Finite Group Databasebeta

← #13523790

Subgroups — #13523790

Order 1792.

1918 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2285C211yesmaximal, characteristic, complemented
2#11739673C211yesmaximal, characteristic
3#11739673C211yesmaximal, characteristic
4#11739672C211yesmaximal, characteristic, complemented
5Dic7×#117C211yesmaximal, characteristic, complemented
6#11739515C211yesmaximal, characteristic
7C2×C4×Dic7⋊C4C211yesmaximal, characteristic, complemented
8#11731778C212yesmaximal, complemented
9#11731777C212yesmaximal, complemented
10#11731618C212yesmaximal, complemented
11#11731617C212yesmaximal, complemented
12#11731373C214yesmaximal, complemented
13#11731355C214yesmaximal, complemented
14#11731224C214yesmaximal, complemented
15C4×#6461C214yesmaximal, complemented
16C22×#1157C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C2×#6489C2211yescharacteristic
21C22×Dic7⋊C4C2212yes—
22C22×C4×Dic7C2211yescharacteristic
23C22×C4×Dic7C2211yescharacteristic
24C2×#6461C2214yes—
25C7×#384C2212yes—
26C7×#382C2212yes—
27C7×#380C2214yes—
28C4×C7×C4⋊C4C2214yes—
29#89985C2214yes—
30C2×C7⋊#95C2212yes—
31#89766C2214yes—
32#89765C2214yes—
33#89762C2214yes—
34#89760C2212yes—
35#89760C2212yes—
36#89758C2214yes—
37#89757C2212yes—
38#89757C2212yes—
39#89754C2214yes—
40#89753C2214yes—
41#89750C2214yes—
42C4⋊C4×Dic7C2212yes—
43C4⋊C4×Dic7C2212yes—
44#89746C2212yes—
45#89746C2212yes—
46#89730C2212yes—
47#89730C2214yes—
48#89730C2212yes—
49#89722C2212yes—
50#89722C2212yes—
51C22⋊C4×Dic7C2214yes—
52#89711C2211yescharacteristic
53#89711C2211yescharacteristic
54C4×(C2×D7)⋊C4C2212yes—
55C4×(C2×D7)⋊C4C2214yes—
56C4×(C2×D7)⋊C4C2214yes—
57C4×(C2×D7)⋊C4C2212yes—
58C4×C4⋊Dic7C2214yes—
59#89704C2212yes—
60C4×Dic7⋊C4C2212yes—
61C4×Dic7⋊C4C2212yes—
62C4×Dic7⋊C4C2212yes—
63C4×Dic7⋊C4C2212yes—
64C42×Dic7C2212yes—
65#89701C2211yescharacteristic
66#89701C2211yescharacteristic
67#89447C2214yes—
68#89446C2214yes—
69#89442C2214yes—
70#89441C2214yes—
71C7⋊#401C2214yes—
72#89432C2214yes—
73#89426C2214yes—
74#89423C2214yes—
75#34816—71nomaximal, Sylow
76C23×C4×C7C2312yes—
77C23×C4×C7C2311yescharacteristic
78C22×C22⋊D7C2312yes—
79C23×Dic7C2311yescharacteristic
80C23×Dic7C2312yes—
81C23×Dic7C2311yescharacteristic
82C2×#1157—24no—
83C2×#1157—24no—
84C2×#1157—24no—
85C22×D28C2311yescharacteristic
86C22×C4×D7C2312yes—
87C22×Dic14C2311yescharacteristic
88C7×#117—24no—
89C7×#117—24no—
90C2×C7×C4⋊C4C2314yes—
91C2×C7×C4⋊C4C2312yes—
92C2×C7×C4⋊C4C2312yes—
93C2×C7×C4⋊C4C2314yes—
94C2×C7×C22⋊C4C2312yes—
95C2×C7×C22⋊C4C2312yes—
96C2×C42×C7—24no—
97C2×C42×C7C2311yescharacteristic
98C2×C42×C7C2311yescharacteristic
99C2×C42×C7C2312yes—
100C2×C42×C7C2312yes—
101C2×C42×C7C2312yes—
102C2×C22⋊Dic7C2312yes—
103C2×(C2×D7)⋊C4C2314yes—
104C2×(C2×D7)⋊C4C2314yes—
105C2×(C2×D7)⋊C4C2314yes—
106C2×(C2×D7)⋊C4C2314yes—
107C2×(C2×D7)⋊C4C2314yes—
108C2×(C2×D7)⋊C4C2314yes—
109#6489—24no—
110C2×C4⋊Dic7C2314yes—
111C2×C4⋊Dic7C2311yescharacteristic
112C2×C4⋊Dic7C2311yescharacteristic
113C2×C4⋊Dic7C2314yes—
114C2×Dic7⋊C4—28no—
115C2×Dic7⋊C4C2312yes—
116C2×Dic7⋊C4C2312yes—
117C2×Dic7⋊C4C2312yes—
118C2×Dic7⋊C4C2312yes—
119C2×C4×Dic7—24no—
120C2×C4×Dic7—24no—
121C2×C4×Dic7C2314yes—
122C2×C4×Dic7C2311yescharacteristic
123C2×C4×Dic7C2312yes—
124C2×C4×Dic7C2311yescharacteristic
125C2×C4×Dic7C2312yes—
126C2×C4×Dic7C2312yes—
127C2×C4×Dic7C2312yes—
128C2×C4×Dic7C2314yes—
129#6461—216no—
130C7×#95C2314yes—
131C7×#95C2314yes—
132C7⋊#95C2312yes—
133C7⋊#95C2314yes—
134C7⋊#95C2314yes—
135C7⋊#95C2314yes—
136C7⋊#95C2314yes—
137C7⋊#95C2314yes—
138C7⋊#95C2314yes—
139C7⋊#95C2314yes—
140C7⋊#95C2312yes—
141C7⋊#95C2314yes—
142C7⋊#95C2314yes—
143C7⋊#95C2314yes—
144C7⋊#95C2312yes—
145C7⋊#95C2312yes—
146#2514—72no—
147#2513—72no—
148#2512—72no—
149#2511—72no—
150#2510—74no—
151#2481—74no—
152#2401—74no—
153C4×#126—74no—
154#2297—71no—
155#2297—71no—
156#2297—71no—
157#2296—71no—
158#2285D711yescharacteristic, complemented
159C4×#117—71no—
160C2×C4×C4⋊C4—71no—
161C24×C7C2411yescharacteristic
162C23×D7C2412yes—
163C22×C4×C7—24no—
164C22×C4×C7—24no—
165C22×C4×C7—42no—
166C22×C4×C7#4114yes—
167C22×C4×C7#4114yes—
168C22×C4×C7#4114yes—
169C22×C4×C7C2412yes—
170C22×C4×C7C2414yes—
171C22×C4×C7C2414yes—
172C22×C4×C7C2412yes—
173C22×C4×C7C2411yescharacteristic
174C22×C4×C7C2411yescharacteristic
175C2×C22⋊D7—24no—
176C2×C22⋊D7—24no—
177C2×C22⋊D7—24no—
178C22×Dic7—22no—
179C22×Dic7—24no—
180C22×Dic7—24no—
181C22×Dic7—22no—
182C22×Dic7—24no—
183C22×Dic7—42no—
184C22×Dic7—42no—
185C22×Dic7C2412yes—
186C22×Dic7C2412yes—
187C22×Dic7C2412yes—
188C22×Dic7C2412yes—
189C22×Dic7C2412yes—
190C22×Dic7C2412yes—
191C22×Dic7C2412yes—
192#1157—416no—
193C2×D28—22no—
194C2×D28—22no—
195C2×D28—22no—
196C2×C4×D7—24no—
197C2×C4×D7—24no—
198C2×C4×D7—24no—
199C2×Dic14—22no—
200C2×Dic14—22no—
201C2×Dic14—22no—
202C7×C4⋊C4—28no—
203C7×C4⋊C4—24no—
204C7×C4⋊C4—24no—
205C7×C4⋊C4—44no—
206C7×C22⋊C4—24no—
207C7×C22⋊C4—24no—
208C42×C7—22no—
209C42×C7—24no—
210C42×C7—22no—
211C42×C7—24no—
212C42×C7—24no—
213C22⋊Dic7—24no—
214(C2×D7)⋊C4—28no—
215(C2×D7)⋊C4—28no—
216(C2×D7)⋊C4—28no—
217(C2×D7)⋊C4—44no—
218(C2×D7)⋊C4—44no—
219(C2×D7)⋊C4—44no—
220C4⋊Dic7—22no—
221C4⋊Dic7—22no—
222C4⋊Dic7—28no—
223C4⋊Dic7—44no—
224Dic7⋊C4—24no—
225Dic7⋊C4—24no—
226Dic7⋊C4—24no—
227Dic7⋊C4#4118yescomplemented
228C4×Dic7—22no—
229C4×Dic7—24no—
230C4×Dic7—22no—
231C4×Dic7—24no—
232C4×Dic7—24no—
233C4×Dic7—24no—
234C4×Dic7—28no—
235C4×Dic7—44no—
236C22×#41—71no—
237C2×#126—74no—
238C2×#117—71no—
239C2×#117C2×D711yescharacteristic
240C2×#117C2×D711yescharacteristic
241C22×C4⋊C4—72no—
242C22×C42—71no—
243C22×C42—71no—
244C22×C42C2×D711yescharacteristic
245#401—74no—
246#400—74no—
247#400—74no—
248#400—74no—
249#397—74no—
250#397—74no—
251#397—74no—
252#396—74no—
253#391—72no—
254#391—72no—
255#391—71no—
256#391—71no—
257#390—72no—
258#390—72no—
259#390—71no—
260#390—71no—
261#389—72no—
262#389—72no—
263#388—74no—
264#388—74no—
265#388—74no—
266#387—74no—
267#387—74no—
268#387—74no—
269#386—74no—
270#386—72no—
271#386—74no—
272#386—72no—
273#385—72no—
274#385—72no—
275#384C2×D712yes—
276#383—74no—
277#382C2×D712yes—
278#380C2×D714yes—
279C4×C4⋊C4—72no—
280C4×C4⋊C4—72no—
281C4×C4⋊C4—72no—
282C4×C4⋊C4—74no—
283C4×C4⋊C4—72no—
284C4×C4⋊C4—72no—
285C4×C4⋊C4—72no—
286C4×C4⋊C4C2×D714yes—
287C4×C22⋊C4—74no—
288C4×C22⋊C4—72no—
289C4×C22⋊C4—74no—
290C4×C22⋊C4—74no—
291C4×C22⋊C4—72no—
292#376—72no—
293C2×#95—72no—
294C43—72no—
295C23×C7—22no—
296C23×C7—22no—
297C23×C7C2×#4112yes—
298C23×C7C2×#4112yes—
299C23×C7C2×#4112yes—
300C23×C7C2511yescharacteristic
301C22×D7—24no—
302C22×D7—22no—
303C22×D7—22no—
304C22×D7—22no—
305C22×D7—24no—
306C2×C4×C7—22no—
307C2×C4×C7—22no—
308C2×C4×C7—24no—
309C2×C4×C7—22no—
310C2×C4×C7—24no—
311C2×C4×C7—22no—
312C2×C4×C7—24no—
313C2×C4×C7—24no—
314C2×C4×C7—24no—
315C2×C4×C7—24no—
316C2×C4×C7—22no—
317C2×C4×C7—48no—
318C2×C4×C7#126116yescomplemented
319C2×C4×C7C2×#4112yes—
320C2×C4×C7C2×#4112yes—
321C2×C4×C7C2×#4112yes—
322C2×C4×C7C2×#4112yes—
323C2×C4×C7C2×#4112yes—
324C2×C4×C7C2×#4112yes—
325C2×C4×C7C2×#4114yes—
326C22⋊D7—48no—
327C2×Dic7—28no—
328C2×Dic7—22no—
329C2×Dic7—22no—
330C2×Dic7—22no—
331C2×Dic7—22no—
332C2×Dic7—22no—
333C2×Dic7—22no—
334C2×Dic7—24no—
335C2×Dic7—22no—
336C2×Dic7—22no—
337C2×Dic7—22no—
338C2×Dic7—22no—
339C2×Dic7—22no—
340C2×Dic7—22no—
341C2×Dic7—24no—
342C2×Dic7—22no—
343C2×Dic7—22no—
344C2×Dic7—22no—
345C2×Dic7—48no—
346C2×Dic7—44no—
347C2×Dic7C2×#4114yes—
348C2×Dic7C2×#4114yes—
349D28—44no—
350C4×D7—48no—
351Dic14—44no—
352C2×#41—144no—
353C2×#41—144no—
354C2×#41—144no—
355C22×Q8—71no—
356C22×D4—72no—
357C22×D4—71no—
358C23×C4—71no—
359C23×C4—72no—
360C23×C4—71no—
361C23×C4—72no—
362C23×C4C22×D712yes—
363C23×C4C22×D711yescharacteristic
364#126—1416no—
365#117—24no—
366#117—24no—
367#117—144no—
368C2×C4⋊C4—72no—
369C2×C4⋊C4—72no—
370C2×C4⋊C4—72no—
371C2×C4⋊C4—72no—
372C2×C4⋊C4—71no—
373C2×C4⋊C4—71no—
374C2×C4⋊C4—74no—
375C2×C4⋊C4—74no—
376C2×C4⋊C4—148no—
377C2×C4⋊C4C22×D712yes—
378C2×C4⋊C4C22×D712yes—
379C2×C4⋊C4C22×D714yes—
380C2×C4⋊C4C22×D714yes—
381C2×C22⋊C4—72no—
382C2×C22⋊C4—74no—
383C2×C22⋊C4—74no—
384C2×C22⋊C4—74no—
385C2×C22⋊C4—74no—
386C2×C22⋊C4—74no—
387C2×C22⋊C4—74no—
388C2×C22⋊C4C22×D712yes—
389C2×C22⋊C4C22×D712yes—
390C2×C42—24no—
391C2×C42—74no—
392C2×C42—72no—
393C2×C42—72no—
394C2×C42—72no—
395C2×C42—71no—
396C2×C42—71no—
397C2×C42—72no—
398C2×C42—74no—
399C2×C42—144no—
400C2×C42—144no—
401C2×C42C22×D711yescharacteristic
402C2×C42C22×D711yescharacteristic
403C2×C42C22×D712yes—
404C2×C42C22×D712yes—
405C2×C42C22×D712yes—
406#95—72no—
407#95—72no—
408#95—72no—
409#95—72no—
410#95—74no—
411#95—74no—
412#95—74no—
413#95—74no—
414#95—74no—
415#95—74no—
416#95—74no—
417#95—74no—
418#95—74no—
419#95—74no—
420#95C22×D714yes—
421#95C22×D714yes—
422C22×C7—22no—
423C22×C7—22no—
424C22×C7—24no—
425C22×C7—24no—
426C22×C7C2×#12614yes—
427C22×C7C22×#4111yescharacteristic
428C22×C7C22×#4111yescharacteristic
429C22×C7C22×#4111yescharacteristic
430C22×C7C22×#4112yes—
431C22×C7#58512yes—
432C2×D7—42no—
433C2×D7—42no—
434C2×D7—42no—
435C2×D7—44no—
436C2×D7—44no—
437C4×C7—24no—
438C4×C7—42no—
439C4×C7—44no—
440C4×C7—44no—
441C4×C7C2×#12614yes—
442C4×C7C2×#12614yes—
443Dic7—24no—
444Dic7—24no—
445Dic7—42no—
446Dic7—42no—
447Dic7—42no—
448Dic7—42no—
449Dic7—42no—
450C24—72no—
451C24C23×D711yescharacteristic
452#41—2816no—
453C2×Q8—142no—
454C2×Q8—142no—
455C2×Q8—142no—
456C2×D4—144no—
457C2×D4—144no—
458C2×D4—144no—
459C2×D4—142no—
460C2×D4—142no—
461C2×D4—142no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—42no—
465C22×C4—72no—
466C22×C4—72no—
467C22×C4—72no—
468C22×C4—72no—
469C22×C4—72no—
470C22×C4—72no—
471C22×C4—72no—
472C22×C4—144no—
473C22×C4—144no—
474C22×C4—142no—
475C22×C4—144no—
476C22×C4—142no—
477C22×C4—144no—
478C22×C4—144no—
479C22×C4—144no—
480C22×C4—282no—
481C22×C4—282no—
482C22×C4#115714yes—
483C22×C4#115914yes—
484C22×C4#116114yes—
485C22×C4C23×D712yes—
486C22×C4C23×D714yes—
487C22×C4C23×D714yes—
488C22×C4C23×D712yes—
489C22×C4C23×D711yescharacteristic
490C22×C4C23×D711yescharacteristic
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—28no—
494C4⋊C4—44no—
495C4⋊C4—78no—
496C4⋊C4—144no—
497C4⋊C4—144no—
498C4⋊C4—148no—
499C4⋊C4—142no—
500C4⋊C4—142no—
501C4⋊C4—144no—
502C4⋊C4—284no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—148no—
506C22⋊C4—148no—
507C22⋊C4—144no—
508C22⋊C4—148no—
509C22⋊C4—284no—
510C22⋊C4—284no—
511C22⋊C4—284no—
512C42—22no—
513C42—24no—
514C42—22no—
515C42—24no—
516C42—24no—
517C42—148no—
518C42—142no—
519C42—144no—
520C42—142no—
521C42—144no—
522C42—144no—
523C42—144no—
524C42—284no—
525C2×C7—24no—
526C2×C7C22×#12611yescharacteristic
527C2×C7#344411yescharacteristic
528C2×C7#345511yescharacteristic
529C2×C7#347512yes—
530C2×C7#348012yes—
531D7—82no—
532C23—22no—
533C23—22no—
534C23—142no—
535C23—142no—
536C23—142no—
537C23—144no—
538C23—144no—
539C23C2×#115712yes—
540C23C2×#115912yes—
541C23C2×#116112yes—
542C23C24×D711yescharacteristic
543Q8—284no—
544D4—288no—
545D4—284no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—48no—
558C2×C4—74no—
559C2×C4—74no—
560C2×C4—148no—
561C2×C4—142no—
562C2×C4—142no—
563C2×C4—142no—
564C2×C4—142no—
565C2×C4—142no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—142no—
569C2×C4—142no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—144no—
573C2×C4—142no—
574C2×C4—144no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—288no—
579C2×C4—284no—
580C2×C4—288no—
581C2×C4#6461116yescomplemented
582C2×C4C2×#115712yes—
583C2×C4C2×#115712yes—
584C2×C4C2×#115912yes—
585C2×C4C2×#115912yes—
586C2×C4C2×#116112yes—
587C2×C4C2×#116112yes—
588C2×C4D7×#4114yes—
589C7#3481611yescharacteristic, complemented, Sylow
590C22—24no—
591C22—24no—
592C22—22no—
593C22—22no—
594C22—282no—
595C22—284no—
596C22—284no—
597C22—282no—
598C22—282no—
599C22C2×#646114yes—
600C22C22×#115711yescharacteristic
601C22C22×#115911yescharacteristic
602C22C22×#116111yescharacteristic
603C22C2×D7×#4112yes—
604C22#9061612yes—
605C4—24no—
606C4—44no—
607C4—44no—
608C4—42no—
609C4—144no—
610C4—144no—
611C4—282no—
612C4—282no—
613C4—282no—
614C4—282no—
615C4—282no—
616C4C2×#646114yes—
617C4C2×#646114yes—
618C2—24no—
619C2—562no—
620C2C22×#646111yescharacteristic
621C2#1174281212yes—
622C2#1174328812yes—
623C2#1174332511yescharacteristic
624C2#1174334111yescharacteristic
subgroupG/H
centerC22×C4C23×D7
commutator, socle, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC7×#2285C2
prime radicalC22×#1157C22
hypercenter#2285D7
hypocenterC7#34816
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ #2285

Complements

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

NH ≅ G/Ntype
1. C7×#2285C2semidirect
4. #11739672C2semidirect
5. Dic7×#117C2semidirect
7. C2×C4×Dic7⋊C4C2semidirect
8. #11731778C2semidirect
9. #11731777C2semidirect
10. #11731618C2semidirect
11. #11731617C2semidirect
12. #11731373C2semidirect
13. #11731355C2semidirect
14. #11731224C2semidirect
15. C4×#6461C2semidirect
158. #2285D7semidirect
227. Dic7⋊C4#41semidirect
318. C2×C4×C7#126semidirect
581. C2×C4#6461semidirect
589. C7#34816semidirect

Maximal subgroups

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

subgroupindex
C7×#22852
#117396732
#117396722
Dic7×#1172
#117395152
C2×C4×Dic7⋊C42
#117317782
#117317772
#117316182
#117316172
#117313732
#117313552
#117312242
C4×#64612
#348167