Finite Group Databasebeta

← #13426539

Subgroups — #13426539

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
1C7×#2497C211yesmaximal, characteristic
2#11730910C211yesmaximal, characteristic
3#11727389C211yesmaximal, characteristic, complemented
4#11727385C211yesmaximal, characteristic, complemented
5#11727373C211yesmaximal, characteristic, complemented
6#11727357C211yesmaximal, characteristic, complemented
7#11727343C211yesmaximal, characteristic, complemented
8#11727321C211yesmaximal, characteristic, complemented
9#11727281C211yesmaximal, characteristic
10#11727281C211yesmaximal, characteristic
11#11727258C211yesmaximal, characteristic
12#11727256C211yesmaximal, characteristic
13C4×#6384C211yesmaximal, characteristic, complemented
14#11727215C211yesmaximal, characteristic, complemented
15#11727205C211yesmaximal, characteristic
16C22×C4⋊Dic7C2211yescharacteristic
17C2×#6487C2211yescharacteristic
18C2×C4×Dic14C2211yescharacteristic
19C7×#400—21no—
20C7×#396—21no—
21C7×#389C2211yescharacteristic
22C7×#389C2211yescharacteristic
23C7×#388—21no—
24C7×#387—21no—
25C7×#384C2211yescharacteristic
26C7×#380C2211yescharacteristic
27C7×#380C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C22⋊C4C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×#6384C2211yescharacteristic
33C2×#6384C2211yescharacteristic
34C2×#6384C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89718—21no—
41C22⋊C4×Dic7—21no—
42C4×C4⋊Dic7C2211yescharacteristic
43#89701C2211yescharacteristic
44#89700C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89427—21no—
48#89419—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#27036—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C22×Dic14C2311yescharacteristic
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4D411yescharacteristic
74C2×C7×C22⋊C4D411yescharacteristic
75C2×C42×C7—21no—
76C2×C42×C7C2311yescharacteristic
77C2×C42×C7C2311yescharacteristic
78C2×C22⋊Dic7—21no—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7—41no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85#6487—22no—
86#6487—22no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C4×Dic14—22no—
97C4×Dic14—41no—
98C7×#95—21no—
99C7×#95—21no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95D411yescharacteristic
104C7×#95D411yescharacteristic
105C7⋊#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C22⋊(C7⋊C8)—22no—
110C22⋊(C7⋊C8)—41no—
111#6384—22no—
112#6384—22no—
113#6384—22no—
114#6384—41no—
115#6381—22no—
116#6381—22no—
117#6381—22no—
118#6381—41no—
119C4⋊(C7⋊C8)—22no—
120C4×C7⋊C8—41no—
121#2497D711yescharacteristic
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#1852—71no—
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×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—42no—
158C22×Dic7—42no—
159C22×Dic7—81no—
160C2×Dic14—21no—
161C2×Dic14—21no—
162C2×Dic14—21no—
163C2×Dic14—21no—
164C2×Dic14—21no—
165C2×Dic14#4112yes—
166C7×C4⋊C4—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—21no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—42no—
175C7×C4⋊C4—81no—
176C7×C4⋊C4—81no—
177C7×C4⋊C4—81no—
178C7×C22⋊C4—41no—
179C7×C22⋊C4—81no—
180C7×C22⋊C4—81no—
181C7×C22⋊C4Q1614yes—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—81no—
187C22⋊Dic7—24no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—42no—
193C4⋊Dic7#4112yes—
194C4⋊Dic7#4112yes—
195Dic7⋊C4—24no—
196Dic7⋊C4—42no—
197Dic7⋊C4—81no—
198C4×Dic7—42no—
199C4×Dic7—81no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8#4112yes—
212C2×C22⋊Q8—71no—
213C2×C4×Q8—71no—
214C22×C4⋊C4—71no—
215C2×#107—71no—
216C2×#107—71no—
217C2×#107—71no—
218C2×#107—71no—
219C2×C4⋊C8—71no—
220C2×#103—71no—
221C2×#103—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×C22⋊C8—71no—
225C2×C22⋊C8—71no—
226C2×C4×C8—71no—
227#400—21no—
228#396—21no—
229#391—71no—
230#389—71no—
231#389—141no—
232#389C2×D711yescharacteristic
233#389C2×D711yescharacteristic
234#388—21no—
235#387—21no—
236#387—141no—
237#385—141no—
238#384C2×D711yescharacteristic
239#380C2×D711yescharacteristic
240#380C2×D711yescharacteristic
241C4×C4⋊C4—71no—
242C4×C4⋊C4C2×D711yescharacteristic
243C4×C22⋊C4—141no—
244C4×C22⋊C4C2×D711yescharacteristic
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
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—21no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
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×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7C2×Q1612yes—
293C2×C4×C7C2×Q1612yes—
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7H(5,𝔽2)11yescharacteristic
299C2×C4×C72−1+411yescharacteristic
300C2×Dic7—22no—
301C2×Dic7—22no—
302C2×Dic7—22no—
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—84no—
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—141no—
323C23×C4C22×D711yescharacteristic
324C22×C8—71no—
325C22×C8—71no—
326C22×C8—71no—
327C22×C8—71no—
328C22⋊Q8—142no—
329C22⋊Q8—142no—
330C4×Q8—142no—
331C4×Q8—281no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—281no—
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C4⋊C4C22×D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—141no—
352C2×C22⋊C4C22⋊D711yescharacteristic
353C2×C22⋊C4C22⋊D711yescharacteristic
354C2×C42—21no—
355C2×C42—141no—
356C2×C42—141no—
357C2×C42C22×D711yescharacteristic
358C2×C42C22×D711yescharacteristic
359#107—142no—
360#107—142no—
361#107—142no—
362#107—281no—
363C4⋊C8—142no—
364#103—142no—
365#103—142no—
366#103—142no—
367#103—281no—
368C22⋊C8—142no—
369C22⋊C8—281no—
370C4×C8—281no—
371#95—21no—
372#95—21no—
373#95—21no—
374#95—21no—
375#95—21no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95C22⋊D711yescharacteristic
381#95C22⋊D711yescharacteristic
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—22no—
397C4×C7—22no—
398C4×C7—41no—
399C4×C7—41no—
400C4×C7—41no—
401C4×C7—81no—
402C4×C7—81no—
403C4×C7—81no—
404C4×C7—81no—
405C4×C7#52911yescharacteristic
406C4×C7#53911yescharacteristic
407C4×C7#55311yescharacteristic
408C4×C7#55611yescharacteristic
409Dic7—42no—
410Dic7—42no—
411Dic7—42no—
412Dic7—81no—
413C24C2×C22⋊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—141no—
431C22×C4—141no—
432C22×C4—141no—
433C22×C4—141no—
434C22×C4—282no—
435C22×C4—282no—
436C22×C4—561no—
437C22×C4C2×C22⋊D711yescharacteristic
438C22×C4C2×C22⋊D711yescharacteristic
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C2×C22⋊D711yescharacteristic
442C22×C4C23×D711yescharacteristic
443C2×C8—72no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C2×C8—141no—
448C2×C8—141no—
449C2×C8—141no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—42no—
464C4⋊C4—72no—
465C4⋊C4—72no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—144no—
470C4⋊C4—141no—
471C4⋊C4—141no—
472C4⋊C4—141no—
473C4⋊C4—141no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—561no—
477C22⋊C4—41no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C22⋊C4—144no—
481C22⋊C4#114414yes—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—81no—
487C42—282no—
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—
503C23C2×#114412yes—
504C23C22×C22⋊D711yescharacteristic
505Q8—142no—
506Q8—142no—
507Q8—281no—
508Q8—281no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—281no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—564no—
551C2×C4C2×#114412yes—
552C2×C4C2×#114412yes—
553C2×C4D7×#4111yescharacteristic
554C2×C4D7×#4111yescharacteristic
555C2×C4D7×#4111yescharacteristic
556C2×C4D7×#4111yescharacteristic
557C2×C4#655311yescharacteristic
558C2×C4#655411yescharacteristic
559C8—142no—
560C8—281no—
561C8—281no—
562C8—281no—
563C7#2703611yescharacteristic, complemented, Sylow
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22C22×#114411yescharacteristic
572C22#9051311yescharacteristic
573C22#9051511yescharacteristic
574C22#9051911yescharacteristic
575C22#9052011yescharacteristic
576C22#9052611yescharacteristic
577C22#9052711yescharacteristic
578C4—22no—
579C4—22no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—81no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—282no—
588C4—282no—
589C4—282no—
590C4—561no—
591C4#9039311yescharacteristic
592C4#9039711yescharacteristic
593C4#9039811yescharacteristic
594C4#9040011yescharacteristic
595C2—42no—
596C2#1174055011yescharacteristic
597C2#1174081311yescharacteristic
598C2#1174087911yescharacteristic
599C2#1174105711yescharacteristic
600C2#1174106811yescharacteristic
601C2#1174108511yescharacteristic
602C2#1174109811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2497C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2497D7
hypocenterC7#27036
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2497

Complements

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

NH ≅ G/Ntype
3. #11727389C2semidirect
4. #11727385C2semidirect
5. #11727373C2semidirect
6. #11727357C2semidirect
7. #11727343C2semidirect
8. #11727321C2semidirect
13. C4×#6384C2semidirect
14. #11727215C2semidirect
563. C7#27036semidirect

Maximal subgroups

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

subgroupindex
C7×#24972
#117309102
#117273892
#117273852
#117273732
#117273572
#117273432
#117273212
#117272812
#117272582
#117272562
C4×#63842
#117272152
#117272052
#270367