Finite Group Databasebeta

← #13412763

Subgroups — #13412763

Order 1792.

714 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11733747C211yesmaximal, characteristic
2#11733519C211yesmaximal, characteristic, complemented
3C7×#2067C211yesmaximal, characteristic
4#11728370C211yesmaximal, characteristic, complemented
5#11728360C211yesmaximal, characteristic, complemented
6#11728209C211yesmaximal, characteristic, complemented
7#11728205C211yesmaximal, characteristic, complemented
8#11727936C211yesmaximal, characteristic
9#11727934C211yesmaximal, characteristic
10#11727907C211yesmaximal, characteristic
11Dic7×#102C211yesmaximal, characteristic
12#11727408C211yesmaximal, characteristic
13#11727357C211yesmaximal, characteristic, complemented
14#11726550C211yesmaximal, characteristic, complemented
15#11726544C211yesmaximal, characteristic, complemented
16C2×D4×Dic7C2211yescharacteristic
17C2×#6440—21no—
18C2×C7×#107C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C2×C7×#102C2211yescharacteristic
21C2×C7×#102C2211yescharacteristic
22C7×#390C2211yescharacteristic
23C7×#389C2211yescharacteristic
24#89993C2211yescharacteristic
25C2×#6406C2211yescharacteristic
26C2×#6406C2211yescharacteristic
27C2×#6392C2211yescharacteristic
28C2×#6391C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89754—21no—
33#89749C2211yescharacteristic
34#89748C2211yescharacteristic
35#89748C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38#89745C2211yescharacteristic
39#89744—21no—
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43#89717—21no—
44#89716—21no—
45#89714—21no—
46#89708C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48C4×Dic7⋊C4C2211yescharacteristic
49#89430—21no—
50#89422—21no—
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#26087—71nomaximal, Sylow
60C22×C7×D4C2311yescharacteristic
61C23×Dic7—21no—
62C22×C7×C8C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C42×C7C2311yescharacteristic
71C2×C22⋊Dic7—21no—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—21no—
74D4×Dic7—22no—
75D4×Dic7Q814yes—
76C2×C4⋊Dic7C2311yescharacteristic
77C2×C4⋊Dic7C2311yescharacteristic
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×Dic7⋊C4—21no—
81C2×Dic7⋊C4—21no—
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4D411yescharacteristic
86C2×Dic7⋊C4D411yescharacteristic
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7C2311yescharacteristic
91C2×C4×Dic7C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94#6440—44no—
95C7×#107—41no—
96C7×C4⋊C8—22no—
97C7×#102—22no—
98C7×#102—41no—
99C7×#95—21no—
100#6406—22no—
101#6406—41no—
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⋊#95D411yescharacteristic
111C7⋊#95D411yescharacteristic
112#6392—41no—
113#6391—41no—
114Dic7⋊C8—22no—
115Dic7⋊C8—22no—
116C8×Dic7—22no—
117#6382—41no—
118#6381—22no—
119#6381—22no—
120#2571—71no—
121#2354—71no—
122#2092—71no—
123#2092—71no—
124#2067—71no—
125#2067D711yescharacteristic
126#1949—71no—
127#1949—71no—
128#1948—71no—
129#1939—71no—
130#1921—71no—
131#1921—71no—
132#1845—71no—
133#1802—71no—
134C4×#102—71no—
135C24×C7—21no—
136C2×C7×D4—21no—
137C2×C7×D4—21no—
138C2×C7×D4—21no—
139C2×C7×D4—21no—
140C2×C7×D4—21no—
141C2×C7×D4C2×Q812yes—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2411yescharacteristic
148C22×Dic7—21no—
149C22×Dic7—24no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—42no—
154C22×Dic7—42no—
155C22×Dic7C2×D411yescharacteristic
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—21no—
163C2×C7×C8—21no—
164C2×C7×C8#4112yes—
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—81no—
172C7×C22⋊C4—42no—
173C7×C22⋊C4—81no—
174C42×C7—21no—
175C42×C7—21no—
176C22⋊Dic7—24no—
177C22⋊Dic7—42no—
178C22⋊Dic7—81no—
179C4⋊Dic7—21no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7C2×Q812yes—
187Dic7⋊C4—22no—
188Dic7⋊C4—22no—
189Dic7⋊C4—42no—
190Dic7⋊C4—42no—
191Dic7⋊C4—81no—
192Dic7⋊C4—81no—
193Dic7⋊C4—81no—
194C4×Dic7—21no—
195C4×Dic7—21no—
196C4×Dic7—42no—
197C4×Dic7—42no—
198C4×Dic7—42no—
199C4×Dic7C2×Q812yes—
200C4×Dic7#4112yes—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8#4112yes—
207C2×C4×D4—71no—
208C2×#117—141no—
209C2×#107—71no—
210C2×#107—71no—
211C2×#107—71no—
212C2×#107C2×D711yescharacteristic
213C2×#106—71no—
214C2×#106—71no—
215C2×C4⋊C8—71no—
216C2×C4⋊C8—71no—
217C2×C4⋊C8C2×D711yescharacteristic
218C2×#102—71no—
219C2×#102—71no—
220C2×#102C2×D711yescharacteristic
221C2×#102C2×D711yescharacteristic
222C2×C4×C8—71no—
223#400—141no—
224#400—141no—
225#390—71no—
226#390C2×D711yescharacteristic
227#389—71no—
228#389—141no—
229#389C2×D711yescharacteristic
230#388—141no—
231#388—141no—
232#384—71no—
233#384—71no—
234#382—71no—
235#382—71no—
236#382—141no—
237#380—141no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340C2×D711yescharacteristic
248#336—71no—
249#336—71no—
250C23×C7—22no—
251C23×C7—41no—
252C23×C7—41no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7C22×D411yescharacteristic
258C7×D4—22no—
259C7×D4—22no—
260C7×D4—41no—
261C7×D4—41no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
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—41no—
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—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—41no—
293C2×Dic7—41no—
294C2×Dic7—48no—
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×Dic7C22⋊Q812yes—
305C2×Dic7C22⋊Q812yes—
306C2×Dic7#13512yes—
307C2×Dic7AGL(1,ℤ/8)12yes—
308C7×C8—22no—
309C7×C8—41no—
310C7⋊C8—22no—
311C7⋊C8—41no—
312C22×D4C22×D711yescharacteristic
313C23×C4—141no—
314C22×C8—71no—
315C22×C8—71no—
316C22×C8C22×D711yescharacteristic
317C22×C8C22×D711yescharacteristic
318C4×D4—74no—
319C4×D4—142no—
320#117—284no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—71no—
323C2×C4⋊C4—71no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—141no—
329C2×C4⋊C4—141no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4C22×D711yescharacteristic
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—141no—
339C2×C22⋊C4—141no—
340C2×C22⋊C4—141no—
341C2×C42—71no—
342C2×C42—71no—
343C2×C42—141no—
344C2×C42—141no—
345C2×C42—141no—
346C2×C42C22×D711yescharacteristic
347#107—41no—
348#107—142no—
349#107—142no—
350#107—281no—
351#106—281no—
352#106—281no—
353C4⋊C8—22no—
354C4⋊C8—142no—
355C4⋊C8—142no—
356#102—22no—
357#102—41no—
358#102—142no—
359#102—281no—
360C4×C8—142no—
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—
372C22×C7—42no—
373C22×C7—42no—
374C22×C7—42no—
375C22×C7—81no—
376C22×C7—81no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7C2×C22⋊Q811yescharacteristic
380C22×C7#54711yescharacteristic
381C22×C7#55011yescharacteristic
382C22×C7C2×#13511yescharacteristic
383C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
384C22×C7#57611yescharacteristic
385C22×C7#57811yescharacteristic
386C4×C7—41no—
387C4×C7—41no—
388C4×C7—81no—
389C4×C7—81no—
390C4×C7—81no—
391C4×C7#53311yescharacteristic
392C4×C7#54411yescharacteristic
393C4×C7#55211yescharacteristic
394C4×C7#55311yescharacteristic
395Dic7—22no—
396Dic7—42no—
397Dic7—41no—
398Dic7—41no—
399Dic7—81no—
400Dic7—81no—
401Dic7—81no—
402Dic7#47814yes—
403C24—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4Q8×D712yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—71no—
414C22×C4—71no—
415C22×C4—71no—
416C22×C4—71no—
417C22×C4—144no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—282no—
423C22×C4—282no—
424C22×C4D4×D711yescharacteristic
425C22×C4D4×D711yescharacteristic
426C22×C4C23×D711yescharacteristic
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—72no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8#115912yes—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—72no—
446C4⋊C4—81no—
447C4⋊C4—142no—
448C4⋊C4—142no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—282no—
457C4⋊C4—282no—
458C4⋊C4—561no—
459C4⋊C4—561no—
460C4⋊C4—561no—
461C22⋊C4—42no—
462C22⋊C4—81no—
463C22⋊C4—144no—
464C22⋊C4—282no—
465C22⋊C4—561no—
466C42—21no—
467C42—21no—
468C42—72no—
469C42—72no—
470C42—141no—
471C42—141no—
472C42—282no—
473C42—282no—
474C42—282no—
475C2×C7—82no—
476C2×C7#244111yescharacteristic
477C2×C7C2×#47811yescharacteristic
478C2×C7#308311yescharacteristic
479C2×C7#334111yescharacteristic
480C2×C7#335311yescharacteristic
481C2×C7#339311yescharacteristic
482C2×C7#339911yescharacteristic
483C23—22no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C2×D4×D711yescharacteristic
491D4—22no—
492D4—22no—
493D4—41no—
494D4—41no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
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—72no—
512C2×C4—72no—
513C2×C4—72no—
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—142no—
523C2×C4—288no—
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×C4C2×#115911yescharacteristic
536C2×C4#654811yescharacteristic
537C2×C4C2×Q8×D711yescharacteristic
538C2×C4D7×#4111yescharacteristic
539C2×C4D7×#4111yescharacteristic
540C2×C4#655411yescharacteristic
541C8—22no—
542C8—41no—
543C8—142no—
544C8—281no—
545C7#2608711yescharacteristic, complemented, Sylow
546C22—42no—
547C22—42no—
548C22—42no—
549C22—81no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22#9029011yescharacteristic
554C22#9030211yescharacteristic
555C22D7×C22⋊Q811yescharacteristic
556C22#9044811yescharacteristic
557C22D7×#13511yescharacteristic
558C22D7×AGL(1,ℤ/8)11yescharacteristic
559C22#9047011yescharacteristic
560C4—41no—
561C4—41no—
562C4—74no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—142no—
567C4—281no—
568C4—281no—
569C4—282no—
570C4—561no—
571C4—561no—
572C4—561no—
573C4#9029211yescharacteristic
574C4#9031311yescharacteristic
575C4#9031411yescharacteristic
576C4#9039911yescharacteristic
577C2—82no—
578C2#1173473411yescharacteristic
579C2#1173777311yescharacteristic
580C2#1173803311yescharacteristic
581C2#1173815511yescharacteristic
582C2#1173824611yescharacteristic
583C2#1173841311yescharacteristic
584C2D7×#47811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2067C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2067D7
hypocenterC7#26087
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
2. #11733519C2semidirect
4. #11728370C2semidirect
5. #11728360C2semidirect
6. #11728209C2semidirect
7. #11728205C2semidirect
13. #11727357C2semidirect
14. #11726550C2semidirect
15. #11726544C2semidirect
545. C7#26087semidirect

Maximal subgroups

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

subgroupindex
#117337472
#117335192
C7×#20672
#117283702
#117283602
#117282092
#117282052
#117279362
#117279342
#117279072
Dic7×#1022
#117274082
#117273572
#117265502
#117265442
#260877