Finite Group Databasebeta

← #13395165

Subgroups — #13395165

Order 1792.

746 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2628C211yesmaximal, characteristic
2C2×#89863C211yesmaximal, characteristic, complemented
3#11739680C211yesmaximal, characteristic
4C2×#89772C211yesmaximal, characteristic
5#11729534C211yesmaximal, characteristic, complemented
6#11727375C211yesmaximal, characteristic, complemented
7#11727373C211yesmaximal, characteristic, complemented
8#11727370C211yesmaximal, characteristic, complemented
9#11727368C211yesmaximal, characteristic, complemented
10#11727313C211yesmaximal, characteristic, complemented
11#11727294C211yesmaximal, characteristic, complemented
12#11727281C211yesmaximal, characteristic
13#11727280C211yesmaximal, characteristic
14#11727273C211yesmaximal, characteristic
15#11727272C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C2×#6487C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C7×#400—21no—
21C7×#399—21no—
22C7×#395—21no—
23C7×#393—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#382C2211yescharacteristic
27C7×#380C2211yescharacteristic
28C7×#380C2211yescharacteristic
29#89984C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32#89863—24no—
33#89772—24no—
34#89750—21no—
35#89744—21no—
36C2×#6384C2211yescharacteristic
37C2×#6384C2211yescharacteristic
38C2×#6384C2211yescharacteristic
39C2×#6384C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89717—21no—
45#89716—21no—
46C4×C4⋊Dic7C2211yescharacteristic
47#89701C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51C2×#6377C2211yescharacteristic
52C2×C4×C7⋊C8C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#24714—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C23×C4×C7C2311yescharacteristic
64C22×Dic14C2311yescharacteristic
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8—22no—
67C7×#117—22no—
68C7×#117—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4D411yescharacteristic
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4D411yescharacteristic
81C2×C7×C22⋊C4D411yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C22⋊Dic7—21no—
85C2×C22⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89#6487—22no—
90#6487—22no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7—21no—
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C4×Dic14—22no—
101C4×Dic14—22no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95—21no—
107C7×#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C22⋊(C7⋊C8)—22no—
114C22⋊(C7⋊C8)—22no—
115#6384—22no—
116#6384—22no—
117#6384—22no—
118#6384—22no—
119#6382—22no—
120#6381—22no—
121#6381—22no—
122#6381—22no—
123C4⋊(C7⋊C8)—22no—
124C4⋊(C7⋊C8)D414yes—
125#6377—22no—
126C4×C7⋊C8—22no—
127C2×#477—71no—
128C2×#433—71no—
129#2628D711yescharacteristic
130#2323—71no—
131#2076—71no—
132#2075—71no—
133#2036—71no—
134#2035—71no—
135#1988—71no—
136#1949—71no—
137#1944—71no—
138#1942—71no—
139#1935—71no—
140#1933—71no—
141#1805—71no—
142C24×C7C2×D411yescharacteristic
143C2×C7×Q8—42no—
144C2×C7×Q8—81no—
145C2×C7×Q8—81no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7—21no—
154C22×C4×C7—21no—
155C22×C4×C7—21no—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2411yescharacteristic
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14#4112yes—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
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—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—42no—
183C7×C4⋊C4—81no—
184C7×C4⋊C4—81no—
185C7×C22⋊C4—22no—
186C7×C22⋊C4—22no—
187C7×C22⋊C4—81no—
188C7×C22⋊C4—81no—
189C42×C7—21no—
190C42×C7—21no—
191C42×C7C2×D412yes—
192C42×C7#4112yes—
193C22⋊Dic7—24no—
194C22⋊Dic7—81no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7#4112yes—
201Dic7⋊C4—24no—
202Dic7⋊C4—42no—
203Dic7⋊C4—81no—
204C4×Dic7—42no—
205C4×Dic7—81no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8C2×D412yes—
217C2×C7⋊C8C2×D412yes—
218C2×C4⋊Q8C2×D711yescharacteristic
219C2×C22⋊Q8—71no—
220C2×C4×Q8—71no—
221C2×#117C2×D711yescharacteristic
222#477—144no—
223#433—144no—
224C2×#107—71no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#106—71no—
228C2×C4⋊C8—71no—
229C2×C4⋊C8—71no—
230C2×#103—71no—
231C2×#103—71no—
232C2×#103—71no—
233C2×#103—71no—
234C2×C22⋊C8—71no—
235C2×C22⋊C8—71no—
236C2×#97—71no—
237C2×C4×C8—71no—
238#400—21no—
239#399—21no—
240#395—21no—
241#393—21no—
242#391—71no—
243#389—71no—
244#389—141no—
245#389C2×D711yescharacteristic
246#389C2×D711yescharacteristic
247#388—141no—
248#388—141no—
249#387—141no—
250#382C2×D711yescharacteristic
251#380—71no—
252#380C2×D711yescharacteristic
253#380C2×D711yescharacteristic
254C4×C4⋊C4—71no—
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#340—71no—
262#340—71no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7C22×D411yescharacteristic
271C7×Q8—84no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7#12112yes—
302C2×C4×C7#12112yes—
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C72−1+411yescharacteristic
308C2×C4×C72−1+411yescharacteristic
309C2×Dic7—22no—
310C2×Dic7—22no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321Dic14—22no—
322Dic14—22no—
323Dic14—41no—
324Dic14—41no—
325C7⋊C8—22no—
326C7⋊C8—22no—
327C7⋊C8—41no—
328C7⋊C8—41no—
329C22×Q8—21no—
330C22×Q8—71no—
331C23×C4C22×D711yescharacteristic
332C22×C8—71no—
333C22×C8—71no—
334C22×C8—71no—
335C22×C8—71no—
336C4⋊Q8—22no—
337C4⋊Q8—22no—
338C22⋊Q8—142no—
339C22⋊Q8—142no—
340C4×Q8—142no—
341C4×Q8—142no—
342#117—22no—
343#117—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4C22⋊D711yescharacteristic
354C2×C4⋊C4C22⋊D711yescharacteristic
355C2×C4⋊C4C22×D711yescharacteristic
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—141no—
362C2×C22⋊C4—141no—
363C2×C22⋊C4C22⋊D711yescharacteristic
364C2×C22⋊C4C22⋊D711yescharacteristic
365C2×C42—141no—
366C2×C42—141no—
367C2×C42C22×D711yescharacteristic
368C2×C42C22×D711yescharacteristic
369#107—142no—
370#107—142no—
371#107—142no—
372#106—142no—
373C4⋊C8—74no—
374C4⋊C8—142no—
375#103—142no—
376#103—142no—
377#103—142no—
378#103—142no—
379C22⋊C8—142no—
380C22⋊C8—142no—
381#97—142no—
382C4×C8—142no—
383#95—21no—
384#95—21no—
385#95—21no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393#95—141no—
394C22×C7—41no—
395C22×C7—41no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7—41no—
400C22×C7—41no—
401C22×C7C2×#12111yescharacteristic
402C22×C7#52511yescharacteristic
403C22×C7#53611yescharacteristic
404C22×C7#57611yescharacteristic
405C22×C7#57711yescharacteristic
406C22×C7#57811yescharacteristic
407C22×C7#57811yescharacteristic
408C4×C7—22no—
409C4×C7—22no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7—41no—
413C4×C7—81no—
414C4×C7—81no—
415C4×C7—81no—
416C4×C7—81no—
417C4×C7#54711yescharacteristic
418C4×C7#54811yescharacteristic
419C4×C7#55211yescharacteristic
420C4×C7#55611yescharacteristic
421Dic7—42no—
422Dic7—42no—
423Dic7—81no—
424Dic7—81no—
425C24C2×C22⋊D711yescharacteristic
426C2×Q8—42no—
427C2×Q8—72no—
428C2×Q8—81no—
429C2×Q8—81no—
430C2×Q8—141no—
431C2×Q8—141no—
432C2×Q8—141no—
433C2×Q8—141no—
434C2×Q8—141no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—141no—
446C22×C4—141no—
447C22×C4—141no—
448C22×C4—141no—
449C22×C4C2×C22⋊D711yescharacteristic
450C22×C4C2×C22⋊D711yescharacteristic
451C22×C4C2×C22⋊D711yescharacteristic
452C22×C4C2×C22⋊D711yescharacteristic
453C22×C4C2×C22⋊D711yescharacteristic
454C22×C4C23×D711yescharacteristic
455C2×C8—72no—
456C2×C8—72no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—42no—
478C4⋊C4—72no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—144no—
482C4⋊C4—141no—
483C4⋊C4—141no—
484C4⋊C4—141no—
485C4⋊C4—141no—
486C4⋊C4—141no—
487C4⋊C4—282no—
488C4⋊C4—561no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—144no—
494C22⋊C4—561no—
495C42—21no—
496C42—21no—
497C42—282no—
498C42—561no—
499C42D4×D712yes—
500C42#115912yes—
501C2×C7—81no—
502C2×C7#241511yescharacteristic
503C2×C7#305911yescharacteristic
504C2×C7#306211yescharacteristic
505C2×C7#312111yescharacteristic
506C2×C7#312411yescharacteristic
507C2×C7#324211yescharacteristic
508C2×C7#324311yescharacteristic
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23C22×C22⋊D711yescharacteristic
517Q8—84no—
518Q8—142no—
519Q8—142no—
520Q8—281no—
521Q8—281no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—281no—
554C2×C4—281no—
555C2×C4—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—281no—
559C2×C4—281no—
560C2×C4—281no—
561C2×C4—281no—
562C2×C4—281no—
563C2×C4#650212yes—
564C2×C4#650212yes—
565C2×C4C2×D4×D711yescharacteristic
566C2×C4C2×#115911yescharacteristic
567C2×C4#655111yescharacteristic
568C2×C4#655111yescharacteristic
569C2×C4D7×#4111yescharacteristic
570C2×C4D7×#4111yescharacteristic
571C8—142no—
572C8—142no—
573C8—281no—
574C8—281no—
575C7#2471411yescharacteristic, complemented, Sylow
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#650211yescharacteristic
584C22#9050711yescharacteristic
585C22#9051911yescharacteristic
586C22#9052611yescharacteristic
587C22#9052711yescharacteristic
588C22#9052911yescharacteristic
589C22#9052911yescharacteristic
590C4—22no—
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—282no—
600C4—282no—
601C4—561no—
602C4—561no—
603C4#9032211yescharacteristic
604C4#9032411yescharacteristic
605C4#9033011yescharacteristic
606C4#9033211yescharacteristic
607C2—81no—
608C2#1174025911yescharacteristic
609C2#1174082411yescharacteristic
610C2#1174084411yescharacteristic
611C2#1174090611yescharacteristic
612C2#1174091211yescharacteristic
613C2#1174092311yescharacteristic
614C2#1174092611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2628C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2628D7
hypocenterC7#24714
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. C2×#89863C2semidirect
5. #11729534C2semidirect
6. #11727375C2semidirect
7. #11727373C2semidirect
8. #11727370C2semidirect
9. #11727368C2semidirect
10. #11727313C2semidirect
11. #11727294C2semidirect
575. C7#24714semidirect

Maximal subgroups

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

subgroupindex
C7×#26282
C2×#898632
#117396802
C2×#897722
#117295342
#117273752
#117273732
#117273702
#117273682
#117273132
#117272942
#117272812
#117272802
#117272732
#117272722
#247147