Finite Group Databasebeta

← #13322386

Subgroups — #13322386

Order 1792.

1178 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×C4⋊Q8C211yesmaximal, characteristic
2#11736776C211yesmaximal, characteristic
3#11736776C211yesmaximal, characteristic
4C7×#1556C211yesmaximal, characteristic, complemented
5#11729530C211yesmaximal, characteristic, complemented
6#11728577C211yesmaximal, characteristic, complemented
7#11728576C211yesmaximal, characteristic, complemented
8#11726176C214yesmaximal, complemented
9#11726174C214yesmaximal, complemented
10C22×Q8×D7—21no—
11C2×C7×C4⋊Q8C2211yescharacteristic
12C2×#6507—21no—
13D7×C4⋊Q8—21no—
14D7×C4⋊Q8—21no—
15D7×C4⋊Q8—21no—
16D7×C4⋊Q8—21no—
17D7×C4⋊Q8C412yes—
18D7×C4⋊Q8C412yes—
19D7×C4⋊Q8C2212yes—
20D7×C4⋊Q8C2212yes—
21C2×D7×C4⋊C4—21no—
22C2×D7×C4⋊C4C412yes—
23C2×#6451—21no—
24C2×#6451C412yes—
25C2×C42×D7C2211yescharacteristic
26C2×#6432C2211yescharacteristic
27C7×#432C2211yescharacteristic
28C7×#432C2211yescharacteristic
29#89698C2211yescharacteristic
30#89698C2211yescharacteristic
31#89609C2212yes—
32#89609C2212yes—
33#89605C2212yes—
34#89605C2212yes—
35#89498—21no—
36#89498—21no—
37#89490—21no—
38#89490—21no—
39#89479—21no—
40#89479—21no—
41#89477—21no—
42#89477—21no—
43C7×#332C2214yescomplemented
44#89342C2214yescomplemented
45#89281C2214yescomplemented
46#89277C2214yescomplemented
47#17742—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C2×Q8×D7—22no—
50C2×Q8×D7—22no—
51C2×Q8×D7—22no—
52C2×Q8×D7—22no—
53C2×Q8×D7—48no—
54C22×C4×D7—22no—
55C22×C4×D7C2×C411yescharacteristic
56C22×C4×D7D411yescharacteristic
57C22×C4×D7D411yescharacteristic
58C22×Dic14—21no—
59C22×Dic14—22no—
60C7×C4⋊Q8—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8C2×C412yes—
63C7×C4⋊Q8C2312yes—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4C2×C412yes—
66C2×C42×C7C2311yescharacteristic
67#6507—22no—
68#6507—22no—
69#6507—22no—
70#6507—22no—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7C2×C412yes—
73C2×Dic7⋊C4—22no—
74C2×Dic7⋊C4—22no—
75C2×C4×Dic7C2×C411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C2×C4×Dic7D411yescharacteristic
78C2×#1131—22no—
79C2×#1131—22no—
80D7×C4⋊C4—22no—
81D7×C4⋊C4—22no—
82D7×C4⋊C4—22no—
83D7×C4⋊C4—22no—
84D7×C4⋊C4—22no—
85D7×C4⋊C4—22no—
86D7×C4⋊C4C2×C414yes—
87D7×C4⋊C4C2×C414yes—
88#6451—22no—
89#6451—22no—
90#6451—22no—
91#6451—22no—
92#6451—22no—
93#6451—22no—
94#6451C2×C414yes—
95#6451C2×C414yes—
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⋊C8C2312yes—
109C7×C4⋊C8C2312yes—
110C7×C22⋊C8—21no—
111C7×C22⋊C8—21no—
112C7×#97—21no—
113C7×#97—21no—
114C22⋊(C7⋊C8)—21no—
115C22⋊(C7⋊C8)—21no—
116(C2×D7)⋊C8—22no—
117(C2×D7)⋊C8—22no—
118Dic7⋊C8—22no—
119Dic7⋊C8—22no—
120C8×Dic7—22no—
121C8×Dic7—22no—
122C4⋊(C7⋊C8)C2312yes—
123C4⋊(C7⋊C8)C2312yes—
124#6377—21no—
125#6377—21no—
126C22×C4⋊Q8—71no—
127#2920—71no—
128#2920—71no—
129#1556—71no—
130#1556—71no—
131#1556—71no—
132#1556D711yescharacteristic, complemented
133#1549—74no—
134#1546—74no—
135C23×D7C22⋊C411yescharacteristic
136C2×C7×Q8—22no—
137C2×C7×Q8—22no—
138C2×C7×Q8—44no—
139C22×C4×C7—22no—
140C22×C4×C7C22×C411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×Dic7—22no—
144C22×Dic7C22⋊C411yescharacteristic
145C22×Dic7C22⋊C411yescharacteristic
146C22×Dic7C22⋊C411yescharacteristic
147Q8×D7—48no—
148Q8×D7—48no—
149Q8×D7—48no—
150Q8×D7—48no—
151C2×C4×D7—24no—
152C2×C4×D7—24no—
153C2×C4×D7—24no—
154C2×C4×D7—24no—
155C2×C4×D7—24no—
156C2×C4×D7—48no—
157C2×C4×D7C22⋊C411yescharacteristic
158C2×C4×D7C22⋊C411yescharacteristic
159C2×C4×D7C22⋊C411yescharacteristic
160C2×C4×D7C22⋊C411yescharacteristic
161C2×C4×D7C22⋊C411yescharacteristic
162C2×C4×D7C22⋊C411yescharacteristic
163C2×C4×D7C22×C411yescharacteristic
164C2×C4×D7C22×C411yescharacteristic
165C2×C4×D7C2×D411yescharacteristic
166C2×C4×D7C2×D411yescharacteristic
167C2×C4×D7C2×D411yescharacteristic
168C2×C4×D7C2×D411yescharacteristic
169C2×Dic14—22no—
170C2×Dic14—22no—
171C2×Dic14—24no—
172C2×Dic14—24no—
173C2×Dic14—48no—
174C2×Dic14—44no—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4C22×C414yes—
181C42×C7C22×C411yescharacteristic
182C42×C7C2×D411yescharacteristic
183C42×C7C2×D411yescharacteristic
184C42×C7C2411yescharacteristic
185C4⋊Dic7—22no—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7C22×C414yes—
189Dic7⋊C4—24no—
190Dic7⋊C4—24no—
191Dic7⋊C4—24no—
192Dic7⋊C4—24no—
193C4×Dic7C22⋊C411yescharacteristic
194C4×Dic7C22⋊C411yescharacteristic
195C4×Dic7C22⋊C411yescharacteristic
196C4×Dic7C22⋊C411yescharacteristic
197C4×Dic7C22⋊C411yescharacteristic
198C4×Dic7C22⋊C411yescharacteristic
199C4×Dic7C22×C411yescharacteristic
200C4×Dic7C22×C411yescharacteristic
201C4×Dic7C2×D411yescharacteristic
202C4×Dic7C2×D411yescharacteristic
203C4×Dic7C2×D411yescharacteristic
204C4×Dic7C2×D411yescharacteristic
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207#1131—44no—
208#1131—44no—
209C23×Q8—141no—
210C2×C4⋊Q8—72no—
211C2×C4⋊Q8—72no—
212C2×C4⋊Q8—72no—
213C2×C4⋊Q8—72no—
214C2×C4⋊Q8—72no—
215C2×C4⋊Q8—71no—
216C2×C4⋊Q8—141no—
217C2×C4⋊Q8—141no—
218C2×C4⋊Q8—141no—
219C2×C4⋊Q8—141no—
220C2×C4⋊Q8—141no—
221C2×C4⋊Q8—141no—
222C2×C4⋊Q8C2×D711yescharacteristic
223C22×C4⋊C4—72no—
224C22×C4⋊C4—141no—
225C22×C42—71no—
226#432—71no—
227#432—71no—
228#432—141no—
229#432—141no—
230#432C2×D711yescharacteristic
231#432C2×D711yescharacteristic
232#431—72no—
233#431—72no—
234#431—141no—
235#431—141no—
236#423—72no—
237#423—72no—
238#407—141no—
239#407—141no—
240C4×M16—141no—
241C4×M16—141no—
242#332—74no—
243#332—74no—
244#332—74no—
245#332C2×D714yescomplemented
246C23×C7C2×C22⋊C411yescharacteristic
247C22×D7—22no—
248C22×D7—22no—
249C22×D7—22no—
250C22×D7C2×C22⋊C411yescharacteristic
251C22×D7C2×C22⋊C411yescharacteristic
252C7×Q8—44no—
253C7×Q8—44no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—24no—
258C2×C4×C7—44no—
259C2×C4×C7C2×C22⋊C411yescharacteristic
260C2×C4×C7C2×C22⋊C411yescharacteristic
261C2×C4×C7C2×C22⋊C411yescharacteristic
262C2×C4×C7C23×C411yescharacteristic
263C2×C4×C7C22×D411yescharacteristic
264C2×C4×C7C22×D411yescharacteristic
265C2×Dic7—22no—
266C2×Dic7—22no—
267C2×Dic7—24no—
268C2×Dic7—44no—
269C2×Dic7#10114yes—
270C2×Dic7C2×C22⋊C411yescharacteristic
271C2×Dic7C2×C22⋊C411yescharacteristic
272C2×Dic7C2×C22⋊C411yescharacteristic
273C2×Dic7C2×C22⋊C411yescharacteristic
274C2×Dic7C2×C22⋊C411yescharacteristic
275C2×Dic7C2×C22⋊C411yescharacteristic
276C4×D7—22no—
277C4×D7—22no—
278C4×D7—22no—
279C4×D7—22no—
280C4×D7—22no—
281C4×D7—22no—
282C4×D7—22no—
283C4×D7—22no—
284C4×D7—22no—
285C4×D7—22no—
286C4×D7—48no—
287C4×D7—48no—
288C4×D7AGL(1,ℤ/8)14yes—
289C4×D7#13714yes—
290Dic14—48no—
291Dic14—44no—
292Dic14—44no—
293Dic14—48no—
294C7×C8—42no—
295C7×C8—42no—
296C7⋊C8—42no—
297C7⋊C8—42no—
298C22×Q8—21no—
299C22×Q8—142no—
300C22×Q8—142no—
301C22×Q8—142no—
302C22×Q8—142no—
303C22×Q8—142no—
304C22×Q8—141no—
305C22×Q8—288no—
306C23×C4—71no—
307C23×C4—71no—
308C23×C4—71no—
309C23×C4—142no—
310C2×M16—142no—
311C2×M16—142no—
312C4⋊Q8—21no—
313C4⋊Q8—21no—
314C4⋊Q8—74no—
315C4⋊Q8—74no—
316C4⋊Q8—72no—
317C4⋊Q8—72no—
318C4⋊Q8—142no—
319C4⋊Q8—142no—
320C4⋊Q8—142no—
321C4⋊Q8—142no—
322C4⋊Q8—142no—
323C4⋊Q8—142no—
324C4⋊Q8—142no—
325C4⋊Q8—142no—
326C4⋊Q8—142no—
327C4⋊Q8—142no—
328C4⋊Q8—141no—
329C4⋊Q8—141no—
330C4⋊Q8C4×D712yes—
331C4⋊Q8C22×D712yes—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—74no—
335C2×C4⋊C4—74no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4—142no—
338C2×C4⋊C4—142no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—142no—
341C2×C4⋊C4—142no—
342C2×C4⋊C4—142no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—141no—
345C2×C4⋊C4C4×D712yes—
346C2×C42—71no—
347C2×C42—71no—
348C2×C42—71no—
349C2×C42—71no—
350C2×C42—71no—
351C2×C42—71no—
352C2×C42—71no—
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—71no—
356C2×C42—71no—
357C2×C42C22×D711yescharacteristic
358C4⋊C8—72no—
359C4⋊C8—72no—
360C4⋊C8—142no—
361C4⋊C8—142no—
362C4⋊C8C22×D712yes—
363C4⋊C8C22×D712yes—
364C22⋊C8—21no—
365C22⋊C8—21no—
366C22⋊C8—142no—
367C22⋊C8—142no—
368C22⋊C8—141no—
369C22⋊C8—141no—
370#97—21no—
371#97—21no—
372#97—141no—
373#97—141no—
374C4×C8—142no—
375C4×C8—142no—
376C22×C7—21no—
377C22×C7—21no—
378C22×C7C2×#10112yes—
379C22×C7C22×C22⋊C411yescharacteristic
380C2×D7—28no—
381C2×D7—21no—
382C2×D7—21no—
383C2×D7—21no—
384C2×D7—21no—
385C2×D7#41812yes—
386C2×D7#41912yes—
387C4×C7—21no—
388C4×C7—21no—
389C4×C7—21no—
390C4×C7—21no—
391C4×C7—44no—
392C4×C7C2×AGL(1,ℤ/8)12yes—
393C4×C7C2×#13712yes—
394Dic7—21no—
395Dic7—21no—
396Dic7—44no—
397Dic7C2×#10112yes—
398Dic7C2×#10112yes—
399Dic7#41812yes—
400Dic7#41912yes—
401C24—71no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—44no—
405C2×Q8—144no—
406C2×Q8—144no—
407C2×Q8—142no—
408C2×Q8—142no—
409C2×Q8—284no—
410C2×Q8—288no—
411C2×Q8—288no—
412C2×Q8—288no—
413C2×Q8—288no—
414C2×Q8—288no—
415C22×C4—22no—
416C22×C4—71no—
417C22×C4—71no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—71no—
422C22×C4—71no—
423C22×C4—71no—
424C22×C4—71no—
425C22×C4—71no—
426C22×C4—71no—
427C22×C4—71no—
428C22×C4—71no—
429C22×C4—71no—
430C22×C4—71no—
431C22×C4—144no—
432C22×C4—144no—
433C22×C4—144no—
434C22×C4—144no—
435C22×C4—144no—
436C22×C4—142no—
437C22×C4—288no—
438C22×C4C2×C4×D711yescharacteristic
439C22×C4D4×D711yescharacteristic
440C22×C4D4×D711yescharacteristic
441M16—284no—
442M16—284no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—142no—
446C2×C8—142no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—74no—
451C4⋊C4—144no—
452C4⋊C4—144no—
453C4⋊C4—144no—
454C4⋊C4—144no—
455C4⋊C4—142no—
456C4⋊C4—142no—
457C4⋊C4—142no—
458C4⋊C4C2×C4×D714yes—
459C42—71no—
460C42—71no—
461C42—71no—
462C42—71no—
463C42—71no—
464C42—71no—
465C42—71no—
466C42—71no—
467C42—71no—
468C42—71no—
469C42—71no—
470C42—71no—
471C42C2×C4×D711yescharacteristic
472C42D4×D711yescharacteristic
473C42D4×D711yescharacteristic
474C42C23×D711yescharacteristic
475C2×C7—22no—
476C2×C7C22×#10111yescharacteristic
477C2×C7C2×#41811yescharacteristic
478C2×C7C2×#41911yescharacteristic
479D7—22no—
480D7—22no—
481C23—71no—
482C23—71no—
483C23—142no—
484C23—142no—
485C23—142no—
486C23D7×C22⋊C411yescharacteristic
487Q8—44no—
488Q8—44no—
489Q8—284no—
490Q8—288no—
491Q8—288no—
492Q8—284no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—44no—
498C2×C4—71no—
499C2×C4—71no—
500C2×C4—71no—
501C2×C4—71no—
502C2×C4—74no—
503C2×C4—74no—
504C2×C4—74no—
505C2×C4—71no—
506C2×C4—71no—
507C2×C4—142no—
508C2×C4—142no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—142no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—142no—
516C2×C4—142no—
517C2×C4—142no—
518C2×C4—142no—
519C2×C4—144no—
520C2×C4—288no—
521C2×C4—288no—
522C2×C4—284no—
523C2×C4D7×C22⋊C411yescharacteristic
524C2×C4D7×C22⋊C411yescharacteristic
525C2×C4D7×C22⋊C411yescharacteristic
526C2×C4C22×C4×D711yescharacteristic
527C2×C4C2×D4×D711yescharacteristic
528C2×C4C2×D4×D711yescharacteristic
529C8—42no—
530C8—42no—
531C8—282no—
532C8—282no—
533C7#1774211yescharacteristic, complemented, Sylow
534C22—21no—
535C22—21no—
536C22—72no—
537C22—72no—
538C22—148no—
539C22—141no—
540C22—141no—
541C22—141no—
542C22—141no—
543C22D7×#10112yes—
544C22C2×D7×C22⋊C411yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—44no—
550C4—72no—
551C4—72no—
552C4—72no—
553C4—72no—
554C4—141no—
555C4—141no—
556C4—284no—
557C4D7×AGL(1,ℤ/8)12yes—
558C4D7×#13712yes—
559C2—22no—
560C2—142no—
561C2—142no—
562C2C2×D7×#10111yescharacteristic
563C2D7×#41811yescharacteristic
564C2D7×#41911yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1556C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1556D7
hypocenterC7#17742
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
4. C7×#1556C2semidirect
5. #11729530C2semidirect
6. #11728577C2semidirect
7. #11728576C2semidirect
8. #11726176C2semidirect
9. #11726174C2semidirect
43. C7×#332C22semidirect
44. #89342C22semidirect
45. #89281C22semidirect
46. #89277C22semidirect
132. #1556D7semidirect
245. #332C2×D7semidirect
533. C7#17742semidirect

Maximal subgroups

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

subgroupindex
C2×D7×C4⋊Q82
#117367762
C7×#15562
#117295302
#117285772
#117285762
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