Finite Group Databasebeta

← #13325193

Subgroups — #13325193

Order 1792.

566 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11737412C211yesmaximal, characteristic
2#11737366C211yesmaximal, characteristic
3#11737167C211yesmaximal, characteristic
4C7×#1584C211yesmaximal, characteristic, complemented
5#11728706C211yesmaximal, characteristic, complemented
6#11728686C211yesmaximal, characteristic, complemented
7Q16×C7⋊C8C211yesmaximal, characteristic, complemented
8#11726327C211yesmaximal, characteristic
9#11726318C211yesmaximal, characteristic, complemented
10#11725997C211yesmaximal, characteristic
11#11725985C211yesmaximal, characteristic
12#11725975C211yesmaximal, characteristic, complemented
13#11725975C211yesmaximal, characteristic, complemented
14#11725840C211yesmaximal, characteristic, complemented
15#11725828C211yesmaximal, characteristic
16#90269C2211yescharacteristic
17C4×Q8×D7C2211yescharacteristic
18C7×#446C2211yescharacteristic
19C7×C8×Q8C2211yescharacteristic
20C4×C7×Q16C2211yescharacteristic
21#89958C411yescharacteristic
22#89954C411yescharacteristic
23C4×#1144C2211yescharacteristic
24C4×#1144C2211yescharacteristic
25Q8×C7⋊C8C2211yescharacteristic
26Q8×C7⋊C8C2211yescharacteristic
27#89656C411yescharacteristic
28#89647C411yescharacteristic
29#89612—21no—
30#89607—21no—
31#89607C2211yescharacteristic
32#89605C2211yescharacteristic
33#89603—21no—
34#89603C2211yescharacteristic
35#89603C2211yescharacteristic
36#89602—21no—
37#89581C411yescharacteristic
38#89580C411yescharacteristic
39#89566C411yescharacteristic
40#89565C411yescharacteristic
41#89481—21no—
42#89473—21no—
43C4×Dic28C2211yescharacteristic
44#89457—21no—
45#89457C2211yescharacteristic
46#89457C2211yescharacteristic
47#89457C2211yescharacteristic
48#89451—21no—
49#89451C2211yescharacteristic
50C8×Dic14C2211yescharacteristic
51C7×#335C2211yescharacteristic
52C7×Q8⋊C8C2211yescharacteristic
53C7×Q8⋊C8C2211yescharacteristic
54C7×#322C2211yescharacteristic
55Q8⋊(C7⋊C8)C2211yescharacteristic
56Q8⋊(C7⋊C8)C2211yescharacteristic
57#89279C2211yescharacteristic
58#89279C2211yescharacteristic
59#89275C2211yescharacteristic
60#89275C2211yescharacteristic
61#89253C2211yescharacteristic
62#89251C2211yescharacteristic
63#89251C2211yescharacteristic
64#89250C2211yescharacteristic
65#89249C2211yescharacteristic
66C8×C7⋊C8C2211yescharacteristic
67#18032—71nomaximal, Sylow
68C2×Q8×D7C2×C411yescharacteristic
69C2×C7×Q16C2×C411yescharacteristic
70C4×C7×Q8C2311yescharacteristic
71C4×C7×Q8C2311yescharacteristic
72#6509C2×C411yescharacteristic
73Q8×Dic7C2×C411yescharacteristic
74#6507C2×C411yescharacteristic
75C2×#1144C2×C411yescharacteristic
76C2×#1144C2×C411yescharacteristic
77C2×Dic28C2×C411yescharacteristic
78#6460C2×C411yescharacteristic
79#6459—21no—
80D7×C4⋊C4—21no—
81D7×C4⋊C4C2×C411yescharacteristic
82#6452—21no—
83#6451C2×C411yescharacteristic
84#6450—21no—
85#6450C2×C411yescharacteristic
86#6435—21no—
87C42×D7—21no—
88C42×D7C2311yescharacteristic
89C4×Dic14—21no—
90C4×Dic14—21no—
91C4×Dic14C2311yescharacteristic
92C4×Dic14C2311yescharacteristic
93C7×#107C2×C411yescharacteristic
94C7×C4⋊C8—21no—
95C7×C4⋊C8—21no—
96C7×C4⋊C8C2311yescharacteristic
97C7×C4⋊C8C2311yescharacteristic
98C7×#103C2×C411yescharacteristic
99C7×#103C2×C411yescharacteristic
100C7×#97—21no—
101C4×C7×C8—21no—
102C4×C7×C8C2311yescharacteristic
103C4×C7×C8C2311yescharacteristic
104C4×C7×C8C2311yescharacteristic
105#6409C2×C411yescharacteristic
106#6409C2×C411yescharacteristic
107(C2×D7)⋊C8—21no—
108(C2×D7)⋊C8—21no—
109(C2×D7)⋊C8—21no—
110(C2×D7)⋊C8C2×C411yescharacteristic
111(C2×D7)⋊C8C2×C411yescharacteristic
112(C2×D7)⋊C8D411yescharacteristic
113(C2×D7)⋊C8D411yescharacteristic
114#6392C2×C411yescharacteristic
115#6390C2×C411yescharacteristic
116#6390C2×C411yescharacteristic
117#6389—21no—
118#6389—21no—
119#6389—21no—
120#6389—21no—
121Dic7⋊C8—21no—
122Dic7⋊C8—21no—
123Dic7⋊C8—21no—
124Dic7⋊C8—21no—
125Dic7⋊C8—21no—
126Dic7⋊C8C2×C411yescharacteristic
127Dic7⋊C8C2×C411yescharacteristic
128Dic7⋊C8D411yescharacteristic
129Dic7⋊C8D411yescharacteristic
130C8×Dic7—21no—
131C8×Dic7—21no—
132#6384C2×C411yescharacteristic
133#6384C2×C411yescharacteristic
134#6381C2×C411yescharacteristic
135#6381C2×C411yescharacteristic
136C4⋊(C7⋊C8)—21no—
137C4⋊(C7⋊C8)—21no—
138C4⋊(C7⋊C8)C2311yescharacteristic
139C4⋊(C7⋊C8)C2311yescharacteristic
140C4×C7⋊C8—21no—
141C4×C7⋊C8—21no—
142C4×C7⋊C8C2311yescharacteristic
143C4×C7⋊C8C2311yescharacteristic
144C4×C7⋊C8C2311yescharacteristic
145#3104—71no—
146#2986—71no—
147#2984—71no—
148#1593—71no—
149#1591—71no—
150#1587—71no—
151#1587—71no—
152#1584—71no—
153#1584D711yescharacteristic, complemented
154#1579—71no—
155C8×Q16—71no—
156#1572—71no—
157#1497—71no—
158#1479—71no—
159#1454—71no—
160C2×C7×Q8C22×C411yescharacteristic
161C2×C7×Q8C22×C411yescharacteristic
162Q8×D7—22no—
163Q8×D7—41no—
164C2×C4×D7—21no—
165C2×C4×D7—21no—
166C2×C4×D7C22×C411yescharacteristic
167C2×C4×D7C2×D411yescharacteristic
168C2×C4×D7#4111yescharacteristic
169C2×Dic14—21no—
170C2×Dic14—21no—
171C2×Dic14C22×C411yescharacteristic
172C2×Dic14C22×C411yescharacteristic
173C7×Q16—22no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8C22×C411yescharacteristic
178C2×C7×C8C22×C411yescharacteristic
179C2×C7×C8C2×D411yescharacteristic
180C2×C7×C8C2×D411yescharacteristic
181C7×C4⋊C4—21no—
182C7×C4⋊C4—21no—
183C7×C4⋊C4C22×C411yescharacteristic
184C7×C4⋊C4C22×C411yescharacteristic
185C42×C7—21no—
186C42×C7—21no—
187C42×C7C2411yescharacteristic
188#1144—22no—
189#1144—41no—
190(C2×D7)⋊C4—21no—
191(C2×D7)⋊C4—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7—21no—
194C4⋊Dic7C22×C411yescharacteristic
195C4⋊Dic7C22×C411yescharacteristic
196Dic7⋊C4—21no—
197Dic7⋊C4—21no—
198Dic7⋊C4—21no—
199Dic7⋊C4—21no—
200Dic7⋊C4—21no—
201Dic7⋊C4—21no—
202Dic7⋊C4—21no—
203Dic7⋊C4—21no—
204C4×Dic7—21no—
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7C22×C411yescharacteristic
209C4×Dic7C2×D411yescharacteristic
210C4×Dic7#4111yescharacteristic
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8C22×C411yescharacteristic
215C2×C7⋊C8C22×C411yescharacteristic
216C2×C7⋊C8C2×D411yescharacteristic
217C2×C7⋊C8C2×D411yescharacteristic
218Dic28—41no—
219#533—71no—
220C2×C4×Q8—71no—
221#484—71no—
222#484—71no—
223#477—71no—
224#477—71no—
225#462—71no—
226#462—71no—
227#451—71no—
228#451—71no—
229#446—71no—
230#446—71no—
231#446—71no—
232#446—141no—
233#446—141no—
234#446—141no—
235#446C2×D711yescharacteristic
236C8×Q8—71no—
237C8×Q8—71no—
238C8×Q8—71no—
239C8×Q8—141no—
240C8×Q8C2×D711yescharacteristic
241C4×Q16—71no—
242C4×Q16—71no—
243C4×Q16—71no—
244C4×Q16C2×D711yescharacteristic
245#433—141no—
246#433—141no—
247#432—71no—
248#432—141no—
249#431—71no—
250#431—71no—
251#431—71no—
252#431—71no—
253#431—141no—
254#335—71no—
255#335C2×D711yescharacteristic
256#334—71no—
257#334—71no—
258Q8⋊C8—71no—
259Q8⋊C8—71no—
260Q8⋊C8—71no—
261Q8⋊C8—71no—
262Q8⋊C8—71no—
263Q8⋊C8—71no—
264Q8⋊C8C2×D711yescharacteristic
265Q8⋊C8C2×D711yescharacteristic
266#322—71no—
267#322—71no—
268#322C2×D711yescharacteristic
269C82—71no—
270C22×D7C4×D411yescharacteristic
271C7×Q8—21no—
272C7×Q8—21no—
273C7×Q8—21no—
274C7×Q8#13112yes—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7C23×C411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×Dic7—21no—
283C2×Dic7—21no—
284C2×Dic7—21no—
285C2×Dic7—21no—
286C2×Dic7C4×D411yescharacteristic
287C2×Dic7C4×D411yescharacteristic
288C2×Dic7C4×D411yescharacteristic
289C4×D7—21no—
290C4×D7—21no—
291C4×D7—21no—
292C4×D7—21no—
293C4×D7—41no—
294C4×D7—42no—
295C4×D7—42no—
296Dic14—21no—
297Dic14—21no—
298Dic14—21no—
299Dic14—21no—
300Dic14—42no—
301Dic14—42no—
302C7×C8—21no—
303C7×C8—21no—
304C7×C8—21no—
305C7×C8—21no—
306C7×C8—41no—
307C7×C8—41no—
308C7×C8—41no—
309C7⋊C8—21no—
310C7⋊C8—21no—
311C7⋊C8—21no—
312C7⋊C8—21no—
313C7⋊C8—41no—
314C7⋊C8—41no—
315C7⋊C8—41no—
316C22×Q8—71no—
317C2×Q16—71no—
318C2×Q16—71no—
319C2×Q16—71no—
320C2×Q16C4×D711yescharacteristic
321C4⋊Q8—71no—
322C4⋊Q8—71no—
323#125—141no—
324C22⋊Q8—71no—
325C22⋊Q8—71no—
326C22⋊Q8—141no—
327C4×Q8—71no—
328C4×Q8—71no—
329C4×Q8—71no—
330C4×Q8—71no—
331C4×Q8—141no—
332C4×Q8—141no—
333C4×Q8—141no—
334C4×Q8C22×D711yescharacteristic
335C4×Q8C22×D711yescharacteristic
336#117—141no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—141no—
339C2×C42—71no—
340C2×C42—141no—
341#107—71no—
342#107—71no—
343#107—71no—
344#107C4×D711yescharacteristic
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—71no—
348C4⋊C8—71no—
349C4⋊C8—71no—
350C4⋊C8—71no—
351C4⋊C8—71no—
352C4⋊C8—71no—
353C4⋊C8—141no—
354C4⋊C8—141no—
355C4⋊C8—141no—
356C4⋊C8—141no—
357C4⋊C8—141no—
358C4⋊C8—141no—
359C4⋊C8—141no—
360C4⋊C8C22×D711yescharacteristic
361C4⋊C8C22×D711yescharacteristic
362#103—71no—
363#103—71no—
364#103—71no—
365#103—71no—
366#103—71no—
367#103—71no—
368#103C4×D711yescharacteristic
369#103C4×D711yescharacteristic
370C22⋊C8—71no—
371C22⋊C8—71no—
372C22⋊C8—71no—
373C22⋊C8—71no—
374C22⋊C8—141no—
375C22⋊C8—141no—
376C22⋊C8—141no—
377#97—21no—
378#97—141no—
379#97—141no—
380#97—141no—
381#97—141no—
382C4×C8—21no—
383C4×C8—71no—
384C4×C8—71no—
385C4×C8—71no—
386C4×C8—141no—
387C4×C8—141no—
388C4×C8—141no—
389C4×C8—141no—
390C4×C8C22×D711yescharacteristic
391C4×C8C22×D711yescharacteristic
392C4×C8C22×D711yescharacteristic
393C22×C7C2×C4×D411yescharacteristic
394C2×D7—21no—
395C2×D7—21no—
396C2×D7—21no—
397C4×C7—22no—
398C4×C7—21no—
399C4×C7—41no—
400C4×C7—41no—
401C4×C7—41no—
402C4×C7C2×#13111yescharacteristic
403C4×C7#56811yescharacteristic
404C4×C7#57811yescharacteristic
405C4×C7#57811yescharacteristic
406Dic7—21no—
407Dic7—21no—
408Dic7—21no—
409Dic7—41no—
410Dic7—41no—
411Dic7—41no—
412Dic7—41no—
413C2×Q8—71no—
414C2×Q8—71no—
415C2×Q8—141no—
416C2×Q8—141no—
417C2×Q8—142no—
418C2×Q8—281no—
419C2×Q8C2×C4×D711yescharacteristic
420C2×Q8C2×C4×D711yescharacteristic
421C22×C4—71no—
422C22×C4—71no—
423C22×C4—71no—
424C22×C4—141no—
425C22×C4—141no—
426Q16—22no—
427Q16—142no—
428Q16—281no—
429Q16—281no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—71no—
434C2×C8—71no—
435C2×C8—71no—
436C2×C8—71no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8C2×C4×D711yescharacteristic
441C2×C8C2×C4×D711yescharacteristic
442C2×C8D4×D711yescharacteristic
443C2×C8D4×D711yescharacteristic
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—71no—
447C4⋊C4—71no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4C2×C4×D711yescharacteristic
459C4⋊C4C2×C4×D711yescharacteristic
460C22⋊C4—141no—
461C22⋊C4—141no—
462C42—21no—
463C42—21no—
464C42—71no—
465C42—71no—
466C42—71no—
467C42—141no—
468C42—141no—
469C42—141no—
470C42—141no—
471C42C23×D711yescharacteristic
472C2×C7#293311yescharacteristic
473C2×C7#295011yescharacteristic
474C2×C7#295511yescharacteristic
475D7—41no—
476C23—71no—
477Q8—21no—
478Q8—21no—
479Q8—21no—
480Q8—141no—
481Q8—141no—
482Q8—141no—
483Q8—141no—
484Q8—282no—
485Q8—282no—
486Q8#647012yes—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—71no—
492C2×C4—71no—
493C2×C4—71no—
494C2×C4—141no—
495C2×C4—141no—
496C2×C4—141no—
497C2×C4—141no—
498C2×C4—141no—
499C2×C4—141no—
500C2×C4—141no—
501C2×C4—141no—
502C2×C4—281no—
503C2×C4—282no—
504C2×C4—282no—
505C2×C4C22×C4×D711yescharacteristic
506C2×C4C2×D4×D711yescharacteristic
507C2×C4D7×#4111yescharacteristic
508C8—21no—
509C8—21no—
510C8—21no—
511C8—21no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—141no—
516C8—141no—
517C8—141no—
518C8—141no—
519C8—281no—
520C8—281no—
521C8—281no—
522C7#1803211yescharacteristic, complemented, Sylow
523C22—141no—
524C22—141no—
525C22—141no—
526C22C4×D4×D711yescharacteristic
527C4—22no—
528C4—21no—
529C4—41no—
530C4—41no—
531C4—41no—
532C4—141no—
533C4—141no—
534C4—141no—
535C4—281no—
536C4—281no—
537C4—281no—
538C4—281no—
539C4C2×#647011yescharacteristic
540C4#9044111yescharacteristic
541C4#9045711yescharacteristic
542C4#9046211yescharacteristic
543C2—281no—
544C2#1173692611yescharacteristic
545C2#1173716511yescharacteristic
546C2#1173734211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1584C2
socleC22×C7C2×C4×D4
prime radicalC22×D7C4×D4
cosocleC42×C7C24
hypercenter#1584D7
hypocenterC7#18032
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
4. C7×#1584C2semidirect
5. #11728706C2semidirect
6. #11728686C2semidirect
7. Q16×C7⋊C8C2semidirect
9. #11726318C2semidirect
12. #11725975C2semidirect
13. #11725975C2semidirect
14. #11725840C2semidirect
153. #1584D7semidirect
522. C7#18032semidirect

Maximal subgroups

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

subgroupindex
#117374122
#117373662
#117371672
C7×#15842
#117287062
#117286862
Q16×C7⋊C82
#117263272
#117263182
#117259972
#117259852
#117259752
#117258402
#117258282
#180327