Finite Group Databasebeta

← #13413158

Subgroups — #13413158

Order 1792.

734 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739778C211yesmaximal, characteristic
2#11731169C211yesmaximal, characteristic, complemented
3C4×C7×#102C211yesmaximal, characteristic
4#11729359C211yesmaximal, characteristic
5#11727923C211yesmaximal, characteristic
6#11727921C211yesmaximal, characteristic
7#11727908C211yesmaximal, characteristic
8#11727907C211yesmaximal, characteristic
9#11727540C211yesmaximal, characteristic, complemented
10#11727517C211yesmaximal, characteristic, complemented
11#11727222C211yesmaximal, characteristic, complemented
12#11726600C211yesmaximal, characteristic, complemented
13#11726597C211yesmaximal, characteristic, complemented
14#11726563C211yesmaximal, characteristic, complemented
15#11726561C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×D4×Dic7C2211yescharacteristic
18C2×D4×Dic7C2211yescharacteristic
19C2×C7×#102C2211yescharacteristic
20C2×C7×#102C2211yescharacteristic
21C2×C4×C7×C8C2211yescharacteristic
22C4×C7×C4⋊C4C2211yescharacteristic
23C2×#6406C2211yescharacteristic
24C2×#6406C2211yescharacteristic
25#89984—21no—
26C2×C7⋊#95—21no—
27C2×Dic7⋊C8C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29#89757C2211yescharacteristic
30#89757C2211yescharacteristic
31#89749C2211yescharacteristic
32#89748C2211yescharacteristic
33#89745C2211yescharacteristic
34#89745C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37#89722—21no—
38#89722—21no—
39#89708C2211yescharacteristic
40#89708C2211yescharacteristic
41C4×C4⋊Dic7C2211yescharacteristic
42#89705—21no—
43#89705—21no—
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45#89432—21no—
46#89429—21no—
47C7×#340C2211yescharacteristic
48C7×#340C2211yescharacteristic
49#89346C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89345C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#26117—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×C4×C7—21no—
62C23×Dic7—21no—
63C23×Dic7—21no—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C4×C7×D4—22no—
67C4×C7×D4Q814yes—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C42×C7—21no—
73C2×C42×C7C2311yescharacteristic
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76D4×Dic7—22no—
77D4×Dic7—22no—
78D4×Dic7—41no—
79D4×Dic7—41no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7D411yescharacteristic
83C2×C4⋊Dic7D411yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C7×#102—22no—
97C7×#102—22no—
98C4×C7×C8—22no—
99C7×#95—21no—
100#6406—22no—
101#6406—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⋊#95D411yescharacteristic
113C7⋊#95D411yescharacteristic
114Dic7⋊C8—41no—
115Dic7⋊C8—41no—
116#6382—41no—
117#6381—41no—
118C4⋊(C7⋊C8)—22no—
119#2570—71no—
120#2349—71no—
121#2101—71no—
122#2100—71no—
123#2088—71no—
124#2086—71no—
125#1949—71no—
126#1939—71no—
127#1864—71no—
128#1862—71no—
129#1835—71no—
130#1804—71no—
131#1804—71no—
132#1802—71no—
133C4×#102D711yescharacteristic
134C24×C7—21no—
135C2×C7×D4—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4C2×Q812yes—
139C2×C7×D4#4112yes—
140C2×C7×D4#4112yes—
141C22×C4×C7—21no—
142C22×C4×C7—24no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—81no—
146C22×C4×C7—81no—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2411yescharacteristic
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—42no—
155C22×Dic7—42no—
156C22×Dic7—42no—
157C22×Dic7—42no—
158C22×Dic7—81no—
159C22×Dic7—81no—
160C22×Dic7C2×D411yescharacteristic
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4C2×Q812yes—
176C7×C22⋊C4—24no—
177C42×C7—81no—
178C42×C7C2×Q812yes—
179C42×C7#4112yes—
180C22⋊Dic7—42no—
181C22⋊Dic7—42no—
182C4⋊Dic7—22no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—41no—
192C4⋊Dic7—81no—
193C4⋊Dic7—81no—
194Dic7⋊C4—42no—
195Dic7⋊C4—42no—
196Dic7⋊C4—81no—
197Dic7⋊C4—81no—
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—21no—
201C4×Dic7—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C4×D4—71no—
209C2×C4×D4—71no—
210C2×C4×D4C2×D711yescharacteristic
211C2×#107—71no—
212C2×#106—71no—
213C2×C4⋊C8—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8—71no—
216C2×#102—71no—
217C2×#102—71no—
218C2×#102C2×D711yescharacteristic
219C2×#102C2×D711yescharacteristic
220C2×C4×C8C2×D711yescharacteristic
221#400—141no—
222#398—141no—
223#389—71no—
224#389—71no—
225#389—71no—
226#389—71no—
227#387—141no—
228#387—141no—
229#385—141no—
230#385—141no—
231#384—71no—
232#382—71no—
233#382—71no—
234#382—71no—
235#380—141no—
236C4×C4⋊C4—71no—
237C4×C4⋊C4C2×D711yescharacteristic
238C2×#95—141no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
249#336—71no—
250#336—71no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7—22no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7C22×D411yescharacteristic
259C7×D4—22no—
260C7×D4—22no—
261C7×D4—22no—
262C7×D4#12514yes—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—22no—
268C2×C4×C7—41no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—84no—
277C2×C4×C7C22⋊Q812yes—
278C2×C4×C7C22⋊Q812yes—
279C2×C4×C7C22×Q811yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7H(5,𝔽2)11yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—41no—
297C2×Dic7—41no—
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—84no—
309C2×Dic7—84no—
310C2×Dic7#13512yes—
311C7×C8—41no—
312C7×C8—41no—
313C7⋊C8—41no—
314C7⋊C8—41no—
315C22×D4C22×D711yescharacteristic
316C23×C4—21no—
317C23×C4—141no—
318C23×C4—141no—
319C22×C8—71no—
320C22×C8—71no—
321C22×C8C22×D711yescharacteristic
322C22×C8C22×D711yescharacteristic
323C4×D4—22no—
324C4×D4—142no—
325C4×D4—142no—
326C4×D4—281no—
327C4×D4—281no—
328C4×D4Dic1414yes—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—141no—
346C2×C22⋊C4—141no—
347C2×C42—21no—
348C2×C42—71no—
349C2×C42—71no—
350C2×C42C22×D711yescharacteristic
351#107—281no—
352#106—281no—
353C4⋊C8—142no—
354C4⋊C8—281no—
355C4⋊C8—281no—
356#102—22no—
357#102—22no—
358#102—142no—
359#102—142no—
360C4×C8—22no—
361#95—21no—
362#95—71no—
363#95—71no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374C22×C7—24no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—42no—
378C22×C7—42no—
379C22×C7—42no—
380C22×C7—42no—
381C22×C7C2×C22⋊Q811yescharacteristic
382C22×C7#54611yescharacteristic
383C22×C7#54811yescharacteristic
384C22×C7C2×#13511yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57611yescharacteristic
387C22×C7#57711yescharacteristic
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—42no—
391C4×C7—81no—
392C4×C7—81no—
393C4×C7C2×#12511yescharacteristic
394C4×C7#54411yescharacteristic
395C4×C7#55211yescharacteristic
396C4×C7#55311yescharacteristic
397Dic7—22no—
398Dic7—41no—
399Dic7—41no—
400Dic7—41no—
401Dic7—81no—
402Dic7—81no—
403Dic7—81no—
404Dic7—81no—
405C24—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4C2×Dic1412yes—
410C2×D4#115712yes—
411C2×D4#115712yes—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—71no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—81no—
421C22×C4—81no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—282no—
427C22×C4—282no—
428C22×C4—282no—
429C22×C4—282no—
430C22×C4—561no—
431C22×C4—561no—
432C22×C4D4×D711yescharacteristic
433C22×C4D4×D711yescharacteristic
434C22×C4C23×D711yescharacteristic
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—141no—
442C2×C8—141no—
443C2×C8—141no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—142no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—281no—
462C4⋊C4—282no—
463C4⋊C4—282no—
464C4⋊C4—561no—
465C4⋊C4—561no—
466C4⋊C4—561no—
467C4⋊C4—561no—
468C4⋊C4C2×Dic1412yes—
469C22⋊C4—24no—
470C22⋊C4—282no—
471C22⋊C4—282no—
472C42—81no—
473C42—141no—
474C42—141no—
475C42—141no—
476C42—141no—
477C42C2×Dic1412yes—
478C42#115912yes—
479C2×C7—44no—
480C2×C7#244211yescharacteristic
481C2×C7#307611yescharacteristic
482C2×C7#308111yescharacteristic
483C2×C7#329711yescharacteristic
484C2×C7#332311yescharacteristic
485C2×C7#336011yescharacteristic
486C2×C7#337011yescharacteristic
487C23—22no—
488C23—22no—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D4×D711yescharacteristic
495D4—22no—
496D4—22no—
497D4—22no—
498D4#643314yes—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—72no—
513C2×C4—84no—
514C2×C4—141no—
515C2×C4—141no—
516C2×C4—141no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—564no—
538C2×C4—564no—
539C2×C4#644112yes—
540C2×C4#644112yes—
541C2×C4C22×Dic1411yescharacteristic
542C2×C4C2×#115711yescharacteristic
543C2×C4C2×#115711yescharacteristic
544C2×C4C2×#115911yescharacteristic
545C2×C4#655311yescharacteristic
546C2×C4#655411yescharacteristic
547C8—41no—
548C8—41no—
549C8—281no—
550C8—281no—
551C7#2611711yescharacteristic, complemented, Sylow
552C22—24no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22C2×#644111yescharacteristic
560C22#9024111yescharacteristic
561C22#9024311yescharacteristic
562C22#9044811yescharacteristic
563C22#9045311yescharacteristic
564C22D7×#13511yescharacteristic
565C22#9045911yescharacteristic
566C4—22no—
567C4—22no—
568C4—42no—
569C4—81no—
570C4—81no—
571C4—142no—
572C4—281no—
573C4—281no—
574C4—281no—
575C4—561no—
576C4—561no—
577C4—561no—
578C4—561no—
579C4C2×#643311yescharacteristic
580C4#9021011yescharacteristic
581C4#9023211yescharacteristic
582C4#9025611yescharacteristic
583C2—44no—
584C2#1173094011yescharacteristic
585C2#1173617611yescharacteristic
586C2#1173626711yescharacteristic
587C2#1173701711yescharacteristic
588C2#1173705411yescharacteristic
589C2#1173706511yescharacteristic
590C2#1173712711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC4×C7×#102C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC4×#102D7
hypocenterC7#26117
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. #11731169C2semidirect
9. #11727540C2semidirect
10. #11727517C2semidirect
11. #11727222C2semidirect
12. #11726600C2semidirect
13. #11726597C2semidirect
14. #11726563C2semidirect
15. #11726561C2semidirect
551. C7#26117semidirect

Maximal subgroups

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

subgroupindex
#117397782
#117311692
C4×C7×#1022
#117293592
#117279232
#117279212
#117279082
#117279072
#117275402
#117275172
#117272222
#117266002
#117265972
#117265632
#117265612
#261177