Finite Group Databasebeta

← #13363486

Subgroups — #13363486

Order 1792.

1166 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90371C211yesmaximal, characteristic
2C2×#89457C211yesmaximal, characteristic, complemented
3D7×#382C211yesmaximal, characteristic, complemented
4C7×#1958C211yesmaximal, characteristic, complemented
5#11729479C211yesmaximal, characteristic, complemented
6#11728022C212yesmaximal, complemented
7#11727976C212yesmaximal, complemented
8#11727954C212yesmaximal
9#11727908C212yesmaximal, complemented
10#11727532C211yesmaximal, characteristic, complemented
11#11727308C211yesmaximal, characteristic, complemented
12C2×C7×#124C2211yescharacteristic
13C2×#6509C2211yescharacteristic
14C2×Q8×Dic7C2211yescharacteristic
15C2×#6501C2211yescharacteristic
16C2×D4×Dic7C2211yescharacteristic
17#90371—24no—
18#90371C418yescomplemented
19C2×D7×C4⋊C4C2212yes—
20C2×#6446—21no—
21C2×(C22⋊D7)⋊C4—21no—
22C2×#6442—21no—
23C2×#6440—21no—
24C2×C42×D7C2211yescharacteristic
25C2×#6433C2211yescharacteristic
26C2×C7×#103C2212yes—
27C2×C7×#102C2212yes—
28C2×C4×C7×C8C2211yescharacteristic
29C7×#382C2211yescharacteristic
30C2×#6409C2212yes—
31C2×#6406C2212yes—
32C2×(C2×D7)⋊C8C2212yes—
33C2×Dic7⋊C8C2212yes—
34#89749C2212yes—
35#89708C2211yescharacteristic
36C2×C4×C7⋊C8C2211yescharacteristic
37#89457—24no—
38D7×#95—21no—
39D7×#95—21no—
40#89422—21no—
41#89422—21no—
42#89346C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21861—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4C2311yescharacteristic
49C22×C22⋊D7—21no—
50C23×Dic7—21no—
51C22×C4×D7—22no—
52C22×C4×D7D411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×C4×D7C2311yescharacteristic
55C22×C7×C8C2312yes—
56C7×#124—22no—
57C7×#124C2×C414yescomplemented
58C2×C7×C4⋊C4C2312yes—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C42×C7C2311yescharacteristic
62C2×C22⋊Dic7—21no—
63C2×C22⋊Dic7—21no—
64#6509—22no—
65#6509C2×C414yes—
66Q8×Dic7—22no—
67Q8×Dic7C2×C414yescomplemented
68#6501—22no—
69#6501C2×C414yes—
70D4×Dic7—22no—
71D4×Dic7C2×C414yescomplemented
72C2×(C2×D7)⋊C4—21no—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7C2312yes—
75C2×C4⋊Dic7C2311yescharacteristic
76C2×C4⋊Dic7C2311yescharacteristic
77C2×Dic7⋊C4—21no—
78C2×Dic7⋊C4—21no—
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—22no—
81C2×C4×Dic7—21no—
82C2×C4×Dic7D411yescharacteristic
83C2×C4×Dic7D411yescharacteristic
84C2×C4×Dic7C2311yescharacteristic
85C22×C7⋊C8C2312yes—
86D7×C4⋊C4—42no—
87D7×C4⋊C4—42no—
88#6446—28no—
89(C22⋊D7)⋊C4—28no—
90#6442—28no—
91#6440—28no—
92C42×D7C2×C414yes—
93C42×D7D414yes—
94#6433—22no—
95#6433C2×C414yescomplemented
96C7×#103—42no—
97C7×#102—42no—
98C4×C7×C8—22no—
99C7×#95—21no—
100C7×#95—21no—
101#6409—42no—
102#6406—42no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—22no—
106C7⋊#95—22no—
107(C2×D7)⋊C8—24no—
108Dic7⋊C8—24no—
109C4×C7⋊C8—22no—
110C2×#529—71no—
111C2×#431—71no—
112C2×#382—71no—
113#1991—71no—
114#1991—71no—
115#1958—71no—
116#1958D711yescharacteristic, complemented
117#1874—72no—
118#1872—72no—
119#1805—72no—
120#1804—72no—
121C24×C7—21no—
122C23×D7C2×D411yescharacteristic
123C2×C7×Q8—21no—
124C2×C7×Q8—22no—
125C2×C7×Q8—22no—
126C2×C7×Q8C22×C412yes—
127C2×C7×D4—21no—
128C2×C7×D4—22no—
129C2×C7×D4—22no—
130C2×C7×D4C22×C412yes—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2411yescharacteristic
136C2×C22⋊D7—24no—
137C2×C22⋊D7—44no—
138C22×Dic7—21no—
139C22×Dic7—21no—
140C22×Dic7—24no—
141C22×Dic7—22no—
142C22×Dic7—44no—
143C22×Dic7C2×D411yescharacteristic
144C22×Dic7C2×D411yescharacteristic
145C22×Dic7C2×D411yescharacteristic
146C2×C4×D7—24no—
147C2×C4×D7—22no—
148C2×C4×D7—22no—
149C2×C4×D7—42no—
150C2×C4×D7—44no—
151C2×C4×D7—44no—
152C2×C4×D7—44no—
153C2×C4×D7C22⋊C414yes—
154C2×C4×D7C22⋊C414yes—
155C2×C4×D7C22×C412yes—
156C2×C4×D7C2×D412yes—
157C2×C7×C8—22no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C7×C4⋊C4—22no—
161C7×C4⋊C4—22no—
162C7×C22⋊C4—24no—
163C7×C22⋊C4—24no—
164C42×C7C22×C412yes—
165C42×C7C2×D412yes—
166C22⋊Dic7—24no—
167C22⋊Dic7—24no—
168(C2×D7)⋊C4—24no—
169C4⋊Dic7—21no—
170C4⋊Dic7—21no—
171C4⋊Dic7—24no—
172C4⋊Dic7—22no—
173C4⋊Dic7—22no—
174C4⋊Dic7C22×C412yes—
175C4⋊Dic7C22×C412yes—
176Dic7⋊C4—24no—
177Dic7⋊C4—24no—
178Dic7⋊C4—24no—
179Dic7⋊C4—44no—
180C4×Dic7—24no—
181C4×Dic7C22⋊C414yes—
182C4×Dic7C22⋊C414yes—
183C4×Dic7C22×C412yes—
184C4×Dic7C2×D412yes—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—22no—
187C2×C7⋊C8—22no—
188#529—78no—
189#529—144no—
190C2×#126—141no—
191C2×#125—71no—
192C2×#124C2×D711yescharacteristic
193C2×#123—141no—
194C2×C22⋊Q8—71no—
195C2×#121—71no—
196C2×C4×Q8—71no—
197C2×C4×D4—71no—
198C2×C4×D4—141no—
199C2×#117—141no—
200C22×C4⋊C4—72no—
201C22×C42—71no—
202#431—144no—
203C2×C4⋊C8—72no—
204C2×#103—72no—
205C2×#103C2×D712yes—
206C2×#102—72no—
207C2×#102C2×D712yes—
208C2×C22⋊C8—72no—
209C2×C4×C8—71no—
210C2×C4×C8C2×D711yescharacteristic
211#382—72no—
212#382—71no—
213#382—141no—
214#382—141no—
215#382C2×D711yescharacteristic
216C2×#95—141no—
217C2×#95—141no—
218#340—72no—
219#340—72no—
220#340—72no—
221#340—72no—
222C23×C7—22no—
223C23×C7—41no—
224C23×C7—41no—
225C23×C7—42no—
226C23×C7—42no—
227C23×C7C22×D411yescharacteristic
228C22×D7—21no—
229C22×D7—21no—
230C22×D7—22no—
231C22×D7C2×C22⋊C412yes—
232C22×D7#13514yes—
233C7×Q8—24no—
234C7×Q8—42no—
235C7×D4—24no—
236C7×D4—42no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7C2×C22⋊C412yes—
246C2×C4×C7C2×C22⋊C412yes—
247C2×C4×C7C23×C411yescharacteristic
248C2×C4×C7C22×D411yescharacteristic
249C2×C4×C7H(5,𝔽2)12yes—
250C2×C4×C72−1+412yes—
251C22⋊D7—48no—
252C2×Dic7—21no—
253C2×Dic7—21no—
254C2×Dic7—22no—
255C2×Dic7—22no—
256C2×Dic7—22no—
257C2×Dic7—22no—
258C2×Dic7—48no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262C2×Dic7—42no—
263C2×Dic7—42no—
264C2×Dic7C2×C22⋊C412yes—
265C2×Dic7C2×C22⋊C412yes—
266C2×Dic7C2×C22⋊C412yes—
267C4×D7—28no—
268C4×D7—24no—
269C4×D7—24no—
270C4×D7—44no—
271C4×D7—88no—
272C7×C8—42no—
273C7⋊C8—42no—
274C22×Q8C22×D711yescharacteristic
275C22×D4—141no—
276C22×D4C22×D711yescharacteristic
277C23×C4—71no—
278C23×C4—71no—
279C23×C4—71no—
280C23×C4—142no—
281C23×C4—141no—
282C22×C8—72no—
283C22×C8C22×D712yes—
284#126—148no—
285#125—74no—
286#125—142no—
287#124—22no—
288#124C4×D714yescomplemented
289#123—148no—
290C22⋊Q8—74no—
291C22⋊Q8—142no—
292#121—74no—
293#121—142no—
294C4×Q8—74no—
295C4×Q8—142no—
296C4×D4—74no—
297C4×D4—142no—
298C4×D4—148no—
299#117—148no—
300C2×C4⋊C4—71no—
301C2×C4⋊C4—71no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—141no—
304C2×C4⋊C4—141no—
305C2×C4⋊C4—141no—
306C2×C4⋊C4—141no—
307C2×C4⋊C4—142no—
308C2×C4⋊C4—282no—
309C2×C4⋊C4—282no—
310C2×C4⋊C4C22×D712yes—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—141no—
314C2×C22⋊C4—141no—
315C2×C22⋊C4—141no—
316C2×C42—71no—
317C2×C42—71no—
318C2×C42—71no—
319C2×C42—74no—
320C2×C42—74no—
321C2×C42—141no—
322C2×C42C22×D711yescharacteristic
323C4⋊C8—144no—
324#103—42no—
325#103—282no—
326#102—42no—
327#102—282no—
328C22⋊C8—144no—
329C4×C8—22no—
330C4×C8—142no—
331#95—21no—
332#95—21no—
333#95—142no—
334#95—142no—
335#95—141no—
336#95—141no—
337C22×C7—42no—
338C22×C7—42no—
339C22×C7—42no—
340C22×C7—82no—
341C22×C7—82no—
342C22×C7C22×C22⋊C411yescharacteristic
343C22×C7#53511yescharacteristic
344C22×C7#53611yescharacteristic
345C22×C7C2×#13512yes—
346C22×C7#57512yes—
347C2×D7—22no—
348C2×D7—22no—
349C2×D7—24no—
350C2×D7—24no—
351C2×D7#41614yes—
352C4×C7—22no—
353C4×C7—22no—
354C4×C7—42no—
355C4×C7—82no—
356C4×C7#52012yes—
357C4×C7#53712yes—
358Dic7—22no—
359Dic7—22no—
360Dic7—22no—
361Dic7—42no—
362Dic7—42no—
363Dic7—82no—
364C24—21no—
365C24—71no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8C2×C4×D712yes—
370C2×D4—21no—
371C2×D4—22no—
372C2×D4—22no—
373C2×D4—144no—
374C2×D4—284no—
375C2×D4C2×C4×D712yes—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—71no—
379C22×C4—71no—
380C22×C4—71no—
381C22×C4—74no—
382C22×C4—72no—
383C22×C4—72no—
384C22×C4—74no—
385C22×C4—144no—
386C22×C4—144no—
387C22×C4—142no—
388C22×C4—142no—
389C22×C4—141no—
390C22×C4—141no—
391C22×C4—142no—
392C22×C4—282no—
393C22×C4—284no—
394C22×C4—284no—
395C22×C4—284no—
396C22×C4—284no—
397C22×C4D4×D711yescharacteristic
398C22×C4D4×D711yescharacteristic
399C22×C4C23×D711yescharacteristic
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—142no—
404C2×C8—142no—
405C2×C8—142no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—72no—
409C4⋊C4—72no—
410C4⋊C4—144no—
411C4⋊C4—144no—
412C4⋊C4—144no—
413C4⋊C4—144no—
414C4⋊C4—141no—
415C4⋊C4—141no—
416C4⋊C4—142no—
417C4⋊C4—142no—
418C4⋊C4—284no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—144no—
422C22⋊C4—144no—
423C22⋊C4—144no—
424C42—74no—
425C42—74no—
426C42—72no—
427C42—72no—
428C42—144no—
429C42C2×C4×D712yes—
430C42D4×D712yes—
431C2×C7—82no—
432C2×C7#231111yescharacteristic
433C2×C7C2×#41611yescharacteristic
434C2×C7#289611yescharacteristic
435C2×C7#321112yes—
436C2×C7#323212yes—
437D7—24no—
438C23—22no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—41no—
443C23—72no—
444C23—74no—
445C23—141no—
446C23—142no—
447C23—141no—
448C23C2×D4×D711yescharacteristic
449Q8—24no—
450Q8—42no—
451D4—24no—
452D4—42no—
453D4—288no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—72no—
463C2×C4—72no—
464C2×C4—72no—
465C2×C4—148no—
466C2×C4—141no—
467C2×C4—142no—
468C2×C4—142no—
469C2×C4—141no—
470C2×C4—144no—
471C2×C4—144no—
472C2×C4—142no—
473C2×C4—142no—
474C2×C4—284no—
475C2×C4—288no—
476C2×C4—282no—
477C2×C4—282no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4—282no—
481C2×C4—568no—
482C2×C4D7×C22⋊C412yes—
483C2×C4D7×C22⋊C412yes—
484C2×C4C22×C4×D711yescharacteristic
485C2×C4C2×D4×D711yescharacteristic
486C2×C4#654812yes—
487C2×C4#655412yes—
488C8—42no—
489C8—282no—
490C7#2186111yescharacteristic, complemented, Sylow
491C22—42no—
492C22—42no—
493C22—42no—
494C22—74no—
495C22—82no—
496C22—82no—
497C22—144no—
498C22—142no—
499C22—144no—
500C22—142no—
501C22C2×D7×C22⋊C411yescharacteristic
502C22#9036611yescharacteristic
503C22#9040611yescharacteristic
504C22D7×#13512yes—
505C22#9045912yes—
506C4—22no—
507C4—22no—
508C4—42no—
509C4—82no—
510C4—142no—
511C4—142no—
512C4—142no—
513C4—282no—
514C4—282no—
515C4—562no—
516C4#9023712yes—
517C4#9029912yes—
518C2—82no—
519C2—144no—
520C2#1173373911yescharacteristic
521C2D7×#41611yescharacteristic
522C2#1173627511yescharacteristic
523C2#1173909312yes—
524C2#1173927012yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1958C2
socleC23×C7C22×D4
prime radicalC2×#6501C22
cosocleC22×C4×C7C24
hypercenter#1958D7
hypocenterC7#21861
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. C2×#89457C2semidirect
3. D7×#382C2semidirect
4. C7×#1958C2semidirect
5. #11729479C2semidirect
6. #11728022C2semidirect
7. #11727976C2semidirect
9. #11727908C2semidirect
10. #11727532C2semidirect
11. #11727308C2semidirect
18. #90371C4semidirect
57. C7×#124C2×C4semidirect
67. Q8×Dic7C2×C4semidirect
71. D4×Dic7C2×C4semidirect
95. #6433C2×C4semidirect
116. #1958D7semidirect
288. #124C4×D7semidirect
490. C7#21861semidirect

Maximal subgroups

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

subgroupindex
C2×#903712
C2×#894572
D7×#3822
C7×#19582
#117294792
#117280222
#117279762
#117279542
#117279082
#117275322
#117273082
#218617