Finite Group Databasebeta

← #13363487

Subgroups — #13363487

Order 1792.

1166 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90271C211yesmaximal, characteristic
2C2×#89605C211yesmaximal, characteristic, complemented
3D7×#382C211yesmaximal, characteristic, complemented
4C7×#1805C211yesmaximal, characteristic, complemented
5#11729381C211yesmaximal, characteristic, complemented
6#11728234C211yesmaximal, characteristic, complemented
7#11728022C212yesmaximal, complemented
8#11727980C212yesmaximal, complemented
9#11727310C211yesmaximal, characteristic, complemented
10#11726634C212yesmaximal
11#11726558C212yesmaximal, complemented
12C2×C4×C7×Q8C2211yescharacteristic
13C2×#6510C2211yescharacteristic
14C2×#6509C2211yescharacteristic
15#90271—24no—
16#90271C418yescomplemented
17C2×#6461—21no—
18C2×#6458C2211yescharacteristic
19C2×#6457—21no—
20C2×D7×C4⋊C4C2212yes—
21C2×#6453C2211yescharacteristic
22C2×C4×D28—21no—
23C2×C4×D28C2211yescharacteristic
24C2×#6435—21no—
25C2×C42×D7C2211yescharacteristic
26C2×C7×C4⋊C8C2211yescharacteristic
27C2×C7×#103C2212yes—
28C7×#382C2211yescharacteristic
29C2×#6409C2212yes—
30C2×#6395C2212yes—
31C2×(C2×D7)⋊C8C2212yes—
32C2×C8×Dic7C2212yes—
33#89749C2212yes—
34C2×#6383C2212yes—
35#89708C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89605—24no—
38D7×#95—21no—
39D7×#95—21no—
40#89422—21no—
41#89422—21no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21861—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C22×D28—21no—
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C22×C4×D7—22no—
52C22×C4×D7D411yescharacteristic
53C22×C4×D7D411yescharacteristic
54C22×C4×D7C2311yescharacteristic
55C22×C7×C8C2312yes—
56C4×C7×Q8—22no—
57C4×C7×Q8C2×C414yescomplemented
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4C2312yes—
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C42×C7—21no—
62C2×C42×C7C2311yescharacteristic
63#6510—22no—
64#6510C2×C414yescomplemented
65#6509—22no—
66#6509C2×C414yes—
67C2×(C2×D7)⋊C4—21no—
68C2×(C2×D7)⋊C4—21no—
69C2×(C2×D7)⋊C4—21no—
70C2×(C2×D7)⋊C4—21no—
71C2×(C2×D7)⋊C4—21no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7C2312yes—
74C2×C4⋊Dic7C2311yescharacteristic
75C2×Dic7⋊C4—21no—
76C2×Dic7⋊C4—21no—
77C2×Dic7⋊C4—22no—
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7C2311yescharacteristic
81C22×C7⋊C8C2312yes—
82#6461—28no—
83#6458—22no—
84#6458C2×C414yes—
85#6457—28no—
86D7×C4⋊C4—42no—
87D7×C4⋊C4—42no—
88#6453—22no—
89#6453C2×C414yescomplemented
90C4×D28—28no—
91C4×D28—22no—
92C4×D28C2×C414yescomplemented
93#6435—28no—
94C42×D7C2×C414yes—
95C42×D7D414yes—
96C7×C4⋊C8—22no—
97C7×#103—42no—
98C7×#95—21no—
99C7×#95—21no—
100#6409—42no—
101C7⋊#95—21no—
102C7⋊#95—21no—
103C7⋊#95—22no—
104C7⋊#95—22no—
105#6395—42no—
106(C2×D7)⋊C8—24no—
107C8×Dic7—24no—
108#6383—42no—
109C4⋊(C7⋊C8)—22no—
110C2×#529—71no—
111C2×#431—71no—
112C2×#382—71no—
113#1991—72no—
114#1958—72no—
115#1874—72no—
116#1872—72no—
117#1805—71no—
118#1805D711yescharacteristic, complemented
119#1804—71no—
120#1804—71no—
121C23×D7—21no—
122C23×D7C2×D411yescharacteristic
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—21no—
128C2×C7×Q8C22×C412yes—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—22no—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2411yescharacteristic
135C22×Dic7—21no—
136C22×Dic7—22no—
137C22×Dic7C2×D411yescharacteristic
138C22×Dic7C2×D411yescharacteristic
139C22×Dic7C2×D411yescharacteristic
140C2×D28—21no—
141C2×D28—21no—
142C2×D28—21no—
143C2×D28—21no—
144C2×D28—24no—
145C2×D28—21no—
146C2×D28—44no—
147C2×D28C22×C412yes—
148C2×C4×D7—24no—
149C2×C4×D7—21no—
150C2×C4×D7—21no—
151C2×C4×D7—21no—
152C2×C4×D7—21no—
153C2×C4×D7—24no—
154C2×C4×D7—42no—
155C2×C4×D7—44no—
156C2×C4×D7—44no—
157C2×C4×D7—44no—
158C2×C4×D7—44no—
159C2×C4×D7C22⋊C414yes—
160C2×C4×D7C22⋊C414yes—
161C2×C4×D7C22×C412yes—
162C2×C4×D7C2×D412yes—
163C2×C7×C8—22no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C7×C4⋊C4—24no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4C22×C412yes—
171C42×C7—24no—
172C42×C7C22×C412yes—
173C42×C7C2×D412yes—
174(C2×D7)⋊C4—24no—
175(C2×D7)⋊C4—24no—
176(C2×D7)⋊C4—24no—
177(C2×D7)⋊C4—24no—
178(C2×D7)⋊C4—24no—
179C4⋊Dic7—21no—
180C4⋊Dic7—24no—
181C4⋊Dic7—22no—
182C4⋊Dic7—22no—
183C4⋊Dic7C22×C412yes—
184Dic7⋊C4—24no—
185Dic7⋊C4—24no—
186Dic7⋊C4—44no—
187C4×Dic7C22⋊C414yes—
188C4×Dic7C22⋊C414yes—
189C4×Dic7C22×C412yes—
190C4×Dic7C2×D412yes—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194#529—78no—
195#529—144no—
196C2×#126—141no—
197C2×#125—71no—
198C2×#124—71no—
199C2×#123—141no—
200C2×C22⋊Q8—71no—
201C2×#121—71no—
202C2×C4×Q8C2×D711yescharacteristic
203C2×C4×D4—71no—
204C2×C4×D4—141no—
205C2×#117—141no—
206C22×C4⋊C4—72no—
207C22×C42—71no—
208#431—144no—
209C2×C4⋊C8—71no—
210C2×C4⋊C8C2×D711yescharacteristic
211C2×#103—72no—
212C2×#103C2×D712yes—
213C2×#102—72no—
214C2×#102—72no—
215C2×C22⋊C8—72no—
216C2×C4×C8—72no—
217#382—72no—
218#382—71no—
219#382—141no—
220#382—141no—
221#382C2×D711yescharacteristic
222C2×#95—141no—
223C2×#95—141no—
224#340—72no—
225#340—72no—
226#340—72no—
227#340C2×D712yes—
228C23×C7C22×D411yescharacteristic
229C22×D7—21no—
230C22×D7—21no—
231C22×D7—22no—
232C22×D7—22no—
233C22×D7—41no—
234C22×D7—41no—
235C22×D7—42no—
236C22×D7—42no—
237C22×D7C2×C22⋊C412yes—
238C22×D7#13514yes—
239C7×Q8—22no—
240C7×Q8—22no—
241C7×Q8—41no—
242C7×Q8—41no—
243C2×C4×C7—22no—
244C2×C4×C7—22no—
245C2×C4×C7—22no—
246C2×C4×C7—22no—
247C2×C4×C7—41no—
248C2×C4×C7—41no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7—42no—
253C2×C4×C7C2×C22⋊C412yes—
254C2×C4×C7C2×C22⋊C412yes—
255C2×C4×C7C23×C411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7H(5,𝔽2)11yescharacteristic
258C2×C4×C7H(5,𝔽2)11yescharacteristic
259C2×C4×C72−1+411yescharacteristic
260C2×C4×C72−1+411yescharacteristic
261C2×Dic7—21no—
262C2×Dic7—21no—
263C2×Dic7—22no—
264C2×Dic7—22no—
265C2×Dic7—22no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7C2×C22⋊C412yes—
271C2×Dic7C2×C22⋊C412yes—
272C2×Dic7C2×C22⋊C412yes—
273D28—22no—
274D28—22no—
275D28—48no—
276D28—41no—
277D28—41no—
278C4×D7—28no—
279C4×D7—22no—
280C4×D7—22no—
281C4×D7—22no—
282C4×D7—22no—
283C4×D7—44no—
284C4×D7—48no—
285C4×D7—88no—
286C7×C8—42no—
287C7⋊C8—42no—
288C22×Q8C22×D711yescharacteristic
289C22×D4—71no—
290C22×D4—141no—
291C23×C4—71no—
292C23×C4—71no—
293C23×C4—71no—
294C23×C4—142no—
295C23×C4—141no—
296C22×C8—72no—
297C22×C8C22×D712yes—
298#126—148no—
299#125—74no—
300#125—142no—
301#124—74no—
302#124—142no—
303#123—148no—
304C22⋊Q8—74no—
305C22⋊Q8—142no—
306#121—74no—
307#121—142no—
308C4×Q8—22no—
309C4×Q8C4×D714yescomplemented
310C4×D4—74no—
311C4×D4—148no—
312C4×D4—142no—
313#117—148no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—71no—
316C2×C4⋊C4—72no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—141no—
319C2×C4⋊C4—141no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—282no—
322C2×C4⋊C4—282no—
323C2×C4⋊C4C22×D712yes—
324C2×C4⋊C4C22×D711yescharacteristic
325C2×C22⋊C4—141no—
326C2×C22⋊C4—141no—
327C2×C22⋊C4—141no—
328C2×C22⋊C4—141no—
329C2×C22⋊C4—141no—
330C2×C42—21no—
331C2×C42—71no—
332C2×C42—71no—
333C2×C42—71no—
334C2×C42—74no—
335C2×C42—74no—
336C2×C42C22×D711yescharacteristic
337C4⋊C8—22no—
338C4⋊C8—142no—
339#103—42no—
340#103—282no—
341#102—282no—
342#102—282no—
343C22⋊C8—144no—
344C4×C8—144no—
345#95—21no—
346#95—21no—
347#95—142no—
348#95—142no—
349#95—141no—
350#95—141no—
351C22×C7C22×C22⋊C411yescharacteristic
352C22×C7#53511yescharacteristic
353C22×C7#53611yescharacteristic
354C22×C7C2×#13512yes—
355C22×C7#57512yes—
356C2×D7—22no—
357C2×D7—22no—
358C2×D7—24no—
359C2×D7—24no—
360C2×D7—42no—
361C2×D7—42no—
362C2×D7—42no—
363C2×D7—82no—
364C2×D7—82no—
365C2×D7#41614yes—
366C4×C7—22no—
367C4×C7—22no—
368C4×C7—42no—
369C4×C7—42no—
370C4×C7—82no—
371C4×C7#52011yescharacteristic
372C4×C7#52011yescharacteristic
373C4×C7#53711yescharacteristic
374C4×C7#53711yescharacteristic
375Dic7—22no—
376Dic7—22no—
377Dic7—22no—
378Dic7—42no—
379Dic7—82no—
380C24—71no—
381C24—141no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8C2×C4×D712yes—
388C2×D4—72no—
389C2×D4—144no—
390C2×D4—141no—
391C2×D4—141no—
392C2×D4—141no—
393C2×D4—141no—
394C2×D4—141no—
395C2×D4—284no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—71no—
400C22×C4—71no—
401C22×C4—71no—
402C22×C4—74no—
403C22×C4—72no—
404C22×C4—72no—
405C22×C4—74no—
406C22×C4—144no—
407C22×C4—141no—
408C22×C4—141no—
409C22×C4—141no—
410C22×C4—141no—
411C22×C4—141no—
412C22×C4—144no—
413C22×C4—142no—
414C22×C4—282no—
415C22×C4—284no—
416C22×C4—284no—
417C22×C4—284no—
418C22×C4—284no—
419C22×C4D4×D711yescharacteristic
420C22×C4D4×D711yescharacteristic
421C22×C4C23×D711yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—142no—
426C2×C8—142no—
427C2×C8—142no—
428C4⋊C4—24no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—72no—
433C4⋊C4—144no—
434C4⋊C4—144no—
435C4⋊C4—144no—
436C4⋊C4—141no—
437C4⋊C4—142no—
438C4⋊C4—142no—
439C4⋊C4—284no—
440C4⋊C4C2×C4×D712yes—
441C22⋊C4—144no—
442C22⋊C4—144no—
443C22⋊C4—144no—
444C22⋊C4—144no—
445C22⋊C4—144no—
446C42—24no—
447C42—74no—
448C42—74no—
449C42—72no—
450C42—72no—
451C42C2×C4×D712yes—
452C42D4×D712yes—
453C2×C7#231111yescharacteristic
454C2×C7C2×#41611yescharacteristic
455C2×C7#289611yescharacteristic
456C2×C7#321112yes—
457C2×C7#323212yes—
458D7—24no—
459D7—82no—
460C23—72no—
461C23—74no—
462C23—142no—
463C23—141no—
464C23—141no—
465C23—142no—
466C23—282no—
467C23—282no—
468C23—281no—
469C23—281no—
470C23C2×D4×D711yescharacteristic
471Q8—22no—
472Q8—22no—
473Q8—41no—
474Q8—41no—
475D4—142no—
476D4—142no—
477D4—288no—
478D4—281no—
479D4—281no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—72no—
491C2×C4—72no—
492C2×C4—72no—
493C2×C4—148no—
494C2×C4—142no—
495C2×C4—142no—
496C2×C4—141no—
497C2×C4—141no—
498C2×C4—142no—
499C2×C4—142no—
500C2×C4—142no—
501C2×C4—142no—
502C2×C4—142no—
503C2×C4—284no—
504C2×C4—288no—
505C2×C4—282no—
506C2×C4—282no—
507C2×C4—282no—
508C2×C4—282no—
509C2×C4—568no—
510C2×C4D7×C22⋊C412yes—
511C2×C4D7×C22⋊C412yes—
512C2×C4C22×C4×D711yescharacteristic
513C2×C4C2×D4×D711yescharacteristic
514C2×C4#654811yescharacteristic
515C2×C4#655111yescharacteristic
516C2×C4#655311yescharacteristic
517C2×C4#655411yescharacteristic
518C8—42no—
519C8—282no—
520C7#2186111yescharacteristic, complemented, Sylow
521C22—74no—
522C22—144no—
523C22—142no—
524C22—144no—
525C22—142no—
526C22—282no—
527C22—282no—
528C22—282no—
529C22—562no—
530C22—562no—
531C22C2×D7×C22⋊C411yescharacteristic
532C22#9029811yescharacteristic
533C22#9031911yescharacteristic
534C22D7×#13512yes—
535C22#9046812yes—
536C4—22no—
537C4—22no—
538C4—42no—
539C4—42no—
540C4—82no—
541C4—142no—
542C4—142no—
543C4—142no—
544C4—282no—
545C4—562no—
546C4#9019511yescharacteristic
547C4#9021411yescharacteristic
548C4#9029911yescharacteristic
549C4#9032011yescharacteristic
550C2—144no—
551C2—562no—
552C2#1173144811yescharacteristic
553C2D7×#41611yescharacteristic
554C2#1173640111yescharacteristic
555C2#1173805012yes—
556C2#1173868212yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1805C2
socleC23×C7C22×D4
prime radicalC2×#6458C22
cosocleC22×C4×C7C24
hypercenter#1805D7
hypocenterC7#21861
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#89605C2semidirect
3. D7×#382C2semidirect
4. C7×#1805C2semidirect
5. #11729381C2semidirect
6. #11728234C2semidirect
7. #11728022C2semidirect
8. #11727980C2semidirect
9. #11727310C2semidirect
11. #11726558C2semidirect
16. #90271C4semidirect
57. C4×C7×Q8C2×C4semidirect
64. #6510C2×C4semidirect
89. #6453C2×C4semidirect
92. C4×D28C2×C4semidirect
118. #1805D7semidirect
309. C4×Q8C4×D7semidirect
520. C7#21861semidirect

Maximal subgroups

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

subgroupindex
C2×#902712
C2×#896052
D7×#3822
C7×#18052
#117293812
#117282342
#117280222
#117279802
#117273102
#117266342
#117265582
#218617