Finite Group Databasebeta

← #13322388

Subgroups — #13322388

Order 1792.

1178 conjugacy classes of subgroups.

The subgroup lattice has 578 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×#1549C211yesmaximal, characteristic, complemented
5#11729529C211yesmaximal, characteristic, complemented
6#11728534C211yesmaximal, characteristic, complemented
7#11728262C211yesmaximal, characteristic, complemented
8#11726182C214yesmaximal, 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×#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—48no—
54C22×C4×D7—22no—
55C22×C4×D7C2×C411yescharacteristic
56C22×C4×D7D411yescharacteristic
57C22×C4×D7D411yescharacteristic
58C22×Dic14—21no—
59C22×Dic14—22no—
60C2×C7×M16—21no—
61C7×C4⋊Q8—21no—
62C7×C4⋊Q8—21no—
63C7×C4⋊Q8C2×C412yes—
64C7×C4⋊Q8C2312yes—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4C2×C412yes—
67C2×C42×C7C2311yescharacteristic
68#6507—22no—
69#6507—22no—
70#6507—22no—
71#6507—22no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7C2×C412yes—
74C2×Dic7⋊C4—22no—
75C2×Dic7⋊C4—22no—
76C2×C4×Dic7C2×C411yescharacteristic
77C2×C4×Dic7D411yescharacteristic
78C2×C4×Dic7D411yescharacteristic
79C2×#1136—21no—
80C2×#1131—22no—
81D7×C4⋊C4—22no—
82D7×C4⋊C4—22no—
83D7×C4⋊C4—22no—
84D7×C4⋊C4—22no—
85D7×C4⋊C4—22no—
86D7×C4⋊C4—22no—
87D7×C4⋊C4C2×C414yes—
88D7×C4⋊C4C2×C414yes—
89#6451—22no—
90#6451—22no—
91#6451—22no—
92#6451—22no—
93#6451—22no—
94#6451—22no—
95#6451C2×C414yes—
96#6451C2×C414yes—
97C42×D7C2×C411yescharacteristic
98C42×D7C2×C411yescharacteristic
99C42×D7D411yescharacteristic
100C42×D7D411yescharacteristic
101C42×D7D411yescharacteristic
102C42×D7D411yescharacteristic
103C42×D7C2311yescharacteristic
104C42×D7C2311yescharacteristic
105#6432—21no—
106#6432—21no—
107#6432C2×C412yes—
108#6432C2312yes—
109C7×C4⋊C8—21no—
110C7×C4⋊C8C2312yes—
111C7×C4⋊C8C2312yes—
112C7×C22⋊C8—21no—
113C4×C7×C8—21no—
114C22⋊(C7⋊C8)—21no—
115(C2×D7)⋊C8—22no—
116(C2×D7)⋊C8—22no—
117(C2×D7)⋊C8—22no—
118#6389—22no—
119#6389—22no—
120Dic7⋊C8—22no—
121C8×Dic7—22no—
122C4⋊(C7⋊C8)—21no—
123C4⋊(C7⋊C8)C2312yes—
124C4⋊(C7⋊C8)C2312yes—
125C4×C7⋊C8—21no—
126C22×C4⋊Q8—71no—
127#2920—71no—
128#2920—71no—
129#1556—74no—
130#1549—71no—
131#1549—71no—
132#1549—71no—
133#1549D711yescharacteristic, complemented
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—22no—
152C2×C4×D7—22no—
153C2×C4×D7—24no—
154C2×C4×D7—24no—
155C2×C4×D7—24no—
156C2×C4×D7—24no—
157C2×C4×D7—48no—
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×D7C22×C411yescharacteristic
166C2×C4×D7C2×D411yescharacteristic
167C2×C4×D7C2×D411yescharacteristic
168C2×C4×D7C2×D411yescharacteristic
169C2×C4×D7C2×D411yescharacteristic
170C2×Dic14—22no—
171C2×Dic14—22no—
172C2×Dic14—24no—
173C2×Dic14—24no—
174C2×Dic14—48no—
175C2×Dic14—44no—
176C7×M16—42no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C7×C4⋊C4—22no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4C22×C414yes—
183C42×C7C22×C411yescharacteristic
184C42×C7C2×D411yescharacteristic
185C42×C7C2×D411yescharacteristic
186C42×C7C2411yescharacteristic
187C4⋊Dic7—22no—
188C4⋊Dic7—22no—
189C4⋊Dic7—22no—
190C4⋊Dic7C22×C414yes—
191Dic7⋊C4—24no—
192Dic7⋊C4—24no—
193Dic7⋊C4—24no—
194Dic7⋊C4—24no—
195C4×Dic7C22⋊C411yescharacteristic
196C4×Dic7C22⋊C411yescharacteristic
197C4×Dic7C22⋊C411yescharacteristic
198C4×Dic7C22⋊C411yescharacteristic
199C4×Dic7C22⋊C411yescharacteristic
200C4×Dic7C22⋊C411yescharacteristic
201C4×Dic7C22×C411yescharacteristic
202C4×Dic7C22×C411yescharacteristic
203C4×Dic7C2×D411yescharacteristic
204C4×Dic7C2×D411yescharacteristic
205C4×Dic7C2×D411yescharacteristic
206C4×Dic7C2×D411yescharacteristic
207#1136—42no—
208C2×C7⋊C8—22no—
209C2×C7⋊C8—22no—
210#1131—44no—
211C23×Q8—141no—
212C2×C4⋊Q8—72no—
213C2×C4⋊Q8—72no—
214C2×C4⋊Q8—72no—
215C2×C4⋊Q8—72no—
216C2×C4⋊Q8—72no—
217C2×C4⋊Q8—71no—
218C2×C4⋊Q8—141no—
219C2×C4⋊Q8—141no—
220C2×C4⋊Q8—141no—
221C2×C4⋊Q8—141no—
222C2×C4⋊Q8—141no—
223C2×C4⋊Q8—141no—
224C2×C4⋊Q8C2×D711yescharacteristic
225C22×C4⋊C4—72no—
226C22×C4⋊C4—141no—
227C22×C42—71no—
228#432—72no—
229#432—72no—
230#432—141no—
231#432—141no—
232#431—71no—
233#431—72no—
234#431—141no—
235#431—141no—
236#431C2×D711yescharacteristic
237#423—71no—
238#423—72no—
239#423C2×D711yescharacteristic
240#407—141no—
241#407—141no—
242C4×M16—141no—
243C4×M16—141no—
244#332—74no—
245#332—74no—
246#332—74no—
247#332C2×D714yescomplemented
248C23×C7C2×C22⋊C411yescharacteristic
249C22×D7—22no—
250C22×D7—22no—
251C22×D7—22no—
252C22×D7C2×C22⋊C411yescharacteristic
253C22×D7C2×C22⋊C411yescharacteristic
254C7×Q8—44no—
255C7×Q8—44no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—24no—
259C2×C4×C7—44no—
260C2×C4×C7C2×C22⋊C411yescharacteristic
261C2×C4×C7C2×C22⋊C411yescharacteristic
262C2×C4×C7C2×C22⋊C411yescharacteristic
263C2×C4×C7#13712yes—
264C2×C4×C7#13712yes—
265C2×C4×C7C23×C411yescharacteristic
266C2×C4×C7C22×D411yescharacteristic
267C2×C4×C7C22×D411yescharacteristic
268C2×Dic7—22no—
269C2×Dic7—22no—
270C2×Dic7—22no—
271C2×Dic7—24no—
272C2×Dic7—44no—
273C2×Dic7C2×C22⋊C411yescharacteristic
274C2×Dic7C2×C22⋊C411yescharacteristic
275C2×Dic7C2×C22⋊C411yescharacteristic
276C2×Dic7C2×C22⋊C411yescharacteristic
277C2×Dic7C2×C22⋊C411yescharacteristic
278C2×Dic7C2×C22⋊C411yescharacteristic
279C4×D7—22no—
280C4×D7—22no—
281C4×D7—22no—
282C4×D7—21no—
283C4×D7—21no—
284C4×D7—21no—
285C4×D7—21no—
286C4×D7—22no—
287C4×D7—22no—
288C4×D7—22no—
289C4×D7—22no—
290C4×D7—22no—
291C4×D7—48no—
292C4×D7—48no—
293C4×D7#10114yes—
294C4×D7AGL(1,ℤ/8)14yes—
295Dic14—48no—
296Dic14—44no—
297Dic14—44no—
298Dic14—48no—
299C7×C8—42no—
300C7×C8—42no—
301C7⋊C8—42no—
302C7⋊C8—42no—
303C22×Q8—21no—
304C22×Q8—142no—
305C22×Q8—142no—
306C22×Q8—142no—
307C22×Q8—142no—
308C22×Q8—142no—
309C22×Q8—141no—
310C22×Q8—288no—
311C23×C4—71no—
312C23×C4—71no—
313C23×C4—71no—
314C23×C4—142no—
315C2×M16—21no—
316C2×M16—142no—
317C2×M16—141no—
318C4⋊Q8—21no—
319C4⋊Q8—21no—
320C4⋊Q8—74no—
321C4⋊Q8—74no—
322C4⋊Q8—72no—
323C4⋊Q8—72no—
324C4⋊Q8—142no—
325C4⋊Q8—142no—
326C4⋊Q8—142no—
327C4⋊Q8—142no—
328C4⋊Q8—142no—
329C4⋊Q8—142no—
330C4⋊Q8—142no—
331C4⋊Q8—142no—
332C4⋊Q8—142no—
333C4⋊Q8—142no—
334C4⋊Q8—141no—
335C4⋊Q8—141no—
336C4⋊Q8C4×D712yes—
337C4⋊Q8C22×D712yes—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—72no—
340C2×C4⋊C4—74no—
341C2×C4⋊C4—74no—
342C2×C4⋊C4—142no—
343C2×C4⋊C4—142no—
344C2×C4⋊C4—142no—
345C2×C4⋊C4—142no—
346C2×C4⋊C4—142no—
347C2×C4⋊C4—142no—
348C2×C4⋊C4—142no—
349C2×C4⋊C4—142no—
350C2×C4⋊C4—141no—
351C2×C4⋊C4C4×D712yes—
352C2×C42—71no—
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—71no—
356C2×C42—71no—
357C2×C42—71no—
358C2×C42—71no—
359C2×C42—71no—
360C2×C42—71no—
361C2×C42—71no—
362C2×C42—71no—
363C2×C42C22×D711yescharacteristic
364C4⋊C8—21no—
365C4⋊C8—72no—
366C4⋊C8—72no—
367C4⋊C8—142no—
368C4⋊C8—141no—
369C4⋊C8C22×D712yes—
370C4⋊C8C22×D712yes—
371C22⋊C8—21no—
372C22⋊C8—142no—
373C22⋊C8—142no—
374C22⋊C8—142no—
375C22⋊C8—141no—
376#97—142no—
377#97—142no—
378C4×C8—21no—
379C4×C8—142no—
380C4×C8—141no—
381C22×C7—21no—
382C22×C7—21no—
383C22×C7#41912yes—
384C22×C7C22×C22⋊C411yescharacteristic
385C2×D7—28no—
386C2×D7—21no—
387C2×D7—21no—
388C2×D7—21no—
389C2×D7—21no—
390C2×D7C2×#10112yes—
391C2×D7#41812yes—
392C4×C7—21no—
393C4×C7—21no—
394C4×C7—44no—
395C4×C7C2×#10112yes—
396C4×C7#41912yes—
397C4×C7C2×AGL(1,ℤ/8)12yes—
398C4×C7C2×#13711yescharacteristic
399C4×C7C2×#13711yescharacteristic
400Dic7—21no—
401Dic7—21no—
402Dic7—21no—
403Dic7—21no—
404Dic7—44no—
405Dic7C2×#10112yes—
406Dic7#41812yes—
407C24—71no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—44no—
411C2×Q8—144no—
412C2×Q8—144no—
413C2×Q8—142no—
414C2×Q8—142no—
415C2×Q8—284no—
416C2×Q8—288no—
417C2×Q8—288no—
418C2×Q8—288no—
419C2×Q8—288no—
420C2×Q8—288no—
421C22×C4—22no—
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—71no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—71no—
437C22×C4—144no—
438C22×C4—144no—
439C22×C4—144no—
440C22×C4—144no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—288no—
445C22×C4C2×C4×D711yescharacteristic
446C22×C4D4×D711yescharacteristic
447C22×C4D4×D711yescharacteristic
448M16—42no—
449M16—284no—
450M16—282no—
451C2×C8—22no—
452C2×C8—22no—
453C2×C8—142no—
454C2×C8—142no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—74no—
459C4⋊C4—144no—
460C4⋊C4—144no—
461C4⋊C4—144no—
462C4⋊C4—144no—
463C4⋊C4—142no—
464C4⋊C4—142no—
465C4⋊C4—142no—
466C4⋊C4C2×C4×D714yes—
467C42—71no—
468C42—71no—
469C42—71no—
470C42—71no—
471C42—71no—
472C42—71no—
473C42—71no—
474C42—71no—
475C42—71no—
476C42—71no—
477C42—71no—
478C42—71no—
479C42C2×C4×D711yescharacteristic
480C42D4×D711yescharacteristic
481C42D4×D711yescharacteristic
482C42C23×D711yescharacteristic
483C2×C7—22no—
484C2×C7C22×#10111yescharacteristic
485C2×C7C2×#41811yescharacteristic
486C2×C7C2×#41911yescharacteristic
487D7—22no—
488D7—22no—
489C23—71no—
490C23—71no—
491C23—142no—
492C23—142no—
493C23—142no—
494C23D7×C22⋊C411yescharacteristic
495Q8—44no—
496Q8—44no—
497Q8—284no—
498Q8—288no—
499Q8—288no—
500Q8—284no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—44no—
505C2×C4—71no—
506C2×C4—71no—
507C2×C4—74no—
508C2×C4—74no—
509C2×C4—71no—
510C2×C4—71no—
511C2×C4—71no—
512C2×C4—71no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—142no—
516C2×C4—142no—
517C2×C4—142no—
518C2×C4—142no—
519C2×C4—142no—
520C2×C4—141no—
521C2×C4—141no—
522C2×C4—141no—
523C2×C4—141no—
524C2×C4—142no—
525C2×C4—142no—
526C2×C4—142no—
527C2×C4—142no—
528C2×C4—144no—
529C2×C4—288no—
530C2×C4—288no—
531C2×C4—284no—
532C2×C4D7×C22⋊C411yescharacteristic
533C2×C4D7×C22⋊C411yescharacteristic
534C2×C4D7×C22⋊C411yescharacteristic
535C2×C4#647812yes—
536C2×C4#648112yes—
537C2×C4C22×C4×D711yescharacteristic
538C2×C4C2×D4×D711yescharacteristic
539C2×C4C2×D4×D711yescharacteristic
540C8—42no—
541C8—42no—
542C8—282no—
543C8—282no—
544C7#1774211yescharacteristic, complemented, Sylow
545C22—21no—
546C22—21no—
547C22—72no—
548C22—72no—
549C22—148no—
550C22—141no—
551C22—141no—
552C22—141no—
553C22—141no—
554C22#8957412yes—
555C22C2×D7×C22⋊C411yescharacteristic
556C4—21no—
557C4—21no—
558C4—44no—
559C4—72no—
560C4—72no—
561C4—141no—
562C4—141no—
563C4—141no—
564C4—141no—
565C4—284no—
566C4D7×#10112yes—
567C4#8957412yes—
568C4C2×#647811yescharacteristic
569C4C2×#648111yescharacteristic
570C4D7×AGL(1,ℤ/8)12yes—
571C2—22no—
572C2—142no—
573C2—142no—
574C2C2×D7×#10111yescharacteristic
575C2C2×#8957411yescharacteristic
576C2D7×#41811yescharacteristic
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#17742
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. #11728534C2semidirect
7. #11728262C2semidirect
8. #11726182C2semidirect
9. #11726174C2semidirect
43. C7×#332C22semidirect
44. #89342C22semidirect
45. #89281C22semidirect
46. #89277C22semidirect
133. #1549D7semidirect
247. #332C2×D7semidirect
544. 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×#15492
#117295292
#117285342
#117282622
#117261822
#117261742
#177427