Finite Group Databasebeta

← #13426540

Subgroups — #13426540

Order 1792.

708 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739607C211yesmaximal, characteristic
2#11731370C211yesmaximal, characteristic
3C7×#1852C211yesmaximal, characteristic
4#11729308C211yesmaximal, characteristic
5#11728397C211yesmaximal, characteristic, complemented
6#11728396C211yesmaximal, characteristic, complemented
7#11728365C211yesmaximal, characteristic, complemented
8#11728364C211yesmaximal, characteristic, complemented
9#11727691C211yesmaximal, characteristic
10#11727667C211yesmaximal, characteristic
11#11727281C211yesmaximal, characteristic
12#11726668C211yesmaximal, characteristic, complemented
13#11726549C211yesmaximal, characteristic, complemented
14#11726544C211yesmaximal, characteristic, complemented
15#11726535C211yesmaximal, characteristic, complemented
16C22×C7×C4⋊C4C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C2×#6450C2211yescharacteristic
19C2×C7×#107C2211yescharacteristic
20C2×C7×#107C2211yescharacteristic
21C2×C7×C22⋊C8C2211yescharacteristic
22C7×#380C2211yescharacteristic
23#89984C2211yescharacteristic
24#89982—21no—
25C4×C22⋊Dic7—21no—
26C4×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊(C7⋊C8)C2211yescharacteristic
28C2×#6392C2211yescharacteristic
29C2×#6390C2211yescharacteristic
30C2×#6390C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89757C2211yescharacteristic
34#89750—21no—
35#89748C2211yescharacteristic
36C4⋊C4×Dic7C2211yescharacteristic
37C4⋊C4×Dic7C2211yescharacteristic
38#89746C2211yescharacteristic
39#89745C2211yescharacteristic
40#89745C2211yescharacteristic
41#89744—21no—
42C2×#6384C2211yescharacteristic
43C2×#6384C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45#89720—21no—
46#89717—21no—
47#89430—21no—
48#89425—21no—
49C7×#340C2211yescharacteristic
50C7×#340C2211yescharacteristic
51#89346C2211yescharacteristic
52#89345C2211yescharacteristic
53#89345C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#27036—71nomaximal, Sylow
60C23×C4×C7—21no—
61C23×C4×C7C2311yescharacteristic
62C22×Dic14C2311yescharacteristic
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—41no—
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C22⋊Dic7—21no—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7D411yescharacteristic
76C2×C22⋊Dic7D411yescharacteristic
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81#6487—22no—
82#6487—22no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
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
96#6450—22no—
97#6450—41no—
98C7×#107—22no—
99C7×#107—41no—
100C7×C22⋊C8—22no—
101C7×#95—21no—
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
112C22⋊(C7⋊C8)—41no—
113#6392—22no—
114#6390—22no—
115#6390—22no—
116Dic7⋊C8—22no—
117C8×Dic7—41no—
118#6384—22no—
119#6384—41no—
120#6381—22no—
121#2497—71no—
122#2360—71no—
123#2072—71no—
124#2068—71no—
125#2036—71no—
126#2036—71no—
127#1947—71no—
128#1942—71no—
129#1923—71no—
130#1921—71no—
131#1921—71no—
132#1863—71no—
133#1852D711yescharacteristic
134#1824—71no—
135C4×#103—71no—
136C24×C7C2×D411yescharacteristic
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—42no—
148C22×C4×C7—42no—
149C22×C4×C7—81no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7C2×D411yescharacteristic
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14—21no—
163C2×Dic14—21no—
164C2×Dic14—21no—
165C2×Dic14#4112yes—
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×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4#4112yes—
181C7×C22⋊C4—81no—
182C22⋊Dic7—24no—
183C22⋊Dic7—41no—
184C22⋊Dic7—81no—
185C22⋊Dic7Q1614yes—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—42no—
192C4⋊Dic7#4112yes—
193Dic7⋊C4—24no—
194Dic7⋊C4—42no—
195Dic7⋊C4—81no—
196Dic7⋊C4—81no—
197Dic7⋊C4—81no—
198Dic7⋊C4—81no—
199C4×Dic7—21no—
200C4×Dic7—21no—
201C4×Dic7—21no—
202C4×Dic7—21no—
203C4×Dic7—42no—
204C4×Dic7—81no—
205C4×Dic7—81no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C22⋊Q8—71no—
213C2×C4×Q8—71no—
214C22×C4⋊C4C2×D711yescharacteristic
215C2×#107—71no—
216C2×#107—71no—
217C2×#107C2×D711yescharacteristic
218C2×#107C2×D711yescharacteristic
219C2×C4⋊C8—71no—
220C2×#103—71no—
221C2×#103—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×C22⋊C8—71no—
225C2×C22⋊C8C2×D711yescharacteristic
226C2×C4×C8—71no—
227#400—141no—
228#396—141no—
229#391—71no—
230#389—71no—
231#389—71no—
232#389—71no—
233#389—141no—
234#388—141no—
235#387—141no—
236#387—141no—
237#385—141no—
238#384—71no—
239#380—71no—
240#380C2×D711yescharacteristic
241C4×C4⋊C4—71no—
242C4×C4⋊C4—71no—
243C4×C22⋊C4—71no—
244C4×C22⋊C4—141no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340C2×D711yescharacteristic
252#340C2×D711yescharacteristic
253#336—71no—
254#336—71no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7C2×Q1612yes—
262C23×C7C22×D411yescharacteristic
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
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—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—84no—
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C7C2×#4111yescharacteristic
285C2×C4×C7C2×#4111yescharacteristic
286C2×C4×C7C2×#4111yescharacteristic
287C2×C4×C7H(5,𝔽2)11yescharacteristic
288C2×C4×C72−1+411yescharacteristic
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—22no—
294C2×Dic7—22no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
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—
310C2×Dic7—41no—
311C2×Dic7C2×Q1612yes—
312C2×Dic7C2×Q1612yes—
313Dic14—22no—
314Dic14—22no—
315Dic14—41no—
316Dic14—41no—
317C7×C8—22no—
318C7×C8—41no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C22×Q8—71no—
322C23×C4—21no—
323C23×C4C22×D711yescharacteristic
324C22×C8—71no—
325C22×C8—71no—
326C22×C8C22×D711yescharacteristic
327C22×C8C22×D711yescharacteristic
328C22⋊Q8—142no—
329C22⋊Q8—142no—
330C4×Q8—142no—
331C4×Q8—281no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—41no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—141no—
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⋊C4C22×D711yescharacteristic
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—71no—
351C2×C22⋊C4—71no—
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C42—71no—
355C2×C42—71no—
356C2×C42—141no—
357C2×C42—141no—
358C2×C42—141no—
359#107—22no—
360#107—41no—
361#107—142no—
362#107—142no—
363C4⋊C8—142no—
364#103—142no—
365#103—142no—
366#103—142no—
367#103—281no—
368C22⋊C8—22no—
369C22⋊C8—281no—
370C4×C8—281no—
371#95—21no—
372#95—71no—
373#95—71no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382C22×C7—22no—
383C22×C7—22no—
384C22×C7—22no—
385C22×C7—41no—
386C22×C7—41no—
387C22×C7—41no—
388C22×C7—41no—
389C22×C7#52511yescharacteristic
390C22×C7#54611yescharacteristic
391C22×C7#54711yescharacteristic
392C22×C7C22×Q1611yescharacteristic
393C22×C7#57511yescharacteristic
394C22×C7#57611yescharacteristic
395C22×C7#57711yescharacteristic
396C4×C7—41no—
397C4×C7—42no—
398C4×C7—81no—
399C4×C7—81no—
400C4×C7—81no—
401C4×C7#52911yescharacteristic
402C4×C7#53911yescharacteristic
403C4×C7#55311yescharacteristic
404C4×C7#55611yescharacteristic
405Dic7—22no—
406Dic7—22no—
407Dic7—41no—
408Dic7—42no—
409Dic7—42no—
410Dic7—41no—
411Dic7—81no—
412Dic7—81no—
413C24D4×D711yescharacteristic
414C2×Q8—72no—
415C2×Q8—141no—
416C2×Q8—141no—
417C2×Q8—141no—
418C2×Q8—141no—
419C2×Q8—141no—
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—21no—
429C22×C4—21no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—81no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—141no—
441C22×C4D4×D711yescharacteristic
442C22×C4C23×D711yescharacteristic
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8#115712yes—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—42no—
463C4⋊C4—72no—
464C4⋊C4—144no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—282no—
471C4⋊C4—282no—
472C4⋊C4—561no—
473C4⋊C4—561no—
474C4⋊C4—561no—
475C4⋊C4—561no—
476C4⋊C4#115912yes—
477C22⋊C4—74no—
478C22⋊C4—81no—
479C22⋊C4—144no—
480C22⋊C4—281no—
481C22⋊C4—561no—
482C42—141no—
483C42—141no—
484C42—141no—
485C42—141no—
486C42—282no—
487C42—561no—
488C42—561no—
489C2×C7—42no—
490C2×C7#246611yescharacteristic
491C2×C7#310411yescharacteristic
492C2×C7#311911yescharacteristic
493C2×C7#329911yescharacteristic
494C2×C7#331911yescharacteristic
495C2×C7#333811yescharacteristic
496C2×C7#335111yescharacteristic
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23D7×Q1612yes—
504C23C2×D4×D711yescharacteristic
505Q8—142no—
506Q8—142no—
507Q8—281no—
508Q8—281no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—21no—
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—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—72no—
529C2×C4—72no—
530C2×C4—84no—
531C2×C4—141no—
532C2×C4—141no—
533C2×C4—141no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—142no—
542C2×C4—142no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4—281no—
552C2×C4—281no—
553C2×C4C2×#115711yescharacteristic
554C2×C4C2×#115911yescharacteristic
555C2×C4#654811yescharacteristic
556C2×C4#655111yescharacteristic
557C2×C4D7×#4111yescharacteristic
558C2×C4D7×#4111yescharacteristic
559C8—22no—
560C8—41no—
561C8—281no—
562C8—281no—
563C7#2703611yescharacteristic, complemented, Sylow
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#9019811yescharacteristic
572C22#9028211yescharacteristic
573C22#9034711yescharacteristic
574C22#9044811yescharacteristic
575C22C2×D7×Q1611yescharacteristic
576C22#9046411yescharacteristic
577C22#9046911yescharacteristic
578C4—42no—
579C4—41no—
580C4—81no—
581C4—81no—
582C4—81no—
583C4—142no—
584C4—142no—
585C4—282no—
586C4—281no—
587C4—281no—
588C4—282no—
589C4—561no—
590C4—561no—
591C4#9023111yescharacteristic
592C4#9024811yescharacteristic
593C4#9032911yescharacteristic
594C4#9033211yescharacteristic
595C2—42no—
596C2#1173188011yescharacteristic
597C2#1173660111yescharacteristic
598C2#1173667611yescharacteristic
599C2#1173746411yescharacteristic
600C2#1173763811yescharacteristic
601C2#1173871711yescharacteristic
602C2#1173889711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1852C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#1852D7
hypocenterC7#27036
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
5. #11728397C2semidirect
6. #11728396C2semidirect
7. #11728365C2semidirect
8. #11728364C2semidirect
12. #11726668C2semidirect
13. #11726549C2semidirect
14. #11726544C2semidirect
15. #11726535C2semidirect
563. C7#27036semidirect

Maximal subgroups

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

subgroupindex
#117396072
#117313702
C7×#18522
#117293082
#117283972
#117283962
#117283652
#117283642
#117276912
#117276672
#117272812
#117266682
#117265492
#117265442
#117265352
#270367