Finite Group Databasebeta

← #13412668

Subgroups — #13412668

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×#89893C211yesmaximal, characteristic
2C2×#89474C211yesmaximal, characteristic, complemented
3#11730940C211yesmaximal, characteristic
4#11730937C211yesmaximal, characteristic
5C7×#1753C211yesmaximal, characteristic
6#11729742C211yesmaximal, characteristic
7#11727683C211yesmaximal, characteristic
8#11727681C211yesmaximal, characteristic
9#11727573C211yesmaximal, characteristic, complemented
10#11727524C211yesmaximal, characteristic, complemented
11#11727520C211yesmaximal, characteristic, complemented
12#11726395C211yesmaximal, characteristic, complemented
13#11726395C211yesmaximal, characteristic, complemented
14#11726388C211yesmaximal, characteristic, complemented
15#11726386C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C2×#6433C2211yescharacteristic
20C2×#6432C2211yescharacteristic
21C2×C7×C22⋊C8C2211yescharacteristic
22C2×C7×#97C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C4×C7×C22⋊C4C2211yescharacteristic
25#89984C2211yescharacteristic
26#89984C2211yescharacteristic
27#89893—24no—
28C2×#6392C2211yescharacteristic
29C2×#6392C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6392C2211yescharacteristic
32C2×#6392C2211yescharacteristic
33C2×#6392C2211yescharacteristic
34C2×#6391C2211yescharacteristic
35C2×#6391C2211yescharacteristic
36C2×#6391C2211yescharacteristic
37C2×#6391C2211yescharacteristic
38#89720—21no—
39#89719—21no—
40#89718—21no—
41#89714—21no—
42#89708C2211yescharacteristic
43#89708C2211yescharacteristic
44#89700C2211yescharacteristic
45#89700C2211yescharacteristic
46#89700C2211yescharacteristic
47#89700C2211yescharacteristic
48#89474—24no—
49#89432—21no—
50#89429—21no—
51#89427—21no—
52#89418—21no—
53C7×#336C2211yescharacteristic
54C7×#336C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#89346C2211yescharacteristic
60#89346C2211yescharacteristic
61#26079—71nomaximal, Sylow
62C23×C4×C7C2311yescharacteristic
63C23×Dic7—21no—
64C22×Dic14—21no—
65C2×C7×M16—22no—
66C2×C7×M16—22no—
67C2×C7×M16Q814yescomplemented
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C22×C7×C8C2311yescharacteristic
72C2×C7×C22⋊C4D411yescharacteristic
73C2×C7×C22⋊C4D411yescharacteristic
74C2×C42×C7C2311yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—21no—
78C2×C22⋊Dic7—21no—
79#6489—22no—
80#6489—41no—
81C2×C4⋊Dic7—22no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7—41no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
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×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×C4×Dic7—21no—
100#6433—22no—
101#6433—41no—
102#6432—22no—
103#6432—41no—
104C7×C22⋊C8Q814yescomplemented
105C7×#97—22no—
106C4×C7×C8—22no—
107C7×#95D411yescharacteristic
108C7×#95D411yescharacteristic
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—
119#6392—22no—
120#6392—22no—
121#6392—22no—
122#6392—41no—
123#6392—41no—
124#6392D414yes—
125#6391—22no—
126#6391—22no—
127#6391—22no—
128#6391—22no—
129C2×#501—71no—
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#1753D711yescharacteristic
144C24×C7C2×D411yescharacteristic
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×Q812yes—
155C22×C4×C7C2×Q812yes—
156C22×C4×C7C2411yescharacteristic
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—21no—
161C22×Dic7—21no—
162C22×Dic7—21no—
163C22×Dic7—21no—
164C22×Dic7—21no—
165C22×Dic7—42no—
166C22×Dic7—42no—
167C22×Dic7—81no—
168C2×Dic14—42no—
169C2×Dic14—42no—
170C2×Dic14—81no—
171C7×M16—28no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8C2×D412yes—
179C2×C7×C8C2×Q812yes—
180C2×C7×C8C2×Q812yes—
181C2×C7×C8C2×Q812yes—
182C2×C7×C8C2×Q812yes—
183C2×C7×C8#4112yes—
184C7×C22⋊C4—41no—
185C7×C22⋊C4—41no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C22⋊Dic7—42no—
191C22⋊Dic7—42no—
192C22⋊Dic7—81no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
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—21no—
204C4⋊Dic7—21no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—42no—
208C4⋊Dic7—42no—
209C4⋊Dic7C2×D412yes—
210C4⋊Dic7C2×D412yes—
211Dic7⋊C4—42no—
212Dic7⋊C4—42no—
213Dic7⋊C4—42no—
214Dic7⋊C4—81no—
215Dic7⋊C4—81no—
216Dic7⋊C4—81no—
217C4×Dic7—42no—
218C22×M16C2×D711yescharacteristic
219C2×C4⋊Q8—71no—
220C2×#125—71no—
221C2×#117—71no—
222C22×C4⋊C4—71no—
223#501—144no—
224#428—144no—
225C2×#107—71no—
226C2×#107—71no—
227C2×#107—71no—
228C2×#107—71no—
229C2×#107—71no—
230C2×#107—71no—
231C2×#106—71no—
232C2×#106—71no—
233C2×#106—71no—
234C2×#106—71no—
235C2×C22⋊C8C2×D711yescharacteristic
236C2×#97C2×D711yescharacteristic
237C2×C4×C8C2×D711yescharacteristic
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⋊C4C2×D711yescharacteristic
255#340—71no—
256#340—71no—
257#340—71no—
258#340—71no—
259#340—71no—
260#340—71no—
261#336C2×D711yescharacteristic
262#336C2×D711yescharacteristic
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—21no—
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×C7C4⋊Q812yes—
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—22no—
297C2×Dic7—22no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
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×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—84no—
321Dic14—84no—
322C7×C8—22no—
323C7×C8—22no—
324C7×C8—22no—
325C7×C8C22⋊Q814yes—
326C22×Q8—141no—
327C23×C4—141no—
328C23×C4C22×D711yescharacteristic
329C2×M16—22no—
330C2×M16—22no—
331C2×M16Dic1414yescomplemented
332C22×C8C22×D711yescharacteristic
333C22×C8C22×D711yescharacteristic
334C22×C8C22×D711yescharacteristic
335C22×C8C22×D711yescharacteristic
336C4⋊Q8—142no—
337C4⋊Q8—281no—
338#125—142no—
339#125—281no—
340#117—142no—
341#117—281no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
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—141no—
357C2×C4⋊C4—141no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4—281no—
360C2×C22⋊C4—141no—
361C2×C22⋊C4—141no—
362C2×C22⋊C4—141no—
363C2×C22⋊C4D2811yescharacteristic
364C2×C22⋊C4D2811yescharacteristic
365C2×C42—141no—
366C2×C42C22×D711yescharacteristic
367C2×C42C22×D711yescharacteristic
368#107—74no—
369#107—142no—
370#107—142no—
371#107—142no—
372#107—281no—
373#107—281no—
374#106—142no—
375#106—142no—
376#106—142no—
377#106—142no—
378C22⋊C8Dic1414yescomplemented
379#97—22no—
380C4×C8—22no—
381#95—141no—
382#95—141no—
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#95D2811yescharacteristic
392#95D2811yescharacteristic
393C22×C7—22no—
394C22×C7—22no—
395C22×C7—22no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7—41no—
400C22×C7C2×C4⋊Q811yescharacteristic
401C22×C7D4211yescharacteristic
402C22×C7#54711yescharacteristic
403C22×C7#57511yescharacteristic
404C22×C7#57511yescharacteristic
405C22×C7#57611yescharacteristic
406C22×C7#57811yescharacteristic
407C4×C7—22no—
408C4×C7—41no—
409C4×C7—41no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7C2×C22⋊Q811yescharacteristic
413C4×C7#54411yescharacteristic
414C4×C7D4×Q811yescharacteristic
415C4×C7#55411yescharacteristic
416Dic7—42no—
417Dic7—42no—
418Dic7—81no—
419Dic7—81no—
420Dic7—81no—
421Dic7—81no—
422Dic7—81no—
423Dic7—81no—
424C24C2×D2811yescharacteristic
425C2×Q8—282no—
426C2×Q8—282no—
427C2×Q8—561no—
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—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—282no—
441C22×C4—282no—
442C22×C4—561no—
443C22×C4C2×Dic1412yes—
444C22×C4C2×Dic1412yes—
445C22×C4C2×D2811yescharacteristic
446C22×C4C2×D2811yescharacteristic
447C22×C4C2×D2811yescharacteristic
448C22×C4C2×D2811yescharacteristic
449C22×C4C2×D2811yescharacteristic
450C22×C4C23×D711yescharacteristic
451M16—28no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8C2×Dic1412yes—
459C2×C8C2×Dic1412yes—
460C2×C8C2×Dic1412yes—
461C2×C8C2×Dic1412yes—
462C2×C8D4×D712yes—
463C2×C8#115912yes—
464C4⋊C4—72no—
465C4⋊C4—72no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—141no—
479C4⋊C4—141no—
480C4⋊C4—282no—
481C4⋊C4—282no—
482C4⋊C4—282no—
483C4⋊C4—282no—
484C4⋊C4—282no—
485C4⋊C4—561no—
486C4⋊C4—561no—
487C4⋊C4—561no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—282no—
491C22⋊C4—282no—
492C22⋊C4—561no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—282no—
498C2×C7—42no—
499C2×C7#244011yescharacteristic
500C2×C7#316411yescharacteristic
501C2×C7#316611yescharacteristic
502C2×C7#327311yescharacteristic
503C2×C7#327711yescharacteristic
504C2×C7#334711yescharacteristic
505C2×C7#334911yescharacteristic
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#643212yes—
513C23C22×D2811yescharacteristic
514Q8—564no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
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—142no—
534C2×C4—142no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
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—281no—
556C2×C4—281no—
557C2×C4—564no—
558C2×C4#643212yes—
559C2×C4C22×Dic1411yescharacteristic
560C2×C4C22×Dic1411yescharacteristic
561C2×C4C2×D4×D711yescharacteristic
562C2×C4C2×#115911yescharacteristic
563C2×C4#655311yescharacteristic
564C2×C4#655411yescharacteristic
565C8—22no—
566C8—22no—
567C8—22no—
568C8#644114yes—
569C7#2607911yescharacteristic, complemented, Sylow
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#643211yescharacteristic
578C22D4×D2811yescharacteristic
579C22#9024611yescharacteristic
580C22#9043911yescharacteristic
581C22#9043911yescharacteristic
582C22#9044311yescharacteristic
583C22#9044411yescharacteristic
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—282no—
590C4—282no—
591C4—561no—
592C4—561no—
593C4—561no—
594C4—561no—
595C4—561no—
596C4—561no—
597C4C2×#644111yescharacteristic
598C4#9021011yescharacteristic
599C4D4×Dic1411yescharacteristic
600C4#9023411yescharacteristic
601C2—42no—
602C2#1173093811yescharacteristic
603C2#1173595211yescharacteristic
604C2#1173599111yescharacteristic
605C2#1173691811yescharacteristic
606C2#1173692011yescharacteristic
607C2#1173699411yescharacteristic
608C2#1173700411yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1753C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1753D7
hypocenterC7#26079
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
2. C2×#89474C2semidirect
9. #11727573C2semidirect
10. #11727524C2semidirect
11. #11727520C2semidirect
12. #11726395C2semidirect
13. #11726395C2semidirect
14. #11726388C2semidirect
15. #11726386C2semidirect
67. C2×C7×M16Q8semidirect
104. C7×C22⋊C8Q8semidirect
331. C2×M16Dic14semidirect
378. C22⋊C8Dic14semidirect
569. C7#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#898932
C2×#894742
#117309402
#117309372
C7×#17532
#117297422
#117276832
#117276812
#117275732
#117275242
#117275202
#117263952
#117263882
#117263862
#260797