Finite Group Databasebeta

← #13322387

Subgroups — #13322387

Order 1792.

1178 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×C4⋊Q8C211yesmaximal, characteristic
2#11736727C211yesmaximal, characteristic
3#11736483C211yesmaximal, characteristic
4C7×#1546C211yesmaximal, characteristic, complemented
5#11729527C211yesmaximal, characteristic, complemented
6#11728532C211yesmaximal, characteristic, complemented
7#11728260C211yesmaximal, characteristic, complemented
8#11726182C214yesmaximal, complemented
9#11726176C214yesmaximal, 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×#431C2211yescharacteristic
28C7×#423C2211yescharacteristic
29#89898—21no—
30M16×Dic7—21no—
31#89697C2211yescharacteristic
32#89696C2211yescharacteristic
33#89609C2212yes—
34#89607—21no—
35#89607C2212yes—
36#89607C2212yes—
37#89605—21no—
38#89605C2212yes—
39#89498—21no—
40#89491—21no—
41C4×#1131—21no—
42#89457—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—44no—
54C2×Q8×D7—44no—
55C22×C4×D7—22no—
56C22×C4×D7C2×C411yescharacteristic
57C22×C4×D7D411yescharacteristic
58C22×C4×D7D411yescharacteristic
59C22×Dic14—21no—
60C22×Dic14—22no—
61C2×C7×M16—21no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8—21no—
64C7×C4⋊Q8C2×C412yes—
65C7×C4⋊Q8C2312yes—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2×C412yes—
68C2×C42×C7C2311yescharacteristic
69#6507—22no—
70#6507—22no—
71#6507—22no—
72#6507—22no—
73C2×C4⋊Dic7—21no—
74C2×C4⋊Dic7C2×C412yes—
75C2×Dic7⋊C4—22no—
76C2×Dic7⋊C4—22no—
77C2×C4×Dic7C2×C411yescharacteristic
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7D411yescharacteristic
80C2×#1136—21no—
81C2×#1131—22no—
82D7×C4⋊C4—22no—
83D7×C4⋊C4—22no—
84D7×C4⋊C4—22no—
85D7×C4⋊C4—22no—
86D7×C4⋊C4—22no—
87D7×C4⋊C4—22no—
88D7×C4⋊C4C2×C414yes—
89D7×C4⋊C4C2×C414yes—
90#6451—22no—
91#6451—22no—
92#6451—22no—
93#6451—22no—
94#6451—22no—
95#6451—22no—
96#6451C2×C414yes—
97#6451C2×C414yes—
98C42×D7C2×C411yescharacteristic
99C42×D7C2×C411yescharacteristic
100C42×D7D411yescharacteristic
101C42×D7D411yescharacteristic
102C42×D7D411yescharacteristic
103C42×D7D411yescharacteristic
104C42×D7C2311yescharacteristic
105C42×D7C2311yescharacteristic
106#6432—21no—
107#6432—21no—
108#6432C2×C412yes—
109#6432C2312yes—
110C7×C4⋊C8—21no—
111C7×C4⋊C8C2312yes—
112C7×C4⋊C8C2312yes—
113C7×C22⋊C8—21no—
114C4×C7×C8—21no—
115C22⋊(C7⋊C8)—21no—
116(C2×D7)⋊C8—22no—
117(C2×D7)⋊C8—22no—
118(C2×D7)⋊C8—22no—
119#6389—22no—
120#6389—22no—
121Dic7⋊C8—22no—
122C8×Dic7—22no—
123C4⋊(C7⋊C8)—21no—
124C4⋊(C7⋊C8)C2312yes—
125C4⋊(C7⋊C8)C2312yes—
126C4×C7⋊C8—21no—
127C22×C4⋊Q8—71no—
128#2920—71no—
129#2920—71no—
130#1556—74no—
131#1549—74no—
132#1546—71no—
133#1546—71no—
134#1546—71no—
135#1546D711yescharacteristic, complemented
136C23×D7C22⋊C411yescharacteristic
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—42no—
140C2×C7×Q8—42no—
141C22×C4×C7—22no—
142C22×C4×C7C22×C411yescharacteristic
143C22×C4×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×Dic7—22no—
146C22×Dic7C22⋊C411yescharacteristic
147C22×Dic7C22⋊C411yescharacteristic
148C22×Dic7C22⋊C411yescharacteristic
149Q8×D7—44no—
150Q8×D7—44no—
151Q8×D7—44no—
152Q8×D7—44no—
153Q8×D7—48no—
154Q8×D7—48no—
155C2×C4×D7—22no—
156C2×C4×D7—22no—
157C2×C4×D7—24no—
158C2×C4×D7—24no—
159C2×C4×D7—24no—
160C2×C4×D7—24no—
161C2×C4×D7—48no—
162C2×C4×D7C22⋊C411yescharacteristic
163C2×C4×D7C22⋊C411yescharacteristic
164C2×C4×D7C22⋊C411yescharacteristic
165C2×C4×D7C22⋊C411yescharacteristic
166C2×C4×D7C22⋊C411yescharacteristic
167C2×C4×D7C22⋊C411yescharacteristic
168C2×C4×D7C22×C411yescharacteristic
169C2×C4×D7C22×C411yescharacteristic
170C2×C4×D7C2×D411yescharacteristic
171C2×C4×D7C2×D411yescharacteristic
172C2×C4×D7C2×D411yescharacteristic
173C2×C4×D7C2×D411yescharacteristic
174C2×Dic14—22no—
175C2×Dic14—22no—
176C2×Dic14—24no—
177C2×Dic14—24no—
178C2×Dic14—48no—
179C2×Dic14—42no—
180C2×Dic14—42no—
181C7×M16—42no—
182C2×C7×C8—22no—
183C2×C7×C8—22no—
184C7×C4⋊C4—22no—
185C7×C4⋊C4—22no—
186C7×C4⋊C4—22no—
187C7×C4⋊C4C22×C414yes—
188C42×C7C22×C411yescharacteristic
189C42×C7C2×D411yescharacteristic
190C42×C7C2×D411yescharacteristic
191C42×C7C2411yescharacteristic
192C4⋊Dic7—22no—
193C4⋊Dic7—22no—
194C4⋊Dic7—22no—
195C4⋊Dic7C22×C414yes—
196Dic7⋊C4—24no—
197Dic7⋊C4—24no—
198Dic7⋊C4—24no—
199Dic7⋊C4—24no—
200C4×Dic7C22⋊C411yescharacteristic
201C4×Dic7C22⋊C411yescharacteristic
202C4×Dic7C22⋊C411yescharacteristic
203C4×Dic7C22⋊C411yescharacteristic
204C4×Dic7C22⋊C411yescharacteristic
205C4×Dic7C22⋊C411yescharacteristic
206C4×Dic7C22×C411yescharacteristic
207C4×Dic7C22×C411yescharacteristic
208C4×Dic7C2×D411yescharacteristic
209C4×Dic7C2×D411yescharacteristic
210C4×Dic7C2×D411yescharacteristic
211C4×Dic7C2×D411yescharacteristic
212#1136—42no—
213C2×C7⋊C8—22no—
214C2×C7⋊C8—22no—
215#1131—44no—
216C23×Q8—141no—
217C2×C4⋊Q8—72no—
218C2×C4⋊Q8—72no—
219C2×C4⋊Q8—72no—
220C2×C4⋊Q8—72no—
221C2×C4⋊Q8—72no—
222C2×C4⋊Q8—71no—
223C2×C4⋊Q8—141no—
224C2×C4⋊Q8—141no—
225C2×C4⋊Q8—141no—
226C2×C4⋊Q8—141no—
227C2×C4⋊Q8—141no—
228C2×C4⋊Q8—141no—
229C2×C4⋊Q8C2×D711yescharacteristic
230C22×C4⋊C4—72no—
231C22×C4⋊C4—141no—
232C22×C42—71no—
233#432—72no—
234#432—72no—
235#432—141no—
236#432—141no—
237#431—71no—
238#431—72no—
239#431—141no—
240#431—141no—
241#431C2×D711yescharacteristic
242#423—71no—
243#423—72no—
244#423C2×D711yescharacteristic
245#407—141no—
246#407—141no—
247C4×M16—141no—
248C4×M16—141no—
249#332—74no—
250#332—74no—
251#332—74no—
252#332C2×D714yescomplemented
253C23×C7C2×C22⋊C411yescharacteristic
254C22×D7—22no—
255C22×D7—22no—
256C22×D7—22no—
257C22×D7C2×C22⋊C411yescharacteristic
258C22×D7C2×C22⋊C411yescharacteristic
259C7×Q8—42no—
260C7×Q8—42no—
261C7×Q8—44no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—24no—
265C2×C4×C7—44no—
266C2×C4×C7C2×C22⋊C411yescharacteristic
267C2×C4×C7C2×C22⋊C411yescharacteristic
268C2×C4×C7C2×C22⋊C411yescharacteristic
269C2×C4×C7AGL(1,ℤ/8)12yes—
270C2×C4×C7AGL(1,ℤ/8)12yes—
271C2×C4×C7C23×C411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—22no—
277C2×Dic7—24no—
278C2×Dic7—44no—
279C2×Dic7C2×C22⋊C411yescharacteristic
280C2×Dic7C2×C22⋊C411yescharacteristic
281C2×Dic7C2×C22⋊C411yescharacteristic
282C2×Dic7C2×C22⋊C411yescharacteristic
283C2×Dic7C2×C22⋊C411yescharacteristic
284C2×Dic7C2×C22⋊C411yescharacteristic
285C4×D7—22no—
286C4×D7—22no—
287C4×D7—21no—
288C4×D7—21no—
289C4×D7—21no—
290C4×D7—21no—
291C4×D7—22no—
292C4×D7—22no—
293C4×D7—22no—
294C4×D7—22no—
295C4×D7—22no—
296C4×D7—22no—
297C4×D7—48no—
298C4×D7—48no—
299C4×D7#10114yes—
300C4×D7#13714yes—
301Dic14—48no—
302Dic14—48no—
303Dic14—42no—
304Dic14—42no—
305Dic14—44no—
306C7×C8—42no—
307C7×C8—42no—
308C7⋊C8—42no—
309C7⋊C8—42no—
310C22×Q8—21no—
311C22×Q8—142no—
312C22×Q8—142no—
313C22×Q8—142no—
314C22×Q8—142no—
315C22×Q8—142no—
316C22×Q8—141no—
317C22×Q8—284no—
318C22×Q8—284no—
319C23×C4—71no—
320C23×C4—71no—
321C23×C4—71no—
322C23×C4—142no—
323C2×M16—21no—
324C2×M16—142no—
325C2×M16—141no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8—74no—
329C4⋊Q8—74no—
330C4⋊Q8—72no—
331C4⋊Q8—72no—
332C4⋊Q8—142no—
333C4⋊Q8—142no—
334C4⋊Q8—142no—
335C4⋊Q8—142no—
336C4⋊Q8—142no—
337C4⋊Q8—142no—
338C4⋊Q8—142no—
339C4⋊Q8—142no—
340C4⋊Q8—142no—
341C4⋊Q8—142no—
342C4⋊Q8—141no—
343C4⋊Q8—141no—
344C4⋊Q8C4×D712yes—
345C4⋊Q8C22×D712yes—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—72no—
348C2×C4⋊C4—74no—
349C2×C4⋊C4—74no—
350C2×C4⋊C4—142no—
351C2×C4⋊C4—142no—
352C2×C4⋊C4—142no—
353C2×C4⋊C4—142no—
354C2×C4⋊C4—142no—
355C2×C4⋊C4—142no—
356C2×C4⋊C4—142no—
357C2×C4⋊C4—142no—
358C2×C4⋊C4—141no—
359C2×C4⋊C4C4×D712yes—
360C2×C42—71no—
361C2×C42—71no—
362C2×C42—71no—
363C2×C42—71no—
364C2×C42—71no—
365C2×C42—71no—
366C2×C42—71no—
367C2×C42—71no—
368C2×C42—71no—
369C2×C42—71no—
370C2×C42—71no—
371C2×C42C22×D711yescharacteristic
372C4⋊C8—21no—
373C4⋊C8—72no—
374C4⋊C8—72no—
375C4⋊C8—142no—
376C4⋊C8—141no—
377C4⋊C8C22×D712yes—
378C4⋊C8C22×D712yes—
379C22⋊C8—21no—
380C22⋊C8—142no—
381C22⋊C8—142no—
382C22⋊C8—142no—
383C22⋊C8—141no—
384#97—142no—
385#97—142no—
386C4×C8—21no—
387C4×C8—142no—
388C4×C8—141no—
389C22×C7—21no—
390C22×C7—21no—
391C22×C7#41812yes—
392C22×C7C22×C22⋊C411yescharacteristic
393C2×D7—28no—
394C2×D7—21no—
395C2×D7—21no—
396C2×D7—21no—
397C2×D7—21no—
398C2×D7C2×#10112yes—
399C2×D7#41912yes—
400C4×C7—21no—
401C4×C7—21no—
402C4×C7—44no—
403C4×C7C2×#10112yes—
404C4×C7#41812yes—
405C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
406C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
407C4×C7C2×#13712yes—
408Dic7—21no—
409Dic7—21no—
410Dic7—21no—
411Dic7—21no—
412Dic7—44no—
413Dic7C2×#10112yes—
414Dic7#41912yes—
415C24—71no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—144no—
421C2×Q8—144no—
422C2×Q8—142no—
423C2×Q8—142no—
424C2×Q8—288no—
425C2×Q8—284no—
426C2×Q8—284no—
427C2×Q8—284no—
428C2×Q8—284no—
429C2×Q8—288no—
430C2×Q8—288no—
431C2×Q8—282no—
432C2×Q8—282no—
433C22×C4—22no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—71no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—71no—
443C22×C4—71no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—144no—
450C22×C4—144no—
451C22×C4—144no—
452C22×C4—144no—
453C22×C4—142no—
454C22×C4—142no—
455C22×C4—142no—
456C22×C4—288no—
457C22×C4C2×C4×D711yescharacteristic
458C22×C4D4×D711yescharacteristic
459C22×C4D4×D711yescharacteristic
460M16—42no—
461M16—284no—
462M16—282no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—142no—
466C2×C8—142no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—74no—
471C4⋊C4—144no—
472C4⋊C4—144no—
473C4⋊C4—144no—
474C4⋊C4—144no—
475C4⋊C4—142no—
476C4⋊C4—142no—
477C4⋊C4—142no—
478C4⋊C4C2×C4×D714yes—
479C42—71no—
480C42—71no—
481C42—71no—
482C42—71no—
483C42—71no—
484C42—71no—
485C42—71no—
486C42—71no—
487C42—71no—
488C42—71no—
489C42—71no—
490C42—71no—
491C42C2×C4×D711yescharacteristic
492C42D4×D711yescharacteristic
493C42D4×D711yescharacteristic
494C42C23×D711yescharacteristic
495C2×C7—22no—
496C2×C7C22×#10111yescharacteristic
497C2×C7C2×#41811yescharacteristic
498C2×C7C2×#41911yescharacteristic
499D7—22no—
500D7—22no—
501C23—71no—
502C23—71no—
503C23—142no—
504C23—142no—
505C23—142no—
506C23D7×C22⋊C411yescharacteristic
507Q8—42no—
508Q8—42no—
509Q8—44no—
510Q8—288no—
511Q8—288no—
512Q8—282no—
513Q8—282no—
514Q8—284no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—44no—
519C2×C4—71no—
520C2×C4—71no—
521C2×C4—74no—
522C2×C4—74no—
523C2×C4—71no—
524C2×C4—71no—
525C2×C4—71no—
526C2×C4—71no—
527C2×C4—142no—
528C2×C4—142no—
529C2×C4—142no—
530C2×C4—142no—
531C2×C4—142no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—141no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—142no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—144no—
543C2×C4—288no—
544C2×C4—288no—
545C2×C4—284no—
546C2×C4D7×C22⋊C411yescharacteristic
547C2×C4D7×C22⋊C411yescharacteristic
548C2×C4D7×C22⋊C411yescharacteristic
549C2×C4#647412yes—
550C2×C4#647712yes—
551C2×C4C22×C4×D711yescharacteristic
552C2×C4C2×D4×D711yescharacteristic
553C2×C4C2×D4×D711yescharacteristic
554C8—42no—
555C8—42no—
556C8—282no—
557C8—282no—
558C7#1774211yescharacteristic, complemented, Sylow
559C22—21no—
560C22—21no—
561C22—72no—
562C22—72no—
563C22—148no—
564C22—141no—
565C22—141no—
566C22—141no—
567C22—141no—
568C22#8954212yes—
569C22C2×D7×C22⋊C411yescharacteristic
570C4—21no—
571C4—21no—
572C4—44no—
573C4—72no—
574C4—72no—
575C4—141no—
576C4—141no—
577C4—141no—
578C4—141no—
579C4—284no—
580C4D7×#10112yes—
581C4#8954212yes—
582C4C2×#647411yescharacteristic
583C4C2×#647711yescharacteristic
584C4D7×#13712yes—
585C2—22no—
586C2—142no—
587C2—142no—
588C2C2×D7×#10111yescharacteristic
589C2C2×#8954211yescharacteristic
590C2D7×#41911yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1546C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1546D7
hypocenterC7#17742
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
4. C7×#1546C2semidirect
5. #11729527C2semidirect
6. #11728532C2semidirect
7. #11728260C2semidirect
8. #11726182C2semidirect
9. #11726176C2semidirect
43. C7×#332C22semidirect
44. #89342C22semidirect
45. #89281C22semidirect
46. #89277C22semidirect
135. #1546D7semidirect
252. #332C2×D7semidirect
558. 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
#117367272
#117364832
C7×#15462
#117295272
#117285322
#117282602
#117261822
#117261762
#177427