Finite Group Databasebeta

← #13412613

Subgroups — #13412613

Order 1792.

774 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2441C211yesmaximal, characteristic
2C2×#89836C211yesmaximal, characteristic, complemented
3C2×#89767C211yesmaximal, characteristic
4#11730940C211yesmaximal, characteristic
5#11729493C211yesmaximal, characteristic, complemented
6#11729330C211yesmaximal, characteristic
7#11727374C211yesmaximal, characteristic, complemented
8#11727372C211yesmaximal, characteristic, complemented
9#11727363C211yesmaximal, characteristic, complemented
10#11727360C211yesmaximal, characteristic, complemented
11#11727332C211yesmaximal, characteristic, complemented
12#11727324C211yesmaximal, characteristic, complemented
13#11727267C211yesmaximal, characteristic
14#11727259C211yesmaximal, characteristic
15#11727202C211yesmaximal, characteristic
16C2×C7×#125C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×#1136C2211yescharacteristic
20C2×#6433C2211yescharacteristic
21C7×#399—21no—
22C7×#396—21no—
23C7×#389—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#387—21no—
27C7×#384C2211yescharacteristic
28C7×#380C2211yescharacteristic
29C4×C7×C22⋊C4C2211yescharacteristic
30#89984C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32#89836—24no—
33#89767—24no—
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6382C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43C2×#6381C2211yescharacteristic
44#89718—21no—
45#89714—21no—
46#89708C2211yescharacteristic
47#89700C2211yescharacteristic
48#89700C2211yescharacteristic
49C2×#6377C2211yescharacteristic
50C2×C4×C7⋊C8C2211yescharacteristic
51#89432—21no—
52#89429—21no—
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#26075—71nomaximal, Sylow
62C23×C4×C7C2311yescharacteristic
63C23×Dic7—21no—
64C7×#125—22no—
65C7×#125—41no—
66C7×#117—22no—
67C7×#117—41no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
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×C7—21no—
83C2×C42×C7C2311yescharacteristic
84C2×C42×C7C2311yescharacteristic
85C2×C22⋊Dic7—21no—
86C2×C4⋊Dic7—22no—
87C2×C4⋊Dic7—21no—
88C2×C4⋊Dic7—41no—
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×C4⋊Dic7C2311yescharacteristic
92C2×C4⋊Dic7C2311yescharacteristic
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×#1136—22no—
98C2×#1136—22no—
99C2×#1136Q814yes—
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C22×C7⋊C8C2311yescharacteristic
104#6433—22no—
105#6433—41no—
106C7×#95—21no—
107C7×#95—21no—
108C7×#95—21no—
109C7×#95D411yescharacteristic
110C7×#95D411yescharacteristic
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C22⋊(C7⋊C8)Q814yes—
117#6382—22no—
118#6382—22no—
119#6382—22no—
120#6382—22no—
121#6382—22no—
122#6382—41no—
123#6381—22no—
124#6381—22no—
125#6381—41no—
126#6381D414yes—
127#6377—22no—
128C4×C7⋊C8—22no—
129C2×#499—71no—
130C2×#428—71no—
131#2442—71no—
132#2441D711yescharacteristic
133#2085—71no—
134#2084—71no—
135#2059—71no—
136#2059—71no—
137#1943—71no—
138#1941—71no—
139#1854—71no—
140#1853—71no—
141#1847—71no—
142#1809—71no—
143#1753—71no—
144C24×C7C2×D411yescharacteristic
145C22×C4×C7—21no—
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×C7C2×D411yescharacteristic
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D411yescharacteristic
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2×D411yescharacteristic
158C22×C4×C7C2×Q812yes—
159C22×C4×C7C2×Q812yes—
160C22×C4×C7C2411yescharacteristic
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—42no—
166C22×Dic7—42no—
167C22×Dic7—81no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—42no—
176C7×C4⋊C4—42no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—81no—
180C7×C4⋊C4—81no—
181C7×C4⋊C4C2×D412yes—
182C7×C22⋊C4—41no—
183C7×C22⋊C4—41no—
184C7×C22⋊C4—42no—
185C7×C22⋊C4—81no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C42×C7—42no—
191C22⋊Dic7—42no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—42no—
200C4⋊Dic7C2×D412yes—
201Dic7⋊C4—42no—
202Dic7⋊C4—42no—
203Dic7⋊C4—81no—
204Dic7⋊C4—81no—
205#1136—28no—
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⋊C8C2×D412yes—
213C2×C7⋊C8C2×Q812yes—
214C2×C7⋊C8C2×Q812yes—
215C2×C7⋊C8C2×Q812yes—
216C2×C7⋊C8C2×Q812yes—
217C2×C7⋊C8#4112yes—
218C22×M16—71no—
219C2×#125—71no—
220C2×#125C2×D711yescharacteristic
221C2×#117C2×D711yescharacteristic
222C22×C4⋊C4—71no—
223#499—144no—
224#428—144no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#107—71no—
228C2×#107—71no—
229C2×#106—71no—
230C2×#106—71no—
231C2×#106—71no—
232C2×#106—71no—
233C2×#106—71no—
234C2×#106—71no—
235C2×C22⋊C8—71no—
236C2×#97—71no—
237C2×C4×C8—71no—
238#400—141no—
239#399—21no—
240#398—141no—
241#396—21no—
242#389—21no—
243#389—71no—
244#389—71no—
245#389C2×D711yescharacteristic
246#389C2×D711yescharacteristic
247#387—21no—
248#385—141no—
249#384C2×D711yescharacteristic
250#382—71no—
251#380—71no—
252#380—141no—
253#380C2×D711yescharacteristic
254C4×C22⋊C4C2×D711yescharacteristic
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#336—71no—
262#336—71no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7—21no—
269C23×C7C4⋊Q812yes—
270C23×C7C22×D411yescharacteristic
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—22no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
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×C7C4⋊Q812yes—
302C2×C4×C7C22×D411yescharacteristic
303C2×C4×C7C22×Q811yescharacteristic
304C2×C4×C7C22×Q811yescharacteristic
305C2×C4×C7C2×#4111yescharacteristic
306C2×C4×C7H(5,𝔽2)11yescharacteristic
307C2×C4×C72−1+411yescharacteristic
308C2×Dic7—22no—
309C2×Dic7—41no—
310C2×Dic7—41no—
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—84no—
321C7⋊C8—22no—
322C7⋊C8—22no—
323C7⋊C8—22no—
324C7⋊C8C22⋊Q814yes—
325C23×C4—141no—
326C23×C4C22×D711yescharacteristic
327C2×M16—74no—
328C2×M16—142no—
329C2×M16—142no—
330C22×C8—71no—
331C22×C8—71no—
332C22×C8—71no—
333C22×C8—71no—
334#125—22no—
335#125—41no—
336#125—142no—
337#125—281no—
338#117—22no—
339#117—41no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
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—142no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—141no—
356C2×C4⋊C4—281no—
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C4⋊C4C22×D711yescharacteristic
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—141no—
364C2×C22⋊C4C22⋊D711yescharacteristic
365C2×C22⋊C4C22⋊D711yescharacteristic
366C2×C42—21no—
367C2×C42C22×D711yescharacteristic
368C2×C42C22×D711yescharacteristic
369#107—74no—
370#107—142no—
371#107—142no—
372#107—281no—
373#106—142no—
374#106—142no—
375#106—142no—
376#106—142no—
377#106—142no—
378#106—281no—
379C22⋊C8—74no—
380#97—142no—
381C4×C8—142no—
382#95—21no—
383#95—21no—
384#95—21no—
385#95—141no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
390#95C22⋊D711yescharacteristic
391#95C22⋊D711yescharacteristic
392C22×C7—22no—
393C22×C7—22no—
394C22×C7—22no—
395C22×C7—41no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7C2×C4⋊Q811yescharacteristic
400C22×C7D4211yescharacteristic
401C22×C7#54711yescharacteristic
402C22×C7#57511yescharacteristic
403C22×C7#57511yescharacteristic
404C22×C7#57711yescharacteristic
405C22×C7#57711yescharacteristic
406C4×C7—22no—
407C4×C7—41no—
408C4×C7—41no—
409C4×C7—41no—
410C4×C7—42no—
411C4×C7—41no—
412C4×C7—81no—
413C4×C7—81no—
414C4×C7—81no—
415C4×C7C2×C22⋊Q811yescharacteristic
416C4×C7#54411yescharacteristic
417C4×C7D4×Q811yescharacteristic
418C4×C7#55411yescharacteristic
419Dic7—42no—
420Dic7—81no—
421Dic7—81no—
422Dic7—81no—
423C24C2×C22⋊D711yescharacteristic
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—282no—
437C22×C4—282no—
438C22×C4—561no—
439C22×C4Q8×D712yes—
440C22×C4Q8×D712yes—
441C22×C4C2×C22⋊D711yescharacteristic
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C23×D711yescharacteristic
447M16—148no—
448C2×C8—72no—
449C2×C8—72no—
450C2×C8—72no—
451C2×C8—72no—
452C2×C8—72no—
453C2×C8—72no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—72no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—141no—
481C4⋊C4—282no—
482C4⋊C4—282no—
483C4⋊C4—282no—
484C4⋊C4—561no—
485C4⋊C4—561no—
486C4⋊C4D4×D712yes—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C22⋊C4—81no—
491C22⋊C4—282no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—42no—
497C2×C7—42no—
498C2×C7#244011yescharacteristic
499C2×C7#316311yescharacteristic
500C2×C7#316611yescharacteristic
501C2×C7#326711yescharacteristic
502C2×C7#327111yescharacteristic
503C2×C7#335111yescharacteristic
504C2×C7#335111yescharacteristic
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23#650712yes—
512C23C22×C22⋊D711yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
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—41no—
533C2×C4—41no—
534C2×C4—41no—
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—142no—
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#650712yes—
557C2×C4C2×D4×D711yescharacteristic
558C2×C4#654811yescharacteristic
559C2×C4C2×Q8×D711yescharacteristic
560C2×C4C2×Q8×D711yescharacteristic
561C2×C4D7×#4111yescharacteristic
562C2×C4#655411yescharacteristic
563C8—74no—
564C8—142no—
565C8—142no—
566C8—142no—
567C7#2607511yescharacteristic, complemented, Sylow
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22D4×C22⋊D711yescharacteristic
576C22C2×#650711yescharacteristic
577C22#9051311yescharacteristic
578C22#9051311yescharacteristic
579C22#9051511yescharacteristic
580C22#9052711yescharacteristic
581C22#9052811yescharacteristic
582C4—22no—
583C4—41no—
584C4—42no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—282no—
592C4—561no—
593C4—561no—
594C4—561no—
595C4#9031411yescharacteristic
596C4D7×C22⋊Q811yescharacteristic
597C4#9032311yescharacteristic
598C4#9032711yescharacteristic
599C2—42no—
600C2#1174020811yescharacteristic
601C2#1174078711yescharacteristic
602C2#1174091711yescharacteristic
603C2#1174096711yescharacteristic
604C2#1174102811yescharacteristic
605C2#1174104111yescharacteristic
606C2#1174105711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2441C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2441D7
hypocenterC7#26075
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
2. C2×#89836C2semidirect
5. #11729493C2semidirect
7. #11727374C2semidirect
8. #11727372C2semidirect
9. #11727363C2semidirect
10. #11727360C2semidirect
11. #11727332C2semidirect
12. #11727324C2semidirect
567. C7#26075semidirect

Maximal subgroups

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

subgroupindex
C7×#24412
C2×#898362
C2×#897672
#117309402
#117294932
#117293302
#117273742
#117273722
#117273632
#117273602
#117273322
#117273242
#117272672
#117272592
#117272022
#260757