Finite Group Databasebeta

← #13411787

Subgroups — #13411787

Order 1792.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11730939C211yesmaximal, characteristic
2#11730901C211yesmaximal, characteristic
3C7×#1851C211yesmaximal, characteristic
4#11729742C211yesmaximal, characteristic
5#11729739C211yesmaximal, characteristic
6#11728218C211yesmaximal, characteristic, complemented
7#11728216C211yesmaximal, characteristic, complemented
8#11728215C211yesmaximal, characteristic, complemented
9#11728208C211yesmaximal, characteristic, complemented
10#11727683C211yesmaximal, characteristic
11#11727679C211yesmaximal, characteristic
12#11726397C211yesmaximal, characteristic, complemented
13#11726388C211yesmaximal, characteristic, complemented
14#11726378C211yesmaximal, characteristic, complemented
15#11726375C211yesmaximal, characteristic, complemented
16C22×C7×M16C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C2×#6440—21no—
20C2×#6433C2211yescharacteristic
21C2×C7×C4⋊C8C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C2×C7×C22⋊C8C2211yescharacteristic
24C4×C7×C22⋊C4C2211yescharacteristic
25#89984C2211yescharacteristic
26#89984C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6391C2211yescharacteristic
30C2×#6391C2211yescharacteristic
31#89721—21no—
32#89720—21no—
33#89715—21no—
34#89707C2211yescharacteristic
35C4×C4⋊Dic7C2211yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37#89700C2211yescharacteristic
38#89700C2211yescharacteristic
39#89700C2211yescharacteristic
40#89700C2211yescharacteristic
41#89426—21no—
42#89425—21no—
43#89425—21no—
44#89419—21no—
45C7×#336C2211yescharacteristic
46C7×#336C2211yescharacteristic
47#89346C2211yescharacteristic
48#89346C2211yescharacteristic
49#89346C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89346C2211yescharacteristic
56#89346C2211yescharacteristic
57#89346C2211yescharacteristic
58#89346C2211yescharacteristic
59#26014—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C2×C7×M16—22no—
63C2×C7×M16—22no—
64C2×C7×M16—41no—
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C2×C7×C22⋊C4D411yescharacteristic
70C2×C7×C22⋊C4D411yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76#6489—22no—
77#6489—41no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7Q814yes—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×Dic7⋊C4—21no—
94C2×Dic7⋊C4—21no—
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—21no—
100#6440—44no—
101#6433—22no—
102#6433—41no—
103C7×C4⋊C8—22no—
104C7×C4⋊C8—41no—
105C7×C22⋊C8—22no—
106C7×#95D411yescharacteristic
107C7×#95D411yescharacteristic
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116#6392—22no—
117#6392—22no—
118#6391—41no—
119#6391—41no—
120#2441—71no—
121#2344—71no—
122#2100—71no—
123#2088—71no—
124#2086—71no—
125#2059—71no—
126#1989—71no—
127#1948—71no—
128#1921—71no—
129#1918—71no—
130#1854—71no—
131#1851D711yescharacteristic
132#1824—71no—
133#1809—71no—
134#1736—71no—
135C24×C7C2×D411yescharacteristic
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×Q812yes—
146C22×C4×C7#4112yes—
147C22×C4×C7C2411yescharacteristic
148C22×Dic7—24no—
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—42no—
158C22×Dic7—42no—
159C7×M16—44no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8#4112yes—
172C7×C22⋊C4—22no—
173C7×C22⋊C4—41no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—21no—
177C42×C7—21no—
178C22⋊Dic7—42no—
179C22⋊Dic7—42no—
180C22⋊Dic7—81no—
181C4⋊Dic7—24no—
182C4⋊Dic7—21no—
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—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7—21no—
195C4⋊Dic7—42no—
196C4⋊Dic7—81no—
197C4⋊Dic7C2×Q812yes—
198C4⋊Dic7C2×Q812yes—
199C4⋊Dic7#4112yes—
200Dic7⋊C4—42no—
201Dic7⋊C4—42no—
202Dic7⋊C4—42no—
203Dic7⋊C4—42no—
204Dic7⋊C4—42no—
205Dic7⋊C4—81no—
206C4×Dic7—42no—
207C4×Dic7—42no—
208C4×Dic7—42no—
209C22×M16C2×D711yescharacteristic
210C2×#125—71no—
211C2×#117—71no—
212C2×#117—141no—
213C22×C4⋊C4—71no—
214C2×#107—71no—
215C2×#107—71no—
216C2×#106—71no—
217C2×#106—71no—
218C2×C4⋊C8C2×D711yescharacteristic
219C2×C4⋊C8C2×D711yescharacteristic
220C2×C22⋊C8C2×D711yescharacteristic
221#399—141no—
222#396—141no—
223#389—71no—
224#389—71no—
225#389—71no—
226#389—71no—
227#389—141no—
228#387—141no—
229#387—141no—
230#387—141no—
231#384—71no—
232#381—141no—
233#380—71no—
234#380—71no—
235C4×C4⋊C4—71no—
236C4×C4⋊C4—71no—
237C4×C22⋊C4C2×D711yescharacteristic
238#340—71no—
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#340—71no—
248#340—71no—
249#340—71no—
250#336C2×D711yescharacteristic
251#336C2×D711yescharacteristic
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7C22⋊Q812yes—
259C23×C7C22×D411yescharacteristic
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7C22⋊Q812yes—
278C2×C4×C7AGL(1,ℤ/8)12yes—
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—22no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—48no—
289C2×Dic7—41no—
290C2×Dic7—41no—
291C2×Dic7—41no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—41no—
295C2×Dic7—41no—
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—41no—
309C2×Dic7—41no—
310C7×C8—22no—
311C7×C8—41no—
312C7×C8—41no—
313C7×C8—41no—
314C23×C4—141no—
315C23×C4C22×D711yescharacteristic
316C2×M16—22no—
317C2×M16—22no—
318C2×M16—41no—
319C22×C8C22×D711yescharacteristic
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322C22×C8C22×D711yescharacteristic
323#125—142no—
324#125—281no—
325#117—142no—
326#117—284no—
327#117—281no—
328C2×C4⋊C4—74no—
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—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—142no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4D2811yescharacteristic
351C2×C22⋊C4D2811yescharacteristic
352C2×C42—141no—
353C2×C42—141no—
354C2×C42—141no—
355C2×C42C22×D711yescharacteristic
356C2×C42C22×D711yescharacteristic
357#107—142no—
358#107—142no—
359#106—281no—
360#106—281no—
361C4⋊C8—22no—
362C4⋊C8—41no—
363C22⋊C8—22no—
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#95D2811yescharacteristic
373#95D2811yescharacteristic
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—41no—
378C22×C7—41no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7C2×C22⋊Q811yescharacteristic
382C22×C7#54611yescharacteristic
383C22×C7#54711yescharacteristic
384C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57711yescharacteristic
387C22×C7#57811yescharacteristic
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—41no—
391C4×C7—41no—
392C4×C7—41no—
393C4×C7#53311yescharacteristic
394C4×C7#54411yescharacteristic
395C4×C7#55111yescharacteristic
396C4×C7#55211yescharacteristic
397Dic7—42no—
398Dic7—42no—
399Dic7—42no—
400Dic7—81no—
401Dic7—81no—
402Dic7—81no—
403Dic7—81no—
404Dic7—81no—
405C24C2×D2811yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—144no—
411C22×C4—141no—
412C22×C4—141no—
413C22×C4—141no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—282no—
420C22×C4—282no—
421C22×C4C2×D2811yescharacteristic
422C22×C4C2×D2811yescharacteristic
423C22×C4C2×D2811yescharacteristic
424C22×C4C2×D2811yescharacteristic
425C22×C4C2×D2811yescharacteristic
426C22×C4#115912yes—
427C22×C4Q8×D712yes—
428C22×C4C23×D711yescharacteristic
429M16—44no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8#115912yes—
442C4⋊C4—72no—
443C4⋊C4—72no—
444C4⋊C4—72no—
445C4⋊C4—144no—
446C4⋊C4—141no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—282no—
460C4⋊C4—282no—
461C4⋊C4—282no—
462C4⋊C4—282no—
463C4⋊C4—282no—
464C4⋊C4—282no—
465C4⋊C4—561no—
466C4⋊C4—561no—
467C22⋊C4—22no—
468C22⋊C4—41no—
469C22⋊C4—282no—
470C22⋊C4—282no—
471C22⋊C4—561no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—282no—
477C42—282no—
478C42—282no—
479C2×C7—42no—
480C2×C7#243811yescharacteristic
481C2×C7#307711yescharacteristic
482C2×C7#308211yescharacteristic
483C2×C7#331711yescharacteristic
484C2×C7#333011yescharacteristic
485C2×C7#334511yescharacteristic
486C2×C7#335211yescharacteristic
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23#646012yes—
494C23C22×D2811yescharacteristic
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—288no—
516C2×C4—281no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4—281no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
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#646012yes—
538C2×C4#647112yes—
539C2×C4C2×#115911yescharacteristic
540C2×C4C2×#115911yescharacteristic
541C2×C4#654811yescharacteristic
542C2×C4C2×Q8×D711yescharacteristic
543C2×C4D7×#4111yescharacteristic
544C2×C4#655411yescharacteristic
545C8—22no—
546C8—41no—
547C8—41no—
548C8—41no—
549C7#2601411yescharacteristic, complemented, Sylow
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C2×#646011yescharacteristic
558C22#9024511yescharacteristic
559C22#9024611yescharacteristic
560C22C2×#647111yescharacteristic
561C22#9043911yescharacteristic
562C22#9044211yescharacteristic
563C22#9044411yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—282no—
570C4—282no—
571C4—282no—
572C4—561no—
573C4—561no—
574C4—561no—
575C4—561no—
576C4—561no—
577C4#9027811yescharacteristic
578C4#9031311yescharacteristic
579C4#9031411yescharacteristic
580C4#9034111yescharacteristic
581C2—42no—
582C2#1173186011yescharacteristic
583C2#1173650411yescharacteristic
584C2#1173654611yescharacteristic
585C2#1173691611yescharacteristic
586C2#1173695911yescharacteristic
587C2#1173697511yescharacteristic
588C2#1173700211yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1851C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1851D7
hypocenterC7#26014
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
6. #11728218C2semidirect
7. #11728216C2semidirect
8. #11728215C2semidirect
9. #11728208C2semidirect
12. #11726397C2semidirect
13. #11726388C2semidirect
14. #11726378C2semidirect
15. #11726375C2semidirect
549. C7#26014semidirect

Maximal subgroups

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

subgroupindex
#117309392
#117309012
C7×#18512
#117297422
#117297392
#117282182
#117282162
#117282152
#117282082
#117276832
#117276792
#117263972
#117263882
#117263782
#117263752
#260147