Finite Group Databasebeta

← #13363523

Subgroups — #13363523

Order 1792.

1114 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90269C211yesmaximal, characteristic
2C2×#89607C211yesmaximal, characteristic
3D7×#382C211yesmaximal, characteristic
4C7×#1805C211yesmaximal, characteristic, complemented
5#11729381C211yesmaximal, characteristic, complemented
6#11728240C211yesmaximal, characteristic, complemented
7#11728022C212yesmaximal
8#11727982C212yesmaximal, complemented
9#11727313C211yesmaximal, characteristic, complemented
10#11726632C212yesmaximal
11#11726560C212yesmaximal, complemented
12C2×C4×C7×Q8C2211yescharacteristic
13C2×#6509C2211yescharacteristic
14C2×#6507C2211yescharacteristic
15#90269—24no—
16#90269C418yescomplemented
17C2×#6460C2211yescharacteristic
18C2×#6459—21no—
19C2×D7×C4⋊C4C2212yes—
20C2×#6452—21no—
21C2×#6451C2211yescharacteristic
22C2×#6435—21no—
23C2×C42×D7C2211yescharacteristic
24C2×C4×Dic14—21no—
25C2×C4×Dic14C2211yescharacteristic
26C2×C7×C4⋊C8C2211yescharacteristic
27C2×C7×#103C2212yes—
28C7×#382C2211yescharacteristic
29C2×#6409C2212yes—
30C2×(C2×D7)⋊C8C2212yes—
31C2×#6390C2212yes—
32C2×#6389C2212yes—
33#89749C2212yes—
34C2×#6384C2212yes—
35#89708C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89607—24no—
38D7×#95—21no—
39D7×#95—21no—
40#89422—21no—
41#89422—21no—
42C7×#340C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21866—71nomaximal, Sylow
47C22×C7×Q8C2311yescharacteristic
48C22×C4×D7—22no—
49C22×C4×D7D411yescharacteristic
50C22×C4×D7D411yescharacteristic
51C22×C4×D7C2311yescharacteristic
52C22×Dic14—21no—
53C22×Dic14C2311yescharacteristic
54C22×C7×C8C2312yes—
55C4×C7×Q8—22no—
56C4×C7×Q8C2×C414yescomplemented
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4C2312yes—
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62#6509—22no—
63#6509C2×C414yes—
64#6507—22no—
65#6507C2×C414yescomplemented
66C2×(C2×D7)⋊C4—21no—
67C2×(C2×D7)⋊C4—21no—
68C2×C4⋊Dic7—21no—
69C2×C4⋊Dic7C2312yes—
70C2×C4⋊Dic7C2311yescharacteristic
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—21no—
73C2×Dic7⋊C4—21no—
74C2×Dic7⋊C4—21no—
75C2×Dic7⋊C4—21no—
76C2×Dic7⋊C4—22no—
77C2×C4×Dic7—21no—
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7C2311yescharacteristic
81C22×C7⋊C8C2312yes—
82#6460—22no—
83#6460C2×C414yes—
84#6459—28no—
85D7×C4⋊C4—42no—
86D7×C4⋊C4—42no—
87#6452—28no—
88#6451—22no—
89#6451C2×C414yescomplemented
90#6435—28no—
91C42×D7C2×C414yes—
92C42×D7D414yes—
93C4×Dic14—22no—
94C4×Dic14—28no—
95C4×Dic14C2×C414yescomplemented
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(C2×D7)⋊C8—24no—
106#6390—42no—
107#6389—24no—
108#6384—42no—
109C4⋊(C7⋊C8)—22no—
110C2×#533—71no—
111C2×#432—71no—
112C2×#382—71no—
113#1993—72no—
114#1960—72no—
115#1872—72no—
116#1872—72no—
117#1805—71no—
118#1805—71no—
119#1805—71no—
120#1805D711yescharacteristic, complemented
121C23×D7C2×D411yescharacteristic
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8C22×C412yes—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7—22no—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2411yescharacteristic
134C22×Dic7—21no—
135C22×Dic7—21no—
136C22×Dic7—22no—
137C22×Dic7C2×D411yescharacteristic
138C22×Dic7C2×D411yescharacteristic
139C22×Dic7C2×D411yescharacteristic
140C2×C4×D7—24no—
141C2×C4×D7—21no—
142C2×C4×D7—21no—
143C2×C4×D7—21no—
144C2×C4×D7—21no—
145C2×C4×D7—42no—
146C2×C4×D7—44no—
147C2×C4×D7—44no—
148C2×C4×D7—44no—
149C2×C4×D7C22⋊C414yes—
150C2×C4×D7C22⋊C414yes—
151C2×C4×D7C22×C412yes—
152C2×C4×D7C2×D412yes—
153C2×Dic14—21no—
154C2×Dic14—21no—
155C2×Dic14—21no—
156C2×Dic14—21no—
157C2×Dic14—21no—
158C2×Dic14—24no—
159C2×Dic14—44no—
160C2×Dic14C22×C412yes—
161C2×C7×C8—22no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C7×C4⋊C4—24no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4C22×C412yes—
169C42×C7—24no—
170C42×C7C22×C412yes—
171C42×C7C2×D412yes—
172(C2×D7)⋊C4—24no—
173(C2×D7)⋊C4—24no—
174C4⋊Dic7—21no—
175C4⋊Dic7—24no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7C22×C412yes—
179Dic7⋊C4—24no—
180Dic7⋊C4—24no—
181Dic7⋊C4—24no—
182Dic7⋊C4—24no—
183Dic7⋊C4—24no—
184Dic7⋊C4—44no—
185C4×Dic7—24no—
186C4×Dic7C22⋊C414yes—
187C4×Dic7C22⋊C414yes—
188C4×Dic7C22×C412yes—
189C4×Dic7C2×D412yes—
190C2×C7⋊C8—22no—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193#533—78no—
194#533—144no—
195C2×C4⋊Q8—71no—
196C2×C4⋊Q8—71no—
197C2×#125—141no—
198C2×C22⋊Q8—71no—
199C2×C22⋊Q8—71no—
200C2×C22⋊Q8—141no—
201C2×C4×Q8—71no—
202C2×C4×Q8—141no—
203C2×C4×Q8C2×D711yescharacteristic
204C2×#117—141no—
205C22×C4⋊C4—72no—
206C22×C42—71no—
207#432—144no—
208C2×C4⋊C8—71no—
209C2×C4⋊C8C2×D711yescharacteristic
210C2×#103—72no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#103C2×D712yes—
214C2×C22⋊C8—72no—
215C2×#97—72no—
216#382—72no—
217#382—71no—
218#382—141no—
219#382—141no—
220#382C2×D711yescharacteristic
221C2×#95—141no—
222C2×#95—141no—
223#340—72no—
224#340—72no—
225#340—72no—
226#340C2×D712yes—
227C23×C7C22×D411yescharacteristic
228C22×D7—21no—
229C22×D7—21no—
230C22×D7—22no—
231C22×D7C2×C22⋊C412yes—
232C22×D7#13714yes—
233C7×Q8—22no—
234C7×Q8—22no—
235C7×Q8—41no—
236C7×Q8—41no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—22no—
240C2×C4×C7—22no—
241C2×C4×C7—41no—
242C2×C4×C7—41no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7C2×C22⋊C412yes—
248C2×C4×C7C2×C22⋊C412yes—
249C2×C4×C7C23×C411yescharacteristic
250C2×C4×C7C22×D411yescharacteristic
251C2×C4×C7H(5,𝔽2)11yescharacteristic
252C2×C4×C7H(5,𝔽2)11yescharacteristic
253C2×C4×C72−1+411yescharacteristic
254C2×C4×C72−1+411yescharacteristic
255C2×Dic7—21no—
256C2×Dic7—21no—
257C2×Dic7—22no—
258C2×Dic7—22no—
259C2×Dic7—22no—
260C2×Dic7—22no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—42no—
264C2×Dic7—42no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7C2×C22⋊C412yes—
268C2×Dic7C2×C22⋊C412yes—
269C2×Dic7C2×C22⋊C412yes—
270C4×D7—28no—
271C4×D7—22no—
272C4×D7—22no—
273C4×D7—22no—
274C4×D7—22no—
275C4×D7—44no—
276C4×D7—88no—
277Dic14—22no—
278Dic14—22no—
279Dic14—48no—
280Dic14—41no—
281Dic14—41no—
282C7×C8—42no—
283C7⋊C8—42no—
284C22×Q8—71no—
285C22×Q8—141no—
286C22×Q8C22×D711yescharacteristic
287C23×C4—71no—
288C23×C4—71no—
289C23×C4—71no—
290C23×C4—142no—
291C22×C8—72no—
292C22×C8C22×D712yes—
293C4⋊Q8—74no—
294C4⋊Q8—74no—
295C4⋊Q8—142no—
296C4⋊Q8—142no—
297#125—148no—
298C22⋊Q8—74no—
299C22⋊Q8—74no—
300C22⋊Q8—142no—
301C22⋊Q8—148no—
302C22⋊Q8—142no—
303C4×Q8—22no—
304C4×Q8—74no—
305C4×Q8—148no—
306C4×Q8—142no—
307C4×Q8C4×D714yescomplemented
308#117—148no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4—72no—
312C2×C4⋊C4—141no—
313C2×C4⋊C4—141no—
314C2×C4⋊C4—141no—
315C2×C4⋊C4—141no—
316C2×C4⋊C4—141no—
317C2×C4⋊C4—141no—
318C2×C4⋊C4—142no—
319C2×C4⋊C4—282no—
320C2×C4⋊C4—282no—
321C2×C4⋊C4C22×D712yes—
322C2×C4⋊C4C22×D711yescharacteristic
323C2×C22⋊C4—141no—
324C2×C22⋊C4—141no—
325C2×C42—21no—
326C2×C42—71no—
327C2×C42—71no—
328C2×C42—71no—
329C2×C42—74no—
330C2×C42—74no—
331C2×C42—141no—
332C2×C42C22×D711yescharacteristic
333C4⋊C8—22no—
334C4⋊C8—142no—
335#103—42no—
336#103—282no—
337#103—282no—
338#103—282no—
339C22⋊C8—144no—
340#97—144no—
341#95—21no—
342#95—21no—
343#95—142no—
344#95—142no—
345#95—141no—
346#95—141no—
347C22×C7C22×C22⋊C411yescharacteristic
348C22×C7#53511yescharacteristic
349C22×C7#53611yescharacteristic
350C22×C7C2×#13712yes—
351C22×C7#57512yes—
352C2×D7—22no—
353C2×D7—22no—
354C2×D7—24no—
355C2×D7—24no—
356C2×D7#41914yes—
357C4×C7—22no—
358C4×C7—22no—
359C4×C7—42no—
360C4×C7—42no—
361C4×C7—82no—
362C4×C7#52011yescharacteristic
363C4×C7#52011yescharacteristic
364C4×C7#53711yescharacteristic
365C4×C7#53711yescharacteristic
366Dic7—22no—
367Dic7—22no—
368Dic7—22no—
369Dic7—42no—
370Dic7—42no—
371Dic7—82no—
372C24—71no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—72no—
379C2×Q8—144no—
380C2×Q8—141no—
381C2×Q8—141no—
382C2×Q8—141no—
383C2×Q8—141no—
384C2×Q8—141no—
385C2×Q8—284no—
386C2×Q8C2×C4×D712yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—71no—
391C22×C4—71no—
392C22×C4—71no—
393C22×C4—74no—
394C22×C4—72no—
395C22×C4—72no—
396C22×C4—74no—
397C22×C4—144no—
398C22×C4—141no—
399C22×C4—141no—
400C22×C4—141no—
401C22×C4—141no—
402C22×C4—141no—
403C22×C4—141no—
404C22×C4—142no—
405C22×C4—282no—
406C22×C4—284no—
407C22×C4—284no—
408C22×C4—284no—
409C22×C4D4×D711yescharacteristic
410C22×C4D4×D711yescharacteristic
411C22×C4C23×D711yescharacteristic
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—142no—
416C2×C8—142no—
417C2×C8—142no—
418C4⋊C4—24no—
419C4⋊C4—21no—
420C4⋊C4—22no—
421C4⋊C4—22no—
422C4⋊C4—72no—
423C4⋊C4—144no—
424C4⋊C4—144no—
425C4⋊C4—144no—
426C4⋊C4—144no—
427C4⋊C4—144no—
428C4⋊C4—144no—
429C4⋊C4—141no—
430C4⋊C4—142no—
431C4⋊C4—142no—
432C4⋊C4—284no—
433C4⋊C4C2×C4×D712yes—
434C22⋊C4—144no—
435C22⋊C4—144no—
436C42—24no—
437C42—74no—
438C42—74no—
439C42—72no—
440C42—72no—
441C42—144no—
442C42C2×C4×D712yes—
443C42D4×D712yes—
444C2×C7#231111yescharacteristic
445C2×C7C2×#41911yescharacteristic
446C2×C7#289611yescharacteristic
447C2×C7#321512yes—
448C2×C7#323612yes—
449D7—24no—
450C23—72no—
451C23—74no—
452C23—142no—
453C23—141no—
454C23—141no—
455C23C2×D4×D711yescharacteristic
456Q8—22no—
457Q8—22no—
458Q8—41no—
459Q8—41no—
460Q8—142no—
461Q8—142no—
462Q8—288no—
463Q8—281no—
464Q8—281no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—72no—
476C2×C4—72no—
477C2×C4—72no—
478C2×C4—148no—
479C2×C4—142no—
480C2×C4—142no—
481C2×C4—141no—
482C2×C4—141no—
483C2×C4—142no—
484C2×C4—142no—
485C2×C4—142no—
486C2×C4—142no—
487C2×C4—142no—
488C2×C4—142no—
489C2×C4—284no—
490C2×C4—282no—
491C2×C4—281no—
492C2×C4—281no—
493C2×C4—282no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4—568no—
497C2×C4D7×C22⋊C412yes—
498C2×C4D7×C22⋊C412yes—
499C2×C4C22×C4×D711yescharacteristic
500C2×C4C2×D4×D711yescharacteristic
501C2×C4#654811yescharacteristic
502C2×C4#655111yescharacteristic
503C2×C4#655311yescharacteristic
504C2×C4#655411yescharacteristic
505C8—42no—
506C8—282no—
507C7#2186611yescharacteristic, complemented, Sylow
508C22—74no—
509C22—144no—
510C22—144no—
511C22—142no—
512C22—142no—
513C22C2×D7×C22⋊C411yescharacteristic
514C22#9029811yescharacteristic
515C22#9031911yescharacteristic
516C22#9045912yes—
517C22D7×#13712yes—
518C4—22no—
519C4—22no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—142no—
524C4—142no—
525C4—142no—
526C4—282no—
527C4—282no—
528C4—562no—
529C4#9019511yescharacteristic
530C4#9021411yescharacteristic
531C4#9029911yescharacteristic
532C4#9032011yescharacteristic
533C2—144no—
534C2#1173144811yescharacteristic
535C2#1173627511yescharacteristic
536C2D7×#41911yescharacteristic
537C2#1173805112yes—
538C2#1173868312yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1805C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1805D7
hypocenterC7#21866
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
4. C7×#1805C2semidirect
5. #11729381C2semidirect
6. #11728240C2semidirect
8. #11727982C2semidirect
9. #11727313C2semidirect
11. #11726560C2semidirect
16. #90269C4semidirect
56. C4×C7×Q8C2×C4semidirect
65. #6507C2×C4semidirect
89. #6451C2×C4semidirect
95. C4×Dic14C2×C4semidirect
120. #1805D7semidirect
307. C4×Q8C4×D7semidirect
507. C7#21866semidirect

Maximal subgroups

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

subgroupindex
C2×#902692
C2×#896072
D7×#3822
C7×#18052
#117293812
#117282402
#117280222
#117279822
#117273132
#117266322
#117265602
#218667