Finite Group Databasebeta

← #13516696

Subgroups — #13516696

Order 1792.

1998 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90269C211yesmaximal, characteristic
2C7×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3#11739909C212yesmaximal
4#11739907C211yesmaximal, characteristic, complemented
5#11739901C212yesmaximal, complemented
6#11733437C214yesmaximal
7#11733429C212yesmaximal
8#11733423C214yesmaximal, complemented
9#11733419C212yesmaximal, complemented
10#11731803C212yesmaximal
11#11731770C211yesmaximal, characteristic, complemented
12#11731770C212yesmaximal, complemented
13#11731763C211yesmaximal, characteristic
14#11731205C212yesmaximal
15#11731197C211yesmaximal, characteristic
16#11731087C212yesmaximal, complemented
17C4⋊C4×Dic14C211yesmaximal, characteristic, complemented
18C2×C4×C7×Q8C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×#6509C2211yescharacteristic
21C2×#6507C2211yescharacteristic
22#90269C4116yescomplemented
23C2×#6460C2211yescharacteristic
24C2×#6459C2212yes—
25C2×#6455C2212yes—
26C2×#6452C2212yes—
27C2×#6451C2211yescharacteristic
28C2×#6435C2212yes—
29C2×C42×D7C2211yescharacteristic
30C2×C4×Dic14C2212yes—
31C2×C4×Dic14C2211yescharacteristic
32C2×C4×Dic14C2212yes—
33C7×#389C2212yes—
34C7×#389C2214yes—
35C7×#384C2211yescharacteristic
36C7×#384C2212yes—
37C4×C7×C4⋊C4C2212yes—
38C4×C7×C4⋊C4C2211yescharacteristic
39#90002C2212yes—
40#89762C2214yes—
41#89762C2214yes—
42#89759C2211yescharacteristic
43#89759C2212yes—
44#89759C2212yes—
45#89759C2211yescharacteristic
46#89753C2214yes—
47#89750C2214yes—
48#89750C2214yes—
49#89750C2214yes—
50#89750C2214yes—
51#89750C2214yes—
52#89749C2212yes—
53#89749C2212yes—
54#89749C2211yescharacteristic
55#89749C2211yescharacteristic
56#89748C2212yes—
57C4⋊C4×Dic7C2212yes—
58C4⋊C4×Dic7C2212yes—
59C4⋊C4×Dic7C2212yes—
60C4⋊C4×Dic7C2212yes—
61C4⋊C4×Dic7C2212yes—
62C4⋊C4×Dic7C2212yes—
63C4⋊C4×Dic7C2214yes—
64#89746C2214yes—
65#89746C2212yes—
66#89712C2214yes—
67C4×(C2×D7)⋊C4C2212yes—
68#89708C2212yes—
69#89707C2211yescharacteristic
70C4×C4⋊Dic7C2212yes—
71C4×C4⋊Dic7C2211yescharacteristic
72#89705C2214yes—
73#89705C2214yes—
74#89704C2212yes—
75C4×Dic7⋊C4C2214yes—
76#89700C2212yes—
77#89441C2214yes—
78#89436C2212yes—
79#89426C2214yes—
80#89424C2214yes—
81#89423C2212yes—
82#89419C2214yes—
83#89418C2214yes—
84#34373—71nomaximal, Sylow
85C22×C7×Q8C2311yescharacteristic
86C22×C4×D7C2311yescharacteristic
87C22×C4×D7C2311yescharacteristic
88C22×C4×D7C2311yescharacteristic
89C22×Dic14C2311yescharacteristic
90C22×Dic14C2312yes—
91C4×C7×Q8—44no—
92C4×C7×Q8C2×C418yes—
93C2×C7×C4⋊C4C2312yes—
94C2×C7×C4⋊C4C2312yes—
95C2×C7×C4⋊C4C2311yescharacteristic
96C2×C7×C4⋊C4C2311yescharacteristic
97C2×C7×C4⋊C4C2314yes—
98C2×C7×C4⋊C4C2312yes—
99C2×C7×C4⋊C4C2314yes—
100C2×C7×C4⋊C4C2311yescharacteristic
101C2×C7×C4⋊C4C2312yes—
102C2×C42×C7C2312yes—
103C2×C42×C7C2314yes—
104C2×C42×C7C2311yescharacteristic
105C2×C42×C7C2312yes—
106#6509C2×C418yes—
107#6507C2×C418yes—
108C2×(C2×D7)⋊C4C2314yes—
109C2×(C2×D7)⋊C4C2312yes—
110C2×(C2×D7)⋊C4C2312yes—
111C2×(C2×D7)⋊C4C2312yes—
112C2×(C2×D7)⋊C4C2312yes—
113C2×C4⋊Dic7C2312yes—
114C2×C4⋊Dic7C2311yescharacteristic
115C2×C4⋊Dic7C2312yes—
116C2×C4⋊Dic7C2311yescharacteristic
117C2×C4⋊Dic7C2312yes—
118C2×C4⋊Dic7C2311yescharacteristic
119C2×C4⋊Dic7C2312yes—
120C2×Dic7⋊C4C2314yes—
121C2×Dic7⋊C4C2314yes—
122C2×Dic7⋊C4C2312yes—
123C2×Dic7⋊C4C2312yes—
124C2×Dic7⋊C4C2312yes—
125C2×Dic7⋊C4C2312yes—
126C2×Dic7⋊C4C2312yes—
127C2×C4×Dic7C2312yes—
128C2×C4×Dic7C2314yes—
129C2×C4×Dic7C2314yes—
130C2×C4×Dic7C2314yes—
131C2×C4×Dic7C2314yes—
132C2×C4×Dic7C2311yescharacteristic
133C2×C4×Dic7C2312yes—
134C2×C4×Dic7C2311yescharacteristic
135C2×C4×Dic7C2311yescharacteristic
136#6460C2×C418yes—
137#6459C2×C4116yes—
138#6455—44no—
139#6452C2×C4116yes—
140#6451C2×C418yes—
141#6435C2×C4116yes—
142C42×D7C2×C418yes—
143C4×Dic14—44no—
144C4×Dic14C2×C4116yes—
145C4×Dic14C2×C418yes—
146C7×#95C2314yes—
147C7×#95C2312yes—
148C7⋊#95C2312yes—
149C7⋊#95C2314yes—
150C7⋊#95C2314yes—
151C7⋊#95C2314yes—
152C7⋊#95C2312yes—
153C7⋊#95C2312yes—
154C7⋊#95C2312yes—
155C7⋊#95C2312yes—
156C7⋊#95C2312yes—
157C7⋊#95C2314yes—
158C7⋊#95C2314yes—
159C7⋊#95C2314yes—
160C7⋊#95C2312yes—
161C7⋊#95C2314yes—
162C7⋊#95C2314yes—
163C7⋊#95C2314yes—
164C2×#533—71no—
165#2378—74no—
166#2362—72no—
167#2362—72no—
168#2355—71no—
169#2355—72no—
170#2355—72no—
171#2355—71no—
172Q8×C4⋊C4—71no—
173Q8×C4⋊C4D711yescharacteristic, complemented
174#2336—74no—
175#2331—72no—
176#2327—72no—
177#2314—72no—
178#2314—72no—
179#2285—71no—
180#2283—71no—
181C23×D7C2411yescharacteristic
182C2×C7×Q8—22no—
183C2×C7×Q8—22no—
184C2×C7×Q8C22×C414yes—
185C22×C4×C7C2412yes—
186C22×C4×C7C2414yes—
187C22×C4×C7C2412yes—
188C22×C4×C7C2412yes—
189C22×C4×C7C2412yes—
190C22×C4×C7C2411yescharacteristic
191C22×C4×C7C2411yescharacteristic
192C22×C4×C7C2411yescharacteristic
193C22×Dic7C2412yes—
194C22×Dic7C2412yes—
195C22×Dic7C2414yes—
196C22×Dic7C2411yescharacteristic
197C22×Dic7C2411yescharacteristic
198C22×Dic7C2411yescharacteristic
199C22×Dic7C2412yes—
200C22×Dic7C2412yes—
201C2×C4×D7—22no—
202C2×C4×D7—22no—
203C2×C4×D7—22no—
204C2×C4×D7—22no—
205C2×C4×D7—41no—
206C2×C4×D7—41no—
207C2×C4×D7C22×C414yes—
208C2×C4×D7C22×C414yes—
209C2×C4×D7C22×C414yes—
210C2×Dic14—22no—
211C2×Dic14—22no—
212C2×Dic14—42no—
213C2×Dic14—42no—
214C2×Dic14C22×C414yes—
215C2×Dic14C22×C418yes—
216C7×C4⋊C4—24no—
217C7×C4⋊C4—22no—
218C7×C4⋊C4—24no—
219C7×C4⋊C4—22no—
220C7×C4⋊C4—44no—
221C7×C4⋊C4—42no—
222C7×C4⋊C4—44no—
223C7×C4⋊C4C22×C414yes—
224C7×C4⋊C4C22×C418yes—
225C42×C7—24no—
226C42×C7—44no—
227C42×C7C22×C414yes—
228C42×C7C22×C418yes—
229(C2×D7)⋊C4—44no—
230(C2×D7)⋊C4—42no—
231(C2×D7)⋊C4—42no—
232(C2×D7)⋊C4C22×C418yes—
233(C2×D7)⋊C4C22×C418yes—
234C4⋊Dic7—24no—
235C4⋊Dic7—24no—
236C4⋊Dic7—22no—
237C4⋊Dic7—22no—
238C4⋊Dic7—42no—
239C4⋊Dic7C22×C414yes—
240C4⋊Dic7C22×C418yes—
241Dic7⋊C4—44no—
242Dic7⋊C4—44no—
243Dic7⋊C4C22×C418yes—
244Dic7⋊C4C22×C418yes—
245Dic7⋊C4C22×C418yes—
246Dic7⋊C4C22×C418yes—
247Dic7⋊C4C22×C418yes—
248C4×Dic7—24no—
249C4×Dic7—44no—
250C4×Dic7—44no—
251C4×Dic7—44no—
252C4×Dic7—44no—
253C4×Dic7C22×C414yes—
254C4×Dic7C22×C418yes—
255C4×Dic7C22×C414yes—
256C4×Dic7C22×C414yes—
257#533—716no—
258C2×C4⋊Q8—71no—
259C2×C4⋊Q8—71no—
260C2×#125—72no—
261C2×C22⋊Q8—71no—
262C2×C22⋊Q8—71no—
263C2×C22⋊Q8—72no—
264C2×C4×Q8—71no—
265C2×C4×Q8—72no—
266C2×C4×Q8—72no—
267C2×C4×Q8C2×D712yes—
268C2×C4×Q8C2×D711yescharacteristic
269C2×#117—72no—
270C2×#117—72no—
271C22×C42—71no—
272#391—72no—
273#391—72no—
274#391—74no—
275#391—74no—
276#389—72no—
277#389—74no—
278#389C2×D712yes—
279#389C2×D714yes—
280#388—74no—
281#388—74no—
282#388—74no—
283#388—74no—
284#387—74no—
285#387—74no—
286#387—74no—
287#387—74no—
288#387—74no—
289#387—74no—
290#387—74no—
291#387—74no—
292#387—74no—
293#387—74no—
294#384—72no—
295#384—71no—
296#384C2×D712yes—
297#384C2×D711yescharacteristic
298#383—72no—
299#382—72no—
300#382—72no—
301#382—71no—
302#382—71no—
303#382—72no—
304#381—72no—
305#380—71no—
306#380—71no—
307#380—72no—
308#380—72no—
309C4×C4⋊C4—72no—
310C4×C4⋊C4—72no—
311C4×C4⋊C4—72no—
312C4×C4⋊C4—72no—
313C4×C4⋊C4—72no—
314C4×C4⋊C4—74no—
315C4×C4⋊C4—72no—
316C4×C4⋊C4—72no—
317C4×C4⋊C4—74no—
318C4×C4⋊C4—71no—
319C4×C4⋊C4C2×D712yes—
320C4×C4⋊C4C2×D711yescharacteristic
321C4×C22⋊C4—72no—
322#376—72no—
323C23×C7C2511yescharacteristic
324C22×D7—21no—
325C22×D7—21no—
326C22×D7—21no—
327C22×D7—21no—
328C22×D7—21no—
329C22×D7—21no—
330C22×D7C23×C412yes—
331C7×Q8—28no—
332C2×C4×C7—22no—
333C2×C4×C7—22no—
334C2×C4×C7—22no—
335C2×C4×C7—21no—
336C2×C4×C7—22no—
337C2×C4×C7—21no—
338C2×C4×C7—22no—
339C2×C4×C7—24no—
340C2×C4×C7—22no—
341C2×C4×C7—22no—
342C2×C4×C7—22no—
343C2×C4×C7—24no—
344C2×C4×C7—24no—
345C2×C4×C7—22no—
346C2×C4×C7—22no—
347C2×C4×C7C23×C412yes—
348C2×C4×C7C23×C414yes—
349C2×C4×C7C23×C414yes—
350C2×C4×C7C23×C412yes—
351C2×C4×C7C23×C412yes—
352C2×C4×C7H(5,𝔽2)12yes—
353C2×C4×C72−1+412yes—
354C2×C4×C72−1+412yes—
355C2×C4×C72−1+412yes—
356C2×Dic7—21no—
357C2×Dic7—22no—
358C2×Dic7—22no—
359C2×Dic7—21no—
360C2×Dic7—21no—
361C2×Dic7—21no—
362C2×Dic7—22no—
363C2×Dic7—22no—
364C2×Dic7—21no—
365C2×Dic7—21no—
366C2×Dic7—24no—
367C2×Dic7—22no—
368C2×Dic7—22no—
369C2×Dic7—24no—
370C2×Dic7—22no—
371C2×Dic7—22no—
372C2×Dic7—22no—
373C2×Dic7—22no—
374C2×Dic7—24no—
375C2×Dic7C23×C412yes—
376C2×Dic7C23×C412yes—
377C2×Dic7C23×C412yes—
378C2×Dic7C23×C414yes—
379C2×Dic7C23×C414yes—
380C4×D7—28no—
381C4×D7—28no—
382C4×D7—44no—
383Dic14—28no—
384Dic14—48no—
385C22×Q8—71no—
386C22×Q8—72no—
387C22×Q8C22×D711yescharacteristic
388C23×C4—71no—
389C23×C4—71no—
390C23×C4—71no—
391C4⋊Q8—78no—
392C4⋊Q8—78no—
393#125—716no—
394C22⋊Q8—78no—
395C22⋊Q8—716no—
396C22⋊Q8—78no—
397C4×Q8—44no—
398C4×Q8—716no—
399C4×Q8—78no—
400C4×Q8—284no—
401C4×Q8C4×D718yes—
402#117—716no—
403#117—284no—
404C2×C4⋊C4—72no—
405C2×C4⋊C4—72no—
406C2×C4⋊C4—72no—
407C2×C4⋊C4—72no—
408C2×C4⋊C4—72no—
409C2×C4⋊C4—72no—
410C2×C4⋊C4—71no—
411C2×C4⋊C4—72no—
412C2×C4⋊C4—71no—
413C2×C4⋊C4—71no—
414C2×C4⋊C4—72no—
415C2×C4⋊C4—74no—
416C2×C4⋊C4—74no—
417C2×C4⋊C4—72no—
418C2×C4⋊C4C22×D712yes—
419C2×C4⋊C4C22×D711yescharacteristic
420C2×C4⋊C4C22×D712yes—
421C2×C4⋊C4C22×D711yescharacteristic
422C2×C4⋊C4C22×D714yes—
423C2×C4⋊C4C22×D712yes—
424C2×C4⋊C4C22×D714yes—
425C2×C4⋊C4C22×D712yes—
426C2×C4⋊C4C22×D711yescharacteristic
427C2×C22⋊C4—72no—
428C2×C22⋊C4—72no—
429C2×C22⋊C4—74no—
430C2×C22⋊C4—72no—
431C2×C22⋊C4—72no—
432C2×C42—72no—
433C2×C42—71no—
434C2×C42—71no—
435C2×C42—71no—
436C2×C42—78no—
437C2×C42—72no—
438C2×C42—74no—
439C2×C42—74no—
440C2×C42—74no—
441C2×C42—74no—
442C2×C42C22×D712yes—
443C2×C42C22×D714yes—
444C2×C42C22×D712yes—
445C2×C42C22×D711yescharacteristic
446#95—72no—
447#95—72no—
448#95—72no—
449#95—72no—
450#95—72no—
451#95—72no—
452#95—74no—
453#95—74no—
454#95—72no—
455#95—74no—
456#95—74no—
457#95—74no—
458#95—74no—
459#95—74no—
460#95—74no—
461#95—74no—
462#95C22×D712yes—
463#95C22×D714yes—
464C22×C7C24×C411yescharacteristic
465C22×C7C2×H(5,𝔽2)11yescharacteristic
466C22×C7C2×2−1+411yescharacteristic
467C22×C7C2×2−1+411yescharacteristic
468C22×C7C2×2−1+411yescharacteristic
469C22×C7#58511yescharacteristic
470C22×C7#58511yescharacteristic
471C2×D7—22no—
472C2×D7—22no—
473C2×D7—22no—
474C2×D7—41no—
475C2×D7—41no—
476C2×D7—41no—
477C2×D7—41no—
478C4×C7—24no—
479C4×C7—24no—
480C4×C7—22no—
481C4×C7—42no—
482C4×C7—42no—
483C4×C7—44no—
484C4×C7#51914yes—
485C4×C7#52014yes—
486Dic7—22no—
487Dic7—24no—
488Dic7—22no—
489Dic7—22no—
490Dic7—24no—
491Dic7—44no—
492Dic7—42no—
493Dic7—42no—
494C24—71no—
495C2×Q8—22no—
496C2×Q8—22no—
497C2×Q8—74no—
498C2×Q8—78no—
499C2×Q8—142no—
500C2×Q8—142no—
501C2×Q8—282no—
502C2×Q8—282no—
503C2×Q8C2×C4×D714yes—
504C22×C4—71no—
505C22×C4—71no—
506C22×C4—71no—
507C22×C4—72no—
508C22×C4—72no—
509C22×C4—74no—
510C22×C4—74no—
511C22×C4—74no—
512C22×C4—72no—
513C22×C4—72no—
514C22×C4—74no—
515C22×C4—142no—
516C22×C4—142no—
517C22×C4—142no—
518C22×C4—142no—
519C22×C4—281no—
520C22×C4—281no—
521C22×C4C23×D712yes—
522C22×C4C23×D714yes—
523C22×C4C23×D712yes—
524C22×C4C23×D712yes—
525C22×C4C23×D712yes—
526C22×C4C23×D711yescharacteristic
527C22×C4C23×D711yescharacteristic
528C22×C4C23×D711yescharacteristic
529C4⋊C4—24no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—44no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—78no—
537C4⋊C4—78no—
538C4⋊C4—78no—
539C4⋊C4—78no—
540C4⋊C4—78no—
541C4⋊C4—78no—
542C4⋊C4—74no—
543C4⋊C4—144no—
544C4⋊C4—142no—
545C4⋊C4—144no—
546C4⋊C4—142no—
547C4⋊C4—284no—
548C4⋊C4—284no—
549C4⋊C4—282no—
550C4⋊C4C2×C4×D718yes—
551C4⋊C4C2×C4×D714yes—
552C22⋊C4—78no—
553C22⋊C4—78no—
554C22⋊C4—282no—
555C22⋊C4—282no—
556C22⋊C4—284no—
557C42—24no—
558C42—44no—
559C42—74no—
560C42—74no—
561C42—74no—
562C42—78no—
563C42—144no—
564C42—284no—
565C42—284no—
566C42—284no—
567C42—284no—
568C42C2×C4×D714yes—
569C42C2×C4×D718yes—
570C2×C7C2×#51911yescharacteristic
571C2×C7C2×#52011yescharacteristic
572C2×C7#342811yescharacteristic
573C2×C7#355411yescharacteristic
574C2×C7#355411yescharacteristic
575C2×C7#355511yescharacteristic
576C2×C7#355511yescharacteristic
577D7—42no—
578C23—72no—
579C23—141no—
580C23—141no—
581C23—141no—
582C23—141no—
583C23—141no—
584C23—141no—
585C23C24×D711yescharacteristic
586Q8—28no—
587Q8—148no—
588Q8—288no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—72no—
605C2×C4—72no—
606C2×C4—72no—
607C2×C4—74no—
608C2×C4—74no—
609C2×C4—141no—
610C2×C4—142no—
611C2×C4—141no—
612C2×C4—141no—
613C2×C4—142no—
614C2×C4—141no—
615C2×C4—142no—
616C2×C4—141no—
617C2×C4—142no—
618C2×C4—141no—
619C2×C4—148no—
620C2×C4—148no—
621C2×C4—144no—
622C2×C4—142no—
623C2×C4—142no—
624C2×C4—144no—
625C2×C4—142no—
626C2×C4—142no—
627C2×C4—142no—
628C2×C4—142no—
629C2×C4—144no—
630C2×C4—284no—
631C2×C4C22×C4×D712yes—
632C2×C4C22×C4×D714yes—
633C2×C4C22×C4×D714yes—
634C2×C4C22×C4×D712yes—
635C2×C4C22×C4×D712yes—
636C2×C4#654812yes—
637C2×C4#655112yes—
638C2×C4#655112yes—
639C2×C4#655412yes—
640C7#3437311yescharacteristic, complemented, Sylow
641C22—142no—
642C22—142no—
643C22—142no—
644C22—281no—
645C22—281no—
646C22—281no—
647C22—281no—
648C22C23×C4×D711yescharacteristic
649C22C2×#654811yescharacteristic
650C22C2×#655111yescharacteristic
651C22C2×#655111yescharacteristic
652C22C2×#655411yescharacteristic
653C22#9061811yescharacteristic
654C22#9062011yescharacteristic
655C4—24no—
656C4—22no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—142no—
662C4—144no—
663C4—142no—
664C4—142no—
665C4—144no—
666C4—284no—
667C4—282no—
668C4—282no—
669C4#9019514yes—
670C4#9026314yes—
671C2—282no—
672C2C2×#9019511yescharacteristic
673C2C2×#9026311yescharacteristic
674C2#1174295311yescharacteristic
675C2#1174383311yescharacteristic
676C2#1174385711yescharacteristic
677C2#1174410411yescharacteristic
678C2#1174411611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×Q8×C4⋊C4C2
socle, cosocleC23×C7C25
prime radicalC23×D7C24
hypercenterQ8×C4⋊C4D7
hypocenterC7#34373
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C7×Q8×C4⋊C4C2semidirect
4. #11739907C2semidirect
5. #11739901C2semidirect
8. #11733423C2semidirect
9. #11733419C2semidirect
11. #11731770C2semidirect
12. #11731770C2semidirect
16. #11731087C2semidirect
17. C4⋊C4×Dic14C2semidirect
22. #90269C4semidirect
173. Q8×C4⋊C4D7semidirect
640. C7#34373semidirect

Maximal subgroups

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

subgroupindex
C2×#902692
C7×Q8×C4⋊C42
#117399092
#117399072
#117399012
#117334372
#117334292
#117334232
#117334192
#117318032
#117317702
#117317632
#117312052
#117311972
#117310872
C4⋊C4×Dic142
#343737