Finite Group Databasebeta

← #13422438

Subgroups — #13422438

Order 1792.

732 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2506C211yesmaximal, characteristic, complemented
2#11731100C211yesmaximal, characteristic
3#11727385C211yesmaximal, characteristic, complemented
4#11727383C211yesmaximal, characteristic
5#11727382C211yesmaximal, characteristic
6#11727382C211yesmaximal, characteristic
7#11727379C211yesmaximal, characteristic
8#11727368C211yesmaximal, characteristic, complemented
9#11727357C211yesmaximal, characteristic, complemented
10#11727356C211yesmaximal, characteristic, complemented
11#11727342C211yesmaximal, characteristic, complemented
12#11727339C211yesmaximal, characteristic
13#11727236C211yesmaximal, characteristic, complemented
14C4×#6383C211yesmaximal, characteristic
15#11727218C211yesmaximal, characteristic, complemented
16C2×C7×#125—21no—
17C2×#6438C2211yescharacteristic
18C2×C4×D28C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C7×#400—21no—
21C7×#398—21no—
22C7×#389C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#387—21no—
27C7×#384C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29C4×C7×C4⋊C4C2211yescharacteristic
30#89762—21no—
31#89761—21no—
32C2×#6384C2211yescharacteristic
33C2×#6384C2211yescharacteristic
34C2×#6384C2211yescharacteristic
35C2×#6384C2211yescharacteristic
36C2×#6383C2211yescharacteristic
37C2×#6383C2211yescharacteristic
38C2×#6383C2211yescharacteristic
39C2×#6383C2211yescharacteristic
40C2×#6382C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42#89711C2211yescharacteristic
43C4×C4⋊Dic7C2211yescharacteristic
44#89701C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89441—21no—
48#89438—21no—
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89319C2211yescharacteristic
56#89319C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26761—71nomaximal, Sylow
60C22×D28C2311yescharacteristic
61C22×C4×D7—21no—
62C22×Dic14C2311yescharacteristic
63C7×#125—44no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4D411yescharacteristic
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×(C2×D7)⋊C4—21no—
81C2×(C2×D7)⋊C4—21no—
82C2×(C2×D7)⋊C4—21no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—21no—
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×Dic7⋊C4—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C22×C7⋊C8C2311yescharacteristic
93C22×C7⋊C8C2311yescharacteristic
94#6438—22no—
95#6438—22no—
96C4×D28—22no—
97C4×D28—41no—
98C4×Dic14—22no—
99C4×Dic14—41no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95D411yescharacteristic
106C7×#95D411yescharacteristic
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110#6384—22no—
111#6384—22no—
112#6384—22no—
113#6384—41no—
114#6383—22no—
115#6383—22no—
116#6383—22no—
117#6383—41no—
118#6382—22no—
119#6381—22no—
120C4⋊(C7⋊C8)—41no—
121C4×C7⋊C8—41no—
122#2506D711yescharacteristic, complemented
123#2367—71no—
124#2071—71no—
125#2065—71no—
126#2042—71no—
127#2042—71no—
128#1933—71no—
129#1927—71no—
130#1921—71no—
131#1920—71no—
132#1916—71no—
133#1863—71no—
134#1862—71no—
135#1803—71no—
136C4×#102—71no—
137C23×D7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—21no—
152C2×D28—21no—
153C2×D28—21no—
154C2×D28—21no—
155C2×D28—21no—
156C2×D28—21no—
157C2×D28#4112yes—
158C2×C4×D7—42no—
159C2×C4×D7—42no—
160C2×C4×D7—81no—
161C2×Dic14—21no—
162C2×Dic14—21no—
163C2×Dic14—21no—
164C2×Dic14—21no—
165C2×Dic14—21no—
166C2×Dic14#4112yes—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—42no—
177C7×C4⋊C4—42no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—42no—
180C7×C4⋊C4—42no—
181C7×C4⋊C4—42no—
182C7×C4⋊C4#4112yes—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—42no—
190(C2×D7)⋊C4—24no—
191(C2×D7)⋊C4—24no—
192(C2×D7)⋊C4—42no—
193(C2×D7)⋊C4—42no—
194(C2×D7)⋊C4—81no—
195C4⋊Dic7—21no—
196C4⋊Dic7—21no—
197C4⋊Dic7—21no—
198C4⋊Dic7#4112yes—
199Dic7⋊C4—42no—
200C4×Dic7—42no—
201C4×Dic7—81no—
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—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×#125—21no—
215C2×#124—71no—
216C2×C4×Q8—71no—
217C2×C4×D4—71no—
218C2×#107—71no—
219C2×#106—71no—
220C2×C4⋊C8—71no—
221C2×#103—71no—
222C2×#103—71no—
223C2×#103—71no—
224C2×#103—71no—
225C2×#102—71no—
226C2×#102—71no—
227C2×#102—71no—
228C2×#102—71no—
229C2×C4×C8—71no—
230#400—21no—
231#398—21no—
232#391—71no—
233#390—71no—
234#389C2×D711yescharacteristic
235#389C2×D711yescharacteristic
236#389C2×D711yescharacteristic
237#389C2×D711yescharacteristic
238#388—141no—
239#388—141no—
240#387—21no—
241#385—141no—
242#384C2×D711yescharacteristic
243C4×C4⋊C4—71no—
244C4×C4⋊C4C2×D711yescharacteristic
245C4×C4⋊C4C2×D711yescharacteristic
246C4×C22⋊C4—141no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#336—71no—
254#336—71no—
255#336—71no—
256#336—71no—
257C23×C7C22×D411yescharacteristic
258C22×D7—22no—
259C22×D7—41no—
260C22×D7—41no—
261C22×D7—41no—
262C22×D7—41no—
263C22×D7—41no—
264C22×D7—41no—
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—22no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
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×C7—41no—
293C2×C4×C7#13512yes—
294C2×C4×C7AGL(1,ℤ/8)12yes—
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×C4×C7C2×#4111yescharacteristic
298C2×C4×C7C2×#4111yescharacteristic
299C2×C4×C7H(5,𝔽2)11yescharacteristic
300C2×C4×C72−1+411yescharacteristic
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—
310D28—22no—
311D28—22no—
312D28—41no—
313D28—41no—
314C4×D7—84no—
315Dic14—22no—
316Dic14—22no—
317Dic14—41no—
318Dic14—41no—
319C7⋊C8—41no—
320C7⋊C8—41no—
321C7⋊C8—41no—
322C7⋊C8—41no—
323C22×Q8—71no—
324C22×D4—71no—
325C23×C4—141no—
326C22×C8—71no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8—71no—
330#125—44no—
331#124—142no—
332#124—142no—
333C4×Q8—142no—
334C4×Q8—281no—
335C4×D4—142no—
336C4×D4—281no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4C22⋊D711yescharacteristic
347C2×C4⋊C4C22⋊D711yescharacteristic
348C2×C4⋊C4C22×D711yescharacteristic
349C2×C4⋊C4C22×D711yescharacteristic
350C2×C4⋊C4C22×D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C22⋊C4—141no—
353C2×C22⋊C4—141no—
354C2×C22⋊C4—141no—
355C2×C22⋊C4—141no—
356C2×C22⋊C4—141no—
357C2×C42—21no—
358C2×C42—141no—
359C2×C42—141no—
360C2×C42C22×D711yescharacteristic
361C2×C42C22×D711yescharacteristic
362C2×C42C22×D711yescharacteristic
363#107—142no—
364#106—142no—
365C4⋊C8—281no—
366#103—142no—
367#103—142no—
368#103—142no—
369#103—281no—
370#102—142no—
371#102—142no—
372#102—142no—
373#102—281no—
374C4×C8—281no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—21no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95C22⋊D711yescharacteristic
384#95C22⋊D711yescharacteristic
385C22×C7#52511yescharacteristic
386C22×C7#54611yescharacteristic
387C22×C7#54811yescharacteristic
388C22×C7C2×#13511yescharacteristic
389C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
390C22×C7#57711yescharacteristic
391C22×C7#57811yescharacteristic
392C2×D7—42no—
393C2×D7—42no—
394C2×D7—42no—
395C2×D7—81no—
396C2×D7—81no—
397C2×D7—81no—
398C2×D7—81no—
399C4×C7—22no—
400C4×C7—22no—
401C4×C7—41no—
402C4×C7—42no—
403C4×C7—41no—
404C4×C7—41no—
405C4×C7—41no—
406C4×C7—81no—
407C4×C7—81no—
408C4×C7—81no—
409C4×C7#52911yescharacteristic
410C4×C7#53911yescharacteristic
411C4×C7#55211yescharacteristic
412C4×C7#55311yescharacteristic
413Dic7—42no—
414Dic7—42no—
415Dic7—81no—
416C24—141no—
417C2×Q8—72no—
418C2×Q8—141no—
419C2×Q8—141no—
420C2×Q8—141no—
421C2×Q8—141no—
422C2×Q8—141no—
423C2×D4—72no—
424C2×D4—141no—
425C2×D4—141no—
426C2×D4—141no—
427C2×D4—141no—
428C2×D4—141no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—141no—
434C22×C4—141no—
435C22×C4—141no—
436C22×C4—282no—
437C22×C4—282no—
438C22×C4—561no—
439C22×C4C2×C22⋊D711yescharacteristic
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C2×C22⋊D711yescharacteristic
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C23×D711yescharacteristic
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—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—42no—
470C4⋊C4—42no—
471C4⋊C4—42no—
472C4⋊C4—42no—
473C4⋊C4—72no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—282no—
478C4⋊C4#116112yes—
479C22⋊C4—144no—
480C22⋊C4—144no—
481C22⋊C4—282no—
482C22⋊C4—282no—
483C22⋊C4—561no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—42no—
491C42—282no—
492C42—561no—
493C2×C7#246011yescharacteristic
494C2×C7#310611yescharacteristic
495C2×C7#312011yescharacteristic
496C2×C7#330011yescharacteristic
497C2×C7#330611yescharacteristic
498C2×C7#336511yescharacteristic
499C2×C7#336711yescharacteristic
500D7—82no—
501C23—142no—
502C23—281no—
503C23—281no—
504C23—281no—
505C23—281no—
506C23—281no—
507C23—281no—
508C23C22×C22⋊D711yescharacteristic
509Q8—142no—
510Q8—142no—
511Q8—281no—
512Q8—281no—
513D4—142no—
514D4—142no—
515D4—281no—
516D4—281no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—142no—
546C2×C4—142no—
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×C4#649512yes—
556C2×C4#651312yes—
557C2×C4C2×#116111yescharacteristic
558C2×C4#655111yescharacteristic
559C2×C4D7×#4111yescharacteristic
560C2×C4D7×#4111yescharacteristic
561C2×C4D7×#4111yescharacteristic
562C2×C4#655311yescharacteristic
563C8—281no—
564C8—281no—
565C8—281no—
566C8—281no—
567C7#2676111yescharacteristic, complemented, Sylow
568C22—282no—
569C22—282no—
570C22—282no—
571C22—561no—
572C22—561no—
573C22—561no—
574C22—561no—
575C22C2×#649511yescharacteristic
576C22#9050811yescharacteristic
577C22C2×#651311yescharacteristic
578C22#9051911yescharacteristic
579C22#9052011yescharacteristic
580C22#9052811yescharacteristic
581C22#9052911yescharacteristic
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—81no—
590C4—81no—
591C4—81no—
592C4—282no—
593C4—282no—
594C4—561no—
595C4#9038211yescharacteristic
596C4#9038311yescharacteristic
597C4#9038411yescharacteristic
598C4#9038611yescharacteristic
599C2—562no—
600C2#1174050411yescharacteristic
601C2#1174074411yescharacteristic
602C2#1174089111yescharacteristic
603C2#1174097111yescharacteristic
604C2#1174100811yescharacteristic
605C2#1174104711yescharacteristic
606C2#1174114211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2506C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2506D7
hypocenterC7#26761
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2506

Complements

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

NH ≅ G/Ntype
1. C7×#2506C2semidirect
3. #11727385C2semidirect
8. #11727368C2semidirect
9. #11727357C2semidirect
10. #11727356C2semidirect
11. #11727342C2semidirect
13. #11727236C2semidirect
15. #11727218C2semidirect
122. #2506D7semidirect
567. C7#26761semidirect

Maximal subgroups

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

subgroupindex
C7×#25062
#117311002
#117273852
#117273832
#117273822
#117273792
#117273682
#117273572
#117273562
#117273422
#117273392
#117272362
C4×#63832
#117272182
#267617