Finite Group Databasebeta

← #13426521

Subgroups — #13426521

Order 1792.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740192C211yesmaximal, characteristic
2#11731366C211yesmaximal, characteristic
3C7×#2035C211yesmaximal, characteristic
4#11729449C211yesmaximal, characteristic
5#11728318C211yesmaximal, characteristic, complemented
6#11728312C211yesmaximal, characteristic, complemented
7#11728016C211yesmaximal, characteristic, complemented
8#11727989C211yesmaximal, characteristic, complemented
9#11727988C211yesmaximal, characteristic, complemented
10Dic7×#103C211yesmaximal, characteristic, complemented
11#11727678C211yesmaximal, characteristic
12#11727666C211yesmaximal, characteristic
13#11727258C211yesmaximal, characteristic
14#11726613C211yesmaximal, characteristic, complemented
15#11726569C211yesmaximal, characteristic, complemented
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C2×C7×#106C2211yescharacteristic
20C2×C7×#103C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×C22⋊C8C2211yescharacteristic
23C7×#380C2211yescharacteristic
24#90002C2211yescharacteristic
25C2×#6409C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27#89984C2211yescharacteristic
28C4×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89757—21no—
33#89757C2211yescharacteristic
34#89757C2211yescharacteristic
35#89753—21no—
36#89749C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38C4⋊C4×Dic7C2211yescharacteristic
39#89745C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42C2×#6381C2211yescharacteristic
43#89724—21no—
44#89723—21no—
45#89722—21no—
46C22⋊C4×Dic7—21no—
47#89431—21no—
48#89423—21no—
49C7×#340C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#27035—71nomaximal, Sylow
60C22×C7×Q8C2311yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—21no—
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C7×C22⋊Q8—22no—
66C7×C22⋊Q8—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7D411yescharacteristic
75C2×C22⋊Dic7D411yescharacteristic
76Q8×Dic7—22no—
77Q8×Dic7—41no—
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—41no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×C4×Dic7—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C7×#106—22no—
97C7×#103—22no—
98C7×#103—41no—
99C7×C22⋊C8—41no—
100C7×#95—21no—
101#6409—22no—
102#6409—22no—
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⋊#95—21no—
113C7⋊#95D411yescharacteristic
114C7⋊#95D411yescharacteristic
115C22⋊(C7⋊C8)—22no—
116Dic7⋊C8—22no—
117C8×Dic7—41no—
118#6381—22no—
119#6381—22no—
120#6381—41no—
121#2495—71no—
122#2360—71no—
123#2074—71no—
124#2068—71no—
125#2036—71no—
126#2035D711yescharacteristic
127#1949—71no—
128#1942—71no—
129#1923—71no—
130#1920—71no—
131#1918—71no—
132#1863—71no—
133#1852—71no—
134#1823—71no—
135C4×#103—71no—
136C24×C7C2×D411yescharacteristic
137C2×C7×Q8—21no—
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8#4112yes—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7—21no—
152C22×C4×C7—21no—
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—42no—
160C22×Dic7—42no—
161C22×Dic7—81no—
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C7×C4⋊C4—24no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4#4112yes—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—42no—
181C22⋊Dic7—22no—
182C22⋊Dic7—22no—
183C22⋊Dic7—81no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—42no—
192C4⋊Dic7—42no—
193C4⋊Dic7—81no—
194C4⋊Dic7—81no—
195C4⋊Dic7#4112yes—
196Dic7⋊C4—81no—
197C4×Dic7—21no—
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—21no—
201C4×Dic7—42no—
202C4×Dic7—42no—
203C4×Dic7—81no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8#4112yes—
210C2×C22⋊Q8C2×D711yescharacteristic
211C2×C4×Q8—71no—
212C22×C4⋊C4—71no—
213C2×#107—71no—
214C2×#107—71no—
215C2×#107—71no—
216C2×#106C2×D711yescharacteristic
217C2×C4⋊C8—71no—
218C2×#103—71no—
219C2×#103—71no—
220C2×#103C2×D711yescharacteristic
221C2×#103C2×D711yescharacteristic
222C2×C22⋊C8—71no—
223C2×C22⋊C8C2×D711yescharacteristic
224C2×C4×C8—71no—
225#400—141no—
226#397—141no—
227#391—71no—
228#389—71no—
229#389—71no—
230#389—71no—
231#389—141no—
232#388—141no—
233#387—141no—
234#385—141no—
235#383—141no—
236#382—71no—
237#380—71no—
238#380C2×D711yescharacteristic
239C4×C4⋊C4—71no—
240C4×C4⋊C4—71no—
241C4×C22⋊C4—71no—
242C4×C22⋊C4—141no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340C2×D711yescharacteristic
251#336—71no—
252#336—71no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7#13512yes—
260C23×C7C22×D411yescharacteristic
261C7×Q8—22no—
262C7×Q8—22no—
263C7×Q8—41no—
264C7×Q8—41no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
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×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7H(5,𝔽2)11yescharacteristic
289C2×C4×C72−1+411yescharacteristic
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—22no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—84no—
314C2×Dic7#13712yes—
315C7×C8—41no—
316C7×C8—41no—
317C7⋊C8—22no—
318C7⋊C8—41no—
319C22×Q8C22×D711yescharacteristic
320C23×C4—141no—
321C23×C4C22×D711yescharacteristic
322C22×C8—71no—
323C22×C8—71no—
324C22×C8C22×D711yescharacteristic
325C22×C8C22×D711yescharacteristic
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C4×Q8—142no—
329C4×Q8—281no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—142no—
336C2×C4⋊C4—141no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—281no—
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—71no—
348C2×C22⋊C4—71no—
349C2×C22⋊C4—141no—
350C2×C42—71no—
351C2×C42—71no—
352C2×C42—141no—
353C2×C42—141no—
354C2×C42—141no—
355#107—142no—
356#107—142no—
357#107—281no—
358#106—22no—
359C4⋊C8—142no—
360#103—22no—
361#103—41no—
362#103—142no—
363#103—142no—
364C22⋊C8—41no—
365C22⋊C8—142no—
366C4×C8—281no—
367#95—21no—
368#95—71no—
369#95—71no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380C22×C7—22no—
381C22×C7—22no—
382C22×C7—22no—
383C22×C7—41no—
384C22×C7—41no—
385C22×C7—41no—
386C22×C7—41no—
387C22×C7#52511yescharacteristic
388C22×C7#54611yescharacteristic
389C22×C7#54711yescharacteristic
390C22×C7C2×#13511yescharacteristic
391C22×C7C2×#13711yescharacteristic
392C22×C7#57611yescharacteristic
393C22×C7#57811yescharacteristic
394C4×C7—42no—
395C4×C7—41no—
396C4×C7—42no—
397C4×C7—81no—
398C4×C7—81no—
399C4×C7#52911yescharacteristic
400C4×C7#53911yescharacteristic
401C4×C7#55311yescharacteristic
402C4×C7#55611yescharacteristic
403Dic7—22no—
404Dic7—41no—
405Dic7—41no—
406Dic7—41no—
407Dic7—42no—
408Dic7—81no—
409Dic7—81no—
410Dic7—81no—
411C24D4×D711yescharacteristic
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8#115912yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—71no—
429C22×C4—71no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—282no—
437C22×C4—282no—
438C22×C4—561no—
439C22×C4D4×D711yescharacteristic
440C22×C4C23×D711yescharacteristic
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—72no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C4⋊C4—24no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—72no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—282no—
468C4⋊C4—282no—
469C4⋊C4—561no—
470C4⋊C4—561no—
471C4⋊C4—561no—
472C4⋊C4#115912yes—
473C22⋊C4—24no—
474C22⋊C4—42no—
475C22⋊C4—142no—
476C22⋊C4—142no—
477C22⋊C4—561no—
478C42—141no—
479C42—141no—
480C42—141no—
481C42—141no—
482C42—282no—
483C42—282no—
484C42—561no—
485C2×C7—42no—
486C2×C7#246611yescharacteristic
487C2×C7#310811yescharacteristic
488C2×C7#312411yescharacteristic
489C2×C7#329711yescharacteristic
490C2×C7#331211yescharacteristic
491C2×C7#333611yescharacteristic
492C2×C7#335711yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23#647912yes—
500C23C2×D4×D711yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—72no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—141no—
528C2×C4—141no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—142no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—564no—
549C2×C4C2×#115911yescharacteristic
550C2×C4C2×#115911yescharacteristic
551C2×C4#654811yescharacteristic
552C2×C4D7×#4111yescharacteristic
553C2×C4D7×#4111yescharacteristic
554C2×C4#655411yescharacteristic
555C8—41no—
556C8—41no—
557C8—142no—
558C8—281no—
559C7#2703511yescharacteristic, complemented, Sylow
560C22—22no—
561C22—22no—
562C22—22no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22#9027711yescharacteristic
568C22#9030111yescharacteristic
569C22#9034611yescharacteristic
570C22C2×#647911yescharacteristic
571C22#9046011yescharacteristic
572C22D7×#13711yescharacteristic
573C22#9047011yescharacteristic
574C4—42no—
575C4—41no—
576C4—42no—
577C4—81no—
578C4—81no—
579C4—142no—
580C4—281no—
581C4—281no—
582C4—282no—
583C4—281no—
584C4—561no—
585C4—561no—
586C4—561no—
587C4#9029311yescharacteristic
588C4#9030511yescharacteristic
589C4#9034011yescharacteristic
590C4#9036211yescharacteristic
591C2—42no—
592C2#1173436311yescharacteristic
593C2#1173747411yescharacteristic
594C2#1173766311yescharacteristic
595C2#1173812611yescharacteristic
596C2#1173836011yescharacteristic
597C2#1173888111yescharacteristic
598C2#1173895811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2035C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2035D7
hypocenterC7#27035
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2035

Complements

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

NH ≅ G/Ntype
5. #11728318C2semidirect
6. #11728312C2semidirect
7. #11728016C2semidirect
8. #11727989C2semidirect
9. #11727988C2semidirect
10. Dic7×#103C2semidirect
14. #11726613C2semidirect
15. #11726569C2semidirect
559. C7#27035semidirect

Maximal subgroups

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

subgroupindex
#117401922
#117313662
C7×#20352
#117294492
#117283182
#117283122
#117280162
#117279892
#117279882
Dic7×#1032
#117276782
#117276662
#117272582
#117266132
#117265692
#270357