Finite Group Databasebeta

← #13520956

Subgroups — #13520956

Order 1792.

1918 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2619C211yesmaximal, characteristic
2#11740625C211yesmaximal, characteristic, complemented
3Dic7×#127C211yesmaximal, characteristic, complemented
4#11740123C212yesmaximal, complemented
5Dic7×#121C212yesmaximal, complemented
6#11739710C211yesmaximal, characteristic, complemented
7Dic7×#117C211yesmaximal, characteristic, complemented
8#11734435C214yesmaximal, complemented
9#11734337C214yesmaximal, complemented
10#11734150C214yesmaximal, complemented
11#11734111C214yesmaximal, complemented
12#11733820C212yesmaximal, complemented
13#11733734C212yesmaximal, complemented
14#11731640C211yesmaximal, characteristic, complemented
15#11731595C211yesmaximal, characteristic, complemented
16C2×C7×#127C2211yescharacteristic
17C2×C7×#121C2212yes—
18C2×C7×#117C2211yescharacteristic
19C2×D4×Dic7C2212yes—
20C2×D4×Dic7C2214yes—
21C2×D4×Dic7C2214yes—
22C2×#6489C2211yescharacteristic
23C22×C4×Dic7C2211yescharacteristic
24C2×#6440C2214yes—
25C2×#6433C2211yescharacteristic
26C7×#394C2214yes—
27C7×#392C2214yes—
28C7×#390C2212yes—
29C7×#382C2211yescharacteristic
30#89994C2214yes—
31#89994C2214yes—
32#89994C2214yes—
33#89994C2214yes—
34#89993C2212yes—
35#89993C2212yes—
36#89993C2214yes—
37#89993C2212yes—
38#89992C2214yes—
39#89984C2212yes—
40#89757C2212yes—
41#89756C2214yes—
42C4⋊C4×Dic7C2212yes—
43C4⋊C4×Dic7C2212yes—
44C4⋊C4×Dic7C2212yes—
45#89745C2212yes—
46#89745C2212yes—
47#89725C2214yes—
48#89725C2214yes—
49#89724C2214yes—
50#89724C2214yes—
51#89723C2214yes—
52#89721C2214yes—
53#89720C2214yes—
54#89720C2214yes—
55#89717C2214yes—
56#89716C2214yes—
57C22⋊C4×Dic7C2212yes—
58C22⋊C4×Dic7C2212yes—
59C22⋊C4×Dic7C2214yes—
60C22⋊C4×Dic7C2214yes—
61#89707C2211yescharacteristic
62#89704C2211yescharacteristic
63#89704C2211yescharacteristic
64C42×Dic7C2211yescharacteristic
65C42×Dic7C2211yescharacteristic
66#89700C2212yes—
67#89433C2214yes—
68#89432C2214yes—
69#89424C2214yes—
70#89421C2214yes—
71#34621—71nomaximal, Sylow
72C22×C7×D4C2312yes—
73C22×C7×D4C2314yes—
74C22×C7×D4C2314yes—
75C23×C4×C7C2311yescharacteristic
76C23×Dic7C2314yes—
77C23×Dic7C2311yescharacteristic
78C23×Dic7C2311yescharacteristic
79C7×#127—24no—
80C7×#121—28no—
81C7×#117—24no—
82C2×C7×C4⋊C4C2312yes—
83C2×C7×C4⋊C4C2312yes—
84C2×C7×C22⋊C4C2312yes—
85C2×C7×C22⋊C4C2314yes—
86C2×C7×C22⋊C4C2314yes—
87C2×C7×C22⋊C4C2314yes—
88C2×C42×C7C2311yescharacteristic
89C2×C42×C7C2311yescharacteristic
90C2×C22⋊Dic7C2312yes—
91C2×C22⋊Dic7C2314yes—
92C2×C22⋊Dic7C2314yes—
93C2×C22⋊Dic7C2314yes—
94C2×C22⋊Dic7C2314yes—
95C2×C22⋊Dic7C2314yes—
96C2×C22⋊Dic7C2314yes—
97C2×C22⋊Dic7C2314yes—
98C2×C22⋊Dic7C2314yes—
99D4×Dic7—28no—
100D4×Dic7—48no—
101D4×Dic7—48no—
102#6489—24no—
103C2×C4⋊Dic7C2312yes—
104C2×C4⋊Dic7C2312yes—
105C2×C4⋊Dic7C2312yes—
106C2×C4⋊Dic7C2314yes—
107C2×C4⋊Dic7C2314yes—
108C2×Dic7⋊C4C2314yes—
109C2×Dic7⋊C4C2314yes—
110C2×Dic7⋊C4C2314yes—
111C2×Dic7⋊C4C2314yes—
112C2×C4×Dic7—24no—
113C2×C4×Dic7C2312yes—
114C2×C4×Dic7C2314yes—
115C2×C4×Dic7C2312yes—
116C2×C4×Dic7C2311yescharacteristic
117C2×C4×Dic7C2311yescharacteristic
118C2×C4×Dic7C2311yescharacteristic
119C2×C4×Dic7C2311yescharacteristic
120C2×C4×Dic7C2311yescharacteristic
121C2×C4×Dic7C2312yes—
122C2×C4×Dic7C2312yes—
123C2×C4×Dic7C2311yescharacteristic
124#6440—48no—
125#6433—24no—
126C7×#95C2314yes—
127C7⋊#95C2314yes—
128C7⋊#95C2314yes—
129C7⋊#95C2312yes—
130C7⋊#95C2314yes—
131C7⋊#95C2314yes—
132C7⋊#95C2312yes—
133C7⋊#95C2314yes—
134C7⋊#95C2312yes—
135C7⋊#95C2312yes—
136C7⋊#95C2314yes—
137#2651—74no—
138#2648—74no—
139#2647—71no—
140#2644—72no—
141#2641—71no—
142#2629—71no—
143#2624—72no—
144#2619D711yescharacteristic
145#2395—74no—
146#2392—74no—
147C4×#127—71no—
148C4×#125—71no—
149C4×#121—72no—
150#2293—72no—
151C4×#117—71no—
152C24×C7C2414yes—
153C24×C7C2411yescharacteristic
154C2×C7×D4—28no—
155C2×C7×D4—24no—
156C2×C7×D4—24no—
157C2×C7×D4—28no—
158C2×C7×D4—24no—
159C2×C7×D4—44no—
160C2×C7×D4—44no—
161C2×C7×D4—44no—
162C2×C7×D4—44no—
163C22×C4×C7—22no—
164C22×C4×C7—22no—
165C22×C4×C7#4114yes—
166C22×C4×C7C2414yes—
167C22×C4×C7C2412yes—
168C22×C4×C7C2411yescharacteristic
169C22×C4×C7C2412yes—
170C22×C4×C7C2411yescharacteristic
171C22×Dic7—22no—
172C22×Dic7—22no—
173C22×Dic7—28no—
174C22×Dic7—28no—
175C22×Dic7—24no—
176C22×Dic7—28no—
177C22×Dic7—42no—
178C22×Dic7C2411yescharacteristic
179C22×Dic7C2412yes—
180C22×Dic7C2412yes—
181C22×Dic7C2411yescharacteristic
182C22×Dic7C2414yes—
183C22×Dic7C2414yes—
184C22×Dic7C2411yescharacteristic
185C22×Dic7C2411yescharacteristic
186C7×C4⋊C4—24no—
187C7×C4⋊C4—24no—
188C7×C22⋊C4—24no—
189C7×C22⋊C4—28no—
190C7×C22⋊C4—44no—
191C7×C22⋊C4—44no—
192C42×C7—22no—
193C42×C7#4114yes—
194C22⋊Dic7—24no—
195C22⋊Dic7—28no—
196C22⋊Dic7—44no—
197C22⋊Dic7—44no—
198C22⋊Dic7—44no—
199C22⋊Dic7—44no—
200C22⋊Dic7—44no—
201C22⋊Dic7—44no—
202C22⋊Dic7—44no—
203C4⋊Dic7—24no—
204C4⋊Dic7—24no—
205C4⋊Dic7—24no—
206C4⋊Dic7—44no—
207C4⋊Dic7—44no—
208Dic7⋊C4—28no—
209Dic7⋊C4—44no—
210Dic7⋊C4—44no—
211Dic7⋊C4—44no—
212C4×Dic7—22no—
213C4×Dic7—22no—
214C4×Dic7—22no—
215C4×Dic7—22no—
216C4×Dic7—22no—
217C4×Dic7—24no—
218C4×Dic7—28no—
219C4×Dic7—24no—
220C4×Dic7—42no—
221C4×Dic7—42no—
222C4×Dic7#4114yes—
223C2×#127C2×D711yescharacteristic
224C2×#125—71no—
225C2×#121C2×D712yes—
226C2×C4×D4—72no—
227C2×C4×D4—74no—
228C2×C4×D4—74no—
229C2×#117—71no—
230C2×#117—74no—
231C2×#117C2×D711yescharacteristic
232C22×C42—71no—
233#400—74no—
234#399—74no—
235#398—74no—
236#398—74no—
237#397—74no—
238#397—74no—
239#397—74no—
240#397—74no—
241#396—74no—
242#396—74no—
243#394—74no—
244#394—74no—
245#394—74no—
246#394—74no—
247#394C2×D714yes—
248#392C2×D714yes—
249#390—72no—
250#390—74no—
251#390—72no—
252#390—72no—
253#390C2×D712yes—
254#389—72no—
255#389—72no—
256#389—72no—
257#389—72no—
258#388—74no—
259#388—74no—
260#388—74no—
261#387—74no—
262#386—74no—
263#384—71no—
264#382C2×D711yescharacteristic
265#380—72no—
266C4×C4⋊C4—72no—
267C4×C4⋊C4—72no—
268C4×C4⋊C4—72no—
269C4×C4⋊C4—74no—
270C4×C22⋊C4—72no—
271C4×C22⋊C4—74no—
272C4×C22⋊C4—72no—
273C4×C22⋊C4—74no—
274#376—71no—
275#376—71no—
276C43—71no—
277C43—71no—
278C23×C7—21no—
279C23×C7—21no—
280C23×C7—22no—
281C23×C7—22no—
282C23×C7—24no—
283C23×C7—24no—
284C23×C7—24no—
285C23×C7—24no—
286C23×C7—24no—
287C23×C7—24no—
288C23×C7—24no—
289C23×C7C2×#4112yes—
290C23×C7C2511yescharacteristic
291C7×D4—416no—
292C7×D4—416no—
293C7×D4—48no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—22no—
297C2×C4×C7—22no—
298C2×C4×C7—28no—
299C2×C4×C7—22no—
300C2×C4×C7—24no—
301C2×C4×C7—22no—
302C2×C4×C7—24no—
303C2×C4×C7—22no—
304C2×C4×C7—24no—
305C2×C4×C7C2×#4114yes—
306C2×C4×C7C2×#4112yes—
307C2×C4×C7C2×#4112yes—
308C2×C4×C7C2×#4112yes—
309C2×C4×C7C2×#4112yes—
310C2×Dic7—22no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—24no—
316C2×Dic7—24no—
317C2×Dic7—22no—
318C2×Dic7—24no—
319C2×Dic7—22no—
320C2×Dic7—24no—
321C2×Dic7—22no—
322C2×Dic7—24no—
323C2×Dic7—22no—
324C2×Dic7—22no—
325C2×Dic7—24no—
326C2×Dic7—22no—
327C2×Dic7—44no—
328C2×Dic7—44no—
329C2×Dic7—416no—
330C2×Dic7C2×#4112yes—
331C2×Dic7C2×#4112yes—
332C2×Dic7H(5,𝔽2)14yes—
333C22×D4C22×D712yes—
334C22×D4C22×D714yes—
335C22×D4C22×D714yes—
336C23×C4—74no—
337C23×C4—71no—
338C23×C4—71no—
339C23×C4C22×D711yescharacteristic
340#127—24no—
341#125—144no—
342#121—28no—
343C4×D4—148no—
344C4×D4—288no—
345C4×D4—288no—
346#117—24no—
347#117—144no—
348#117—288no—
349C2×C4⋊C4—72no—
350C2×C4⋊C4—72no—
351C2×C4⋊C4—74no—
352C2×C4⋊C4—72no—
353C2×C4⋊C4—74no—
354C2×C4⋊C4—74no—
355C2×C4⋊C4—74no—
356C2×C4⋊C4—74no—
357C2×C4⋊C4—74no—
358C2×C4⋊C4C22×D712yes—
359C2×C4⋊C4C22×D712yes—
360C2×C22⋊C4—72no—
361C2×C22⋊C4—74no—
362C2×C22⋊C4—74no—
363C2×C22⋊C4—74no—
364C2×C22⋊C4—74no—
365C2×C22⋊C4—74no—
366C2×C22⋊C4—74no—
367C2×C22⋊C4—74no—
368C2×C22⋊C4—74no—
369C2×C22⋊C4C22×D712yes—
370C2×C22⋊C4C22×D714yes—
371C2×C22⋊C4C22×D714yes—
372C2×C22⋊C4C22×D714yes—
373C2×C42—71no—
374C2×C42—72no—
375C2×C42—74no—
376C2×C42—71no—
377C2×C42—71no—
378C2×C42—72no—
379C2×C42—71no—
380C2×C42—71no—
381C2×C42—71no—
382C2×C42—72no—
383C2×C42—72no—
384C2×C42—144no—
385C2×C42C22×D711yescharacteristic
386C2×C42C22×D711yescharacteristic
387#95—72no—
388#95—72no—
389#95—72no—
390#95—72no—
391#95—74no—
392#95—74no—
393#95—74no—
394#95—74no—
395#95—74no—
396#95—74no—
397#95C22×D714yes—
398C22×C7—22no—
399C22×C7—24no—
400C22×C7—41no—
401C22×C7—41no—
402C22×C7—42no—
403C22×C7—44no—
404C22×C7—44no—
405C22×C7—44no—
406C22×C7—44no—
407C22×C7—44no—
408C22×C7—44no—
409C22×C7—44no—
410C22×C7C22×#4111yescharacteristic
411C22×C7C22×#4111yescharacteristic
412C22×C7C22×#4111yescharacteristic
413C22×C7C2×H(5,𝔽2)12yes—
414C22×C7#58512yes—
415C4×C7—22no—
416C4×C7—24no—
417C4×C7—42no—
418C4×C7—44no—
419C4×C7#52914yes—
420Dic7—22no—
421Dic7—41no—
422Dic7—44no—
423Dic7—41no—
424Dic7—44no—
425Dic7—42no—
426Dic7—42no—
427Dic7#54014yes—
428C24C23×D714yes—
429C24C23×D711yescharacteristic
430C2×D4—28no—
431C2×D4—24no—
432C2×D4—28no—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—44no—
436C2×D4—44no—
437C2×D4—44no—
438C2×D4—44no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—71no—
442C22×C4—71no—
443C22×C4—71no—
444C22×C4—72no—
445C22×C4—72no—
446C22×C4—71no—
447C22×C4—74no—
448C22×C4—74no—
449C22×C4—142no—
450C22×C4—142no—
451C22×C4—148no—
452C22×C4—148no—
453C22×C4—144no—
454C22×C4—148no—
455C22×C4—282no—
456C22×C4#115914yes—
457C22×C4C23×D714yes—
458C22×C4C23×D712yes—
459C22×C4C23×D711yescharacteristic
460C22×C4C23×D712yes—
461C22×C4C23×D711yescharacteristic
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—148no—
465C4⋊C4—144no—
466C4⋊C4—144no—
467C4⋊C4—144no—
468C4⋊C4—284no—
469C4⋊C4—284no—
470C4⋊C4—284no—
471C4⋊C4—284no—
472C4⋊C4—284no—
473C22⋊C4—24no—
474C22⋊C4—28no—
475C22⋊C4—44no—
476C22⋊C4—44no—
477C22⋊C4—148no—
478C22⋊C4—144no—
479C22⋊C4—284no—
480C22⋊C4—284no—
481C22⋊C4—284no—
482C22⋊C4—284no—
483C22⋊C4—284no—
484C22⋊C4—284no—
485C22⋊C4—284no—
486C42—22no—
487C42—74no—
488C42—142no—
489C42—142no—
490C42—144no—
491C42—148no—
492C42—144no—
493C42—142no—
494C42—142no—
495C42—142no—
496C42—282no—
497C42—282no—
498C42#115914yes—
499C2×C7—42no—
500C2×C7—84no—
501C2×C7C2×#52911yescharacteristic
502C2×C7C2×#54011yescharacteristic
503C2×C7#345511yescharacteristic
504C2×C7#346012yes—
505C2×C7#346012yes—
506C23—21no—
507C23—21no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23C2×#115912yes—
518C23C24×D711yescharacteristic
519D4—416no—
520D4—416no—
521D4—48no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—72no—
534C2×C4—74no—
535C2×C4—72no—
536C2×C4—142no—
537C2×C4—142no—
538C2×C4—142no—
539C2×C4—144no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—144no—
543C2×C4—142no—
544C2×C4—144no—
545C2×C4—142no—
546C2×C4—142no—
547C2×C4—144no—
548C2×C4—144no—
549C2×C4—142no—
550C2×C4—142no—
551C2×C4—144no—
552C2×C4—142no—
553C2×C4—284no—
554C2×C4—2816no—
555C2×C4—284no—
556C2×C4C2×#115914yes—
557C2×C4C2×#115912yes—
558C2×C4C2×#115912yes—
559C2×C4C2×#115912yes—
560C2×C4D7×#4112yes—
561C7#3462111yescharacteristic, complemented, Sylow
562C22—22no—
563C22—24no—
564C22—41no—
565C22—41no—
566C22—42no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22C22×#115911yescharacteristic
575C22C22×#115911yescharacteristic
576C22C2×D7×#4111yescharacteristic
577C22D7×H(5,𝔽2)12yes—
578C22#9061812yes—
579C4—22no—
580C4—24no—
581C4—44no—
582C4—42no—
583C4—74no—
584C4—142no—
585C4—281no—
586C4—281no—
587C4—284no—
588C4—284no—
589C4—282no—
590C4—282no—
591C4#9029114yes—
592C2—42no—
593C2—84no—
594C2C2×#9029111yescharacteristic
595C2#1174338212yes—
596C2#1174338212yes—
597C2#1174353511yescharacteristic
598C2D7×#54011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
Fitting, prime radicalC7×#2619C2
hypercenter#2619D7
hypocenterC7#34621
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2619

Complements

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

NH ≅ G/Ntype
2. #11740625C2semidirect
3. Dic7×#127C2semidirect
4. #11740123C2semidirect
5. Dic7×#121C2semidirect
6. #11739710C2semidirect
7. Dic7×#117C2semidirect
8. #11734435C2semidirect
9. #11734337C2semidirect
10. #11734150C2semidirect
11. #11734111C2semidirect
12. #11733820C2semidirect
13. #11733734C2semidirect
14. #11731640C2semidirect
15. #11731595C2semidirect
561. C7#34621semidirect

Maximal subgroups

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

subgroupindex
C7×#26192
#117406252
Dic7×#1272
#117401232
Dic7×#1212
#117397102
Dic7×#1172
#117344352
#117343372
#117341502
#117341112
#117338202
#117337342
#117316402
#117315952
#346217