Finite Group Databasebeta

← #13412960

Subgroups — #13412960

Order 1792.

708 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#11733748C211yesmaximal, characteristic
2#11733511C211yesmaximal, characteristic, complemented
3C7×#2067C211yesmaximal, characteristic
4#11728369C211yesmaximal, characteristic, complemented
5#11728362C211yesmaximal, characteristic, complemented
6#11728208C211yesmaximal, characteristic, complemented
7#11728206C211yesmaximal, characteristic, complemented
8#11727938C211yesmaximal, characteristic
9#11727933C211yesmaximal, characteristic
10#11727907C211yesmaximal, characteristic
11#11727906C211yesmaximal, characteristic
12#11727409C211yesmaximal, characteristic
13#11727355C211yesmaximal, characteristic, complemented
14#11726551C211yesmaximal, characteristic, complemented
15#11726544C211yesmaximal, characteristic, complemented
16C2×D4×Dic7C2211yescharacteristic
17C2×#6452—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×#6391C2211yescharacteristic
28C2×#6391C2211yescharacteristic
29C2×#6389C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32#89755—21no—
33#89749C2211yescharacteristic
34#89748C2211yescharacteristic
35#89748C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38#89745C2211yescharacteristic
39C2×#6382C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42#89717—21no—
43#89716—21no—
44#89715—21no—
45C22⋊C4×Dic7—21no—
46#89707C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48C4×Dic7⋊C4C2211yescharacteristic
49#89429—21no—
50#89419—21no—
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#26102—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⋊Dic7—21no—
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⋊C4D411yescharacteristic
88C2×Dic7⋊C4D411yescharacteristic
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96#6452—44no—
97C7×#107—41no—
98C7×C4⋊C8—22no—
99C7×#102—22no—
100C7×#102—41no—
101C7×#95—21no—
102#6406—22no—
103#6406—41no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95D411yescharacteristic
111C7⋊#95D411yescharacteristic
112#6391—41no—
113#6391—41no—
114#6389—22no—
115Dic7⋊C8—22no—
116Dic7⋊C8—22no—
117#6382—22no—
118#6381—22no—
119#6381—41no—
120#2552—71no—
121#2357—71no—
122#2094—71no—
123#2093—71no—
124#2067D711yescharacteristic
125#2065—71no—
126#1949—71no—
127#1948—71no—
128#1947—71no—
129#1940—71no—
130#1921—71no—
131#1919—71no—
132#1847—71no—
133#1802—71no—
134#1762—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—81no—
154C22×Dic7—81no—
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—42no—
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⋊Dic7—42no—
187C4⋊Dic7—81no—
188C4⋊Dic7C2×Q812yes—
189Dic7⋊C4—22no—
190Dic7⋊C4—22no—
191Dic7⋊C4—42no—
192Dic7⋊C4—42no—
193Dic7⋊C4—42no—
194Dic7⋊C4—81no—
195Dic7⋊C4—81no—
196C4×Dic7—21no—
197C4×Dic7—21no—
198C4×Dic7—42no—
199C4×Dic7—81no—
200C4×Dic7—81no—
201C4×Dic7C2×Q812yes—
202C4×Dic7#4112yes—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8#4112yes—
209C2×#125—141no—
210C2×C4×D4—71no—
211C2×#107—71no—
212C2×#107—71no—
213C2×#107C2×D711yescharacteristic
214C2×#106—71no—
215C2×#106—71no—
216C2×#106—71no—
217C2×C4⋊C8—71no—
218C2×C4⋊C8—71no—
219C2×C4⋊C8C2×D711yescharacteristic
220C2×#102—71no—
221C2×#102—71no—
222C2×#102C2×D711yescharacteristic
223C2×#102C2×D711yescharacteristic
224C2×#97—71no—
225#398—141no—
226#398—141no—
227#390—71no—
228#390C2×D711yescharacteristic
229#389—71no—
230#389—141no—
231#389C2×D711yescharacteristic
232#388—141no—
233#388—141no—
234#384—71no—
235#384—71no—
236#384—71no—
237#382—71no—
238#381—141no—
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C4⋊C4—71no—
242C4×C4⋊C4—71no—
243C4×C22⋊C4—141no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340C2×D711yescharacteristic
250#336—71no—
251#336—71no—
252C23×C7—22no—
253C23×C7—41no—
254C23×C7—41no—
255C23×C7—41no—
256C23×C7—41no—
257C23×C7—41no—
258C23×C7—41no—
259C23×C7C22×D411yescharacteristic
260C7×D4—22no—
261C7×D4—22no—
262C7×D4—41no—
263C7×D4—41no—
264C2×C4×C7—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
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×C7—41no—
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—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—22no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
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—84no—
307C2×Dic7C22⋊Q812yes—
308C2×Dic7C22⋊Q812yes—
309C2×Dic7#13712yes—
310C7×C8—22no—
311C7×C8—41no—
312C7⋊C8—22no—
313C7⋊C8—41no—
314C22×D4C22×D711yescharacteristic
315C23×C4—141no—
316C22×C8—71no—
317C22×C8—71no—
318C22×C8C22×D711yescharacteristic
319C22×C8C22×D711yescharacteristic
320#125—284no—
321C4×D4—74no—
322C4×D4—142no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—71no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—141no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4—141no—
345C2×C42—71no—
346C2×C42—71no—
347C2×C42—141no—
348C2×C42—141no—
349C2×C42—141no—
350C2×C42C22×D711yescharacteristic
351#107—41no—
352#107—142no—
353#107—281no—
354#106—142no—
355#106—281no—
356#106—281no—
357C4⋊C8—22no—
358C4⋊C8—142no—
359C4⋊C8—142no—
360#102—22no—
361#102—41no—
362#102—142no—
363#102—281no—
364#97—142no—
365#95—21no—
366#95—71no—
367#95—71no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374C22×C7—42no—
375C22×C7—42no—
376C22×C7—42no—
377C22×C7—81no—
378C22×C7—81no—
379C22×C7—81no—
380C22×C7—81no—
381C22×C7C2×C22⋊Q811yescharacteristic
382C22×C7#54711yescharacteristic
383C22×C7#55011yescharacteristic
384C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C7C2×#13711yescharacteristic
386C22×C7#57611yescharacteristic
387C22×C7#57711yescharacteristic
388C4×C7—22no—
389C4×C7—41no—
390C4×C7—81no—
391C4×C7—81no—
392C4×C7—81no—
393C4×C7#53311yescharacteristic
394C4×C7#54411yescharacteristic
395C4×C7#55211yescharacteristic
396C4×C7#55311yescharacteristic
397Dic7—22no—
398Dic7—22no—
399Dic7—22no—
400Dic7—42no—
401Dic7—41no—
402Dic7—81no—
403Dic7—81no—
404Dic7—81no—
405C24—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4Q8×D712yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—71no—
416C22×C4—71no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—144no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—561no—
425C22×C4—561no—
426C22×C4D4×D711yescharacteristic
427C22×C4D4×D711yescharacteristic
428C22×C4C23×D711yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—72no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8#115912yes—
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—142no—
450C4⋊C4—142no—
451C4⋊C4—141no—
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—282no—
462C4⋊C4—561no—
463C4⋊C4—561no—
464C4⋊C4—561no—
465C22⋊C4—42no—
466C22⋊C4—81no—
467C22⋊C4—144no—
468C22⋊C4—282no—
469C22⋊C4—561no—
470C42—21no—
471C42—21no—
472C42—72no—
473C42—72no—
474C42—141no—
475C42—141no—
476C42—282no—
477C42—561no—
478C42—561no—
479C2×C7—82no—
480C2×C7#244111yescharacteristic
481C2×C7#307911yescharacteristic
482C2×C7#308111yescharacteristic
483C2×C7#334511yescharacteristic
484C2×C7#334611yescharacteristic
485C2×C7#339911yescharacteristic
486C2×C7#340011yescharacteristic
487C23—22no—
488C23—41no—
489C23—41no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23C2×D4×D711yescharacteristic
495D4—22no—
496D4—22no—
497D4—41no—
498D4—41no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
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—41no—
513C2×C4—72no—
514C2×C4—72no—
515C2×C4—72no—
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—142no—
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#647412yes—
539C2×C4C2×#115911yescharacteristic
540C2×C4#654811yescharacteristic
541C2×C4C2×Q8×D711yescharacteristic
542C2×C4D7×#4111yescharacteristic
543C2×C4D7×#4111yescharacteristic
544C2×C4#655411yescharacteristic
545C8—22no—
546C8—41no—
547C8—142no—
548C8—281no—
549C7#2610211yescharacteristic, complemented, Sylow
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#9029011yescharacteristic
558C22#9030211yescharacteristic
559C22D7×C22⋊Q811yescharacteristic
560C22C2×#647411yescharacteristic
561C22#9046111yescharacteristic
562C22#9046511yescharacteristic
563C22D7×#13711yescharacteristic
564C4—22no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—142no—
570C4—142no—
571C4—142no—
572C4—281no—
573C4—282no—
574C4—561no—
575C4—561no—
576C4—561no—
577C4#9029211yescharacteristic
578C4#9031311yescharacteristic
579C4#9031411yescharacteristic
580C4#9039911yescharacteristic
581C2—82no—
582C2#1173473411yescharacteristic
583C2#1173777211yescharacteristic
584C2#1173803611yescharacteristic
585C2#1173815611yescharacteristic
586C2#1173827511yescharacteristic
587C2#1173841111yescharacteristic
588C2#1173861311yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2067C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenter#2067D7
hypocenterC7#26102
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. #11733511C2semidirect
4. #11728369C2semidirect
5. #11728362C2semidirect
6. #11728208C2semidirect
7. #11728206C2semidirect
13. #11727355C2semidirect
14. #11726551C2semidirect
15. #11726544C2semidirect
549. C7#26102semidirect

Maximal subgroups

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

subgroupindex
#117337482
#117335112
C7×#20672
#117283692
#117283622
#117282082
#117282062
#117279382
#117279332
#117279072
#117279062
#117274092
#117273552
#117265512
#117265442
#261027