Finite Group Databasebeta

← #13322201

Subgroups — #13322201

Order 1792.

972 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2D7×#431C211yesmaximal, characteristic
3#11736483C211yesmaximal, characteristic
4C7×#1549C211yesmaximal, characteristic, complemented
5#11729529C211yesmaximal, characteristic, complemented
6#11728532C211yesmaximal, characteristic, complemented
7#11728260C211yesmaximal, characteristic, complemented
8#11726179C214yesmaximal, complemented
9#11726173C214yesmaximal, complemented
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×#6509—21no—
12C2×Q8×Dic7—21no—
13#90419—21no—
14#90419—21no—
15#90419—21no—
16#90419—21no—
17#90419C412yes—
18#90419C412yes—
19#90419C2212yes—
20#90419C2212yes—
21C2×#6460—21no—
22C2×#6455C412yes—
23C2×#6453C412yes—
24C2×#6450—21no—
25C2×C42×D7C2211yescharacteristic
26C2×#6432C2211yescharacteristic
27C7×#431C2211yescharacteristic
28C7×#423C2211yescharacteristic
29#89898—21no—
30M16×Dic7—21no—
31#89697C2211yescharacteristic
32#89696C2211yescharacteristic
33#89607—21no—
34#89607C2212yes—
35#89605—21no—
36#89605C2212yes—
37#89605C2212yes—
38D7×C4⋊C8C2212yes—
39D7×C22⋊C8—21no—
40#89490—21no—
41#89457—21no—
42C4×C8×D7—21no—
43C7×#332C2214yescomplemented
44#89342C2214yescomplemented
45#89281C2214yescomplemented
46#89277C2214yescomplemented
47#17715—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C4×D7C2×C411yescharacteristic
50C22×C4×D7D411yescharacteristic
51C22×C4×D7D411yescharacteristic
52C22×Dic14—21no—
53C2×C7×M16—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8C2×C412yes—
57C7×C4⋊Q8C2312yes—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4C2×C412yes—
60C2×C42×C7C2311yescharacteristic
61#6509—22no—
62#6509—22no—
63#6509—22no—
64#6509—22no—
65Q8×Dic7—22no—
66Q8×Dic7—22no—
67Q8×Dic7—22no—
68Q8×Dic7—22no—
69C2×(C2×D7)⋊C4—22no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7—22no—
72C2×C4⋊Dic7C2×C412yes—
73C2×Dic7⋊C4—22no—
74C2×C4×Dic7—22no—
75C2×C4×Dic7C2×C411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C2×C4×Dic7D411yescharacteristic
78C2×#1136—21no—
79C2×C8×D7—22no—
80#6460—22no—
81#6460—22no—
82#6460—22no—
83#6460—22no—
84#6455—22no—
85#6455—22no—
86#6455C2×C414yes—
87#6455C2×C414yes—
88#6453—22no—
89#6453—22no—
90#6453C2×C414yes—
91#6453C2×C414yes—
92#6450—22no—
93#6450—22no—
94#6450—22no—
95#6450—22no—
96C42×D7C2×C411yescharacteristic
97C42×D7C2×C411yescharacteristic
98C42×D7D411yescharacteristic
99C42×D7D411yescharacteristic
100C42×D7D411yescharacteristic
101C42×D7D411yescharacteristic
102C42×D7C2311yescharacteristic
103C42×D7C2311yescharacteristic
104#6432—21no—
105#6432—21no—
106#6432C2×C412yes—
107#6432C2312yes—
108C7×C4⋊C8—21no—
109C7×C4⋊C8C2312yes—
110C7×C4⋊C8C2312yes—
111C7×C22⋊C8—21no—
112C4×C7×C8—21no—
113C22⋊(C7⋊C8)—21no—
114(C2×D7)⋊C8—22no—
115(C2×D7)⋊C8—22no—
116(C2×D7)⋊C8—22no—
117#6389—22no—
118Dic7⋊C8—22no—
119C8×Dic7—22no—
120C8×Dic7—22no—
121C4⋊(C7⋊C8)—21no—
122C4⋊(C7⋊C8)C2312yes—
123C4⋊(C7⋊C8)C2312yes—
124C4×C7⋊C8—21no—
125C2×#533—71no—
126#2920—71no—
127C2×#431—71no—
128#1552—74no—
129#1549—71no—
130#1549D711yescharacteristic, complemented
131#1546—71no—
132#1546—71no—
133#1544—74no—
134C23×D7C22⋊C411yescharacteristic
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—44no—
138C22×C4×C7—22no—
139C22×C4×C7C22×C411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×Dic7—22no—
143C22×Dic7C22⋊C411yescharacteristic
144C22×Dic7C22⋊C411yescharacteristic
145C22×Dic7C22⋊C411yescharacteristic
146C2×C4×D7—22no—
147C2×C4×D7—22no—
148C2×C4×D7—24no—
149C2×C4×D7—42no—
150C2×C4×D7C22⋊C411yescharacteristic
151C2×C4×D7C22⋊C411yescharacteristic
152C2×C4×D7C22⋊C411yescharacteristic
153C2×C4×D7C22⋊C411yescharacteristic
154C2×C4×D7C22⋊C411yescharacteristic
155C2×C4×D7C22⋊C411yescharacteristic
156C2×C4×D7C22×C411yescharacteristic
157C2×C4×D7C22×C411yescharacteristic
158C2×C4×D7C2×D411yescharacteristic
159C2×C4×D7C2×D411yescharacteristic
160C2×C4×D7C2×D411yescharacteristic
161C2×C4×D7C2×D411yescharacteristic
162C2×Dic14—22no—
163C2×Dic14—22no—
164C2×Dic14—42no—
165C2×Dic14—42no—
166C7×M16—42no—
167C2×C7×C8—22no—
168C2×C7×C8—22no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4C22×C414yes—
173C42×C7C22×C411yescharacteristic
174C42×C7C2×D411yescharacteristic
175C42×C7C2×D411yescharacteristic
176C42×C7C2411yescharacteristic
177(C2×D7)⋊C4—24no—
178(C2×D7)⋊C4—24no—
179C4⋊Dic7—22no—
180C4⋊Dic7—22no—
181C4⋊Dic7—22no—
182C4⋊Dic7—24no—
183C4⋊Dic7—24no—
184C4⋊Dic7C22×C414yes—
185Dic7⋊C4—24no—
186Dic7⋊C4—24no—
187C4×Dic7—24no—
188C4×Dic7—24no—
189C4×Dic7C22⋊C411yescharacteristic
190C4×Dic7C22⋊C411yescharacteristic
191C4×Dic7C22⋊C411yescharacteristic
192C4×Dic7C22⋊C411yescharacteristic
193C4×Dic7C22⋊C411yescharacteristic
194C4×Dic7C22⋊C411yescharacteristic
195C4×Dic7C22×C411yescharacteristic
196C4×Dic7C22×C411yescharacteristic
197C4×Dic7C2×D411yescharacteristic
198C4×Dic7C2×D411yescharacteristic
199C4×Dic7C2×D411yescharacteristic
200C4×Dic7C2×D411yescharacteristic
201#1136—42no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C8×D7—44no—
205#533—72no—
206#533—72no—
207#533—72no—
208#533—72no—
209#533—141no—
210#533—141no—
211#533—141no—
212#533—141no—
213C2×C4⋊Q8—71no—
214C2×C4⋊Q8C2×D711yescharacteristic
215C2×#125—72no—
216C2×C22⋊Q8—141no—
217C2×C22⋊Q8—141no—
218C2×C4×Q8—141no—
219C2×C4×Q8—141no—
220C2×#117—72no—
221C22×C42—71no—
222#432—72no—
223#432—141no—
224#431—71no—
225#431—72no—
226#431—72no—
227#431—141no—
228#431—141no—
229#431—141no—
230#431C2×D711yescharacteristic
231#423—71no—
232#423C2×D711yescharacteristic
233C2×C4⋊C8—72no—
234#407—141no—
235C2×C22⋊C8—141no—
236C4×M16—141no—
237C2×C4×C8—141no—
238#332—74no—
239#332—74no—
240#332—74no—
241#332C2×D714yescomplemented
242C23×C7C2×C22⋊C411yescharacteristic
243C22×D7—22no—
244C22×D7—22no—
245C22×D7—41no—
246C22×D7C2×C22⋊C411yescharacteristic
247C22×D7C2×C22⋊C411yescharacteristic
248C7×Q8—44no—
249C7×Q8—44no—
250C2×C4×C7—22no—
251C2×C4×C7—22no—
252C2×C4×C7—24no—
253C2×C4×C7—44no—
254C2×C4×C7C2×C22⋊C411yescharacteristic
255C2×C4×C7C2×C22⋊C411yescharacteristic
256C2×C4×C7C2×C22⋊C411yescharacteristic
257C2×C4×C7#13512yes—
258C2×C4×C7#13512yes—
259C2×C4×C7C23×C411yescharacteristic
260C2×C4×C7C22×D411yescharacteristic
261C2×C4×C7C22×D411yescharacteristic
262C2×Dic7—22no—
263C2×Dic7—22no—
264C2×Dic7—24no—
265C2×Dic7—41no—
266C2×Dic7—44no—
267C2×Dic7C2×C22⋊C411yescharacteristic
268C2×Dic7C2×C22⋊C411yescharacteristic
269C2×Dic7C2×C22⋊C411yescharacteristic
270C2×Dic7C2×C22⋊C411yescharacteristic
271C2×Dic7C2×C22⋊C411yescharacteristic
272C2×Dic7C2×C22⋊C411yescharacteristic
273C4×D7—22no—
274C4×D7—21no—
275C4×D7—21no—
276C4×D7—21no—
277C4×D7—21no—
278C4×D7—22no—
279C4×D7—22no—
280C4×D7—22no—
281C4×D7—22no—
282C4×D7—22no—
283C4×D7—22no—
284C4×D7—41no—
285C4×D7—41no—
286C4×D7#10114yes—
287Dic14—42no—
288Dic14—42no—
289Dic14—44no—
290C7×C8—42no—
291C7×C8—42no—
292C7⋊C8—42no—
293C7⋊C8—42no—
294C22×Q8—21no—
295C22×Q8—141no—
296C23×C4—71no—
297C23×C4—71no—
298C23×C4—71no—
299C2×M16—21no—
300C2×M16—141no—
301C22×C8—142no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—72no—
305C4⋊Q8—72no—
306C4⋊Q8—141no—
307C4⋊Q8—141no—
308C4⋊Q8C4×D712yes—
309C4⋊Q8C22×D712yes—
310#125—74no—
311#125—74no—
312#125—142no—
313#125—142no—
314C22⋊Q8—142no—
315C22⋊Q8—142no—
316C22⋊Q8—142no—
317C22⋊Q8—142no—
318C22⋊Q8—142no—
319C22⋊Q8—142no—
320C22⋊Q8—142no—
321C22⋊Q8—142no—
322C4×Q8—142no—
323C4×Q8—142no—
324C4×Q8—142no—
325C4×Q8—142no—
326C4×Q8—142no—
327C4×Q8—142no—
328C4×Q8—142no—
329C4×Q8—142no—
330#117—74no—
331#117—74no—
332#117—142no—
333#117—142no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—72no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4—142no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4C4×D712yes—
340C2×C22⋊C4—142no—
341C2×C42—71no—
342C2×C42—71no—
343C2×C42—71no—
344C2×C42—71no—
345C2×C42—71no—
346C2×C42—71no—
347C2×C42—71no—
348C2×C42—71no—
349C2×C42—71no—
350C2×C42—71no—
351C2×C42—71no—
352C2×C42—142no—
353C2×C42C22×D711yescharacteristic
354C4⋊C8—21no—
355C4⋊C8—72no—
356C4⋊C8—72no—
357C4⋊C8—142no—
358C4⋊C8—141no—
359C4⋊C8C22×D712yes—
360C4⋊C8C22×D712yes—
361C22⋊C8—21no—
362C22⋊C8—142no—
363C22⋊C8—142no—
364C22⋊C8—142no—
365C22⋊C8—141no—
366#97—142no—
367C4×C8—21no—
368C4×C8—142no—
369C4×C8—142no—
370C4×C8—141no—
371C22×C7—21no—
372C22×C7—21no—
373C22×C7#41612yes—
374C22×C7C22×C22⋊C411yescharacteristic
375C2×D7—21no—
376C2×D7—21no—
377C2×D7—21no—
378C2×D7—21no—
379C2×D7—21no—
380C2×D7—44no—
381C2×D7C2×#10112yes—
382C4×C7—21no—
383C4×C7—21no—
384C4×C7—44no—
385C4×C7C2×#10112yes—
386C4×C7#41612yes—
387C4×C7C2×#13511yescharacteristic
388C4×C7C2×#13511yescharacteristic
389C4×C7#57512yes—
390Dic7—21no—
391Dic7—21no—
392Dic7—21no—
393Dic7—21no—
394Dic7—21no—
395Dic7—44no—
396Dic7C2×#10112yes—
397C24—71no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—142no—
402C2×Q8—142no—
403C2×Q8—282no—
404C2×Q8—282no—
405C22×C4—22no—
406C22×C4—71no—
407C22×C4—71no—
408C22×C4—71no—
409C22×C4—71no—
410C22×C4—71no—
411C22×C4—71no—
412C22×C4—71no—
413C22×C4—71no—
414C22×C4—71no—
415C22×C4—71no—
416C22×C4—71no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—144no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—282no—
426C22×C4C2×C4×D711yescharacteristic
427C22×C4D4×D711yescharacteristic
428C22×C4D4×D711yescharacteristic
429M16—42no—
430M16—282no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—142no—
434C2×C8—142no—
435C2×C8—284no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—74no—
440C4⋊C4—144no—
441C4⋊C4—144no—
442C4⋊C4—144no—
443C4⋊C4—144no—
444C4⋊C4—142no—
445C4⋊C4—142no—
446C4⋊C4—142no—
447C4⋊C4C2×C4×D714yes—
448C22⋊C4—144no—
449C22⋊C4—144no—
450C42—71no—
451C42—71no—
452C42—71no—
453C42—71no—
454C42—71no—
455C42—71no—
456C42—71no—
457C42—71no—
458C42—71no—
459C42—71no—
460C42—71no—
461C42—71no—
462C42—144no—
463C42—144no—
464C42C2×C4×D711yescharacteristic
465C42D4×D711yescharacteristic
466C42D4×D711yescharacteristic
467C42C23×D711yescharacteristic
468C2×C7—22no—
469C2×C7C22×#10111yescharacteristic
470C2×C7C2×#41611yescharacteristic
471C2×C7#289611yescharacteristic
472D7—22no—
473D7—41no—
474C23—71no—
475C23—71no—
476C23—142no—
477C23—142no—
478C23—281no—
479C23D7×C22⋊C411yescharacteristic
480Q8—44no—
481Q8—44no—
482Q8—282no—
483Q8—282no—
484Q8—284no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—44no—
489C2×C4—71no—
490C2×C4—74no—
491C2×C4—71no—
492C2×C4—71no—
493C2×C4—71no—
494C2×C4—71no—
495C2×C4—71no—
496C2×C4—142no—
497C2×C4—142no—
498C2×C4—142no—
499C2×C4—142no—
500C2×C4—141no—
501C2×C4—141no—
502C2×C4—141no—
503C2×C4—141no—
504C2×C4—142no—
505C2×C4—142no—
506C2×C4—142no—
507C2×C4—142no—
508C2×C4—142no—
509C2×C4—144no—
510C2×C4—281no—
511C2×C4—281no—
512C2×C4—281no—
513C2×C4—284no—
514C2×C4D7×C22⋊C411yescharacteristic
515C2×C4D7×C22⋊C411yescharacteristic
516C2×C4D7×C22⋊C411yescharacteristic
517C2×C4#647912yes—
518C2×C4#648212yes—
519C2×C4C22×C4×D711yescharacteristic
520C2×C4C2×D4×D711yescharacteristic
521C2×C4C2×D4×D711yescharacteristic
522C8—42no—
523C8—42no—
524C8—282no—
525C8—282no—
526C7#1771511yescharacteristic, complemented, Sylow
527C22—21no—
528C22—21no—
529C22—72no—
530C22—141no—
531C22—141no—
532C22—141no—
533C22—141no—
534C22—141no—
535C22—284no—
536C22#8957612yes—
537C22C2×D7×C22⋊C411yescharacteristic
538C4—21no—
539C4—21no—
540C4—44no—
541C4—72no—
542C4—141no—
543C4—141no—
544C4—141no—
545C4—141no—
546C4—141no—
547C4—284no—
548C4D7×#10112yes—
549C4#8957612yes—
550C4C2×#647911yescharacteristic
551C4C2×#648211yescharacteristic
552C4#9046412yes—
553C2—22no—
554C2—142no—
555C2—281no—
556C2C2×D7×#10111yescharacteristic
557C2C2×#8957611yescharacteristic
558C2#1173636911yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1549C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1549D7
hypocenterC7#17715
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
4. C7×#1549C2semidirect
5. #11729529C2semidirect
6. #11728532C2semidirect
7. #11728260C2semidirect
8. #11726179C2semidirect
9. #11726173C2semidirect
43. C7×#332C22semidirect
44. #89342C22semidirect
45. #89281C22semidirect
46. #89277C22semidirect
130. #1549D7semidirect
241. #332C2×D7semidirect
526. C7#17715semidirect

Maximal subgroups

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

subgroupindex
C2×#904192
D7×#4312
#117364832
C7×#15492
#117295292
#117285322
#117282602
#117261792
#117261732
#177157