Finite Group Databasebeta

← #13520952

Subgroups — #13520952

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×#2647C211yesmaximal, characteristic, complemented
2#11735662C214yesmaximal, complemented
3#11735574C214yesmaximal, complemented
4#11735274C214yesmaximal, complemented
5#11735182C214yesmaximal, complemented
6#11733697C212yesmaximal, complemented
7#11733694C212yesmaximal, complemented
8#11733409C212yesmaximal, complemented
9#11733406C212yesmaximal, complemented
10#11731789C211yesmaximal, characteristic, complemented
11#11731764C211yesmaximal, characteristic, complemented
12#11731689C211yesmaximal, characteristic
13#11731595C211yesmaximal, characteristic, complemented
14#11730874C211yesmaximal, characteristic, complemented
15#11730827C211yesmaximal, characteristic, complemented
16C2×#6458C2212yes—
17C2×#6456C2212yes—
18C2×#6456C2214yes—
19C2×#6456C2214yes—
20C2×#6455C2214yes—
21C2×#6437C2211yescharacteristic
22C2×#6435C2211yescharacteristic
23C2×#6435C2211yescharacteristic
24C2×C42×D7C2211yescharacteristic
25C2×#6433C2211yescharacteristic
26C7×#400C2214yes—
27C7×#398C2214yes—
28C7×#389C2212yes—
29C7×#389C2212yes—
30C7×#384C2211yescharacteristic
31C7×#382C2211yescharacteristic
32C7×#376C2211yescharacteristic
33#89766C2214yes—
34#89766C2214yes—
35#89765C2214yes—
36#89764C2214yes—
37#89763C2214yes—
38#89762C2214yes—
39#89762C2214yes—
40#89760C2214yes—
41#89760C2212yes—
42#89760C2212yes—
43#89759C2212yes—
44#89757C2212yes—
45#89755C2214yes—
46C4⋊C4×Dic7C2212yes—
47#89745C2212yes—
48#89711C2212yes—
49#89711C2212yes—
50#89704C2211yescharacteristic
51C42×Dic7C2211yescharacteristic
52C42×Dic7C2211yescharacteristic
53#89447C2214yes—
54#89447C2214yes—
55#89446C2214yes—
56#89444C2214yes—
57#89444C2214yes—
58#89444C2214yes—
59#89444C2214yes—
60#89443C2214yes—
61#89442C2214yes—
62#89438C2214yes—
63#89438C2214yes—
64#89438C2212yes—
65#89438C2212yes—
66#89437C2214yes—
67#89424C2214yes—
68#89421C2212yes—
69#89421C2214yes—
70#89421C2212yes—
71#34621—71nomaximal, Sylow
72C22×D28C2312yes—
73C22×D28C2314yes—
74C22×D28C2314yes—
75C22×C4×D7C2311yescharacteristic
76C22×C4×D7C2314yes—
77C22×C4×D7C2311yescharacteristic
78C22×C4×D7C2311yescharacteristic
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C4⋊C4C2312yes—
81C2×C7×C4⋊C4C2314yes—
82C2×C7×C4⋊C4C2314yes—
83C2×C7×C4⋊C4C2314yes—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×(C2×D7)⋊C4C2312yes—
88C2×(C2×D7)⋊C4C2312yes—
89C2×(C2×D7)⋊C4C2314yes—
90C2×(C2×D7)⋊C4C2314yes—
91C2×(C2×D7)⋊C4C2314yes—
92C2×(C2×D7)⋊C4C2314yes—
93C2×(C2×D7)⋊C4C2314yes—
94C2×(C2×D7)⋊C4C2314yes—
95C2×(C2×D7)⋊C4C2314yes—
96C2×(C2×D7)⋊C4C2314yes—
97C2×(C2×D7)⋊C4C2314yes—
98C2×(C2×D7)⋊C4C2314yes—
99C2×(C2×D7)⋊C4C2314yes—
100C2×C4⋊Dic7C2312yes—
101C2×C4⋊Dic7C2314yes—
102C2×Dic7⋊C4C2314yes—
103C2×Dic7⋊C4C2312yes—
104C2×Dic7⋊C4C2312yes—
105C2×Dic7⋊C4C2314yes—
106C2×C4×Dic7C2311yescharacteristic
107C2×C4×Dic7C2311yescharacteristic
108C2×C4×Dic7C2311yescharacteristic
109C2×C4×Dic7C2311yescharacteristic
110C2×C4×Dic7C2312yes—
111C2×C4×Dic7C2314yes—
112C2×C4×Dic7C2312yes—
113C2×C4×Dic7C2312yes—
114C2×C4×Dic7C2312yes—
115C2×C4×Dic7C2311yescharacteristic
116#6458—28no—
117#6456—28no—
118#6456—48no—
119#6456—48no—
120#6455—48no—
121#6437—24no—
122#6435—24no—
123#6435—24no—
124C42×D7—24no—
125#6433—24no—
126C7×#95C2314yes—
127C7×#95C2312yes—
128C7×#95C2312yes—
129C7×#95C2314yes—
130C7×#95C2314yes—
131C7⋊#95C2314yes—
132C7⋊#95C2312yes—
133C7⋊#95C2312yes—
134C7⋊#95C2314yes—
135C7⋊#95C2314yes—
136C7⋊#95C2314yes—
137#2651—74no—
138#2648—74no—
139#2647D711yescharacteristic, complemented
140#2644—72no—
141#2641—71no—
142#2629—71no—
143#2624—72no—
144#2619—71no—
145#2395—74no—
146#2392—74no—
147C4×#127—71no—
148C4×#125—71no—
149C4×#121—72no—
150#2293—72no—
151C4×#117—71no—
152C23×D7C2414yes—
153C23×D7C2411yescharacteristic
154C22×C4×C7C2411yescharacteristic
155C22×C4×C7C2414yes—
156C22×C4×C7C2411yescharacteristic
157C22×C4×C7C2414yes—
158C22×C4×C7C2412yes—
159C22×C4×C7C2412yes—
160C22×C4×C7C2411yescharacteristic
161C22×Dic7C2412yes—
162C22×Dic7C2412yes—
163C22×Dic7C2414yes—
164C22×Dic7C2411yescharacteristic
165C22×Dic7C2411yescharacteristic
166C22×Dic7C2411yescharacteristic
167C2×D28—28no—
168C2×D28—28no—
169C2×D28—24no—
170C2×D28—24no—
171C2×D28—24no—
172C2×D28—44no—
173C2×D28—44no—
174C2×D28—44no—
175C2×D28—44no—
176C2×C4×D7—22no—
177C2×C4×D7—22no—
178C2×C4×D7—22no—
179C2×C4×D7—22no—
180C2×C4×D7—28no—
181C2×C4×D7—28no—
182C2×C4×D7—24no—
183C2×C4×D7—28no—
184C2×C4×D7—42no—
185C2×C4×D7#4114yes—
186C7×C4⋊C4—24no—
187C7×C4⋊C4—24no—
188C7×C4⋊C4—44no—
189C7×C4⋊C4—44no—
190C7×C4⋊C4—44no—
191C42×C7—22no—
192C42×C7—22no—
193C42×C7#4114yes—
194(C2×D7)⋊C4—24no—
195(C2×D7)⋊C4—24no—
196(C2×D7)⋊C4—28no—
197(C2×D7)⋊C4—28no—
198(C2×D7)⋊C4—44no—
199(C2×D7)⋊C4—44no—
200(C2×D7)⋊C4—44no—
201(C2×D7)⋊C4—44no—
202(C2×D7)⋊C4—44no—
203(C2×D7)⋊C4—44no—
204(C2×D7)⋊C4—44no—
205(C2×D7)⋊C4—44no—
206(C2×D7)⋊C4—44no—
207C4⋊Dic7—24no—
208C4⋊Dic7—44no—
209Dic7⋊C4—28no—
210Dic7⋊C4—24no—
211Dic7⋊C4—24no—
212Dic7⋊C4—44no—
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×#127—71no—
224C2×#125—71no—
225C2×#121—72no—
226C2×C4×D4—72no—
227C2×C4×D4—74no—
228C2×C4×D4—74no—
229C2×#117—74no—
230C2×#117—71no—
231C2×#117—71no—
232C22×C42—71no—
233#400C2×D714yes—
234#399—74no—
235#398—74no—
236#398C2×D714yes—
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#394—74no—
248#392—74no—
249#390—72no—
250#390—74no—
251#390—72no—
252#390—72no—
253#390—72no—
254#389—72no—
255#389—72no—
256#389C2×D712yes—
257#389C2×D712yes—
258#388—74no—
259#388—74no—
260#388—74no—
261#387—74no—
262#386—74no—
263#384C2×D711yescharacteristic
264#382C2×D711yescharacteristic
265#380—72no—
266C4×C4⋊C4—72no—
267C4×C4⋊C4—74no—
268C4×C4⋊C4—72no—
269C4×C4⋊C4—72no—
270C4×C22⋊C4—72no—
271C4×C22⋊C4—74no—
272C4×C22⋊C4—72no—
273C4×C22⋊C4—74no—
274#376—71no—
275#376C2×D711yescharacteristic
276C43—71no—
277C43—71no—
278C23×C7C2511yescharacteristic
279C22×D7—21no—
280C22×D7—21no—
281C22×D7—22no—
282C22×D7—22no—
283C22×D7—24no—
284C22×D7—24no—
285C22×D7—24no—
286C22×D7—24no—
287C22×D7—24no—
288C22×D7—24no—
289C22×D7—24no—
290C22×D7C2×#4112yes—
291C2×C4×C7—22no—
292C2×C4×C7—22no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—22no—
296C2×C4×C7—24no—
297C2×C4×C7—24no—
298C2×C4×C7—22no—
299C2×C4×C7—24no—
300C2×C4×C7—24no—
301C2×C4×C7—22no—
302C2×C4×C7—24no—
303C2×C4×C7—22no—
304C2×C4×C7—24no—
305C2×C4×C7—22no—
306C2×C4×C7C2×#4114yes—
307C2×C4×C7C2×#4112yes—
308C2×C4×C7C2×#4112yes—
309C2×C4×C7C2×#4112yes—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—22no—
313C2×Dic7—22no—
314C2×Dic7—22no—
315C2×Dic7—24no—
316C2×Dic7—22no—
317C2×Dic7—24no—
318C2×Dic7—22no—
319C2×Dic7—22no—
320C2×Dic7—22no—
321C2×Dic7—24no—
322C2×Dic7C2×#4112yes—
323C2×Dic7C2×#4112yes—
324C2×Dic7C2×#4112yes—
325C2×Dic7H(5,𝔽2)14yes—
326D28—416no—
327D28—416no—
328D28—48no—
329C4×D7—28no—
330C4×D7—44no—
331C4×D7—44no—
332C4×D7—416no—
333C22×D4—72no—
334C22×D4—74no—
335C22×D4—74no—
336C23×C4—71no—
337C23×C4—74no—
338C23×C4—71no—
339C23×C4—71no—
340#127—144no—
341#125—144no—
342#121—148no—
343C4×D4—148no—
344C4×D4—288no—
345C4×D4—288no—
346#117—144no—
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⋊C4C22×D712yes—
356C2×C4⋊C4C22×D712yes—
357C2×C4⋊C4C22×D714yes—
358C2×C4⋊C4C22×D714yes—
359C2×C4⋊C4C22×D714yes—
360C2×C22⋊C4—72no—
361C2×C22⋊C4—72no—
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⋊C4—74no—
370C2×C22⋊C4—74no—
371C2×C22⋊C4—74no—
372C2×C22⋊C4—74no—
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—72no—
382C2×C42—72no—
383C2×C42—144no—
384C2×C42C22×D711yescharacteristic
385C2×C42C22×D711yescharacteristic
386C2×C42C22×D711yescharacteristic
387#95—72no—
388#95—72no—
389#95—74no—
390#95—74no—
391#95—74no—
392#95—74no—
393#95C22×D712yes—
394#95C22×D712yes—
395#95C22×D714yes—
396#95C22×D714yes—
397#95C22×D714yes—
398C22×C7C22×#4111yescharacteristic
399C22×C7C22×#4111yescharacteristic
400C22×C7C22×#4111yescharacteristic
401C22×C7C2×H(5,𝔽2)12yes—
402C22×C7#58512yes—
403C2×D7—22no—
404C2×D7—24no—
405C2×D7—41no—
406C2×D7—41no—
407C2×D7—42no—
408C2×D7—44no—
409C2×D7—44no—
410C2×D7—44no—
411C2×D7—44no—
412C2×D7—44no—
413C2×D7—44no—
414C2×D7—44no—
415C4×C7—24no—
416C4×C7—41no—
417C4×C7—44no—
418C4×C7—41no—
419C4×C7—44no—
420C4×C7—42no—
421C4×C7#52914yes—
422Dic7—22no—
423Dic7—22no—
424Dic7—42no—
425Dic7—42no—
426Dic7—44no—
427Dic7#54014yes—
428C24—71no—
429C24—74no—
430C2×D4—148no—
431C2×D4—148no—
432C2×D4—144no—
433C2×D4—144no—
434C2×D4—144no—
435C2×D4—284no—
436C2×D4—284no—
437C2×D4—284no—
438C2×D4—284no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—74no—
443C22×C4—72no—
444C22×C4—72no—
445C22×C4—74no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—148no—
449C22×C4—148no—
450C22×C4—144no—
451C22×C4—148no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—282no—
455C22×C4C23×D711yescharacteristic
456C22×C4C23×D714yes—
457C22×C4C23×D711yescharacteristic
458C22×C4C23×D714yes—
459C22×C4C23×D712yes—
460C22×C4C23×D712yes—
461C22×C4C23×D711yescharacteristic
462C4⋊C4—24no—
463C4⋊C4—24no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—144no—
468C4⋊C4—144no—
469C4⋊C4—148no—
470C4⋊C4—144no—
471C4⋊C4—284no—
472C4⋊C4—284no—
473C22⋊C4—148no—
474C22⋊C4—148no—
475C22⋊C4—144no—
476C22⋊C4—144no—
477C22⋊C4—284no—
478C22⋊C4—284no—
479C22⋊C4—284no—
480C22⋊C4—284no—
481C22⋊C4—284no—
482C22⋊C4—284no—
483C22⋊C4—284no—
484C22⋊C4—284no—
485C22⋊C4—284no—
486C42—22no—
487C42—22no—
488C42—74no—
489C42—142no—
490C42—142no—
491C42—144no—
492C42—148no—
493C42—144no—
494C42—142no—
495C42—142no—
496C42—282no—
497C42—282no—
498C42#116114yes—
499C2×C7C2×#52911yescharacteristic
500C2×C7C2×#54011yescharacteristic
501C2×C7#345511yescharacteristic
502C2×C7#346012yes—
503C2×C7#346012yes—
504D7—42no—
505D7—84no—
506C23—72no—
507C23—141no—
508C23—141no—
509C23—142no—
510C23—142no—
511C23—144no—
512C23—144no—
513C23—144no—
514C23—144no—
515C23—144no—
516C23—144no—
517C23—144no—
518C23C24×D711yescharacteristic
519D4—2816no—
520D4—2816no—
521D4—288no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—72no—
538C2×C4—74no—
539C2×C4—72no—
540C2×C4—72no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—142no—
544C2×C4—148no—
545C2×C4—142no—
546C2×C4—144no—
547C2×C4—142no—
548C2×C4—142no—
549C2×C4—144no—
550C2×C4—142no—
551C2×C4—142no—
552C2×C4—142no—
553C2×C4—144no—
554C2×C4—284no—
555C2×C4—2816no—
556C2×C4—284no—
557C2×C4C2×#116114yes—
558C2×C4C2×#116112yes—
559C2×C4D7×#4112yes—
560C2×C4D7×#4112yes—
561C7#3462111yescharacteristic, complemented, Sylow
562C22—144no—
563C22—142no—
564C22—281no—
565C22—281no—
566C22—282no—
567C22—284no—
568C22—284no—
569C22—284no—
570C22—284no—
571C22—284no—
572C22—284no—
573C22—284no—
574C22C22×#116111yescharacteristic
575C22C2×D7×#4111yescharacteristic
576C22C2×D7×#4111yescharacteristic
577C22D7×H(5,𝔽2)12yes—
578C22#9062012yes—
579C4—24no—
580C4—41no—
581C4—44no—
582C4—41no—
583C4—44no—
584C4—42no—
585C4—74no—
586C4—142no—
587C4—142no—
588C4—282no—
589C4—282no—
590C4—284no—
591C4#9038214yes—
592C2—282no—
593C2—564no—
594C2C2×#9038211yescharacteristic
595C2D7×#54011yescharacteristic
596C2#1174356311yescharacteristic
597C2#1174359212yes—
598C2#1174365012yes—
subgroupG/H
center, FrattiniC23C24×D7
commutator, socle, cosocleC23×C7C25
FittingC7×#2647C2
prime radical#11731689C2
hypercenter#2647D7
hypocenterC7#34621
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2647

Complements

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

NH ≅ G/Ntype
1. C7×#2647C2semidirect
2. #11735662C2semidirect
3. #11735574C2semidirect
4. #11735274C2semidirect
5. #11735182C2semidirect
6. #11733697C2semidirect
7. #11733694C2semidirect
8. #11733409C2semidirect
9. #11733406C2semidirect
10. #11731789C2semidirect
11. #11731764C2semidirect
13. #11731595C2semidirect
14. #11730874C2semidirect
15. #11730827C2semidirect
139. #2647D7semidirect
561. C7#34621semidirect

Maximal subgroups

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

subgroupindex
C7×#26472
#117356622
#117355742
#117352742
#117351822
#117336972
#117336942
#117334092
#117334062
#117317892
#117317642
#117316892
#117315952
#117308742
#117308272
#346217