Finite Group Databasebeta

← #13354001

Subgroups — #13354001

Order 1792.

894 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89480C211yesmaximal, characteristic
2#11731804C211yesmaximal, characteristic
3#11731670C211yesmaximal, characteristic
4C7×#1836C211yesmaximal, characteristic, complemented
5#11727569C211yesmaximal, characteristic, complemented
6#11727328C211yesmaximal, characteristic, complemented
7#11727301C211yesmaximal, characteristic, complemented
8#11726621C211yesmaximal, characteristic
9#11726621C211yesmaximal, characteristic
10#11726620C211yesmaximal, characteristic
11D7×#340C211yesmaximal, characteristic
12#11726603C211yesmaximal, characteristic, complemented
13#11726603C211yesmaximal, characteristic, complemented
14#11726602C211yesmaximal, characteristic, complemented
15#11726601C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C22×C8×D7C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4C2211yescharacteristic
21C2×D7×C4⋊C4C2211yescharacteristic
22C2×#6435C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C7×#384C2211yescharacteristic
25C7×#382C2211yescharacteristic
26C2×#6389C2211yescharacteristic
27C2×C8×Dic7C2211yescharacteristic
28#89761—21no—
29#89761—21no—
30#89749C2211yescharacteristic
31#89749C2211yescharacteristic
32#89748C2211yescharacteristic
33#89748C2211yescharacteristic
34#89744—21no—
35#89744—21no—
36#89708C2211yescharacteristic
37#89707C2211yescharacteristic
38C2×C4×C7⋊C8C2211yescharacteristic
39#89480C418yescomplemented
40#89440—21no—
41#89440—21no—
42#89419—21no—
43#89419—21no—
44C7×#340C2211yescharacteristic
45C7×#340C2211yescharacteristic
46C7×#340C2211yescharacteristic
47C7×#340C2211yescharacteristic
48#89346C2211yescharacteristic
49#89346C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#20763—71nomaximal, Sylow
61C22×C4×D7—21no—
62C22×C4×D7—21no—
63C22×C4×D7C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4D411yescharacteristic
76C2×(C2×D7)⋊C4D411yescharacteristic
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4D411yescharacteristic
90C2×Dic7⋊C4D411yescharacteristic
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C2×#1131—22no—
97C2×#1131—22no—
98C2×#1131C2×C414yes—
99C2×C8×D7—22no—
100C2×C8×D7—22no—
101C2×C8×D7C2×C414yes—
102#6455—22no—
103#6455—22no—
104#6455—22no—
105#6455—22no—
106D7×C4⋊C4—22no—
107D7×C4⋊C4—22no—
108D7×C4⋊C4—22no—
109D7×C4⋊C4—22no—
110#6435C2×C414yes—
111#6435Q814yes—
112C7×#97C2×C414yescomplemented
113C7×#95—21no—
114C7×#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121#6389C2×C414yescomplemented
122C8×Dic7C2×C414yescomplemented
123C4×C7⋊C8C2×C414yescomplemented
124C2×#405—71no—
125#2317—71no—
126#2317—71no—
127#1836—71no—
128#1836—71no—
129#1836—71no—
130#1836D711yescharacteristic, complemented
131#1835—71no—
132#1835—71no—
133#1834—71no—
134#1834—71no—
135#1736—71no—
136#1736—71no—
137#1735—71no—
138C2×#340—71no—
139C23×D7C2×D411yescharacteristic
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2411yescharacteristic
147C22×Dic7—21no—
148C22×Dic7—21no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7C2×D411yescharacteristic
152C22×Dic7C2×D411yescharacteristic
153C22×Dic7C2×D411yescharacteristic
154C2×C4×D7—42no—
155C2×C4×D7—42no—
156C2×C4×D7—81no—
157C2×C4×D7—81no—
158C2×C4×D7—81no—
159C2×C4×D7—81no—
160C2×C4×D7C22×C412yes—
161C2×C4×D7C2×Q812yes—
162C2×C4×D7#4112yes—
163C2×C4×D7#4112yes—
164C2×C4×D7#4112yes—
165C2×C4×D7#4112yes—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8C22×C412yes—
171C2×C7×C8C22×C412yes—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—81no—
181C7×C4⋊C4—81no—
182C42×C7C22×C412yes—
183C42×C7C2×Q812yes—
184(C2×D7)⋊C4—42no—
185(C2×D7)⋊C4—42no—
186(C2×D7)⋊C4C4⋊C414yes—
187(C2×D7)⋊C4C4⋊C414yes—
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—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200Dic7⋊C4—81no—
201Dic7⋊C4—81no—
202Dic7⋊C4—81no—
203Dic7⋊C4—81no—
204Dic7⋊C4C4⋊C414yes—
205Dic7⋊C4C4⋊C414yes—
206C4×Dic7—42no—
207C4×Dic7—42no—
208C4×Dic7C22×C412yes—
209C4×Dic7C2×Q812yes—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8C22×C412yes—
215C2×C7⋊C8C22×C412yes—
216#1131—28no—
217C8×D7—28no—
218C22×M16—71no—
219C23×C8—71no—
220C2×#117—71no—
221C2×#117—71no—
222C2×#117—71no—
223C22×C4⋊C4—71no—
224C22×C4⋊C4—71no—
225#405—78no—
226C2×#97—71no—
227C2×#97C2×D711yescharacteristic
228C2×C4×C8—71no—
229C2×C4×C8—71no—
230#389—141no—
231#389—141no—
232#389—141no—
233#389—141no—
234#385—141no—
235#385—141no—
236#385—141no—
237#385—141no—
238#384—71no—
239#384—71no—
240#384—71no—
241#384C2×D711yescharacteristic
242#382—71no—
243#382—71no—
244#382—71no—
245#382C2×D711yescharacteristic
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
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#340C2×D711yescharacteristic
259#340C2×D711yescharacteristic
260#340C2×D711yescharacteristic
261#340C2×D711yescharacteristic
262C23×C7C22×D411yescharacteristic
263C22×D7—21no—
264C22×D7—21no—
265C22×D7—21no—
266C22×D7—21no—
267C22×D7C2×C4⋊C412yes—
268C22×D7#12412yes—
269C22×D7#12412yes—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7C2×C4⋊C412yes—
287C2×C4×C7C2×C4⋊C412yes—
288C2×C4×C7C23×C411yescharacteristic
289C2×C4×C7C22×Q811yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
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×Dic7C2×C4⋊C412yes—
311C2×Dic7C2×C4⋊C412yes—
312C2×Dic7C2×C4⋊C412yes—
313C2×Dic7#12412yes—
314C2×Dic7#12412yes—
315C4×D7—84no—
316C4×D7—84no—
317C4×D7#11714yes—
318C4×D7#11714yes—
319C4×D7#12514yes—
320C4×D7#12514yes—
321C7×C8—22no—
322C7×C8—22no—
323C7⋊C8—22no—
324C7⋊C8—22no—
325C23×C4—71no—
326C23×C4—141no—
327C23×C4—141no—
328C2×M16—74no—
329C2×M16—142no—
330C2×M16—142no—
331C22×C8—71no—
332C22×C8—71no—
333C22×C8—74no—
334C22×C8—142no—
335C22×C8—142no—
336C22×C8C22×D711yescharacteristic
337C22×C8C22×D711yescharacteristic
338#117—74no—
339#117—74no—
340#117—142no—
341#117—142no—
342#117—142no—
343#117—142no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—142no—
353C2×C4⋊C4—142no—
354C2×C4⋊C4—142no—
355C2×C4⋊C4—142no—
356C2×C4⋊C4—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—141no—
360C2×C4⋊C4—141no—
361C2×C4⋊C4—141no—
362C2×C4⋊C4—141no—
363C2×C4⋊C4—141no—
364C2×C4⋊C4C22×D711yescharacteristic
365C2×C4⋊C4C22×D711yescharacteristic
366C2×C4⋊C4C22×D711yescharacteristic
367C2×C4⋊C4C22×D711yescharacteristic
368C2×C22⋊C4—71no—
369C2×C22⋊C4—71no—
370C2×C22⋊C4—141no—
371C2×C22⋊C4—141no—
372C2×C42—71no—
373C2×C42—141no—
374C2×C42—141no—
375C2×C42C22×D711yescharacteristic
376#97—74no—
377#97C4×D714yescomplemented
378C4×C8—74no—
379C4×C8—74no—
380#95—21no—
381#95—21no—
382#95—141no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387#95—141no—
388C22×C7C22×C4⋊C411yescharacteristic
389C22×C7C2×#12411yescharacteristic
390C22×C7C2×#12411yescharacteristic
391C22×C7#57611yescharacteristic
392C22×C7#57711yescharacteristic
393C22×C7#57711yescharacteristic
394C22×C7#57811yescharacteristic
395C2×D7—22no—
396C2×D7—22no—
397C2×D7—22no—
398C2×D7—22no—
399C2×D7—22no—
400C2×D7—22no—
401C2×D7#38214yes—
402C4×C7—22no—
403C4×C7—22no—
404C4×C7—81no—
405C4×C7—81no—
406C4×C7—81no—
407C4×C7—81no—
408C4×C7C2×#11711yescharacteristic
409C4×C7C2×#11711yescharacteristic
410C4×C7C2×#12511yescharacteristic
411C4×C7C2×#12511yescharacteristic
412Dic7—22no—
413Dic7—22no—
414Dic7—81no—
415Dic7—81no—
416Dic7—81no—
417Dic7—81no—
418Dic7#38214yes—
419C24—71no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—71no—
425C22×C4—72no—
426C22×C4—72no—
427C22×C4—72no—
428C22×C4—72no—
429C22×C4—72no—
430C22×C4—72no—
431C22×C4—71no—
432C22×C4—71no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—141no—
437C22×C4—282no—
438C22×C4—282no—
439C22×C4—561no—
440C22×C4—561no—
441C22×C4—561no—
442C22×C4—561no—
443C22×C4D4×D711yescharacteristic
444C22×C4D4×D711yescharacteristic
445C22×C4C23×D711yescharacteristic
446M16—148no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—72no—
452C2×C8—72no—
453C2×C8—148no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8C2×C4×D712yes—
459C2×C8C2×C4×D712yes—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—74no—
469C4⋊C4—74no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—282no—
481C4⋊C4—282no—
482C4⋊C4—561no—
483C4⋊C4—561no—
484C4⋊C4—561no—
485C4⋊C4—561no—
486C4⋊C4—561no—
487C4⋊C4—561no—
488C22⋊C4—74no—
489C22⋊C4—74no—
490C22⋊C4—282no—
491C22⋊C4—282no—
492C42—72no—
493C42—72no—
494C42—282no—
495C42—282no—
496C42C2×C4×D712yes—
497C42Q8×D712yes—
498C2×C7C2×#38211yescharacteristic
499C2×C7#291211yescharacteristic
500C2×C7#291311yescharacteristic
501C2×C7#313611yescharacteristic
502C2×C7#313611yescharacteristic
503C2×C7#313711yescharacteristic
504C2×C7#313711yescharacteristic
505D7—24no—
506C23—72no—
507C23—72no—
508C23—72no—
509C23—141no—
510C23—141no—
511C23—141no—
512C23—141no—
513C23C2×D4×D711yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
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—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—72no—
531C2×C4—72no—
532C2×C4—72no—
533C2×C4—74no—
534C2×C4—74no—
535C2×C4—74no—
536C2×C4—74no—
537C2×C4—72no—
538C2×C4—72no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
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—281no—
552C2×C4—281no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—564no—
556C2×C4—564no—
557C2×C4D7×C4⋊C412yes—
558C2×C4D7×C4⋊C412yes—
559C2×C4C22×C4×D711yescharacteristic
560C2×C4C2×Q8×D711yescharacteristic
561C2×C4D7×#4111yescharacteristic
562C2×C4D7×#4111yescharacteristic
563C2×C4D7×#4111yescharacteristic
564C2×C4D7×#4111yescharacteristic
565C8—22no—
566C8—22no—
567C8—142no—
568C8—142no—
569C7#2076311yescharacteristic, complemented, Sylow
570C22—74no—
571C22—142no—
572C22—142no—
573C22—142no—
574C22—142no—
575C22—142no—
576C22—142no—
577C22C2×D7×C4⋊C411yescharacteristic
578C22D7×#12411yescharacteristic
579C22D7×#12411yescharacteristic
580C22#9046511yescharacteristic
581C22#9046611yescharacteristic
582C22#9046911yescharacteristic
583C22#9047011yescharacteristic
584C4—22no—
585C4—22no—
586C4—74no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—142no—
592C4—142no—
593C4—561no—
594C4—561no—
595C4—561no—
596C4—561no—
597C4D7×#11711yescharacteristic
598C4D7×#11711yescharacteristic
599C4D7×#12511yescharacteristic
600C4D7×#12511yescharacteristic
601C2—144no—
602C2D7×#38211yescharacteristic
603C2#1173665411yescharacteristic
604C2#1173665511yescharacteristic
605C2#1173912511yescharacteristic
606C2#1173912611yescharacteristic
607C2#1173915911yescharacteristic
608C2#1173918311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1836C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1836D7
hypocenterC7#20763
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

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

NH ≅ G/Ntype
4. C7×#1836C2semidirect
5. #11727569C2semidirect
6. #11727328C2semidirect
7. #11727301C2semidirect
12. #11726603C2semidirect
13. #11726603C2semidirect
14. #11726602C2semidirect
15. #11726601C2semidirect
39. #89480C4semidirect
112. C7×#97C2×C4semidirect
121. #6389C2×C4semidirect
122. C8×Dic7C2×C4semidirect
123. C4×C7⋊C8C2×C4semidirect
130. #1836D7semidirect
377. #97C4×D7semidirect
569. C7#20763semidirect

Maximal subgroups

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

subgroupindex
C2×#894802
#117318042
#117316702
C7×#18362
#117275692
#117273282
#117273012
#117266212
#117266202
D7×#3402
#117266032
#117266022
#117266012
#207637