Finite Group Databasebeta

← #13245237

Subgroups — #13245237

Order 1792.

1620 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
1C22×C7×#121C211yesmaximal, characteristic
2#11729881C211yesmaximal, characteristic, complemented
3#11729880C211yesmaximal, characteristic, complemented
4C2×#89332C214yesmaximal, complemented
5C23×C7×D4—21no—
6C23×C7×D4C411yescharacteristic, complemented
7C24×C4×C7C2211yescharacteristic
8C2×C7×#121—21no—
9C2×C7×#121—21no—
10C2×C7×#121—28no—
11C2×C7×#121C412yescomplemented
12C2×C7×#121C2212yes—
13C22×C7×C4⋊C4C411yescharacteristic, complemented
14C22×C7×C22⋊C4—21no—
15C22×C22⋊Dic7—21no—
16C2×C7⋊#95—21no—
17C2×C7⋊#95C2212yescomplemented
18C2×C22⋊(C7⋊C8)—21no—
19C2×C22⋊(C7⋊C8)—21no—
20C2×C22⋊(C7⋊C8)C2212yescomplemented
21#89332—28no—
22#8328—71nomaximal, Sylow
23C25×C7—21no—
24C25×C7D411yescharacteristic
25C22×C7×D4—28no—
26C22×C7×D4—22no—
27C22×C7×D4—22no—
28C22×C7×D4—24no—
29C22×C7×D4—21no—
30C22×C7×D4—42no—
31C22×C7×D4—42no—
32C22×C7×D4—44no—
33C22×C7×D4—42no—
34C22×C7×D4—42no—
35C22×C7×D4C2×C412yes—
36C23×C4×C7—21no—
37C23×C4×C7—24no—
38C23×C4×C7—42no—
39C23×C4×C7—42no—
40C23×C4×C7C2×C411yescharacteristic
41C23×C4×C7C2×C411yescharacteristic
42C23×C4×C7D411yescharacteristic
43C23×C4×C7D411yescharacteristic
44C23×C4×C7D411yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×Dic7—22no—
47C7×#121—28no—
48C7×#121—28no—
49C7×#121—24no—
50C7×#121—24no—
51C7×#121—24no—
52C7×#121D418yes—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—24no—
55C2×C7×C4⋊C4C2×C412yes—
56C2×C7×C22⋊C4—28no—
57C2×C7×C22⋊C4—22no—
58C2×C7×C22⋊C4—22no—
59C2×C22⋊Dic7—24no—
60C2×C22⋊Dic7—42no—
61C2×C22⋊Dic7—42no—
62C22×C7⋊C8—22no—
63C7⋊#95—24no—
64C7⋊#95—42no—
65C22⋊(C7⋊C8)—24no—
66C22⋊(C7⋊C8)—42no—
67C22⋊(C7⋊C8)—42no—
68C22×#121D711yescharacteristic
69#1779—71no—
70C2×#327—74no—
71C22⋊#95—71no—
72C24×C7—22no—
73C24×C7—22no—
74C24×C7—22no—
75C24×C7—22no—
76C24×C7—24no—
77C24×C7—22no—
78C24×C7—41no—
79C24×C7—41no—
80C24×C7—42no—
81C24×C7—42no—
82C24×C7—42no—
83C24×C7—42no—
84C24×C7—42no—
85C24×C7—42no—
86C24×C7C22⋊C411yescharacteristic
87C24×C7C22⋊C411yescharacteristic
88C24×C7D812yes—
89C24×C7SD1612yes—
90C24×C7C2×D411yescharacteristic
91C2×C7×D4—24no—
92C2×C7×D4—28no—
93C2×C7×D4—28no—
94C2×C7×D4—22no—
95C2×C7×D4—416no—
96C2×C7×D4—42no—
97C2×C7×D4—42no—
98C2×C7×D4—42no—
99C2×C7×D4—416no—
100C2×C7×D4—416no—
101C2×C7×D4—44no—
102C2×C7×D4—44no—
103C2×C7×D4—42no—
104C2×C7×D4—42no—
105C2×C7×D4—44no—
106C2×C7×D4—42no—
107C2×C7×D4C22⋊C414yescomplemented
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—24no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—41no—
121C22×C4×C7—41no—
122C22×C4×C7—41no—
123C22×C4×C7—41no—
124C22×C4×C7—41no—
125C22×C4×C7—41no—
126C22×C4×C7—41no—
127C22×C4×C7—416no—
128C22×C4×C7—42no—
129C22×C4×C7—42no—
130C22×C4×C7C22⋊C411yescharacteristic
131C22×C4×C7C22⋊C411yescharacteristic
132C22×C4×C7C22×C411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D414yes—
135C22×Dic7—24no—
136C22×Dic7—42no—
137C22×Dic7—42no—
138C22×Dic7—42no—
139C22×Dic7—42no—
140C22×Dic7—42no—
141C22×Dic7—42no—
142C7×C4⋊C4—24no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4C22⋊C414yescomplemented
145C7×C22⋊C4—28no—
146C7×C22⋊C4—28no—
147C22⋊Dic7—48no—
148C2×C7⋊C8—42no—
149C2×C7⋊C8—42no—
150C2×C7⋊C8—42no—
151C23×D4—21no—
152C23×D4Dic711yescharacteristic, complemented
153C24×C4C2×D711yescharacteristic
154C2×#121—21no—
155C2×#121—21no—
156C2×#121—28no—
157C2×#121Dic712yescomplemented
158C2×#121C2×D712yes—
159C22×C4⋊C4Dic711yescharacteristic, complemented
160C22×C22⋊C4—21no—
161C22×C22⋊C4—141no—
162C2×C22⋊C8—72no—
163C2×C22⋊C8—141no—
164C2×C22⋊C8—141no—
165C2×#95—72no—
166C2×#95—141no—
167#327—148no—
168C23×C7—21no—
169C23×C7—21no—
170C23×C7—21no—
171C23×C7—21no—
172C23×C7—21no—
173C23×C7—21no—
174C23×C7—21no—
175C23×C7—21no—
176C23×C7—24no—
177C23×C7—24no—
178C23×C7—21no—
179C23×C7—21no—
180C23×C7—21no—
181C23×C7—21no—
182C23×C7—21no—
183C23×C7—21no—
184C23×C7—24no—
185C23×C7—24no—
186C23×C7—24no—
187C23×C7—42no—
188C23×C7—42no—
189C23×C7—44no—
190C23×C7—44no—
191C23×C7—44no—
192C23×C7—44no—
193C23×C7—48no—
194C23×C7—48no—
195C23×C7—48no—
196C23×C7—42no—
197C23×C7—42no—
198C23×C7—42no—
199C23×C7—42no—
200C23×C7—42no—
201C23×C7—42no—
202C23×C7—84no—
203C23×C7—84no—
204C23×C7—84no—
205C23×C7—84no—
206C23×C7#9912yes—
207C23×C7#10212yes—
208C23×C7#10212yes—
209C23×C7#10412yes—
210C23×C7C2×C22⋊C411yescharacteristic
211C23×C7#12012yes—
212C23×C7C2×D811yescharacteristic
213C23×C7C2×SD1611yescharacteristic
214C23×C7AGL(1,ℤ/8)12yes—
215C7×D4—44no—
216C7×D4—48no—
217C7×D4—44no—
218C7×D4—416no—
219C7×D4—48no—
220C7×D4—48no—
221C2×C4×C7—28no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—24no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—44no—
233C2×C4×C7—44no—
234C2×C4×C7—44no—
235C2×C4×C7—44no—
236C2×C4×C7—44no—
237C2×C4×C7—48no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7#9912yes—
241C2×C4×C7C2×C22⋊C412yes—
242C2×C4×C7#12012yes—
243C2×Dic7—44no—
244C2×Dic7—44no—
245C2×Dic7—44no—
246C2×Dic7—82no—
247C2×Dic7—82no—
248C2×Dic7—82no—
249C2×Dic7—82no—
250C7⋊C8—82no—
251C25—21no—
252C25C22⋊D711yescharacteristic
253C22×D4—28no—
254C22×D4—22no—
255C22×D4—22no—
256C22×D4—24no—
257C22×D4—21no—
258C22×D4—42no—
259C22×D4—42no—
260C22×D4—44no—
261C22×D4—42no—
262C22×D4—42no—
263C22×D4C2×Dic712yes—
264C23×C4—21no—
265C23×C4—24no—
266C23×C4—42no—
267C23×C4—42no—
268C23×C4—142no—
269C23×C4C2×Dic711yescharacteristic
270C23×C4C2×Dic711yescharacteristic
271C23×C4C22⋊D711yescharacteristic
272C23×C4C22⋊D711yescharacteristic
273C23×C4C22⋊D711yescharacteristic
274C23×C4C22×D711yescharacteristic
275C22×C8—142no—
276#121—28no—
277#121—28no—
278#121—24no—
279#121—24no—
280#121—24no—
281#121C22⋊D718yes—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4C2×Dic712yes—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—144no—
289C2×C22⋊C4—282no—
290C2×C22⋊C4—282no—
291C22⋊C8—144no—
292C22⋊C8—282no—
293C22⋊C8—282no—
294#95—144no—
295#95—282no—
296C22×C7—21no—
297C22×C7—21no—
298C22×C7—22no—
299C22×C7—22no—
300C22×C7—22no—
301C22×C7—22no—
302C22×C7—21no—
303C22×C7—22no—
304C22×C7—21no—
305C22×C7—22no—
306C22×C7—22no—
307C22×C7—22no—
308C22×C7—22no—
309C22×C7—22no—
310C22×C7—22no—
311C22×C7—22no—
312C22×C7—22no—
313C22×C7—48no—
314C22×C7—48no—
315C22×C7—41no—
316C22×C7—41no—
317C22×C7—41no—
318C22×C7—41no—
319C22×C7—44no—
320C22×C7—44no—
321C22×C7—44no—
322C22×C7—816no—
323C22×C7—82no—
324C22×C7—82no—
325C22×C7—82no—
326C22×C7—82no—
327C22×C7#32714yescomplemented
328C22×C7#37911yescharacteristic
329C22×C7C2×#9911yescharacteristic
330C22×C7C2×#9911yescharacteristic
331C22×C7C2×#10211yescharacteristic
332C22×C7#41811yescharacteristic
333C22×C7C2×#10411yescharacteristic
334C22×C7#42111yescharacteristic
335C22×C7#44712yes—
336C22×C7#45012yes—
337C4×C7—22no—
338C4×C7—22no—
339C4×C7—22no—
340C4×C7—41no—
341C4×C7—44no—
342Dic7—84no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—41no—
350C24—41no—
351C24—42no—
352C24—42no—
353C24—42no—
354C24—42no—
355C24—42no—
356C24—42no—
357C24#114112yes—
358C24#114212yes—
359C24C22⋊Dic711yescharacteristic
360C24C22⋊Dic711yescharacteristic
361C24C2×C22⋊D711yescharacteristic
362C2×D4—24no—
363C2×D4—28no—
364C2×D4—28no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—416no—
369C2×D4—42no—
370C2×D4—416no—
371C2×D4—416no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—42no—
375C2×D4—44no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4C22⋊Dic714yescomplemented
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—416no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—144no—
402C22×C4—282no—
403C22×C4—282no—
404C22×C4—282no—
405C22×C4—282no—
406C22×C4—282no—
407C22×C4—282no—
408C22×C4C22⋊Dic711yescharacteristic
409C22×C4C22⋊Dic711yescharacteristic
410C22×C4C22×Dic711yescharacteristic
411C22×C4C2×C22⋊D711yescharacteristic
412C22×C4C2×C22⋊D714yes—
413C2×C8—282no—
414C2×C8—282no—
415C2×C8—282no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4C22⋊Dic714yescomplemented
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—288no—
422C2×C7—22no—
423C2×C7—24no—
424C2×C7—22no—
425C2×C7—42no—
426C2×C7—84no—
427C2×C7C2×#32711yescharacteristic
428C2×C7#150711yescharacteristic
429C2×C7#151011yescharacteristic
430C2×C7#151911yescharacteristic
431C2×C7#178111yescharacteristic
432C2×C7#178611yescharacteristic
433C2×C7#178911yescharacteristic
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—24no—
444C23—24no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—48no—
460C23—48no—
461C23—48no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23#640612yes—
473C23#640612yes—
474C23#640812yes—
475C23#641112yes—
476C23C2×#114111yescharacteristic
477C23#649512yes—
478C23C2×#114211yescharacteristic
479C23C2×C22⋊Dic711yescharacteristic
480C23#651612yes—
481D4—44no—
482D4—48no—
483D4—44no—
484D4—416no—
485D4—48no—
486D4—48no—
487C2×C4—28no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—284no—
507C2×C4—284no—
508C2×C4—284no—
509C2×C4—562no—
510C2×C4—562no—
511C2×C4—562no—
512C2×C4—562no—
513C2×C4#640812yes—
514C2×C4C2×C22⋊Dic712yes—
515C2×C4#651612yes—
516C8—562no—
517C7#832811yescharacteristic, complemented, Sylow
518C22—21no—
519C22—21no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—21no—
525C22—22no—
526C22—21no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—48no—
536C22—48no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22—816no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22#8933214yescomplemented
550C22C2×#640611yescharacteristic
551C22#8998711yescharacteristic
552C22#8998912yes—
553C22#8999012yes—
554C22C2×#640811yescharacteristic
555C22C2×#640811yescharacteristic
556C22C2×#641111yescharacteristic
557C22#9000811yescharacteristic
558C22#9001911yescharacteristic
559C4—22no—
560C4—22no—
561C4—22no—
562C4—44no—
563C4—41no—
564C4—564no—
565C2—22no—
566C2—24no—
567C2—22no—
568C2—42no—
569C2—84no—
570C2C2×#8933211yescharacteristic
571C2#1172942711yescharacteristic
572C2#1172942811yescharacteristic
573C2#1172942911yescharacteristic
574C2#1172988211yescharacteristic
575C2#1172988411yescharacteristic
576C2#1172989211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
Fitting, prime radicalC22×C7×#121C2
socleC23×C7C2×C22⋊C4
cosocleC23×C4×C7C23
hypercenterC22×#121D7
hypocenterC7#8328
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#121

Complements

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

NH ≅ G/Ntype
2. #11729881C2semidirect
3. #11729880C2semidirect
4. C2×#89332C2semidirect
6. C23×C7×D4C4semidirect
11. C2×C7×#121C4semidirect
13. C22×C7×C4⋊C4C4semidirect
17. C2×C7⋊#95C22semidirect
20. C2×C22⋊(C7⋊C8)C22semidirect
107. C2×C7×D4C22⋊C4semidirect
144. C7×C4⋊C4C22⋊C4semidirect
152. C23×D4Dic7semidirect
157. C2×#121Dic7semidirect
159. C22×C4⋊C4Dic7semidirect
327. C22×C7#327semidirect
378. C2×D4C22⋊Dic7semidirect
418. C4⋊C4C22⋊Dic7semidirect
517. C7#8328semidirect
549. C22#89332semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1212
#117298812
#117298802
C2×#893322
#83287