Finite Group Databasebeta

← #13358703

Subgroups — #13358703

Order 1792.

826 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2274C211yesmaximal, characteristic
2#11739676C211yesmaximal, characteristic
3C2×#89772C211yesmaximal, characteristic
4#11729330C211yesmaximal, characteristic
5#11729330C211yesmaximal, characteristic
6#11729310C211yesmaximal, characteristic
7#11729309C211yesmaximal, characteristic
8#11727235C211yesmaximal, characteristic, complemented
9#11727235C211yesmaximal, characteristic, complemented
10C4×#6381C211yesmaximal, characteristic, complemented
11C4×#6382C211yesmaximal, characteristic, complemented
12#11727222C211yesmaximal, characteristic, complemented
13#11727222C211yesmaximal, characteristic, complemented
14#11727221C211yesmaximal, characteristic, complemented
15#11727220C211yesmaximal, characteristic, complemented
16C2×C7×#117C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C22×C7×C4⋊C4C2211yescharacteristic
19C4×C7×C4⋊C4—21no—
20C4×C7×C4⋊C4—21no—
21C4×C7×C4⋊C4C2211yescharacteristic
22C4×C7×C4⋊C4C2211yescharacteristic
23C4×C7×C4⋊C4C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C4×C7×C22⋊C4—21no—
26C4×C7×C22⋊C4—21no—
27#89984C2211yescharacteristic
28#89984C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31#89772C418yescomplemented
32#89753—21no—
33#89753—21no—
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42#89723—21no—
43#89723—21no—
44#89708C2211yescharacteristic
45#89707C2211yescharacteristic
46C4×C4⋊Dic7C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×#6377C2211yescharacteristic
51C2×C4×C7⋊C8C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21367—71nomaximal, Sylow
61C23×C4×C7—21no—
62C23×C4×C7C2311yescharacteristic
63C7×#117—22no—
64C7×#117—22no—
65C7×#117C2×C414yes—
66C7×#117Q814yes—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4D411yescharacteristic
78C2×C7×C22⋊C4D411yescharacteristic
79C2×C42×C7—21no—
80C2×C42×C7—21no—
81C2×C42×C7—21no—
82C2×C42×C7—21no—
83C2×C42×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C22⋊(C7⋊C8)C2×C414yes—
111C22⋊(C7⋊C8)C2×C414yes—
112#6382—22no—
113#6382—22no—
114#6382—22no—
115#6382—22no—
116#6381—22no—
117#6381—22no—
118#6381—22no—
119#6381—22no—
120C4⋊(C7⋊C8)C2×C414yescomplemented
121C4⋊(C7⋊C8)C2×C414yescomplemented
122#6377C2×C414yescomplemented
123C4×C7⋊C8C2×C414yescomplemented
124C2×#433—71no—
125#2314—71no—
126#2274D711yescharacteristic
127#1949—71no—
128#1949—71no—
129#1948—71no—
130#1947—71no—
131#1854—71no—
132#1854—71no—
133#1853—71no—
134#1852—71no—
135#1776—71no—
136#1776—71no—
137C4×#107—71no—
138C4×#106—71no—
139C24×C7C2×D411yescharacteristic
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—24no—
149C22×C4×C7—81no—
150C22×C4×C7—81no—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×Q812yes—
158C22×C4×C7C2411yescharacteristic
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C7×C4⋊C4—24no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4C4⋊C414yes—
169C7×C4⋊C4C4⋊C414yes—
170C7×C4⋊C4#4112yes—
171C7×C4⋊C4#4112yes—
172C7×C4⋊C4#4112yes—
173C7×C4⋊C4#4112yes—
174C7×C22⋊C4—24no—
175C7×C22⋊C4C4⋊C414yes—
176C7×C22⋊C4C4⋊C414yes—
177C42×C7—24no—
178C42×C7—81no—
179C42×C7—81no—
180C42×C7—81no—
181C42×C7—81no—
182C42×C7C22×C412yes—
183C42×C7C22×C412yes—
184C42×C7C2×Q812yes—
185C42×C7C2×Q812yes—
186C22⋊Dic7—81no—
187C22⋊Dic7—81no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—81no—
197C4⋊Dic7—81no—
198C4×Dic7—81no—
199C4×Dic7—81no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8C22×C412yes—
209C2×C7⋊C8C22×C412yes—
210C2×C7⋊C8C22×C412yes—
211C2×C7⋊C8C22×C412yes—
212C2×#117C2×D711yescharacteristic
213C2×#117C2×D711yescharacteristic
214C22×C4⋊C4C2×D711yescharacteristic
215#433—78no—
216C2×#107—71no—
217C2×#107—71no—
218C2×#107—71no—
219C2×#107—71no—
220C2×#106—71no—
221C2×#106—71no—
222C2×#106—71no—
223C2×#106—71no—
224C2×C4⋊C8—71no—
225C2×C4⋊C8—71no—
226C2×C22⋊C8—71no—
227C2×C22⋊C8—71no—
228C2×#97—71no—
229C2×C4×C8—71no—
230#388—141no—
231#388—141no—
232#387—141no—
233#387—141no—
234#384—71no—
235#382—71no—
236#380—71no—
237#380—71no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4—71no—
242C4×C4⋊C4C2×D711yescharacteristic
243C4×C4⋊C4C2×D711yescharacteristic
244C4×C4⋊C4C2×D711yescharacteristic
245C4×C4⋊C4C2×D711yescharacteristic
246C4×C22⋊C4—21no—
247C4×C22⋊C4—21no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340—71no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—21no—
262C23×C7C2×C4⋊C412yes—
263C23×C7C22×D411yescharacteristic
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—84no—
291C2×C4×C7C2×C4⋊C412yes—
292C2×C4×C7C2×C4⋊C412yes—
293C2×C4×C7C2×C4⋊C412yes—
294C2×C4×C7C2×C4⋊C412yes—
295C2×C4×C7C2×C4⋊C412yes—
296C2×C4×C7C23×C411yescharacteristic
297C2×C4×C7C22×Q811yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7C2×#4111yescharacteristic
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C7⋊C8—22no—
315C7⋊C8—22no—
316C7⋊C8—22no—
317C7⋊C8—22no—
318C23×C4—21no—
319C23×C4C22×D711yescharacteristic
320C22×C8—71no—
321C22×C8—71no—
322C22×C8—71no—
323C22×C8—71no—
324#117—22no—
325#117—22no—
326#117Dic1414yes—
327#117C4×D714yes—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4C22⋊D711yescharacteristic
338C2×C4⋊C4C22⋊D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—141no—
345C2×C22⋊C4—141no—
346C2×C22⋊C4C22⋊D711yescharacteristic
347C2×C22⋊C4C22⋊D711yescharacteristic
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—141no—
354C2×C42—141no—
355C2×C42C22×D711yescharacteristic
356C2×C42C22×D711yescharacteristic
357#107—142no—
358#107—142no—
359#107—142no—
360#107—142no—
361#106—142no—
362#106—142no—
363#106—142no—
364#106—142no—
365C4⋊C8—74no—
366C4⋊C8—74no—
367C22⋊C8—74no—
368C22⋊C8—74no—
369#97—74no—
370C4×C8—74no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—22no—
384C22×C7—41no—
385C22×C7—41no—
386C22×C7—41no—
387C22×C7—41no—
388C22×C7C22×C4⋊C411yescharacteristic
389C22×C7#52511yescharacteristic
390C22×C7#52511yescharacteristic
391C22×C7#57611yescharacteristic
392C22×C7#57711yescharacteristic
393C22×C7#57711yescharacteristic
394C22×C7#57811yescharacteristic
395C4×C7—22no—
396C4×C7—22no—
397C4×C7—22no—
398C4×C7—22no—
399C4×C7—22no—
400C4×C7—42no—
401C4×C7—42no—
402C4×C7—42no—
403C4×C7—42no—
404C4×C7C4×#4111yescharacteristic
405C4×C7C4×#4111yescharacteristic
406C4×C7#53311yescharacteristic
407C4×C7#53311yescharacteristic
408Dic7—81no—
409Dic7—81no—
410Dic7—81no—
411Dic7—81no—
412C24C2×C22⋊D711yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—81no—
423C22×C4—81no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4C2×Dic1412yes—
429C22×C4C2×C4×D712yes—
430C22×C4C2×C22⋊D711yescharacteristic
431C22×C4C2×C22⋊D711yescharacteristic
432C22×C4C2×C22⋊D711yescharacteristic
433C22×C4C2×C22⋊D711yescharacteristic
434C22×C4C2×C22⋊D711yescharacteristic
435C22×C4C23×D711yescharacteristic
436C2×C8—72no—
437C2×C8—72no—
438C2×C8—72no—
439C2×C8—72no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C2×C8—141no—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—561no—
462C4⋊C4—561no—
463C4⋊C4Dic7⋊C414yes—
464C4⋊C4Dic7⋊C414yes—
465C4⋊C4#115712yes—
466C4⋊C4#115712yes—
467C4⋊C4#115712yes—
468C4⋊C4#115712yes—
469C22⋊C4—24no—
470C22⋊C4—561no—
471C22⋊C4—561no—
472C22⋊C4Dic7⋊C414yes—
473C22⋊C4Dic7⋊C414yes—
474C42—24no—
475C42—81no—
476C42—81no—
477C42—81no—
478C42—81no—
479C42—561no—
480C42—561no—
481C42C2×Dic1412yes—
482C42C2×Dic1412yes—
483C42C2×C4×D712yes—
484C42C2×C4×D712yes—
485C2×C7—42no—
486C2×C7#229611yescharacteristic
487C2×C7#291211yescharacteristic
488C2×C7#291311yescharacteristic
489C2×C7#311811yescharacteristic
490C2×C7#311911yescharacteristic
491C2×C7#312011yescharacteristic
492C2×C7#312111yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23C2×Dic7⋊C412yes—
500C23C22×C22⋊D711yescharacteristic
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—84no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4C2×Dic7⋊C412yes—
541C2×C4C2×Dic7⋊C412yes—
542C2×C4C2×Dic7⋊C412yes—
543C2×C4C2×Dic7⋊C412yes—
544C2×C4C2×Dic7⋊C412yes—
545C2×C4C22×Dic1411yescharacteristic
546C2×C4C22×C4×D711yescharacteristic
547C2×C4C2×#115711yescharacteristic
548C2×C4C2×#115711yescharacteristic
549C2×C4C2×#115711yescharacteristic
550C2×C4C2×#115711yescharacteristic
551C8—142no—
552C8—142no—
553C8—142no—
554C8—142no—
555C7#2136711yescharacteristic, complemented, Sylow
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22C22×Dic7⋊C411yescharacteristic
564C22#9048211yescharacteristic
565C22#9048211yescharacteristic
566C22#9052611yescharacteristic
567C22#9052711yescharacteristic
568C22#9052811yescharacteristic
569C22#9052911yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—22no—
574C4—22no—
575C4—42no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—561no—
580C4—561no—
581C4—561no—
582C4—561no—
583C4#9016111yescharacteristic
584C4#9016111yescharacteristic
585C4C4×#115711yescharacteristic
586C4C4×#115711yescharacteristic
587C2—42no—
588C2#1173951411yescharacteristic
589C2#1173998611yescharacteristic
590C2#1173999711yescharacteristic
591C2#1174000611yescharacteristic
592C2#1174000711yescharacteristic
593C2#1174001311yescharacteristic
594C2#1174002011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2274C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#2274D7
hypocenterC7#21367
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2274

Complements

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

NH ≅ G/Ntype
8. #11727235C2semidirect
9. #11727235C2semidirect
10. C4×#6381C2semidirect
11. C4×#6382C2semidirect
12. #11727222C2semidirect
13. #11727222C2semidirect
14. #11727221C2semidirect
15. #11727220C2semidirect
31. #89772C4semidirect
120. C4⋊(C7⋊C8)C2×C4semidirect
121. C4⋊(C7⋊C8)C2×C4semidirect
122. #6377C2×C4semidirect
123. C4×C7⋊C8C2×C4semidirect
555. C7#21367semidirect

Maximal subgroups

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

subgroupindex
C7×#22742
#117396762
C2×#897722
#117293302
#117293102
#117293092
#117272352
C4×#63812
C4×#63822
#117272222
#117272212
#117272202
#213677