Finite Group Databasebeta

← #13426508

Subgroups — #13426508

Order 1792.

728 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740192C211yesmaximal, characteristic
2#11731367C211yesmaximal, characteristic
3C7×#2035C211yesmaximal, characteristic
4#11729448C211yesmaximal, characteristic
5#11728317C211yesmaximal, characteristic, complemented
6#11728314C211yesmaximal, characteristic, complemented
7#11728001C211yesmaximal, characteristic, complemented
8#11727992C211yesmaximal, characteristic, complemented
9#11727987C211yesmaximal, characteristic, complemented
10Dic7×#103C211yesmaximal, characteristic, complemented
11#11727682C211yesmaximal, characteristic
12#11727667C211yesmaximal, characteristic
13#11727256C211yesmaximal, characteristic
14#11726573C211yesmaximal, characteristic, complemented
15#11726570C211yesmaximal, characteristic, complemented
16C2×C7×C22⋊Q8C2211yescharacteristic
17C2×Q8×Dic7C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C2×#6442—21no—
20C2×C7×#106C2211yescharacteristic
21C2×C7×#103C2211yescharacteristic
22C2×C7×#103C2211yescharacteristic
23C2×C7×C22⋊C8C2211yescharacteristic
24C7×#380C2211yescharacteristic
25#90002C2211yescharacteristic
26C2×#6409C2211yescharacteristic
27C2×#6409C2211yescharacteristic
28#89984C2211yescharacteristic
29C4×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊(C7⋊C8)C2211yescharacteristic
31C2×#6391C2211yescharacteristic
32C2×#6391C2211yescharacteristic
33C2×Dic7⋊C8C2211yescharacteristic
34C2×C8×Dic7C2211yescharacteristic
35#89757—21no—
36#89757C2211yescharacteristic
37#89757C2211yescharacteristic
38#89753—21no—
39#89748C2211yescharacteristic
40C4⋊C4×Dic7C2211yescharacteristic
41C4⋊C4×Dic7C2211yescharacteristic
42#89745C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44#89724—21no—
45#89722—21no—
46C22⋊C4×Dic7—21no—
47#89430—21no—
48#89427—21no—
49C7×#340C2211yescharacteristic
50#89346C2211yescharacteristic
51#89346C2211yescharacteristic
52#89345C2211yescharacteristic
53#89345C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#27034—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×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7C2311yescharacteristic
95C2×C4×Dic7C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98#6442—44no—
99C7×#106—22no—
100C7×#103—22no—
101C7×#103—22no—
102C7×C22⋊C8—22no—
103C7×#95—21no—
104#6409—22no—
105#6409—41no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95D411yescharacteristic
115C7⋊#95D411yescharacteristic
116C22⋊(C7⋊C8)—41no—
117#6391—22no—
118#6391—41no—
119Dic7⋊C8—22no—
120C8×Dic7—41no—
121#6382—22no—
122#2496—71no—
123#2360—71no—
124#2072—71no—
125#2066—71no—
126#2035—71no—
127#2035D711yescharacteristic
128#1948—71no—
129#1941—71no—
130#1923—71no—
131#1920—71no—
132#1919—71no—
133#1865—71no—
134#1853—71no—
135#1824—71no—
136C4×#103—71no—
137C24×C7C2×D411yescharacteristic
138C2×C7×Q8—21no—
139C2×C7×Q8—21no—
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8#4112yes—
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×C7—21no—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—21no—
160C22×Dic7—42no—
161C22×Dic7—42no—
162C22×Dic7—81no—
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C22×Dic7C2×D411yescharacteristic
166C22×Dic7C2×D411yescharacteristic
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8#4112yes—
173C7×C4⋊C4—24no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—21no—
177C7×C4⋊C4—21no—
178C7×C4⋊C4—21no—
179C7×C4⋊C4#4112yes—
180C7×C22⋊C4—24no—
181C7×C22⋊C4—42no—
182C22⋊Dic7—41no—
183C22⋊Dic7—42no—
184C22⋊Dic7SD1614yes—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—42no—
193C4⋊Dic7—42no—
194C4⋊Dic7—42no—
195C4⋊Dic7—81no—
196C4⋊Dic7#4112yes—
197Dic7⋊C4—42no—
198Dic7⋊C4—42no—
199Dic7⋊C4—42no—
200C4×Dic7—21no—
201C4×Dic7—21no—
202C4×Dic7—21no—
203C4×Dic7—21no—
204C4×Dic7—42no—
205C4×Dic7—42no—
206C4×Dic7—81no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×#126—141no—
214C2×C22⋊Q8C2×D711yescharacteristic
215C2×C4×Q8—71no—
216C22×C4⋊C4—71no—
217C2×#106—71no—
218C2×#106—71no—
219C2×#106—71no—
220C2×#106C2×D711yescharacteristic
221C2×C4⋊C8—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×#103C2×D711yescharacteristic
225C2×#103C2×D711yescharacteristic
226C2×C22⋊C8—71no—
227C2×C22⋊C8C2×D711yescharacteristic
228C2×C4×C8—71no—
229#400—141no—
230#397—141no—
231#391—71no—
232#389—71no—
233#389—71no—
234#389—71no—
235#389—141no—
236#387—141no—
237#387—141no—
238#385—141no—
239#384—71no—
240#380—71no—
241#380C2×D711yescharacteristic
242C4×C4⋊C4—71no—
243C4×C4⋊C4—71no—
244C4×C22⋊C4—71no—
245C4×C22⋊C4—141no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340C2×D711yescharacteristic
254#336—71no—
255#336—71no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7—21no—
261C23×C7—21no—
262C23×C7C2×SD1612yes—
263C23×C7C22×D411yescharacteristic
264C7×Q8—22no—
265C7×Q8—22no—
266C7×Q8—41no—
267C7×Q8—41no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
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×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7C2×#4111yescharacteristic
288C2×C4×C7C2×#4111yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7H(5,𝔽2)11yescharacteristic
292C2×C4×C72−1+411yescharacteristic
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—22no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
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—41no—
314C2×Dic7—41no—
315C2×Dic7—84no—
316C2×Dic7C2×SD1612yes—
317C2×Dic7C2×SD1612yes—
318C7×C8—22no—
319C7×C8—41no—
320C7⋊C8—41no—
321C7⋊C8—41no—
322C22×Q8C22×D711yescharacteristic
323C23×C4—141no—
324C23×C4C22×D711yescharacteristic
325C22×C8—71no—
326C22×C8—71no—
327C22×C8C22×D711yescharacteristic
328C22×C8C22×D711yescharacteristic
329#126—284no—
330C22⋊Q8—22no—
331C22⋊Q8—22no—
332C4×Q8—142no—
333C4×Q8—281no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—71no—
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—141no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—281no—
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—71no—
354C2×C22⋊C4—71no—
355C2×C22⋊C4—141no—
356C2×C42—71no—
357C2×C42—71no—
358C2×C42—141no—
359C2×C42—141no—
360C2×C42—141no—
361#106—22no—
362#106—142no—
363#106—142no—
364#106—281no—
365C4⋊C8—142no—
366#103—22no—
367#103—22no—
368#103—142no—
369#103—281no—
370C22⋊C8—22no—
371C22⋊C8—281no—
372C4×C8—281no—
373#95—21no—
374#95—71no—
375#95—71no—
376#95—141no—
377#95—141no—
378#95—141no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384C22×C7—22no—
385C22×C7—22no—
386C22×C7—22no—
387C22×C7—41no—
388C22×C7—41no—
389C22×C7—41no—
390C22×C7—41no—
391C22×C7#52511yescharacteristic
392C22×C7#54611yescharacteristic
393C22×C7#54711yescharacteristic
394C22×C7C22×SD1611yescharacteristic
395C22×C7#57511yescharacteristic
396C22×C7#57711yescharacteristic
397C22×C7#57811yescharacteristic
398C4×C7—42no—
399C4×C7—41no—
400C4×C7—42no—
401C4×C7—81no—
402C4×C7—81no—
403C4×C7#52911yescharacteristic
404C4×C7#53911yescharacteristic
405C4×C7#55311yescharacteristic
406C4×C7#55611yescharacteristic
407Dic7—22no—
408Dic7—22no—
409Dic7—41no—
410Dic7—41no—
411Dic7—42no—
412Dic7—81no—
413Dic7—81no—
414Dic7—81no—
415C24D4×D711yescharacteristic
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8#115912yes—
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—21no—
431C22×C4—21no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4—282no—
441C22×C4—282no—
442C22×C4—561no—
443C22×C4D4×D711yescharacteristic
444C22×C4C23×D711yescharacteristic
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8#115912yes—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—72no—
464C4⋊C4—141no—
465C4⋊C4—141no—
466C4⋊C4—141no—
467C4⋊C4—141no—
468C4⋊C4—141no—
469C4⋊C4—141no—
470C4⋊C4—141no—
471C4⋊C4—282no—
472C4⋊C4—282no—
473C4⋊C4—282no—
474C4⋊C4—282no—
475C4⋊C4—282no—
476C4⋊C4—282no—
477C4⋊C4—561no—
478C4⋊C4#115912yes—
479C22⋊C4—24no—
480C22⋊C4—42no—
481C22⋊C4—74no—
482C22⋊C4—281no—
483C22⋊C4—282no—
484C42—141no—
485C42—141no—
486C42—141no—
487C42—141no—
488C42—282no—
489C42—282no—
490C42—561no—
491C2×C7—42no—
492C2×C7#246611yescharacteristic
493C2×C7#310311yescharacteristic
494C2×C7#312011yescharacteristic
495C2×C7#329811yescharacteristic
496C2×C7#331111yescharacteristic
497C2×C7#333511yescharacteristic
498C2×C7#335211yescharacteristic
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23D7×SD1612yes—
506C23C2×D4×D711yescharacteristic
507Q8—22no—
508Q8—22no—
509Q8—41no—
510Q8—41no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
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—41no—
529C2×C4—41no—
530C2×C4—72no—
531C2×C4—72no—
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—141no—
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×C4—281no—
554C2×C4—564no—
555C2×C4C2×#115911yescharacteristic
556C2×C4C2×#115911yescharacteristic
557C2×C4C2×#115911yescharacteristic
558C2×C4D7×#4111yescharacteristic
559C2×C4#655311yescharacteristic
560C2×C4#655411yescharacteristic
561C8—22no—
562C8—41no—
563C8—281no—
564C8—281no—
565C7#2703411yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22#9027711yescharacteristic
574C22#9030211yescharacteristic
575C22#9034511yescharacteristic
576C22C2×D7×SD1611yescharacteristic
577C22#9046111yescharacteristic
578C22#9046811yescharacteristic
579C22#9047011yescharacteristic
580C4—42no—
581C4—41no—
582C4—42no—
583C4—81no—
584C4—81no—
585C4—142no—
586C4—142no—
587C4—281no—
588C4—282no—
589C4—281no—
590C4—561no—
591C4—561no—
592C4—561no—
593C4#9029111yescharacteristic
594C4#9030911yescharacteristic
595C4#9035611yescharacteristic
596C4#9036211yescharacteristic
597C2—42no—
598C2#1173436411yescharacteristic
599C2#1173747311yescharacteristic
600C2#1173766111yescharacteristic
601C2#1173809311yescharacteristic
602C2#1173834511yescharacteristic
603C2#1173889511yescharacteristic
604C2#1173897111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2035C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2035D7
hypocenterC7#27034
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. #11728317C2semidirect
6. #11728314C2semidirect
7. #11728001C2semidirect
8. #11727992C2semidirect
9. #11727987C2semidirect
10. Dic7×#103C2semidirect
14. #11726573C2semidirect
15. #11726570C2semidirect
565. C7#27034semidirect

Maximal subgroups

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

subgroupindex
#117401922
#117313672
C7×#20352
#117294482
#117283172
#117283142
#117280012
#117279922
#117279872
Dic7×#1032
#117276822
#117276672
#117272562
#117265732
#117265702
#270347