Finite Group Databasebeta

← #13360151

Subgroups — #13360151

Order 1792.

980 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#464C211yesmaximal, characteristic
2C2×#89956C211yesmaximal, characteristic, complemented
3C2×SD16×Dic7C211yesmaximal, characteristic
4#11740398C211yesmaximal, characteristic
5#11739895C211yesmaximal, characteristic, complemented
6#11729819C211yesmaximal, characteristic, complemented
7#11729802C211yesmaximal, characteristic
8#11729475C211yesmaximal, characteristic
9#11729380C211yesmaximal, characteristic, complemented
10#11728208C211yesmaximal, characteristic, complemented
11#11728196C211yesmaximal, characteristic, complemented
12#11727977C211yesmaximal, characteristic, complemented
13Dic7×#103C211yesmaximal, characteristic, complemented
14#11727910C211yesmaximal, characteristic
15#11727906C211yesmaximal, characteristic
16C22×C7×Q16C2211yescharacteristic
17C22×C7×SD16C2211yescharacteristic
18C2×C7×#124C2211yescharacteristic
19C2×C4×C7×Q8C2211yescharacteristic
20C2×Q8×Dic7C2211yescharacteristic
21C2×Q8×Dic7C2211yescharacteristic
22C2×Q8×Dic7C2211yescharacteristic
23C2×D4×Dic7C2211yescharacteristic
24C7×#464C418yescomplemented
25C2×C7×C4⋊C8C2211yescharacteristic
26C2×C7×#103C2211yescharacteristic
27C2×C7×#102C2211yescharacteristic
28#90002C2211yescharacteristic
29C2×#6409C2211yescharacteristic
30C2×#6409C2211yescharacteristic
31C2×#6409C2211yescharacteristic
32C2×#6409C2211yescharacteristic
33#89993C2211yescharacteristic
34C2×#6406C2211yescharacteristic
35C2×#6406C2211yescharacteristic
36#89956—24no—
37SD16×Dic7—24no—
38C2×#6391C2211yescharacteristic
39C2×#6391C2211yescharacteristic
40C2×#6389C2211yescharacteristic
41C2×C8×Dic7C2211yescharacteristic
42C4⋊C4×Dic7—21no—
43C4⋊C4×Dic7—21no—
44C4⋊C4×Dic7C2211yescharacteristic
45C4⋊C4×Dic7C2211yescharacteristic
46#89723—21no—
47#89723—21no—
48#89722—21no—
49#89722—21no—
50#89707C2211yescharacteristic
51C4×C4⋊Dic7—21no—
52C4×C4⋊Dic7C2211yescharacteristic
53#89704—21no—
54#89704C2211yescharacteristic
55C2×C4⋊(C7⋊C8)C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89345C2211yescharacteristic
59#89345C2211yescharacteristic
60#89324C2211yescharacteristic
61#89324C2211yescharacteristic
62#21511—71nomaximal, Sylow
63C22×C7×Q8C2311yescharacteristic
64C22×C7×Q8C2311yescharacteristic
65C22×C7×D4C2311yescharacteristic
66C23×Dic7—21no—
67C2×C7×Q16—22no—
68C2×C7×Q16—22no—
69C2×C7×Q16C2×C414yescomplemented
70C2×C7×SD16—22no—
71C2×C7×SD16—22no—
72C2×C7×SD16C2×C414yes—
73C22×C7×C8C2311yescharacteristic
74C22×C7×C8C2311yescharacteristic
75C7×#124—22no—
76C7×#124C2×C414yes—
77C4×C7×Q8—22no—
78C4×C7×Q8C2×C414yescomplemented
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C42×C7—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C22⋊Dic7—21no—
86C2×C22⋊Dic7—21no—
87C2×C22⋊Dic7—21no—
88Q8×Dic7—22no—
89Q8×Dic7—22no—
90Q8×Dic7—22no—
91Q8×Dic7—22no—
92Q8×Dic7—22no—
93Q8×Dic7D414yes—
94D4×Dic7—22no—
95D4×Dic7—22no—
96C2×C4⋊Dic7—21no—
97C2×C4⋊Dic7—21no—
98C2×C4⋊Dic7—21no—
99C2×C4⋊Dic7—21no—
100C2×C4⋊Dic7—21no—
101C2×C4⋊Dic7C2311yescharacteristic
102C2×C4⋊Dic7C2311yescharacteristic
103C2×C4⋊Dic7C2311yescharacteristic
104C2×C4⋊Dic7C2311yescharacteristic
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7—21no—
108C2×C4×Dic7—21no—
109C2×C4×Dic7—21no—
110C2×C4×Dic7—21no—
111C2×C4×Dic7C2311yescharacteristic
112C2×C4×Dic7C2311yescharacteristic
113C22×C7⋊C8C2311yescharacteristic
114C22×C7⋊C8C2311yescharacteristic
115C7×C4⋊C8C2×C414yescomplemented
116C7×#103C2×C414yescomplemented
117C7×#102C2×C414yes—
118#6409—22no—
119#6409—22no—
120#6409—22no—
121#6409—22no—
122#6406—22no—
123#6406—22no—
124C7⋊#95—21no—
125C7⋊#95—21no—
126C7⋊#95—21no—
127C7⋊#95—21no—
128C7⋊#95—21no—
129C7⋊#95D411yescharacteristic
130C7⋊#95D411yescharacteristic
131C7⋊#95D411yescharacteristic
132C7⋊#95D411yescharacteristic
133#6391—22no—
134#6391—22no—
135#6389—22no—
136C8×Dic7—22no—
137C4⋊(C7⋊C8)—22no—
138C2×#464D711yescharacteristic
139C2×#441—71no—
140C2×C4×SD16—71no—
141#2326—71no—
142#2276—71no—
143#1983—71no—
144#1948—71no—
145#1927—71no—
146#1926—71no—
147#1880—71no—
148#1878—71no—
149#1803—71no—
150#1768—71no—
151#1762—71no—
152C4×#103—71no—
153C24×C7—21no—
154C2×C7×Q8—21no—
155C2×C7×Q8—21no—
156C2×C7×Q8—21no—
157C2×C7×Q8—21no—
158C2×C7×Q8—21no—
159C2×C7×Q8—21no—
160C2×C7×Q8—21no—
161C2×C7×Q8—21no—
162C2×C7×Q8C22×C412yes—
163C2×C7×Q8C22×C412yes—
164C2×C7×Q8C2×D412yes—
165C2×C7×Q8#4112yes—
166C2×C7×D4—21no—
167C2×C7×D4—21no—
168C2×C7×D4—21no—
169C2×C7×D4—21no—
170C2×C7×D4—21no—
171C2×C7×D4C22×C412yes—
172C22×C4×C7—21no—
173C22×C4×C7—21no—
174C22×C4×C7—21no—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2411yescharacteristic
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7—21no—
181C22×Dic7—21no—
182C22×Dic7—42no—
183C22×Dic7—81no—
184C22×Dic7—81no—
185C22×Dic7C2×D411yescharacteristic
186C22×Dic7C2×D411yescharacteristic
187C22×Dic7C2×D411yescharacteristic
188C22×Dic7C2×D411yescharacteristic
189C7×Q16—28no—
190C7×SD16—28no—
191C2×C7×C8—21no—
192C2×C7×C8—21no—
193C2×C7×C8—21no—
194C2×C7×C8—21no—
195C2×C7×C8C22×C412yes—
196C2×C7×C8C22×C412yes—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—21no—
199C7×C4⋊C4C22×C412yes—
200C7×C22⋊C4—24no—
201C7×C22⋊C4—24no—
202C42×C7—24no—
203C42×C7C22×C412yes—
204C42×C7#4112yes—
205C22⋊Dic7—42no—
206C22⋊Dic7—81no—
207C22⋊Dic7—81no—
208C4⋊Dic7—24no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—24no—
212C4⋊Dic7—21no—
213C4⋊Dic7—21no—
214C4⋊Dic7—21no—
215C4⋊Dic7—21no—
216C4⋊Dic7—42no—
217C4⋊Dic7—81no—
218C4⋊Dic7—81no—
219C4⋊Dic7C2×D412yes—
220C4⋊Dic7#4112yes—
221C4×Dic7—24no—
222C4×Dic7—21no—
223C4×Dic7—21no—
224C4×Dic7—24no—
225C4×Dic7—42no—
226C4×Dic7—81no—
227C4×Dic7—81no—
228C4×Dic7—81no—
229C4×Dic7C2×D412yes—
230C4×Dic7#4112yes—
231C2×C7⋊C8—21no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—21no—
234C2×C7⋊C8—21no—
235C2×C7⋊C8—21no—
236C2×C7⋊C8—21no—
237C22×Q16C2×D711yescharacteristic
238C22×SD16C2×D711yescharacteristic
239C2×#124C2×D711yescharacteristic
240C2×C4×Q8—71no—
241C2×C4×Q8—71no—
242C2×C4×Q8—71no—
243C2×C4×Q8C2×D711yescharacteristic
244C2×C4×D4—71no—
245#464Dic718yescomplemented
246#441—144no—
247C4×SD16—144no—
248C2×#106—71no—
249C2×#106—71no—
250C2×C4⋊C8—71no—
251C2×C4⋊C8C2×D711yescharacteristic
252C2×#103—71no—
253C2×#103—71no—
254C2×#103—71no—
255C2×#103—71no—
256C2×#103C2×D711yescharacteristic
257C2×#102—71no—
258C2×#102—71no—
259C2×#102C2×D711yescharacteristic
260C2×#97—71no—
261C2×C4×C8—71no—
262#391—71no—
263#390—71no—
264#388—141no—
265#388—141no—
266#385—141no—
267#385—141no—
268#384—71no—
269C4×C4⋊C4—71no—
270C4×C4⋊C4—71no—
271C4×C4⋊C4—71no—
272C4×C4⋊C4—141no—
273C4×C4⋊C4—141no—
274C4×C4⋊C4—141no—
275#376—71no—
276#376—141no—
277#340—71no—
278#340—71no—
279#340—71no—
280#340—71no—
281#336—71no—
282#336—71no—
283C23×C7—22no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7—41no—
289C23×C7—41no—
290C23×C7C22×D411yescharacteristic
291C7×Q8—22no—
292C7×Q8—22no—
293C7×Q8—22no—
294C7×Q8—22no—
295C7×Q8—22no—
296C7×Q8—41no—
297C7×Q8—41no—
298C7×Q8C4×D414yes—
299C7×D4—22no—
300C7×D4—22no—
301C7×D4—41no—
302C7×D4—41no—
303C2×C4×C7—22no—
304C2×C4×C7—22no—
305C2×C4×C7—22no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—21no—
310C2×C4×C7—22no—
311C2×C4×C7—22no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7C4×D412yes—
317C2×C4×C7C4×D412yes—
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—22no—
329C2×Dic7—21no—
330C2×Dic7—21no—
331C2×Dic7—22no—
332C2×Dic7—21no—
333C2×Dic7—21no—
334C2×Dic7—21no—
335C2×Dic7—21no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341C2×Dic7—41no—
342C2×Dic7—41no—
343C2×Dic7—41no—
344C2×Dic7—41no—
345C2×Dic7—41no—
346C2×Dic7—84no—
347C2×Dic7#12112yes—
348C2×Dic7#12112yes—
349C7×C8—22no—
350C7×C8—22no—
351C7⋊C8—41no—
352C7⋊C8—41no—
353C22×Q8C22×D711yescharacteristic
354C22×Q8C22×D711yescharacteristic
355C22×D4C22×D711yescharacteristic
356C23×C4—141no—
357C2×Q16—22no—
358C2×Q16—22no—
359C2×Q16C2×Dic714yescomplemented
360C2×SD16—22no—
361C2×SD16—22no—
362C2×SD16C2×Dic714yes—
363C22×C8—71no—
364C22×C8—71no—
365C22×C8C22×D711yescharacteristic
366C22×C8C22×D711yescharacteristic
367#124—22no—
368#124C2×Dic714yes—
369C4×Q8—22no—
370C4×Q8—74no—
371C4×Q8—142no—
372C4×Q8—142no—
373C4×Q8—142no—
374C4×Q8—142no—
375C4×Q8—142no—
376C4×Q8C2×Dic714yescomplemented
377C4×D4—142no—
378C4×D4—142no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—71no—
381C2×C4⋊C4—71no—
382C2×C4⋊C4—71no—
383C2×C4⋊C4—71no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4—141no—
388C2×C4⋊C4—141no—
389C2×C4⋊C4C22×D711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—141no—
393C2×C22⋊C4—141no—
394C2×C22⋊C4—141no—
395C2×C42—21no—
396C2×C42—71no—
397C2×C42—71no—
398C2×C42—141no—
399C2×C42—141no—
400C2×C42—141no—
401C2×C42—141no—
402C2×C42—141no—
403C2×C42—141no—
404C2×C42C22×D711yescharacteristic
405#106—142no—
406#106—142no—
407C4⋊C8—142no—
408C4⋊C8C2×Dic714yescomplemented
409#103—142no—
410#103—142no—
411#103—142no—
412#103—142no—
413#103C2×Dic714yescomplemented
414#102—142no—
415#102—142no—
416#102C2×Dic714yes—
417#97—142no—
418C4×C8—142no—
419#95—71no—
420#95—71no—
421#95—71no—
422#95—71no—
423#95—141no—
424#95—141no—
425#95—141no—
426#95—141no—
427#95—141no—
428C22×C7—42no—
429C22×C7—42no—
430C22×C7—42no—
431C22×C7—81no—
432C22×C7—81no—
433C22×C7—81no—
434C22×C7—81no—
435C22×C7C2×C4×D411yescharacteristic
436C22×C7C2×#12111yescharacteristic
437C22×C7#52511yescharacteristic
438C22×C7#57711yescharacteristic
439C22×C7#57711yescharacteristic
440C22×C7#57711yescharacteristic
441C22×C7#57811yescharacteristic
442C4×C7—22no—
443C4×C7—22no—
444C4×C7—24no—
445C4×C7—42no—
446C4×C7—42no—
447C4×C7C2×C4×D411yescharacteristic
448C4×C7C4×#4111yescharacteristic
449C4×C7#52911yescharacteristic
450C4×C7#53211yescharacteristic
451Dic7—22no—
452Dic7—22no—
453Dic7—41no—
454Dic7—41no—
455Dic7—42no—
456Dic7—42no—
457Dic7—81no—
458Dic7—81no—
459C24—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8D4×D712yes—
469C2×Q8#115912yes—
470C2×Q8C22×Dic712yes—
471C2×Q8C22×Dic712yes—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4C22×Dic712yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—71no—
482C22×C4—71no—
483C22×C4—71no—
484C22×C4—71no—
485C22×C4—141no—
486C22×C4—141no—
487C22×C4—141no—
488C22×C4—141no—
489C22×C4—282no—
490C22×C4—561no—
491C22×C4—561no—
492C22×C4D4×D711yescharacteristic
493C22×C4D4×D711yescharacteristic
494C22×C4C23×D711yescharacteristic
495Q16—28no—
496SD16—28no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—141no—
502C2×C8—141no—
503C2×C8—141no—
504C2×C8—141no—
505C2×C8—141no—
506C2×C8—141no—
507C2×C8C22×Dic712yes—
508C2×C8C22×Dic712yes—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—72no—
512C4⋊C4—72no—
513C4⋊C4—144no—
514C4⋊C4—144no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—141no—
521C4⋊C4—282no—
522C4⋊C4—561no—
523C4⋊C4—561no—
524C4⋊C4C22×Dic712yes—
525C22⋊C4—24no—
526C22⋊C4—24no—
527C22⋊C4—282no—
528C22⋊C4—561no—
529C22⋊C4—561no—
530C42—24no—
531C42—72no—
532C42—72no—
533C42—144no—
534C42—144no—
535C42—141no—
536C42—141no—
537C42—282no—
538C42—561no—
539C42—561no—
540C42—561no—
541C42#115912yes—
542C42C22×Dic712yes—
543C2×C7—82no—
544C2×C7C4×#12111yescharacteristic
545C2×C7#294711yescharacteristic
546C2×C7#295211yescharacteristic
547C2×C7#304311yescharacteristic
548C2×C7#304611yescharacteristic
549C2×C7#312111yescharacteristic
550C2×C7#312511yescharacteristic
551C23—22no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23C2×D4×D711yescharacteristic
559Q8—22no—
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—22no—
564Q8—41no—
565Q8—41no—
566Q8D4×Dic714yes—
567D4—22no—
568D4—22no—
569D4—41no—
570D4—41no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—72no—
585C2×C4—72no—
586C2×C4—141no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—141no—
593C2×C4—141no—
594C2×C4—142no—
595C2×C4—141no—
596C2×C4—142no—
597C2×C4—141no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—281no—
602C2×C4—281no—
603C2×C4—281no—
604C2×C4—281no—
605C2×C4—281no—
606C2×C4—281no—
607C2×C4—281no—
608C2×C4—564no—
609C2×C4D4×Dic712yes—
610C2×C4D4×Dic712yes—
611C2×C4C2×D4×D711yescharacteristic
612C2×C4C2×#115911yescharacteristic
613C2×C4C2×#115911yescharacteristic
614C2×C4D7×#4111yescharacteristic
615C2×C4D7×#4111yescharacteristic
616C2×C4C23×Dic711yescharacteristic
617C8—22no—
618C8—22no—
619C8—281no—
620C8—281no—
621C7#2151111yescharacteristic, complemented, Sylow
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22D7×#12111yescharacteristic
630C22#9029711yescharacteristic
631C22#9046111yescharacteristic
632C22#9046111yescharacteristic
633C22#9046911yescharacteristic
634C22#9047011yescharacteristic
635C22C2×D4×Dic711yescharacteristic
636C4—22no—
637C4—24no—
638C4—22no—
639C4—42no—
640C4—42no—
641C4—142no—
642C4—142no—
643C4—281no—
644C4—281no—
645C4—282no—
646C4—282no—
647C4—561no—
648C4—561no—
649C4#9028711yescharacteristic
650C4#9029111yescharacteristic
651C4C2×D4×Dic711yescharacteristic
652C4#41×Dic711yescharacteristic
653C2—82no—
654C2#1173801711yescharacteristic
655C2#1173802011yescharacteristic
656C2#1173804511yescharacteristic
657C2#1173804811yescharacteristic
658C2Dic7×#12111yescharacteristic
659C2#1174134411yescharacteristic
660C2#1174141011yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#464C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenterC2×#464D7
hypocenterC7#21511
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#464

Complements

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

NH ≅ G/Ntype
2. C2×#89956C2semidirect
5. #11739895C2semidirect
6. #11729819C2semidirect
9. #11729380C2semidirect
10. #11728208C2semidirect
11. #11728196C2semidirect
12. #11727977C2semidirect
13. Dic7×#103C2semidirect
24. C7×#464C4semidirect
69. C2×C7×Q16C2×C4semidirect
78. C4×C7×Q8C2×C4semidirect
115. C7×C4⋊C8C2×C4semidirect
116. C7×#103C2×C4semidirect
245. #464Dic7semidirect
359. C2×Q16C2×Dic7semidirect
376. C4×Q8C2×Dic7semidirect
408. C4⋊C8C2×Dic7semidirect
413. #103C2×Dic7semidirect
621. C7#21511semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4642
C2×#899562
C2×SD16×Dic72
#117403982
#117398952
#117298192
#117298022
#117294752
#117293802
#117282082
#117281962
#117279772
Dic7×#1032
#117279102
#117279062
#215117