic
274C7×D4—22no—
275C7×D4—22no—
276C7×D4—41no—
277C7×D4—41no—
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—22no—
285C2×C4×C7—21no—
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—48no—
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×C7C4×D412yes—
304C2×C4×C7C4×D412yes—
305C2×C4×C7AGL(1,ℤ/8)12yes—
306C2×C4×C7C23×C411yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×C4×C7C2×#4111yescharacteristic
309C2×C4×C7C2×#4111yescharacteristic
310C2×C4×C7H(5,𝔽2)11yescharacteristic
311C2×C4×C72−1+411yescharacteristic
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7—41no—
323C2×Dic7—41no—
324Dic14—22no—
325Dic14—22no—
326Dic14—22no—
327Dic14#11714yescomplemented
328C7⋊C8—22no—
329C7⋊C8—22no—
330C7⋊C8—41no—
331C7⋊C8—41no—
332C22×Q8—71no—
333C22×D4C22×D711yescharacteristic
334C23×C4—21no—
335C2×SD16—74no—
336C2×SD16—142no—
337C2×SD16—142no—
338C22×C8—71no—
339C22×C8—71no—
340C22×C8—71no—
341C22×C8—71no—
342C4×Q8—74no—
343C4×Q8—142no—
344C4×Q8—142no—
345C4×Q8—142no—
346C4×Q8—281no—
347C4×Q8—281no—
348C4×D4—22no—
349C4×D4C4×D714yes—
350#117—44no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4C22⋊D711yescharacteristic
362C2×C4⋊C4C22⋊D711yescharacteristic
363C2×C4⋊C4C22×D711yescharacteristic
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C42—21no—
372C2×C42—21no—
373C2×C42—141no—
374C2×C42—141no—
375C2×C42—141no—
376C2×C42C22×D711yescharacteristic
377C2×C42C22×D711yescharacteristic
378C2×C42C22×D711yescharacteristic
379#107—74no—
380C4⋊C8—142no—
381#103—74no—
382#103—142no—
383#103—142no—
384#102—74no—
385#102—281no—
386#102—281no—
387#97—74no—
388#97—281no—
389C4×C8—281no—
390#95—21no—
391#95—21no—
392#95—141no—
393#95—141no—
394#95—141no—
395#95—141no—
396#95C22⋊D711yescharacteristic
397#95C22⋊D711yescharacteristic
398C22×C7—42no—
399C22×C7—42no—
400C22×C7—42no—
401C22×C7—81no—
402C22×C7—81no—
403C22×C7—81no—
404C22×C7—81no—
405C22×C7C2×C4×D411yescharacteristic
406C22×C7#54611yescharacteristic
407C22×C7#54811yescharacteristic
408C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
409C22×C7#57511yescharacteristic
410C22×C7#57711yescharacteristic
411C22×C7#57811yescharacteristic
412C4×C7—22no—
413C4×C7—22no—
414C4×C7—22no—
415C4×C7—42no—
416C4×C7—41no—
417C4×C7—41no—
418C4×C7—41no—
419C4×C7—81no—
420C4×C7—81no—
421C4×C7C2×#11711yescharacteristic
422C4×C7#52011yescharacteristic
423C4×C7#55211yescharacteristic
424C4×C7#55311yescharacteristic
425Dic7—24no—
426Dic7—42no—
427Dic7—81no—
428Dic7—81no—
429C24—21no—
430C2×Q8—72no—
431C2×Q8—72no—
432C2×Q8—72no—
433C2×Q8—141no—
434C2×Q8—141no—
435C2×Q8—141no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4C2×C4×D712yes—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—141no—
449C22×C4—141no—
450C22×C4—141no—
451C22×C4—141no—
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C2×C22⋊D711yescharacteristic
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C2×C22⋊D711yescharacteristic
457C22×C4C2×C22⋊D711yescharacteristic
458C22×C4C23×D711yescharacteristic
459SD16—148no—
460C2×C8—72no—
461C2×C8—72no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—41no—
479C4⋊C4—42no—
480C4⋊C4—72no—
481C4⋊C4—81no—
482C4⋊C4—144no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—141no—
488C4⋊C4—282no—
489C4⋊C4—282no—
490C4⋊C4—282no—
491C4⋊C4—282no—
492C4⋊C4C2×C4×D712yes—
493C22⋊C4—24no—
494C22⋊C4—42no—
495C22⋊C4—42no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—42no—
504C42—144no—
505C42—282no—
506C42—282no—
507C42C2×C4×D712yes—
508C42#115712yes—
509C2×C7—82no—
510C2×C7#235211yescharacteristic
511C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
512C2×C7#294711yescharacteristic
513C2×C7#330811yescharacteristic
514C2×C7#332111yescharacteristic
515C2×C7#336811yescharacteristic
516C2×C7#337711yescharacteristic
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C22×C22⋊D711yescharacteristic
525Q8—74no—
526Q8—142no—
527Q8—142no—
528Q8—142no—
529D4—22no—
530D4—22no—
531D4—41no—
532D4—41no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
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—48no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—142no—
559C2×C4—142no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—281no—
563C2×C4—281no—
564C2×C4—281no—
565C2×C4—281no—
566C2×C4—281no—
567C2×C4—281no—
568C2×C4—281no—
569C2×C4—281no—
570C2×C4C4×C22⋊D712yes—
571C2×C4C4×C22⋊D712yes—
572C2×C4#649512yes—
573C2×C4C22×C4×D711yescharacteristic
574C2×C4C2×#115711yescharacteristic
575C2×C4D7×#4111yescharacteristic
576C2×C4D7×#4111yescharacteristic
577C2×C4#655311yescharacteristic
578C2×C4#655411yescharacteristic
579C8—142no—
580C8—142no—
581C8—281no—
582C8—281no—
583C7#2278111yescharacteristic, complemented, Sylow
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22C2×C4×C22⋊D711yescharacteristic
592C22C2×#649511yescharacteristic
593C22#9051311yescharacteristic
594C22#9052011yescharacteristic
595C22#9052211yescharacteristic
596C22#9052711yescharacteristic
597C22#9052911yescharacteristic
598C4—22no—
599C4—22no—
600C4—22no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—144no—
608C4—282no—
609C4—561no—
610C4—561no—
611C4D7×#11711yescharacteristic
612C4#9021411yescharacteristic
613C4#9022011yescharacteristic
614C4#9022111yescharacteristic
615C2—82no—
616C2#1173971711yescharacteristic
617C2C4×#649511yescharacteristic
618C2#1174001611yescharacteristic
619C2#1174097711yescharacteristic
620C2#1174103211yescharacteristic
621C2#1174110111yescharacteristic
622C2#1174112311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2354C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#2354D7
hypocenterC7#22781
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
3. #11731082C2semidirect
9. #11727342C2semidirect
10. #11727329C2semidirect
11. #11727232C2semidirect
12. #11727229C2semidirect
13. C4×#6384C2semidirect
14. #11727215C2semidirect
15. #11727212C2semidirect
34. #89790C4semidirect
327. Dic14#117semidirect
583. C7#22781semidirect

Maximal subgroups

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

subgroupindex
C7×#23542
C2×#897902
#117310822
#117293592
#117274142
#117273992
#117273982
#117273422
#117273292
#117272322
#117272292
C4×#63842
#117272152
#117272122
#227817