Finite Group Databasebeta

← #13526534

Subgroups — #13526534

Order 1792.

2042 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2315C211yesmaximal, characteristic
2#11739540C211yesmaximal, characteristic
3#11739540C211yesmaximal, characteristic
4#11739540C211yesmaximal, characteristic
5C2×#89722C211yesmaximal, characteristic
6C2×#89718C211yesmaximal, characteristic
7C2×C22⋊C4×Dic7C211yesmaximal, characteristic
8#11732031C214yesmaximal, complemented
9#11731949C214yesmaximal, complemented
10#11731908C214yesmaximal, complemented
11#11731886C214yesmaximal, complemented
12#11731372C214yesmaximal, complemented
13#11731353C214yesmaximal, complemented
14C24×Dic7C2211yescharacteristic
15C22×C7×C22⋊C4C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C22⋊Dic7C2211yescharacteristic
19C22×C22⋊Dic7C2211yescharacteristic
20C22×C22⋊Dic7C2211yescharacteristic
21C22×C4⋊Dic7C2212yes—
22C22×Dic7⋊C4C2212yes—
23C22×C4×Dic7C2212yes—
24C2×#6452C2214yescomplemented
25C7×#385C2214yes—
26C7×#385C2214yes—
27C7×#380C2212yes—
28C7×#380C2212yes—
29#89982C2214yes—
30#89982C2212yes—
31#89982C2212yes—
32#89982C2214yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35C2×C7⋊#95C2212yes—
36#89756C2214yescomplemented
37#89755C2214yescomplemented
38#89754C2214yescomplemented
39#89753C2214yescomplemented
40#89752C2214yescomplemented
41#89750C2214yescomplemented
42#89744C2214yescomplemented
43#89722C2212yes—
44#89722C2214yes—
45#89722C2212yes—
46#89718C2212yes—
47#89718C2214yes—
48#89718C2212yes—
49#89718C2214yes—
50#89718C2214yes—
51#89718C2212yes—
52#89718C2212yes—
53#89714C2214yes—
54#89714C2214yes—
55#89714C2212yes—
56#89714C2212yes—
57C22⋊C4×Dic7C2214yes—
58C22⋊C4×Dic7C2214yes—
59#89431C2214yescomplemented
60#89430C2214yescomplemented
61#89430C2214yescomplemented
62#89429C2214yescomplemented
63#89425C2214yescomplemented
64#89424C2214yescomplemented
65#89422C2214yescomplemented
66#89419C2214yescomplemented
67#34994—71nomaximal, Sylow
68C25×C7C2311yescharacteristic
69C23×C4×C7C2312yes—
70C23×C4×C7C2311yescharacteristic
71C23×C4×C7C2312yes—
72C23×C4×C7C2311yescharacteristic
73C23×Dic7—24no—
74C23×Dic7—44no—
75C23×Dic7C2311yescharacteristic
76C23×Dic7C2312yes—
77C23×Dic7C2312yes—
78C23×Dic7C2312yes—
79C23×Dic7C2311yescharacteristic
80C23×Dic7C2311yescharacteristic
81C23×Dic7C2311yescharacteristic
82C23×Dic7C2312yes—
83C2×C7×C4⋊C4C2314yes—
84C2×C7×C4⋊C4C2314yes—
85C2×C7×C22⋊C4—24no—
86C2×C7×C22⋊C4—24no—
87C2×C7×C22⋊C4—24no—
88C2×C7×C22⋊C4C2312yes—
89C2×C7×C22⋊C4C2312yes—
90C2×C7×C22⋊C4C2312yes—
91C2×C7×C22⋊C4C2312yes—
92C2×C7×C22⋊C4C2312yes—
93C2×C7×C22⋊C4C2312yes—
94C2×C22⋊Dic7—24no—
95C2×C22⋊Dic7—24no—
96C2×C22⋊Dic7Q818yes—
97C2×C22⋊Dic7C2312yes—
98C2×C22⋊Dic7C2312yes—
99C2×C22⋊Dic7C2312yes—
100C2×C22⋊Dic7C2312yes—
101C2×C22⋊Dic7C2312yes—
102C2×C22⋊Dic7C2312yes—
103C2×C4⋊Dic7—44no—
104C2×C4⋊Dic7C2314yes—
105C2×C4⋊Dic7C2314yes—
106C2×Dic7⋊C4—44no—
107C2×Dic7⋊C4C2314yes—
108C2×Dic7⋊C4C2314yes—
109C2×Dic7⋊C4C2314yes—
110C2×Dic7⋊C4C2314yes—
111C2×Dic7⋊C4C2314yes—
112C2×Dic7⋊C4C2314yes—
113C2×Dic7⋊C4C2314yes—
114C2×Dic7⋊C4C2314yes—
115C2×C4×Dic7—44no—
116C2×C4×Dic7C2314yes—
117C2×C4×Dic7C2314yes—
118#6452—48no—
119C7×#95C2314yes—
120C7×#95C2314yes—
121C7⋊#95C2314yes—
122C7⋊#95C2314yes—
123C7⋊#95C2314yes—
124C7⋊#95C2314yes—
125C7⋊#95C2314yes—
126C7⋊#95C2314yes—
127C7⋊#95C2314yes—
128C7⋊#95C2314yes—
129C7⋊#95C2314yes—
130C7⋊#95C2314yes—
131C7⋊#95C2314yes—
132C7⋊#95C2314yes—
133#2851—74no—
134#2646—74no—
135#2505—74no—
136#2505—74no—
137#2473—74no—
138#2473—74no—
139#2315—71no—
140#2315—71no—
141#2315—71no—
142#2315D711yescharacteristic
143C2×#385—71no—
144C2×#385—71no—
145C2×C4×C22⋊C4—71no—
146C24×C7—42no—
147C24×C7—42no—
148C24×C7C2×Q814yes—
149C24×C7#4114yes—
150C24×C7#4114yes—
151C24×C7C2411yescharacteristic
152C24×C7C2411yescharacteristic
153C24×C7C2411yescharacteristic
154C22×C4×C7—24no—
155C22×C4×C7—24no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7—42no—
159C22×C4×C7—42no—
160C22×C4×C7—42no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7—42no—
164C22×C4×C7C2412yes—
165C22×C4×C7C2414yes—
166C22×C4×C7C2414yes—
167C22×C4×C7C2412yes—
168C22×Dic7—22no—
169C22×Dic7—24no—
170C22×Dic7—24no—
171C22×Dic7—24no—
172C22×Dic7—24no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—22no—
176C22×Dic7—24no—
177C22×Dic7—24no—
178C22×Dic7—42no—
179C22×Dic7—42no—
180C22×Dic7—42no—
181C22×Dic7—42no—
182C22×Dic7—42no—
183C22×Dic7—42no—
184C22×Dic7—42no—
185C22×Dic7—42no—
186C22×Dic7—42no—
187C22×Dic7—416no—
188C22×Dic7C2×Q818yes—
189C22×Dic7C2412yes—
190C22×Dic7C2412yes—
191C22×Dic7C2414yes—
192C22×Dic7C2412yes—
193C22×Dic7C2414yes—
194C22×Dic7C2412yes—
195C7×C4⋊C4—44no—
196C7×C4⋊C4—44no—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—24no—
199C7×C22⋊C4—24no—
200C7×C22⋊C4—24no—
201C7×C22⋊C4—24no—
202C7×C22⋊C4—24no—
203C22⋊Dic7—24no—
204C22⋊Dic7—24no—
205C22⋊Dic7—24no—
206C22⋊Dic7—24no—
207C22⋊Dic7C2×Q818yes—
208C22⋊Dic7C2×Q818yes—
209C4⋊Dic7—28no—
210C4⋊Dic7—44no—
211Dic7⋊C4—28no—
212Dic7⋊C4—44no—
213Dic7⋊C4—44no—
214Dic7⋊C4—44no—
215Dic7⋊C4—44no—
216Dic7⋊C4—44no—
217Dic7⋊C4—44no—
218Dic7⋊C4—44no—
219C4×Dic7—28no—
220C4×Dic7—44no—
221C24×C4—71no—
222C2×#125—74no—
223C22×C4⋊C4—72no—
224C22×C4⋊C4—72no—
225C22×C22⋊C4—71no—
226C22×C22⋊C4—71no—
227C22×C22⋊C4—71no—
228C22×C22⋊C4C2×D711yescharacteristic
229C22×C22⋊C4C2×D711yescharacteristic
230C22×C22⋊C4C2×D711yescharacteristic
231C22×C42—72no—
232#400—74no—
233#400—74no—
234#400—74no—
235#400—74no—
236#398—74no—
237#398—74no—
238#398—74no—
239#398—74no—
240#389—74no—
241#389—74no—
242#387—74no—
243#387—74no—
244#387—74no—
245#387—74no—
246#385—74no—
247#385—72no—
248#385—74no—
249#385—72no—
250#385—72no—
251#385—74no—
252#385—74no—
253#385—72no—
254#385—72no—
255#385—72no—
256#385—72no—
257#385—72no—
258#385—74no—
259#385—74no—
260#385C2×D714yes—
261#385C2×D714yes—
262#382—74no—
263#380—74no—
264#380—74no—
265#380—72no—
266#380—72no—
267#380C2×D712yes—
268#380C2×D712yes—
269C4×C22⋊C4—74no—
270C4×C22⋊C4—74no—
271C2×#95—72no—
272C2×#95—72no—
273C2×#95—72no—
274C23×C7—22no—
275C23×C7—22no—
276C23×C7—22no—
277C23×C7—22no—
278C23×C7—22no—
279C23×C7—22no—
280C23×C7—44no—
281C23×C7—48no—
282C23×C7—44no—
283C23×C7—48no—
284C23×C7#125116yescomplemented
285C23×C7C22×Q812yes—
286C23×C7C22×Q812yes—
287C23×C7C22×Q812yes—
288C23×C7C2×#4112yes—
289C23×C7C2×#4112yes—
290C23×C7C2×#4112yes—
291C23×C7C2×#4112yes—
292C23×C7C2×#4112yes—
293C23×C7C2×#4112yes—
294C23×C7C2511yescharacteristic
295C2×C4×C7—22no—
296C2×C4×C7—22no—
297C2×C4×C7—24no—
298C2×C4×C7—24no—
299C2×C4×C7—24no—
300C2×C4×C7—24no—
301C2×C4×C7—24no—
302C2×C4×C7—24no—
303C2×C4×C7—24no—
304C2×C4×C7—24no—
305C2×C4×C7—48no—
306C2×C4×C7—48no—
307C2×C4×C7—82no—
308C2×C4×C7—82no—
309C2×Dic7—28no—
310C2×Dic7—22no—
311C2×Dic7—24no—
312C2×Dic7—22no—
313C2×Dic7—24no—
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—24no—
317C2×Dic7—22no—
318C2×Dic7—24no—
319C2×Dic7—24no—
320C2×Dic7—22no—
321C2×Dic7—22no—
322C2×Dic7—24no—
323C2×Dic7—24no—
324C2×Dic7—48no—
325C2×Dic7—48no—
326C2×Dic7—48no—
327C2×Dic7—48no—
328C2×Dic7—44no—
329C2×Dic7—82no—
330C2×Dic7—82no—
331C2×Dic7C22×Q818yes—
332C2×Dic7H(5,𝔽2)14yes—
333C25C22×D711yescharacteristic
334C23×C4—72no—
335C23×C4—71no—
336C23×C4—72no—
337C23×C4—72no—
338C23×C4—71no—
339C23×C4—72no—
340C23×C4—71no—
341C23×C4—71no—
342C23×C4—144no—
343C23×C4—284no—
344C23×C4C22×D711yescharacteristic
345C23×C4C22×D712yes—
346C23×C4C22×D711yescharacteristic
347C23×C4C22×D712yes—
348#125—288no—
349C2×C4⋊C4—74no—
350C2×C4⋊C4—74no—
351C2×C4⋊C4—74no—
352C2×C4⋊C4—74no—
353C2×C4⋊C4—74no—
354C2×C4⋊C4—74no—
355C2×C4⋊C4—74no—
356C2×C4⋊C4—74no—
357C2×C4⋊C4—74no—
358C2×C4⋊C4—74no—
359C2×C4⋊C4—284no—
360C2×C4⋊C4—284no—
361C2×C4⋊C4C22×D714yes—
362C2×C4⋊C4C22×D714yes—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4—24no—
366C2×C22⋊C4—78no—
367C2×C22⋊C4—72no—
368C2×C22⋊C4—72no—
369C2×C22⋊C4—72no—
370C2×C22⋊C4—72no—
371C2×C22⋊C4—72no—
372C2×C22⋊C4—72no—
373C2×C22⋊C4—144no—
374C2×C22⋊C4—144no—
375C2×C22⋊C4C22×D712yes—
376C2×C22⋊C4C22×D712yes—
377C2×C22⋊C4C22×D712yes—
378C2×C22⋊C4C22×D712yes—
379C2×C22⋊C4C22×D712yes—
380C2×C22⋊C4C22×D712yes—
381C2×C42—74no—
382C2×C42—74no—
383C2×C42—284no—
384#95—74no—
385#95—74no—
386#95—74no—
387#95—74no—
388#95—74no—
389#95—74no—
390#95—74no—
391#95—74no—
392#95—74no—
393#95—74no—
394#95—74no—
395#95—74no—
396#95C22×D714yes—
397#95C22×D714yes—
398C22×C7—24no—
399C22×C7—24no—
400C22×C7—24no—
401C22×C7—22no—
402C22×C7—22no—
403C22×C7—22no—
404C22×C7—22no—
405C22×C7—22no—
406C22×C7—22no—
407C22×C7—24no—
408C22×C7—24no—
409C22×C7—24no—
410C22×C7—416no—
411C22×C7C2×#12514yes—
412C22×C7C2×#12514yes—
413C22×C7C2×#12514yes—
414C22×C7C23×Q811yescharacteristic
415C22×C7C22×#4111yescharacteristic
416C22×C7C22×#4111yescharacteristic
417C22×C7C2×H(5,𝔽2)12yes—
418C22×C7#58512yes—
419C4×C7—44no—
420C4×C7—44no—
421C4×C7—42no—
422C4×C7—42no—
423Dic7—24no—
424Dic7—28no—
425Dic7—44no—
426Dic7—42no—
427Dic7—42no—
428Dic7—42no—
429C24—42no—
430C24—42no—
431C24#115714yes—
432C24#115914yes—
433C24Q8×D714yes—
434C24C23×D711yescharacteristic
435C24C23×D711yescharacteristic
436C24C23×D711yescharacteristic
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—74no—
448C22×C4—72no—
449C22×C4—78no—
450C22×C4—72no—
451C22×C4—72no—
452C22×C4—74no—
453C22×C4—72no—
454C22×C4—144no—
455C22×C4—144no—
456C22×C4—144no—
457C22×C4—142no—
458C22×C4—144no—
459C22×C4—144no—
460C22×C4—142no—
461C22×C4—144no—
462C22×C4—142no—
463C22×C4—142no—
464C22×C4—282no—
465C22×C4—282no—
466C22×C4—282no—
467C22×C4—282no—
468C22×C4—282no—
469C22×C4—282no—
470C22×C4—282no—
471C22×C4—282no—
472C22×C4—282no—
473C22×C4—2816no—
474C22×C4C23×D712yes—
475C22×C4C23×D714yes—
476C22×C4C23×D714yes—
477C22×C4C23×D712yes—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—148no—
481C4⋊C4—148no—
482C4⋊C4—284no—
483C4⋊C4—284no—
484C4⋊C4—284no—
485C4⋊C4—284no—
486C4⋊C4—284no—
487C4⋊C4—284no—
488C4⋊C4—284no—
489C4⋊C4—284no—
490C22⋊C4—24no—
491C22⋊C4—24no—
492C22⋊C4—24no—
493C22⋊C4—24no—
494C22⋊C4—24no—
495C22⋊C4—24no—
496C22⋊C4—78no—
497C22⋊C4—78no—
498C22⋊C4—144no—
499C22⋊C4—144no—
500C22⋊C4—144no—
501C22⋊C4—144no—
502C42—148no—
503C42—284no—
504C2×C7—24no—
505C2×C7—24no—
506C2×C7—24no—
507C2×C7C22×#12511yescharacteristic
508C2×C7#345912yes—
509C2×C7#349011yescharacteristic
510C2×C7#349111yescharacteristic
511C2×C7#350112yes—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—44no—
519C23—44no—
520C23—48no—
521C23—48no—
522C23#6452116yescomplemented
523C23C2×#115712yes—
524C23C2×#115712yes—
525C23C2×#115712yes—
526C23C2×#115912yes—
527C23C2×#115912yes—
528C23C2×#115912yes—
529C23C2×Q8×D712yes—
530C23C2×Q8×D712yes—
531C23C2×Q8×D712yes—
532C23C24×D711yescharacteristic
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—48no—
544C2×C4—48no—
545C2×C4—78no—
546C2×C4—74no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—148no—
550C2×C4—142no—
551C2×C4—144no—
552C2×C4—142no—
553C2×C4—144no—
554C2×C4—142no—
555C2×C4—142no—
556C2×C4—142no—
557C2×C4—144no—
558C2×C4—144no—
559C2×C4—144no—
560C2×C4—142no—
561C2×C4—142no—
562C2×C4—144no—
563C2×C4—144no—
564C2×C4—288no—
565C2×C4—288no—
566C2×C4—288no—
567C2×C4—288no—
568C2×C4—284no—
569C2×C4—562no—
570C2×C4—562no—
571C7#3499411yescharacteristic, complemented, Sylow
572C22—24no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—416no—
585C22C2×#645214yes—
586C22C2×#645214yes—
587C22C2×#645214yes—
588C22C22×#115711yescharacteristic
589C22C22×#115911yescharacteristic
590C22C22×Q8×D711yescharacteristic
591C22D7×H(5,𝔽2)12yes—
592C22#9061812yes—
593C4—44no—
594C4—44no—
595C4—42no—
596C4—42no—
597C4—148no—
598C4—144no—
599C4—284no—
600C4—282no—
601C4—282no—
602C4—282no—
603C2—24no—
604C2—24no—
605C2—24no—
606C2C22×#645211yescharacteristic
607C2#1174292711yescharacteristic
608C2#1174293611yescharacteristic
609C2#1174336312yes—
610C2#1174364212yes—
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC7×#2315C2
prime radicalC25×C7C23
hypercenter#2315D7
hypocenterC7#34994
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2315

Complements

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

NH ≅ G/Ntype
8. #11732031C2semidirect
9. #11731949C2semidirect
10. #11731908C2semidirect
11. #11731886C2semidirect
12. #11731372C2semidirect
13. #11731353C2semidirect
24. C2×#6452C22semidirect
36. #89756C22semidirect
37. #89755C22semidirect
38. #89754C22semidirect
39. #89753C22semidirect
40. #89752C22semidirect
41. #89750C22semidirect
42. #89744C22semidirect
59. #89431C22semidirect
60. #89430C22semidirect
61. #89430C22semidirect
62. #89429C22semidirect
63. #89425C22semidirect
64. #89424C22semidirect
65. #89422C22semidirect
66. #89419C22semidirect
284. C23×C7#125semidirect
522. C23#6452semidirect
571. C7#34994semidirect

Maximal subgroups

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

subgroupindex
C7×#23152
#117395402
C2×#897222
C2×#897182
C2×C22⋊C4×Dic72
#117320312
#117319492
#117319082
#117318862
#117313722
#117313532
#349947