Finite Group Databasebeta

← #13373213

Subgroups — #13373213

Order 1792.

942 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2349C211yesmaximal, characteristic, complemented
2C2×#89786C211yesmaximal, characteristic
3#11731083C211yesmaximal, characteristic, complemented
4#11729367C211yesmaximal, characteristic
5#11729367C211yesmaximal, characteristic
6#11729363C211yesmaximal, characteristic, complemented
7C4×#6406C211yesmaximal, characteristic, complemented
8#11729358C211yesmaximal, characteristic, complemented
9#11727371C211yesmaximal, characteristic, complemented
10#11727369C211yesmaximal, characteristic, complemented
11#11727356C211yesmaximal, characteristic, complemented
12#11727355C211yesmaximal, characteristic, complemented
13#11727310C211yesmaximal, characteristic, complemented
14#11727302C211yesmaximal, characteristic, complemented
15#11727232C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C22×#1141C2211yescharacteristic
20C2×#6455—21no—
21C2×C4×D28C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#385—21no—
25C7×#385—21no—
26C7×#382C2211yescharacteristic
27C7×#380—21no—
28C4×C7×C4⋊C4C2211yescharacteristic
29C2×C7×#95—21no—
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33#89786C418yescomplemented
34#89759—21no—
35C2×#6383C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39#89711C2211yescharacteristic
40#89711C2211yescharacteristic
41#89707C2211yescharacteristic
42C4×C4⋊Dic7C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×#6377C2211yescharacteristic
46C2×#6377C2211yescharacteristic
47C2×C4×C7⋊C8C2211yescharacteristic
48#89442—21no—
49#89442—21no—
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22780—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C22×D28C2311yescharacteristic
66C22×C4×D7—21no—
67C4×C7×D4—22no—
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C4×C7×D4—41no—
71C4×C7×D4—41no—
72C4×C7×D4C2×C414yes—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C22⋊C4—21no—
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×#1141—22no—
87C2×#1141—22no—
88C2×#1141C2×C414yes—
89C2×(C2×D7)⋊C4—21no—
90C2×(C2×D7)⋊C4—21no—
91C2×(C2×D7)⋊C4—21no—
92C2×(C2×D7)⋊C4—21no—
93C2×(C2×D7)⋊C4—21no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—21no—
96C2×C4⋊Dic7C2311yescharacteristic
97C2×C4⋊Dic7C2311yescharacteristic
98C2×C4⋊Dic7C2311yescharacteristic
99C2×C4×Dic7—21no—
100C2×C4×Dic7—21no—
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105#6455—44no—
106C4×D28—22no—
107C4×D28C2×C414yescomplemented
108C7×#95—21no—
109C7×#95—21no—
110C7×#95—21no—
111C7×#95—21no—
112C7×#95D411yescharacteristic
113C7×#95D411yescharacteristic
114#6406—22no—
115#6406—22no—
116#6406C2×C414yes—
117C7⋊#95—21no—
118C7⋊#95—21no—
119#6383—41no—
120#6383—41no—
121#6383C2×C414yescomplemented
122#6382C2×C414yescomplemented
123C4⋊(C7⋊C8)—22no—
124#6377—41no—
125#6377C2×C414yescomplemented
126C4×C7⋊C8—41no—
127C2×#442—71no—
128#2354—71no—
129#2349D711yescharacteristic, complemented
130#1936—71no—
131#1934—71no—
132#1920—71no—
133#1919—71no—
134#1879—71no—
135#1879—71no—
136#1836—71no—
137#1804—71no—
138#1802—71no—
139#1771—71no—
140#1762—71no—
141C4×#102—71no—
142C24×C7—21no—
143C23×D7—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4C22×C412yes—
148C2×C7×D4#4112yes—
149C2×C7×D4#4112yes—
150C22×C4×C7—21no—
151C22×C4×C7—24no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—42no—
155C22×C4×C7—42no—
156C22×C4×C7—42no—
157C22×C4×C7—42no—
158C22×C4×C7—81no—
159C22×C4×C7—81no—
160C22×C4×C7—81no—
161C22×C4×C7—81no—
162C22×C4×C7C2×D411yescharacteristic
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—
172C2×D28—21no—
173C2×D28—21no—
174C2×D28—21no—
175C2×D28—21no—
176C2×D28—21no—
177C2×D28C22×C412yes—
178C2×C4×D7—24no—
179C2×C4×D7—42no—
180C2×C4×D7—42no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—21no—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—41no—
188C7×C4⋊C4—81no—
189C7×C4⋊C4—81no—
190C7×C4⋊C4C22×C412yes—
191C7×C22⋊C4—24no—
192C7×C22⋊C4—42no—
193C7×C22⋊C4—42no—
194C42×C7—21no—
195C42×C7—21no—
196C42×C7—21no—
197C42×C7—21no—
198C42×C7C22×C412yes—
199C42×C7#4112yes—
200#1141—28no—
201(C2×D7)⋊C4—24no—
202(C2×D7)⋊C4—42no—
203(C2×D7)⋊C4—42no—
204(C2×D7)⋊C4—81no—
205(C2×D7)⋊C4—81no—
206C4⋊Dic7—21no—
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—42no—
212C4⋊Dic7—81no—
213C4⋊Dic7C22×C412yes—
214C4×Dic7—42no—
215C4×Dic7—42no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8C22×C412yes—
227C2×C7⋊C8C22×C412yes—
228C22×D8—71no—
229C2×C4×D4—71no—
230C2×C4×D4C2×D711yescharacteristic
231C2×C4×D4C2×D711yescharacteristic
232C2×C4×D4C2×D711yescharacteristic
233C2×#117—141no—
234#442—78no—
235C2×#106—71no—
236C2×C4⋊C8—71no—
237C2×#102—71no—
238C2×#102—71no—
239C2×#102—71no—
240C2×#102—71no—
241C2×#102—71no—
242C2×#102—71no—
243C2×#97—71no—
244C2×#97—71no—
245C2×C4×C8—71no—
246#390—71no—
247#390—71no—
248#389C2×D711yescharacteristic
249#389C2×D711yescharacteristic
250#388—141no—
251#388—141no—
252#385—21no—
253#385—21no—
254#384—71no—
255#382C2×D711yescharacteristic
256#380—21no—
257#380—141no—
258C4×C4⋊C4—71no—
259C4×C4⋊C4—71no—
260C4×C4⋊C4C2×D711yescharacteristic
261C2×#95—21no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#336—71no—
271#336—71no—
272C23×C7—22no—
273C23×C7—22no—
274C23×C7—22no—
275C23×C7—41no—
276C23×C7—41no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7C22×D411yescharacteristic
280C22×D7—22no—
281C22×D7—41no—
282C22×D7—41no—
283C22×D7—41no—
284C22×D7—41no—
285C22×D7—41no—
286C22×D7—41no—
287C7×D4—22no—
288C7×D4—22no—
289C7×D4—22no—
290C7×D4#11714yescomplemented
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—22no—
298C2×C4×C7—21no—
299C2×C4×C7—21no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—84no—
316C2×C4×C7—84no—
317C2×C4×C7—84no—
318C2×C4×C7C4×D412yes—
319C2×C4×C7C4×D412yes—
320C2×C4×C7AGL(1,ℤ/8)12yes—
321C2×C4×C7C23×C411yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×C4×C7C2×#4111yescharacteristic
325C2×C4×C7H(5,𝔽2)11yescharacteristic
326C2×C4×C72−1+411yescharacteristic
327C2×Dic7—22no—
328C2×Dic7—41no—
329C2×Dic7—41no—
330C2×Dic7—41no—
331C2×Dic7—41no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336D28—22no—
337D28—22no—
338D28—41no—
339D28—41no—
340C4×D7—48no—
341C7⋊C8—22no—
342C7⋊C8—22no—
343C7⋊C8—41no—
344C7⋊C8—41no—
345C22×D4—71no—
346C22×D4C22×D711yescharacteristic
347C23×C4—21no—
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—141no—
351C2×D8—74no—
352C2×D8—142no—
353C2×D8—142no—
354C22×C8—71no—
355C22×C8—71no—
356C22×C8—71no—
357C22×C8—71no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—41no—
362C4×D4—41no—
363C4×D4—74no—
364C4×D4—142no—
365C4×D4C4×D714yes—
366#117—284no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—71no—
370C2×C4⋊C4—71no—
371C2×C4⋊C4—71no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4C22⋊D711yescharacteristic
375C2×C4⋊C4C22⋊D711yescharacteristic
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—141no—
383C2×C22⋊C4—141no—
384C2×C22⋊C4—141no—
385C2×C22⋊C4—141no—
386C2×C22⋊C4—141no—
387C2×C42—141no—
388C2×C42—141no—
389C2×C42C22×D711yescharacteristic
390C2×C42C22×D711yescharacteristic
391C2×C42C22×D711yescharacteristic
392#106—74no—
393C4⋊C8—142no—
394#102—74no—
395#102—74no—
396#102—142no—
397#102—142no—
398#102—281no—
399#102—281no—
400#97—74no—
401#97—281no—
402C4×C8—281no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—141no—
408#95—141no—
409#95C22⋊D711yescharacteristic
410#95C22⋊D711yescharacteristic
411C22×C7—24no—
412C22×C7—42no—
413C22×C7—42no—
414C22×C7—42no—
415C22×C7—42no—
416C22×C7—42no—
417C22×C7—42no—
418C22×C7C2×C4×D411yescharacteristic
419C22×C7#54611yescharacteristic
420C22×C7#54811yescharacteristic
421C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
422C22×C7#57511yescharacteristic
423C22×C7#57611yescharacteristic
424C22×C7#57711yescharacteristic
425C2×D7—42no—
426C2×D7—42no—
427C2×D7—42no—
428C2×D7—81no—
429C2×D7—81no—
430C2×D7—81no—
431C2×D7—81no—
432C4×C7—22no—
433C4×C7—22no—
434C4×C7—22no—
435C4×C7—42no—
436C4×C7—41no—
437C4×C7—41no—
438C4×C7—41no—
439C4×C7—81no—
440C4×C7—81no—
441C4×C7C2×#11711yescharacteristic
442C4×C7#52011yescharacteristic
443C4×C7#55211yescharacteristic
444C4×C7#55311yescharacteristic
445Dic7—42no—
446Dic7—81no—
447Dic7—81no—
448C24—21no—
449C24—141no—
450C2×D4—21no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—72no—
454C2×D4—141no—
455C2×D4—141no—
456C2×D4—141no—
457C2×D4—141no—
458C2×D4—141no—
459C2×D4C2×C4×D712yes—
460C2×D4#115912yes—
461C2×D4#116112yes—
462C22×C4—21no—
463C22×C4—24no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—141no—
475C22×C4—144no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—282no—
479C22×C4—282no—
480C22×C4C2×C22⋊D711yescharacteristic
481C22×C4C2×C22⋊D711yescharacteristic
482C22×C4C2×C22⋊D711yescharacteristic
483C22×C4C2×C22⋊D711yescharacteristic
484C22×C4C2×C22⋊D711yescharacteristic
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C23×D711yescharacteristic
487D8—148no—
488C2×C8—72no—
489C2×C8—72no—
490C2×C8—141no—
491C2×C8—141no—
492C2×C8—141no—
493C2×C8—141no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C2×C8—141no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—41no—
507C4⋊C4—72no—
508C4⋊C4—81no—
509C4⋊C4—81no—
510C4⋊C4—141no—
511C4⋊C4—141no—
512C4⋊C4—141no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—282no—
516C4⋊C4—561no—
517C4⋊C4C2×C4×D712yes—
518C22⋊C4—24no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—144no—
522C22⋊C4—282no—
523C22⋊C4—282no—
524C22⋊C4—561no—
525C22⋊C4—561no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—282no—
531C42—282no—
532C42C2×C4×D712yes—
533C42#115712yes—
534C2×C7—44no—
535C2×C7#235211yescharacteristic
536C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
537C2×C7#294611yescharacteristic
538C2×C7#330611yescharacteristic
539C2×C7#331911yescharacteristic
540C2×C7#336611yescharacteristic
541C2×C7#337611yescharacteristic
542D7—82no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—142no—
551C23—281no—
552C23—281no—
553C23—281no—
554C23—281no—
555C23—281no—
556C23—281no—
557C23C22×C22⋊D711yescharacteristic
558D4—22no—
559D4—22no—
560D4—22no—
561D4—142no—
562D4—142no—
563D4—281no—
564D4—281no—
565D4#645514yescomplemented
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—84no—
592C2×C4—84no—
593C2×C4—142no—
594C2×C4—288no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4C4×C22⋊D712yes—
604C2×C4C4×C22⋊D712yes—
605C2×C4#651212yes—
606C2×C4C22×C4×D711yescharacteristic
607C2×C4C2×#115711yescharacteristic
608C2×C4C2×#115911yescharacteristic
609C2×C4#654811yescharacteristic
610C2×C4C2×#116111yescharacteristic
611C2×C4#655111yescharacteristic
612C8—142no—
613C8—142no—
614C8—281no—
615C8—281no—
616C7#2278011yescharacteristic, complemented, Sylow
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—282no—
625C22—282no—
626C22—282no—
627C22—561no—
628C22—561no—
629C22—561no—
630C22—561no—
631C22C2×C4×C22⋊D711yescharacteristic
632C22#9049711yescharacteristic
633C22#9050811yescharacteristic
634C22C2×#651211yescharacteristic
635C22#9051311yescharacteristic
636C22#9052611yescharacteristic
637C22#9052711yescharacteristic
638C4—22no—
639C4—22no—
640C4—22no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—42no—
645C4—81no—
646C4—81no—
647C4—282no—
648C4—561no—
649C4—561no—
650C4C2×#645511yescharacteristic
651C4#9019511yescharacteristic
652C4#9020411yescharacteristic
653C4#9020511yescharacteristic
654C2—44no—
655C2—562no—
656C2#1173960611yescharacteristic
657C2C4×#651211yescharacteristic
658C2#1173998211yescharacteristic
659C2#1174094111yescharacteristic
660C2#1174094411yescharacteristic
661C2#1174108311yescharacteristic
662C2#1174109011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2349C2
socleC23×C7C22×D4
prime radicalC22×#1141C22
cosocleC22×C4×C7C24
hypercenter#2349D7
hypocenterC7#22780
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
1. C7×#2349C2semidirect
3. #11731083C2semidirect
6. #11729363C2semidirect
7. C4×#6406C2semidirect
8. #11729358C2semidirect
9. #11727371C2semidirect
10. #11727369C2semidirect
11. #11727356C2semidirect
12. #11727355C2semidirect
13. #11727310C2semidirect
14. #11727302C2semidirect
15. #11727232C2semidirect
33. #89786C4semidirect
107. C4×D28C2×C4semidirect
121. #6383C2×C4semidirect
122. #6382C2×C4semidirect
125. #6377C2×C4semidirect
129. #2349D7semidirect
290. C7×D4#117semidirect
565. D4#6455semidirect
616. C7#22780semidirect

Maximal subgroups

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

subgroupindex
C7×#23492
C2×#897862
#117310832
#117293672
#117293632
C4×#64062
#117293582
#117273712
#117273692
#117273562
#117273552
#117273102
#117273022
#117272322
#227807