Finite Group Databasebeta

← #13412664

Subgroups — #13412664

Order 1792.

782 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×C7×#501C211yesmaximal, characteristic, complemented
2#11740701C211yesmaximal, characteristic
3#11740473C211yesmaximal, characteristic
4C2×#89653C211yesmaximal, characteristic
5#11729536C211yesmaximal, characteristic, complemented
6#11729493C211yesmaximal, characteristic, complemented
7#11728386C211yesmaximal, characteristic
8#11728374C211yesmaximal, characteristic, complemented
9#11728366C211yesmaximal, characteristic, complemented
10#11728334C211yesmaximal, characteristic
11#11728332C211yesmaximal, characteristic
12#11728325C211yesmaximal, characteristic, complemented
13#11728309C211yesmaximal, characteristic, complemented
14#11727573C211yesmaximal, characteristic, complemented
15#11727562C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C7×#125C2211yescharacteristic
18C22×#1131C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4C2211yescharacteristic
21C7×#501—24no—
22C2×C7×#107C2211yescharacteristic
23C2×C7×#107C2211yescharacteristic
24C2×C7×#106C2211yescharacteristic
25C2×C7×#106C2211yescharacteristic
26C2×C7×#97C2211yescharacteristic
27#90003—21no—
28#90002—21no—
29C2×(C2×D7)⋊C8C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89766—21no—
34#89761—21no—
35#89759—21no—
36#89759C2211yescharacteristic
37#89759C2211yescharacteristic
38#89757C2211yescharacteristic
39#89757C2211yescharacteristic
40#89756—21no—
41#89754—21no—
42#89750—21no—
43#89749C2211yescharacteristic
44#89749C2211yescharacteristic
45C2×#6382C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47C2×#6381C2211yescharacteristic
48C2×#6381C2211yescharacteristic
49C4×(C2×D7)⋊C4C2211yescharacteristic
50#89700C2211yescharacteristic
51#89700C2211yescharacteristic
52#89653—24no—
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89319C2211yescharacteristic
61#26079—71nomaximal, Sylow
62C22×C7×Q8—21no—
63C22×C4×D7—21no—
64C22×C4×D7C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C7×C4⋊Q8—22no—
68C7×C4⋊Q8—41no—
69C7×#125—22no—
70C7×#125—41no—
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×C42×C7C2311yescharacteristic
79C2×(C2×D7)⋊C4—21no—
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4D411yescharacteristic
83C2×(C2×D7)⋊C4D411yescharacteristic
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—21no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C2×#1131—22no—
98C2×#1131—22no—
99C2×#1131Q814yescomplemented
100#6455—22no—
101#6455—41no—
102D7×C4⋊C4—22no—
103D7×C4⋊C4—41no—
104C7×#107—22no—
105C7×#107D414yes—
106C7×#106—22no—
107C7×#106—22no—
108C7×#97—22no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116C7⋊#95—21no—
117C7⋊#95—21no—
118C7⋊#95—21no—
119C7⋊#95D411yescharacteristic
120C7⋊#95D411yescharacteristic
121(C2×D7)⋊C8Q814yescomplemented
122#6392—22no—
123#6391—22no—
124C8×Dic7—22no—
125#6382—22no—
126#6381—22no—
127#6381—41no—
128#6381—41no—
129C2×#501D711yescharacteristic, complemented
130C2×#428—71no—
131#2442—71no—
132#2439—71no—
133#2085—71no—
134#2085—71no—
135#2059—71no—
136#2057—71no—
137#1943—71no—
138#1942—71no—
139#1854—71no—
140#1852—71no—
141#1846—71no—
142#1809—71no—
143#1753—71no—
144C23×D7C2×D411yescharacteristic
145C2×C7×Q8—42no—
146C2×C7×Q8—42no—
147C2×C7×Q8—81no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7C2×D411yescharacteristic
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C2×C4×D7—21no—
163C2×C4×D7—21no—
164C2×C4×D7—21no—
165C2×C4×D7—21no—
166C2×C4×D7—42no—
167C2×C4×D7—42no—
168C2×C4×D7—81no—
169C2×C4×D7C2×Q812yes—
170C2×C4×D7C2×Q812yes—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8C2×D412yes—
174C2×C7×C8C2×Q812yes—
175C2×C7×C8C2×Q812yes—
176C2×C7×C8#4112yes—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—21no—
180C7×C4⋊C4—21no—
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4—42no—
184C7×C4⋊C4—42no—
185C7×C4⋊C4—42no—
186C7×C4⋊C4C2×D412yes—
187C7×C4⋊C4C2×D412yes—
188C42×C7—21no—
189C42×C7—21no—
190(C2×D7)⋊C4—41no—
191(C2×D7)⋊C4—41no—
192(C2×D7)⋊C4—42no—
193(C2×D7)⋊C4—42no—
194(C2×D7)⋊C4—81no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—81no—
204C4⋊Dic7—81no—
205Dic7⋊C4—42no—
206Dic7⋊C4—42no—
207Dic7⋊C4—81no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7—42no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8C2×Q812yes—
216C2×C7⋊C8C2×Q812yes—
217#1131—28no—
218C22×M16—71no—
219C2×C4⋊Q8C2×D711yescharacteristic
220C2×#125C2×D711yescharacteristic
221C2×#117—71no—
222C22×C4⋊C4—71no—
223#501—24no—
224#428—144no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#107—71no—
228C2×#107—71no—
229C2×#107C2×D711yescharacteristic
230C2×#107C2×D711yescharacteristic
231C2×#106—71no—
232C2×#106—71no—
233C2×#106C2×D711yescharacteristic
234C2×#106C2×D711yescharacteristic
235C2×C22⋊C8—71no—
236C2×#97C2×D711yescharacteristic
237C2×C4×C8—71no—
238#400—141no—
239#398—141no—
240#396—141no—
241#393—141no—
242#391—141no—
243#389—71no—
244#389—71no—
245#389—71no—
246#389—71no—
247#387—141no—
248#385—141no—
249#382—71no—
250#382—71no—
251#380—71no—
252#380—71no—
253#380—141no—
254C4×C22⋊C4—71no—
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×C7C22×D411yescharacteristic
264C22×D7—21no—
265C22×D7—21no—
266C22×D7—21no—
267C22×D7—21no—
268C22×D7—21no—
269C22×D7—21no—
270C22×D7C4⋊Q812yes—
271C7×Q8—84no—
272C2×C4×C7—22no—
273C2×C4×C7—22no—
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—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7C22×D411yescharacteristic
291C2×C4×C7C22×Q811yescharacteristic
292C2×C4×C7C22×Q811yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7H(5,𝔽2)11yescharacteristic
295C2×C4×C72−1+411yescharacteristic
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7C4⋊Q812yes—
317C4×D7—22no—
318C4×D7—22no—
319C4×D7—22no—
320C4×D7—22no—
321C4×D7—84no—
322C7×C8—22no—
323C7×C8C22⋊Q814yes—
324C7⋊C8—22no—
325C7⋊C8—22no—
326C22×Q8—21no—
327C23×C4—71no—
328C23×C4—141no—
329C2×M16—74no—
330C2×M16—142no—
331C2×M16—142no—
332C22×C8—71no—
333C22×C8—71no—
334C22×C8C22×D711yescharacteristic
335C22×C8C22×D711yescharacteristic
336C4⋊Q8—22no—
337C4⋊Q8—41no—
338#125—22no—
339#125—41no—
340#117—142no—
341#117—281no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—71no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—142no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4—141no—
352C2×C4⋊C4—141no—
353C2×C4⋊C4—141no—
354C2×C4⋊C4—141no—
355C2×C4⋊C4—281no—
356C2×C4⋊C4C22×D711yescharacteristic
357C2×C4⋊C4C22×D711yescharacteristic
358C2×C4⋊C4C22×D711yescharacteristic
359C2×C4⋊C4C22×D711yescharacteristic
360C2×C22⋊C4—71no—
361C2×C22⋊C4—71no—
362C2×C22⋊C4—141no—
363C2×C22⋊C4—141no—
364C2×C22⋊C4—141no—
365C2×C42—71no—
366C2×C42—141no—
367C2×C42C22×D711yescharacteristic
368#107—22no—
369#107—142no—
370#107—142no—
371#107—281no—
372#107—281no—
373#107C22⋊D714yes—
374#106—22no—
375#106—22no—
376#106—142no—
377#106—142no—
378C22⋊C8—74no—
379#97—22no—
380C4×C8—142no—
381#95—71no—
382#95—71no—
383#95—141no—
384#95—141no—
385#95—141no—
386#95—141no—
387#95—141no—
388#95—141no—
389#95—141no—
390#95—141no—
391#95—141no—
392#95—141no—
393C22×C7C2×C4⋊Q811yescharacteristic
394C22×C7D4211yescharacteristic
395C22×C7#54711yescharacteristic
396C22×C7#57511yescharacteristic
397C22×C7#57511yescharacteristic
398C22×C7#57611yescharacteristic
399C22×C7#57811yescharacteristic
400C2×D7—22no—
401C2×D7—22no—
402C2×D7—22no—
403C2×D7—41no—
404C2×D7—41no—
405C2×D7—41no—
406C2×D7—41no—
407C4×C7—42no—
408C4×C7—41no—
409C4×C7—42no—
410C4×C7—41no—
411C4×C7—81no—
412C4×C7—81no—
413C4×C7C2×C22⋊Q811yescharacteristic
414C4×C7#54411yescharacteristic
415C4×C7D4×Q811yescharacteristic
416C4×C7#55411yescharacteristic
417Dic7—22no—
418Dic7—41no—
419Dic7—41no—
420Dic7—81no—
421Dic7—81no—
422Dic7—81no—
423Dic7—81no—
424C24—71no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—81no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—71no—
433C22×C4—72no—
434C22×C4—72no—
435C22×C4—71no—
436C22×C4—71no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4—141no—
442C22×C4—141no—
443C22×C4—141no—
444C22×C4—141no—
445C22×C4—282no—
446C22×C4—282no—
447C22×C4—561no—
448C22×C4D4×D711yescharacteristic
449C22×C4D4×D711yescharacteristic
450C22×C4C23×D711yescharacteristic
451M16—148no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—72no—
455C2×C8—72no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C2×C8—141no—
460C2×C8Q8×D712yes—
461C2×C8Q8×D712yes—
462C2×C8#116112yes—
463C2×C8C2×C22⋊D712yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—141no—
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—561no—
484C4⋊C4—561no—
485C4⋊C4—561no—
486C4⋊C4C2×C22⋊D712yes—
487C4⋊C4C2×C22⋊D712yes—
488C22⋊C4—281no—
489C22⋊C4—281no—
490C22⋊C4—282no—
491C22⋊C4—282no—
492C22⋊C4—561no—
493C42—21no—
494C42—21no—
495C42—141no—
496C42—141no—
497C42—282no—
498C2×C7#244011yescharacteristic
499C2×C7#316411yescharacteristic
500C2×C7#316611yescharacteristic
501C2×C7#327311yescharacteristic
502C2×C7#327711yescharacteristic
503C2×C7#334711yescharacteristic
504C2×C7#334911yescharacteristic
505D7—42no—
506C23—72no—
507C23—141no—
508C23—141no—
509C23—141no—
510C23—141no—
511C23—141no—
512C23—141no—
513C23C2×D4×D711yescharacteristic
514Q8—84no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—72no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
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—281no—
556C2×C4—281no—
557C2×C4—281no—
558C2×C4—564no—
559C2×C4#654811yescharacteristic
560C2×C4C2×Q8×D711yescharacteristic
561C2×C4C2×Q8×D711yescharacteristic
562C2×C4C2×#116111yescharacteristic
563C2×C4#655111yescharacteristic
564C2×C4C22×C22⋊D711yescharacteristic
565C8—22no—
566C8—142no—
567C8—142no—
568C8#650914yes—
569C7#2607911yescharacteristic, complemented, Sylow
570C22—142no—
571C22—142no—
572C22—142no—
573C22—281no—
574C22—281no—
575C22—281no—
576C22—281no—
577C22D7×C4⋊Q811yescharacteristic
578C22#9041711yescharacteristic
579C22#9046411yescharacteristic
580C22#9046511yescharacteristic
581C22#9046811yescharacteristic
582C22#9047011yescharacteristic
583C22D4×C22⋊D711yescharacteristic
584C4—41no—
585C4—42no—
586C4—42no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—142no—
591C4—281no—
592C4—281no—
593C4—561no—
594C4—561no—
595C4—561no—
596C4—561no—
597C4#9042011yescharacteristic
598C4#9042111yescharacteristic
599C4C2×#650911yescharacteristic
600C4Q8×C22⋊D711yescharacteristic
601C2—282no—
602C2#1173942011yescharacteristic
603C2#1173942211yescharacteristic
604C2#1173943911yescharacteristic
605C2#1173944111yescharacteristic
606C2#1174067811yescharacteristic
607C2#1174140611yescharacteristic
608C2#1174143911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#501C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenterC2×#501D7
hypocenterC7#26079
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#501

Complements

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

NH ≅ G/Ntype
1. C2×C7×#501C2semidirect
5. #11729536C2semidirect
6. #11729493C2semidirect
8. #11728374C2semidirect
9. #11728366C2semidirect
12. #11728325C2semidirect
13. #11728309C2semidirect
14. #11727573C2semidirect
99. C2×#1131Q8semidirect
121. (C2×D7)⋊C8Q8semidirect
129. C2×#501D7semidirect
569. C7#26079semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5012
#117407012
#117404732
C2×#896532
#117295362
#117294932
#117283862
#117283742
#117283662
#117283342
#117283322
#117283252
#117283092
#117275732
#117275622
#260797