Finite Group Databasebeta

← #13359078

Subgroups — #13359078

Order 1792.

1070 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90274C211yesmaximal, characteristic
2C2×#89480C211yesmaximal, characteristic
3#11731205C211yesmaximal, characteristic
4C7×#1763C211yesmaximal, characteristic, complemented
5C4×#6409C211yesmaximal, characteristic, complemented
6#11728025C211yesmaximal, characteristic
7#11728024C211yesmaximal, characteristic
8#11727982C211yesmaximal, characteristic, complemented
9#11727979C211yesmaximal, characteristic, complemented
10#11727567C211yesmaximal, characteristic, complemented
11C4×#6384C211yesmaximal, characteristic, complemented
12#11726638C211yesmaximal, characteristic
13#11726636C211yesmaximal, characteristic
14#11726559C211yesmaximal, characteristic, complemented
15#11726558C211yesmaximal, characteristic, complemented
16C2×C4×C7×Q8C2211yescharacteristic
17C2×#6509C2211yescharacteristic
18C2×#6507C2211yescharacteristic
19C22×#1131C2211yescharacteristic
20C22×C8×D7C2211yescharacteristic
21#90274—24no—
22#90274C418yescomplemented
23C2×#6461—21no—
24C2×#6460C2211yescharacteristic
25C2×#6453C2211yescharacteristic
26C2×#6452—21no—
27C2×#6439—21no—
28C2×#6435C2211yescharacteristic
29C2×#6433—21no—
30C2×C4×Dic14C2211yescharacteristic
31C2×C7×#103C2211yescharacteristic
32C2×C7×#103C2211yescharacteristic
33C2×C7×#97C2211yescharacteristic
34C4×C7×C4⋊C4C2211yescharacteristic
35C2×#6409C2211yescharacteristic
36C2×#6409C2211yescharacteristic
37C2×#6390C2211yescharacteristic
38C2×#6390C2211yescharacteristic
39C2×#6389C2211yescharacteristic
40C2×C8×Dic7C2211yescharacteristic
41#89759C2211yescharacteristic
42#89759C2211yescharacteristic
43#89749C2211yescharacteristic
44#89748C2211yescharacteristic
45C2×#6384C2211yescharacteristic
46C2×#6384C2211yescharacteristic
47#89712—21no—
48C4×C4⋊Dic7C2211yescharacteristic
49#89705—21no—
50C2×C4×C7⋊C8C2211yescharacteristic
51#89480—24no—
52#89441—21no—
53#89424—21no—
54C7×#340C2211yescharacteristic
55C7×#340C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21413—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×C4×D7C2311yescharacteristic
65C22×Dic14C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C4×C7×Q8—22no—
69C4×C7×Q8C2×C414yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C42×C7—21no—
75C2×C42×C7—21no—
76C2×C42×C7C2311yescharacteristic
77#6509—22no—
78#6509C2×C414yes—
79#6507—22no—
80#6507C2×C414yes—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×(C2×D7)⋊C4D411yescharacteristic
86C2×(C2×D7)⋊C4D411yescharacteristic
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4D411yescharacteristic
100C2×Dic7⋊C4D411yescharacteristic
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C2×#1131—22no—
107C2×#1131—22no—
108C2×#1131—22no—
109C2×C8×D7—22no—
110C2×C8×D7—22no—
111C2×C8×D7—22no—
112#6461—28no—
113#6460—22no—
114#6460C2×C414yes—
115#6453—22no—
116#6453C2×C414yes—
117#6452—28no—
118#6439—28no—
119#6435C2×C414yes—
120#6435D414yes—
121#6433—28no—
122C4×Dic14—22no—
123C4×Dic14C2×C414yes—
124C7×#103—22no—
125C7×#103—22no—
126C7×#97—22no—
127C7×#95—21no—
128#6409—22no—
129#6409—22no—
130C7⋊#95—21no—
131C7⋊#95—21no—
132C7⋊#95—21no—
133C7⋊#95—21no—
134C7⋊#95—21no—
135#6390—22no—
136#6390—22no—
137#6389—22no—
138C8×Dic7—22no—
139#6384—22no—
140#6384—22no—
141C4×C7⋊C8—22no—
142C2×#541—71no—
143C2×#405—71no—
144#2314—71no—
145#1992—71no—
146#1991—71no—
147#1960—71no—
148#1956—71no—
149#1894—71no—
150#1894—71no—
151#1892—71no—
152#1892—71no—
153#1763—71no—
154#1763D711yescharacteristic, complemented
155C4×#103—71no—
156C4×#103—71no—
157C23×D7C2×D411yescharacteristic
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8—21no—
163C2×C7×Q8C22×C412yes—
164C22×C4×C7—21no—
165C22×C4×C7—21no—
166C22×C4×C7—21no—
167C22×C4×C7—21no—
168C22×C4×C7C2×D411yescharacteristic
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2411yescharacteristic
171C22×Dic7—21no—
172C22×Dic7—21no—
173C22×Dic7—21no—
174C22×Dic7—21no—
175C22×Dic7C2×D411yescharacteristic
176C22×Dic7C2×D411yescharacteristic
177C22×Dic7C2×D411yescharacteristic
178C2×C4×D7—21no—
179C2×C4×D7—21no—
180C2×C4×D7—21no—
181C2×C4×D7—21no—
182C2×C4×D7C22×C412yes—
183C2×C4×D7C2×D412yes—
184C2×Dic14—21no—
185C2×Dic14—21no—
186C2×Dic14—21no—
187C2×Dic14—21no—
188C2×Dic14—21no—
189C2×Dic14C22×C412yes—
190C2×C7×C8—21no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8#4112yes—
195C2×C7×C8#4112yes—
196C7×C4⋊C4—24no—
197C7×C4⋊C4—21no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4—21no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4—21no—
202C7×C4⋊C4C22×C412yes—
203C42×C7—24no—
204C42×C7—81no—
205C42×C7C22×C412yes—
206C42×C7C2×D412yes—
207(C2×D7)⋊C4—24no—
208(C2×D7)⋊C4—24no—
209(C2×D7)⋊C4—81no—
210(C2×D7)⋊C4—81no—
211(C2×D7)⋊C4C22⋊C414yes—
212(C2×D7)⋊C4C22⋊C414yes—
213C4⋊Dic7—21no—
214C4⋊Dic7—24no—
215C4⋊Dic7—24no—
216C4⋊Dic7—21no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7—21no—
220C4⋊Dic7C22×C412yes—
221Dic7⋊C4—24no—
222Dic7⋊C4—24no—
223Dic7⋊C4—24no—
224Dic7⋊C4—24no—
225Dic7⋊C4—24no—
226Dic7⋊C4—81no—
227Dic7⋊C4—81no—
228Dic7⋊C4C22⋊C414yes—
229Dic7⋊C4C22⋊C414yes—
230C4×Dic7—24no—
231C4×Dic7—81no—
232C4×Dic7C22×C412yes—
233C4×Dic7C2×D412yes—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C2×C7⋊C8—21no—
238C2×C7⋊C8#4112yes—
239C2×C7⋊C8#4112yes—
240#1131—28no—
241C8×D7—28no—
242C22×M16—71no—
243C23×C8—71no—
244#541—78no—
245#541—144no—
246C2×C4⋊Q8—71no—
247C2×#126—141no—
248C2×#126—141no—
249C2×#125—71no—
250C2×#125—141no—
251C2×#125—141no—
252C2×C22⋊Q8—71no—
253C2×C22⋊Q8—71no—
254C2×C4×Q8—71no—
255C2×C4×Q8C2×D711yescharacteristic
256C2×#117—71no—
257C2×#103—71no—
258C2×#103—71no—
259C2×#103—71no—
260C2×#103—71no—
261C2×#103—71no—
262C2×#103—71no—
263C2×#103C2×D711yescharacteristic
264C2×#103C2×D711yescharacteristic
265#405—144no—
266C2×#97—71no—
267C2×#97C2×D711yescharacteristic
268C2×C4×C8—71no—
269C2×C4×C8—71no—
270#388—141no—
271#388—141no—
272#387—141no—
273#387—141no—
274#384—71no—
275#382—71no—
276#380—71no—
277#380—71no—
278C4×C4⋊C4—71no—
279C4×C4⋊C4C2×D711yescharacteristic
280#340—71no—
281#340—71no—
282#340—71no—
283#340—71no—
284#340—71no—
285#340—71no—
286#340C2×D711yescharacteristic
287#340C2×D711yescharacteristic
288C23×C7C22×D411yescharacteristic
289C22×D7—21no—
290C22×D7—21no—
291C22×D7—21no—
292C22×D7—21no—
293C22×D7—21no—
294C22×D7—21no—
295C22×D7C2×C22⋊C412yes—
296C7×Q8—22no—
297C7×Q8—22no—
298C7×Q8—41no—
299C7×Q8—41no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—41no—
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—41no—
316C2×C4×C7C2×C22⋊C412yes—
317C2×C4×C7C2×C22⋊C412yes—
318C2×C4×C7C23×C411yescharacteristic
319C2×C4×C7C22×D411yescharacteristic
320C2×C4×C7C2×#4111yescharacteristic
321C2×C4×C7C2×#4111yescharacteristic
322C2×C4×C7C2×#4111yescharacteristic
323C2×C4×C7C2×#4111yescharacteristic
324C2×Dic7—21no—
325C2×Dic7—21no—
326C2×Dic7—21no—
327C2×Dic7—21no—
328C2×Dic7—21no—
329C2×Dic7—21no—
330C2×Dic7—22no—
331C2×Dic7—22no—
332C2×Dic7—41no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7C2×C22⋊C412yes—
343C2×Dic7C2×C22⋊C412yes—
344C2×Dic7C2×C22⋊C412yes—
345C4×D7—22no—
346C4×D7—22no—
347C4×D7—22no—
348C4×D7—22no—
349Dic14—22no—
350Dic14—22no—
351Dic14—41no—
352Dic14—41no—
353C7×C8—22no—
354C7×C8—22no—
355C7⋊C8—22no—
356C7⋊C8—22no—
357C22×Q8—71no—
358C22×Q8C22×D711yescharacteristic
359C23×C4—71no—
360C2×M16—142no—
361C2×M16—142no—
362C2×M16—142no—
363C22×C8—71no—
364C22×C8—71no—
365C22×C8—142no—
366C22×C8—142no—
367C22×C8—142no—
368C22×C8C22×D711yescharacteristic
369C22×C8C22×D711yescharacteristic
370C4⋊Q8—74no—
371C4⋊Q8—142no—
372#126—148no—
373#126—148no—
374#125—74no—
375#125—148no—
376#125—142no—
377#125—148no—
378C22⋊Q8—74no—
379C22⋊Q8—74no—
380C22⋊Q8—142no—
381C22⋊Q8—142no—
382C4×Q8—22no—
383C4×Q8—74no—
384C4×Q8—142no—
385C4×Q8C4×D714yes—
386#117—74no—
387#117—74no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—71no—
390C2×C4⋊C4—71no—
391C2×C4⋊C4—71no—
392C2×C4⋊C4—71no—
393C2×C4⋊C4—71no—
394C2×C4⋊C4—141no—
395C2×C4⋊C4—141no—
396C2×C4⋊C4—141no—
397C2×C4⋊C4—141no—
398C2×C4⋊C4—141no—
399C2×C4⋊C4—141no—
400C2×C4⋊C4—141no—
401C2×C4⋊C4—141no—
402C2×C4⋊C4—141no—
403C2×C4⋊C4C22×D711yescharacteristic
404C2×C4⋊C4C22×D711yescharacteristic
405C2×C4⋊C4C22×D711yescharacteristic
406C2×C22⋊C4—71no—
407C2×C22⋊C4—71no—
408C2×C22⋊C4—141no—
409C2×C22⋊C4—141no—
410C2×C22⋊C4—141no—
411C2×C22⋊C4—141no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—71no—
415C2×C42—141no—
416C2×C42—141no—
417C2×C42C22×D711yescharacteristic
418#103—22no—
419#103—22no—
420#103—142no—
421#103—142no—
422#103—142no—
423#103—142no—
424#103—142no—
425#103—142no—
426#97—22no—
427#97—142no—
428C4×C8—142no—
429C4×C8—142no—
430#95—21no—
431#95—141no—
432#95—141no—
433#95—141no—
434#95—141no—
435#95—141no—
436C22×C7C22×C22⋊C411yescharacteristic
437C22×C7#53211yescharacteristic
438C22×C7#53211yescharacteristic
439C22×C7#57711yescharacteristic
440C22×C7#57711yescharacteristic
441C22×C7#57811yescharacteristic
442C22×C7#57811yescharacteristic
443C2×D7—22no—
444C2×D7—22no—
445C2×D7—22no—
446C2×D7—41no—
447C2×D7—41no—
448C2×D7—41no—
449C2×D7—41no—
450C4×C7—22no—
451C4×C7—22no—
452C4×C7—42no—
453C4×C7—42no—
454C4×C7—81no—
455C4×C7—81no—
456C4×C7C4×#4111yescharacteristic
457C4×C7C4×#4111yescharacteristic
458C4×C7#52511yescharacteristic
459C4×C7#52511yescharacteristic
460Dic7—22no—
461Dic7—22no—
462Dic7—22no—
463Dic7—42no—
464Dic7—42no—
465Dic7—81no—
466Dic7—81no—
467C24—71no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—72no—
474C2×Q8—141no—
475C2×Q8—141no—
476C2×Q8—141no—
477C2×Q8—141no—
478C2×Q8—141no—
479C2×Q8C2×C4×D712yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—71no—
485C22×C4—71no—
486C22×C4—71no—
487C22×C4—72no—
488C22×C4—72no—
489C22×C4—141no—
490C22×C4—141no—
491C22×C4—141no—
492C22×C4—141no—
493C22×C4—141no—
494C22×C4—141no—
495C22×C4—141no—
496C22×C4—141no—
497C22×C4D4×D711yescharacteristic
498C22×C4D4×D711yescharacteristic
499C22×C4C23×D711yescharacteristic
500M16—148no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—72no—
506C2×C8—72no—
507C2×C8—148no—
508C2×C8—141no—
509C2×C8—141no—
510C2×C8—141no—
511C2×C8—141no—
512C2×C8#115712yes—
513C2×C8#115712yes—
514C4⋊C4—24no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—74no—
521C4⋊C4—74no—
522C4⋊C4—72no—
523C4⋊C4—144no—
524C4⋊C4—144no—
525C4⋊C4—144no—
526C4⋊C4—144no—
527C4⋊C4—144no—
528C4⋊C4—144no—
529C4⋊C4—144no—
530C4⋊C4—141no—
531C4⋊C4—141no—
532C4⋊C4—141no—
533C4⋊C4—141no—
534C4⋊C4—141no—
535C4⋊C4—561no—
536C4⋊C4—561no—
537C4⋊C4C2×C4×D712yes—
538C22⋊C4—74no—
539C22⋊C4—74no—
540C22⋊C4—144no—
541C22⋊C4—144no—
542C22⋊C4—561no—
543C22⋊C4—561no—
544C42—24no—
545C42—72no—
546C42—72no—
547C42—81no—
548C42—144no—
549C42—561no—
550C42C2×C4×D712yes—
551C42D4×D712yes—
552C2×C7#229711yescharacteristic
553C2×C7#289811yescharacteristic
554C2×C7#289811yescharacteristic
555C2×C7#317411yescharacteristic
556C2×C7#317711yescharacteristic
557C2×C7#317811yescharacteristic
558C2×C7#317811yescharacteristic
559D7—42no—
560C23—72no—
561C23—141no—
562C23—141no—
563C23—141no—
564C23—141no—
565C23—141no—
566C23—141no—
567C23C2×D4×D711yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—142no—
573Q8—142no—
574Q8—281no—
575Q8—281no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
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—41no—
591C2×C4—41no—
592C2×C4—72no—
593C2×C4—72no—
594C2×C4—72no—
595C2×C4—141no—
596C2×C4—141no—
597C2×C4—141no—
598C2×C4—141no—
599C2×C4—141no—
600C2×C4—141no—
601C2×C4—142no—
602C2×C4—142no—
603C2×C4—142no—
604C2×C4—142no—
605C2×C4—142no—
606C2×C4—142no—
607C2×C4—281no—
608C2×C4—281no—
609C2×C4—281no—
610C2×C4—281no—
611C2×C4—281no—
612C2×C4—281no—
613C2×C4—281no—
614C2×C4—281no—
615C2×C4—281no—
616C2×C4—281no—
617C2×C4D7×C22⋊C412yes—
618C2×C4D7×C22⋊C412yes—
619C2×C4C22×C4×D711yescharacteristic
620C2×C4C2×#115711yescharacteristic
621C2×C4C2×#115711yescharacteristic
622C2×C4C2×D4×D711yescharacteristic
623C2×C4D7×#4111yescharacteristic
624C2×C4D7×#4111yescharacteristic
625C8—22no—
626C8—22no—
627C8—142no—
628C8—142no—
629C7#2141311yescharacteristic, complemented, Sylow
630C22—142no—
631C22—142no—
632C22—142no—
633C22—281no—
634C22—281no—
635C22—281no—
636C22—281no—
637C22C2×D7×C22⋊C411yescharacteristic
638C22#9023911yescharacteristic
639C22#9023911yescharacteristic
640C22#9046611yescharacteristic
641C22#9046911yescharacteristic
642C22#9047011yescharacteristic
643C22#9047011yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—142no—
651C4—142no—
652C4—142no—
653C4—282no—
654C4—282no—
655C4—561no—
656C4—561no—
657C4C4×#115711yescharacteristic
658C4#9021311yescharacteristic
659C4#9023811yescharacteristic
660C4#9023811yescharacteristic
661C2—282no—
662C2#1173100811yescharacteristic
663C2#1173632811yescharacteristic
664C2#1173640211yescharacteristic
665C2#1173721211yescharacteristic
666C2#1173721411yescharacteristic
667C2#1173726911yescharacteristic
668C2#1173727011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1763C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1763D7
hypocenterC7#21413
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
4. C7×#1763C2semidirect
5. C4×#6409C2semidirect
8. #11727982C2semidirect
9. #11727979C2semidirect
10. #11727567C2semidirect
11. C4×#6384C2semidirect
14. #11726559C2semidirect
15. #11726558C2semidirect
22. #90274C4semidirect
154. #1763D7semidirect
629. C7#21413semidirect

Maximal subgroups

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

subgroupindex
C2×#902742
C2×#894802
#117312052
C7×#17632
C4×#64092
#117280252
#117280242
#117279822
#117279792
#117275672
C4×#63842
#117266382
#117266362
#117265592
#117265582
#214137