Finite Group Databasebeta

← #13411856

Subgroups — #13411856

Order 1792.

704 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731772C211yesmaximal, characteristic
2#11731213C211yesmaximal, characteristic
3C7×#1947C211yesmaximal, characteristic, complemented
4#11728387C211yesmaximal, characteristic
5#11728385C211yesmaximal, characteristic
6#11728375C211yesmaximal, characteristic, complemented
7#11728372C211yesmaximal, characteristic, complemented
8#11728232C211yesmaximal, characteristic
9#11728218C211yesmaximal, characteristic, complemented
10#11727326C211yesmaximal, characteristic, complemented
11#11727213C211yesmaximal, characteristic, complemented
12#11726625C211yesmaximal, characteristic
13#11726621C211yesmaximal, characteristic
14#11726572C211yesmaximal, characteristic, complemented
15#11726556C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C2×#6455C2211yescharacteristic
18C2×D7×C4⋊C4C2211yescharacteristic
19C2×#6453C2211yescharacteristic
20C2×C7×#107C2211yescharacteristic
21C2×C7×#107C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C7×#384C2211yescharacteristic
24C4×C7×C4⋊C4C2211yescharacteristic
25C2×(C2×D7)⋊C8C2211yescharacteristic
26C2×#6391C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28#89766—21no—
29#89764—21no—
30#89759C2211yescharacteristic
31#89759C2211yescharacteristic
32#89753—21no—
33C4⋊C4×Dic7C2211yescharacteristic
34#89745C2211yescharacteristic
35#89745C2211yescharacteristic
36#89744—21no—
37C2×#6382C2211yescharacteristic
38C4×(C2×D7)⋊C4—21no—
39C4×(C2×D7)⋊C4C2211yescharacteristic
40#89705—21no—
41#89700C2211yescharacteristic
42#89700C2211yescharacteristic
43#89439—21no—
44#89424—21no—
45C7×#340C2211yescharacteristic
46C7×#340C2211yescharacteristic
47#89346C2211yescharacteristic
48#89346C2211yescharacteristic
49#89346C2211yescharacteristic
50#89345C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#26018—71nomaximal, Sylow
60C22×C4×D7—21no—
61C22×C4×D7C2311yescharacteristic
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C42×C7—21no—
71C2×C42×C7C2311yescharacteristic
72C2×(C2×D7)⋊C4—21no—
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4D411yescharacteristic
76C2×(C2×D7)⋊C4D411yescharacteristic
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C2×#1131—22no—
94C2×#1131—22no—
95C2×#1131—41no—
96#6455—22no—
97#6455—41no—
98D7×C4⋊C4—22no—
99D7×C4⋊C4Q814yes—
100#6453—22no—
101#6453—41no—
102C7×#107—41no—
103C7×#107—41no—
104C7×C4⋊C8—22no—
105C7×#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115(C2×D7)⋊C8—22no—
116#6391—22no—
117Dic7⋊C8—41no—
118#6382—22no—
119#2441—71no—
120#2345—71no—
121#2100—71no—
122#2088—71no—
123#2086—71no—
124#2059—71no—
125#1989—71no—
126#1947D711yescharacteristic, complemented
127#1920—71no—
128#1919—71no—
129#1854—71no—
130#1851—71no—
131#1823—71no—
132#1809—71no—
133#1736—71no—
134C23×D7C2×D411yescharacteristic
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2411yescharacteristic
142C22×Dic7—21no—
143C22×Dic7—21no—
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7C2×D411yescharacteristic
147C22×Dic7C2×D411yescharacteristic
148C22×Dic7C2×D411yescharacteristic
149C2×C4×D7—21no—
150C2×C4×D7—21no—
151C2×C4×D7—21no—
152C2×C4×D7—21no—
153C2×C4×D7—24no—
154C2×C4×D7—81no—
155C2×C4×D7—81no—
156C2×C4×D7C2×Q812yes—
157C2×C4×D7#4112yes—
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8#4112yes—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—42no—
172C7×C4⋊C4—81no—
173C7×C4⋊C4C2×Q812yes—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—81no—
177(C2×D7)⋊C4—22no—
178(C2×D7)⋊C4—41no—
179(C2×D7)⋊C4—42no—
180(C2×D7)⋊C4—42no—
181(C2×D7)⋊C4—81no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—42no—
189C4⋊Dic7C2×Q812yes—
190C4⋊Dic7#4112yes—
191Dic7⋊C4—24no—
192Dic7⋊C4—42no—
193Dic7⋊C4—42no—
194Dic7⋊C4—81no—
195Dic7⋊C4—81no—
196Dic7⋊C4—81no—
197C4×Dic7—21no—
198C4×Dic7—21no—
199C4×Dic7—42no—
200C4×Dic7—81no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207#1131—44no—
208C22×M16—71no—
209C2×#125—71no—
210C2×#117—71no—
211C22×C4⋊C4—71no—
212C2×#107C2×D711yescharacteristic
213C2×#107C2×D711yescharacteristic
214C2×#106—71no—
215C2×#106—71no—
216C2×C4⋊C8—71no—
217C2×C4⋊C8C2×D711yescharacteristic
218C2×C22⋊C8—71no—
219#399—141no—
220#396—141no—
221#389—71no—
222#389—71no—
223#389—71no—
224#389—71no—
225#389—141no—
226#387—141no—
227#387—141no—
228#387—141no—
229#384C2×D711yescharacteristic
230#380—71no—
231#380—71no—
232#380—141no—
233C4×C4⋊C4—71no—
234C4×C4⋊C4C2×D711yescharacteristic
235C4×C22⋊C4—71no—
236C4×C22⋊C4—141no—
237#340—71no—
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#340C2×D711yescharacteristic
248#340C2×D711yescharacteristic
249#336—71no—
250#336—71no—
251C23×C7C22×D411yescharacteristic
252C22×D7—21no—
253C22×D7—21no—
254C22×D7—21no—
255C22×D7—21no—
256C22×D7—21no—
257C22×D7—21no—
258C22×D7C22⋊Q812yes—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—22no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7—41no—
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#13712yes—
277C2×C4×C7C22×Q811yescharacteristic
278C2×C4×C7C2×#4111yescharacteristic
279C2×C4×C7C2×#4111yescharacteristic
280C2×C4×C7C2×#4111yescharacteristic
281C2×C4×C7H(5,𝔽2)11yescharacteristic
282C2×C4×C72−1+411yescharacteristic
283C2×Dic7—21no—
284C2×Dic7—21no—
285C2×Dic7—22no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—22no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
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×Dic7C22⋊Q812yes—
304C4×D7—22no—
305C4×D7—22no—
306C4×D7—22no—
307C4×D7—22no—
308C4×D7—84no—
309C7×C8—22no—
310C7×C8—41no—
311C7⋊C8—41no—
312C7⋊C8—41no—
313C23×C4—71no—
314C23×C4—141no—
315C2×M16—142no—
316C2×M16—142no—
317C2×M16—281no—
318C22×C8—71no—
319C22×C8—71no—
320C22×C8C22×D711yescharacteristic
321C22×C8C22×D711yescharacteristic
322#125—142no—
323#125—281no—
324#117—142no—
325#117—281no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—74no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—142no—
334C2×C4⋊C4—141no—
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×C4⋊C4C22×D711yescharacteristic
345C2×C22⋊C4—71no—
346C2×C22⋊C4—71no—
347C2×C22⋊C4—141no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C42—21no—
351C2×C42—71no—
352C2×C42—141no—
353C2×C42—141no—
354C2×C42C22×D711yescharacteristic
355#107—41no—
356#107—41no—
357#106—142no—
358#106—142no—
359C4⋊C8—22no—
360C4⋊C8—281no—
361C22⋊C8—142no—
362#95—21no—
363#95—71no—
364#95—71no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372C22×C7C2×C22⋊Q811yescharacteristic
373C22×C7#54611yescharacteristic
374C22×C7#54711yescharacteristic
375C22×C7C2×#13711yescharacteristic
376C22×C7#57511yescharacteristic
377C22×C7#57611yescharacteristic
378C22×C7#57711yescharacteristic
379C2×D7—22no—
380C2×D7—22no—
381C2×D7—22no—
382C2×D7—41no—
383C2×D7—41no—
384C2×D7—41no—
385C2×D7—41no—
386C4×C7—22no—
387C4×C7—42no—
388C4×C7—41no—
389C4×C7—81no—
390C4×C7—81no—
391C4×C7—81no—
392C4×C7#53311yescharacteristic
393C4×C7#54411yescharacteristic
394C4×C7#55111yescharacteristic
395C4×C7#55211yescharacteristic
396Dic7—22no—
397Dic7—41no—
398Dic7—42no—
399Dic7—42no—
400Dic7—41no—
401Dic7—81no—
402Dic7—81no—
403C24—71no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—71no—
409C22×C4—72no—
410C22×C4—72no—
411C22×C4—71no—
412C22×C4—71no—
413C22×C4—144no—
414C22×C4—141no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—561no—
423C22×C4—561no—
424C22×C4D4×D711yescharacteristic
425C22×C4D4×D711yescharacteristic
426C22×C4C23×D711yescharacteristic
427M16—284no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8#115712yes—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—42no—
448C4⋊C4—72no—
449C4⋊C4—72no—
450C4⋊C4—81no—
451C4⋊C4—144no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—282no—
459C4⋊C4—282no—
460C4⋊C4—282no—
461C4⋊C4—561no—
462C4⋊C4—561no—
463C4⋊C4—561no—
464C4⋊C4Q8×D712yes—
465C22⋊C4—142no—
466C22⋊C4—281no—
467C22⋊C4—282no—
468C22⋊C4—282no—
469C22⋊C4—561no—
470C42—21no—
471C42—21no—
472C42—81no—
473C42—141no—
474C42—141no—
475C42—282no—
476C42—561no—
477C2×C7#243811yescharacteristic
478C2×C7#307811yescharacteristic
479C2×C7#308111yescharacteristic
480C2×C7#330911yescharacteristic
481C2×C7#333111yescharacteristic
482C2×C7#334611yescharacteristic
483C2×C7#335111yescharacteristic
484D7—42no—
485C23—72no—
486C23—141no—
487C23—141no—
488C23—141no—
489C23—141no—
490C23—141no—
491C23—141no—
492C23C2×D4×D711yescharacteristic
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—72no—
511C2×C4—141no—
512C2×C4—141no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—142no—
516C2×C4—142no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—141no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—142no—
524C2×C4—142no—
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—564no—
536C2×C4#648112yes—
537C2×C4C2×#115711yescharacteristic
538C2×C4#654811yescharacteristic
539C2×C4C2×Q8×D711yescharacteristic
540C2×C4#655111yescharacteristic
541C2×C4D7×#4111yescharacteristic
542C2×C4D7×#4111yescharacteristic
543C8—22no—
544C8—41no—
545C8—281no—
546C8—281no—
547C7#2601811yescharacteristic, complemented, Sylow
548C22—142no—
549C22—142no—
550C22—142no—
551C22—281no—
552C22—281no—
553C22—281no—
554C22—281no—
555C22#9019811yescharacteristic
556C22#9030311yescharacteristic
557C22D7×C22⋊Q811yescharacteristic
558C22#9044811yescharacteristic
559C22C2×#648111yescharacteristic
560C22#9046411yescharacteristic
561C22#9046911yescharacteristic
562C4—22no—
563C4—42no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—142no—
569C4—282no—
570C4—281no—
571C4—282no—
572C4—281no—
573C4—561no—
574C4—561no—
575C4#9024711yescharacteristic
576C4#9026911yescharacteristic
577C4#9038411yescharacteristic
578C4#9042011yescharacteristic
579C2—282no—
580C2#1173366711yescharacteristic
581C2#1173660411yescharacteristic
582C2#1173667611yescharacteristic
583C2#1173779911yescharacteristic
584C2#1173794111yescharacteristic
585C2#1173841211yescharacteristic
586C2#1173855311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1947C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenter#1947D7
hypocenterC7#26018
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
3. C7×#1947C2semidirect
6. #11728375C2semidirect
7. #11728372C2semidirect
9. #11728218C2semidirect
10. #11727326C2semidirect
11. #11727213C2semidirect
14. #11726572C2semidirect
15. #11726556C2semidirect
126. #1947D7semidirect
547. C7#26018semidirect

Maximal subgroups

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

subgroupindex
#117317722
#117312132
C7×#19472
#117283872
#117283852
#117283752
#117283722
#117282322
#117282182
#117273262
#117272132
#117266252
#117266212
#117265722
#117265562
#260187