Finite Group Databasebeta

← #13372157

Subgroups — #13372157

Order 1792.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2349C211yesmaximal, characteristic
2C2×C4×#1142C211yesmaximal, characteristic
3C2×#89790C211yesmaximal, characteristic
4C2×#89790C211yesmaximal, characteristic
5C4⋊C4×Dic14C211yesmaximal, characteristic, complemented
6#11729361C211yesmaximal, characteristic
7#11729360C211yesmaximal, characteristic
8#11729358C211yesmaximal, characteristic
9#11727386C211yesmaximal, characteristic, complemented
10#11727385C211yesmaximal, characteristic, complemented
11#11727374C211yesmaximal, characteristic, complemented
12#11727373C211yesmaximal, characteristic, complemented
13#11727313C211yesmaximal, characteristic, complemented
14#11727304C211yesmaximal, characteristic, complemented
15#11727232C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C22×#1142C2211yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C4×Dic14C2211yescharacteristic
22C2×C4×Dic14C2211yescharacteristic
23C7×#389C2211yescharacteristic
24C7×#389C2211yescharacteristic
25C7×#385—21no—
26C7×#385—21no—
27C7×#382C2211yescharacteristic
28C7×#380—21no—
29C4×C7×C4⋊C4C2211yescharacteristic
30C2×C7×#95—21no—
31C2×#6406C2211yescharacteristic
32C2×#6406C2211yescharacteristic
33C2×#6406C2211yescharacteristic
34#89790—24no—
35#89790C418yescomplemented
36C4×#1142—24no—
37#89748—21no—
38C4⋊C4×Dic7—21no—
39C2×#6384C2211yescharacteristic
40C2×#6384C2211yescharacteristic
41C2×#6384C2211yescharacteristic
42C2×#6382C2211yescharacteristic
43C2×#6382C2211yescharacteristic
44C2×#6381C2211yescharacteristic
45C2×#6381C2211yescharacteristic
46C2×#6381C2211yescharacteristic
47#89707C2211yescharacteristic
48#89700C2211yescharacteristic
49#89700C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51C2×#6377C2211yescharacteristic
52C2×#6377C2211yescharacteristic
53C2×C4×C7⋊C8C2211yescharacteristic
54#89419—21no—
55#89419—21no—
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89319C2211yescharacteristic
61#89319C2211yescharacteristic
62#22678—71nomaximal, Sylow
63C22×C7×D4C2311yescharacteristic
64C23×C4×C7—21no—
65C23×C4×C7—21no—
66C23×C4×C7—21no—
67C22×Dic14C2311yescharacteristic
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C4×C7×D4—22no—
71C4×C7×D4—41no—
72C4×C7×D4—41no—
73C4×C7×D4C2×C414yes—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4—21no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C7×C4⋊C4C2311yescharacteristic
79C2×C7×C4⋊C4C2311yescharacteristic
80C2×C7×C4⋊C4C2311yescharacteristic
81C2×C7×C22⋊C4—21no—
82C2×C7×C22⋊C4—21no—
83C2×C7×C22⋊C4—21no—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×C42×C7C2311yescharacteristic
87C2×#1142C2×C414yes—
88C2×#1142D414yes—
89C2×#1142Q814yes—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7C2311yescharacteristic
93C2×C4⋊Dic7C2311yescharacteristic
94C2×C4⋊Dic7C2311yescharacteristic
95C2×Dic7⋊C4—21no—
96C2×Dic7⋊C4—21no—
97C2×Dic7⋊C4—21no—
98C2×Dic7⋊C4—21no—
99C2×Dic7⋊C4—21no—
100C2×C4×Dic7—21no—
101C2×C4×Dic7—21no—
102C2×C4×Dic7—21no—
103C22×C7⋊C8C2311yescharacteristic
104C22×C7⋊C8C2311yescharacteristic
105C22×C7⋊C8C2311yescharacteristic
106C22×C7⋊C8C2311yescharacteristic
107C4×Dic14—22no—
108C4×Dic14—22no—
109C4×Dic14—22no—
110C4×Dic14—41no—
111C4×Dic14—41no—
112C4×Dic14C2×C414yescomplemented
113C7×#95—21no—
114C7×#95—21no—
115C7×#95—21no—
116C7×#95—21no—
117C7×#95D411yescharacteristic
118C7×#95D411yescharacteristic
119#6406—22no—
120#6406—22no—
121#6406C2×C414yes—
122C7⋊#95—21no—
123C7⋊#95—21no—
124#6384—22no—
125#6384—22no—
126#6384C2×C414yescomplemented
127#6382—22no—
128#6382—41no—
129#6381—22no—
130#6381—41no—
131#6381C2×C414yescomplemented
132C4⋊(C7⋊C8)—22no—
133#6377—22no—
134#6377C2×C414yescomplemented
135C4×C7⋊C8—22no—
136C2×#440—71no—
137C2×#440—71no—
138C2×C4×SD16—71no—
139Q8×C4⋊C4—71no—
140#2349D711yescharacteristic
141#1943—71no—
142#1942—71no—
143#1865—71no—
144#1864—71no—
145#1863—71no—
146#1862—71no—
147#1847—71no—
148#1805—71no—
149#1802—71no—
150#1771—71no—
151C24×C7—21no—
152C2×C7×D4—21no—
153C2×C7×D4—21no—
154C2×C7×D4—21no—
155C2×C7×D4C22×C412yes—
156C2×C7×D4C2×D412yes—
157C2×C7×D4C2×Q812yes—
158C22×C4×C7—21no—
159C22×C4×C7—24no—
160C22×C4×C7—21no—
161C22×C4×C7—21no—
162C22×C4×C7—42no—
163C22×C4×C7—42no—
164C22×C4×C7—42no—
165C22×C4×C7—42no—
166C22×C4×C7—81no—
167C22×C4×C7—81no—
168C22×C4×C7—81no—
169C22×C4×C7—81no—
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2×D411yescharacteristic
172C22×C4×C7C2×D411yescharacteristic
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D411yescharacteristic
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2411yescharacteristic
177C22×Dic7—21no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7—21no—
181C2×Dic14—21no—
182C2×Dic14—21no—
183C2×Dic14—21no—
184C2×Dic14C22×C412yes—
185C2×Dic14C2×D412yes—
186C2×Dic14C2×Q812yes—
187C7×C4⋊C4—22no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—21no—
191C7×C4⋊C4—21no—
192C7×C4⋊C4—21no—
193C7×C4⋊C4—41no—
194C7×C4⋊C4—81no—
195C7×C4⋊C4—81no—
196C7×C4⋊C4C22×C412yes—
197C7×C22⋊C4—24no—
198C7×C22⋊C4—42no—
199C7×C22⋊C4—42no—
200C42×C7—21no—
201C42×C7—21no—
202C42×C7—21no—
203C42×C7—21no—
204C42×C7C22×C412yes—
205C42×C7#4112yes—
206#1142C4⋊C4116yescomplemented
207C4⋊Dic7—21no—
208C4⋊Dic7—21no—
209C4⋊Dic7—21no—
210C4⋊Dic7—21no—
211C4⋊Dic7—21no—
212C4⋊Dic7—42no—
213C4⋊Dic7—81no—
214C4⋊Dic7C22×C412yes—
215Dic7⋊C4—24no—
216Dic7⋊C4—42no—
217Dic7⋊C4—42no—
218Dic7⋊C4—81no—
219Dic7⋊C4—81no—
220C4×Dic7—24no—
221C4×Dic7—42no—
222C4×Dic7—42no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8—21no—
226C2×C7⋊C8—21no—
227C2×C7⋊C8—21no—
228C2×C7⋊C8—21no—
229C2×C7⋊C8—21no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8C22×C412yes—
232C2×C7⋊C8C22×C412yes—
233C2×C7⋊C8C2×D412yes—
234C2×C7⋊C8C2×Q812yes—
235C22×SD16—71no—
236C2×C4×Q8—71no—
237C2×C4×Q8—71no—
238C2×C4×Q8—71no—
239C2×C4×D4C2×D711yescharacteristic
240C2×C4×D4C2×D711yescharacteristic
241C2×C4×D4C2×D711yescharacteristic
242#440—78no—
243#440—144no—
244C4×SD16—144no—
245C2×#107—71no—
246C2×#107—71no—
247C2×#107—71no—
248C2×#106—71no—
249C2×#106—71no—
250C2×C4⋊C8—71no—
251C2×#103—71no—
252C2×#103—71no—
253C2×#103—71no—
254C2×#102—71no—
255C2×#102—71no—
256C2×#102—71no—
257C2×#97—71no—
258C2×#97—71no—
259C2×C4×C8—71no—
260#389—71no—
261#389—71no—
262#389—141no—
263#389—141no—
264#389C2×D711yescharacteristic
265#389C2×D711yescharacteristic
266#385—21no—
267#385—21no—
268#384—71no—
269#384—141no—
270#382C2×D711yescharacteristic
271#380—21no—
272C4×C4⋊C4—141no—
273C4×C4⋊C4C2×D711yescharacteristic
274C2×#95—21no—
275#340—71no—
276#340—71no—
277#340—71no—
278#340—71no—
279#336—71no—
280#336—71no—
281C23×C7—22no—
282C23×C7—22no—
283C23×C7—22no—
284C23×C7—41no—
285C23×C7—41no—
286C23×C7—41no—
287C23×C7—41no—
288C23×C7C22×D411yescharacteristic
289C7×D4—22no—
290C7×D4—22no—
291C7×D4—22no—
292C7×D4C2×C4⋊C414yes—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7—21no—
297C2×C4×C7—21no—
298C2×C4×C7—21no—
299C2×C4×C7—22no—
300C2×C4×C7—21no—
301C2×C4×C7—21no—
302C2×C4×C7—21no—
303C2×C4×C7—21no—
304C2×C4×C7—21no—
305C2×C4×C7—21no—
306C2×C4×C7—21no—
307C2×C4×C7—21no—
308C2×C4×C7—21no—
309C2×C4×C7—41no—
310C2×C4×C7—41no—
311C2×C4×C7—41no—
312C2×C4×C7—41no—
313C2×C4×C7—41no—
314C2×C4×C7—41no—
315C2×C4×C7—41no—
316C2×C4×C7—41no—
317C2×C4×C7—84no—
318C2×C4×C7—84no—
319C2×C4×C7—84no—
320C2×C4×C7C4×D412yes—
321C2×C4×C7C4×D412yes—
322C2×C4×C7#13712yes—
323C2×C4×C7C23×C411yescharacteristic
324C2×C4×C7C22×D411yescharacteristic
325C2×C4×C7C22×Q811yescharacteristic
326C2×C4×C7C2×#4111yescharacteristic
327C2×C4×C7H(5,𝔽2)11yescharacteristic
328C2×C4×C72−1+411yescharacteristic
329C2×Dic7—22no—
330C2×Dic7—22no—
331C2×Dic7—22no—
332C2×Dic7—22no—
333C2×Dic7—41no—
334C2×Dic7—41no—
335C2×Dic7—41no—
336C2×Dic7—41no—
337C2×Dic7—41no—
338C2×Dic7—41no—
339C2×Dic7—41no—
340C2×Dic7—41no—
341Dic14—22no—
342Dic14—22no—
343Dic14—22no—
344Dic14C2×C4⋊C414yes—
345C7⋊C8—22no—
346C7⋊C8—41no—
347C7⋊C8—41no—
348C7⋊C8C2×C4⋊C414yes—
349C22×Q8—71no—
350C22×D4C22×D711yescharacteristic
351C23×C4—21no—
352C23×C4—21no—
353C23×C4—21no—
354C2×SD16—74no—
355C2×SD16—74no—
356C2×SD16—74no—
357C22×C8—71no—
358C22×C8—71no—
359C22×C8—71no—
360C22×C8—71no—
361C4×Q8—74no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×Q8—281no—
366C4×Q8—281no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—41no—
371C4×D4—41no—
372C4×D4C4×D714yes—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—71no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—141no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—141no—
382C2×C4⋊C4—141no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4C22⋊D711yescharacteristic
386C2×C4⋊C4C22⋊D711yescharacteristic
387C2×C4⋊C4C22×D711yescharacteristic
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C4⋊C4C22×D711yescharacteristic
390C2×C22⋊C4—21no—
391C2×C22⋊C4—21no—
392C2×C22⋊C4—21no—
393C2×C42—141no—
394C2×C42—141no—
395C2×C42—141no—
396C2×C42C22×D711yescharacteristic
397C2×C42C22×D711yescharacteristic
398C2×C42C22×D711yescharacteristic
399#107—74no—
400#107—142no—
401#107—281no—
402#106—142no—
403#106—281no—
404C4⋊C8—142no—
405#103—74no—
406#103—142no—
407#103—142no—
408#102—74no—
409#102—142no—
410#102—142no—
411#97—74no—
412#97—142no—
413C4×C8—142no—
414#95—21no—
415#95—21no—
416#95—21no—
417#95—21no—
418#95—141no—
419#95—141no—
420#95C22⋊D711yescharacteristic
421#95C22⋊D711yescharacteristic
422C22×C7—24no—
423C22×C7—42no—
424C22×C7—42no—
425C22×C7—42no—
426C22×C7—42no—
427C22×C7—42no—
428C22×C7—42no—
429C22×C7C2×C4×D411yescharacteristic
430C22×C7D4211yescharacteristic
431C22×C7D4×Q811yescharacteristic
432C22×C7C2×#13711yescharacteristic
433C22×C7#57511yescharacteristic
434C22×C7#57611yescharacteristic
435C22×C7#57711yescharacteristic
436C4×C7—22no—
437C4×C7—22no—
438C4×C7—22no—
439C4×C7—42no—
440C4×C7—41no—
441C4×C7—41no—
442C4×C7—41no—
443C4×C7—81no—
444C4×C7—81no—
445C4×C7C22×C4⋊C411yescharacteristic
446C4×C7#52011yescharacteristic
447C4×C7#54711yescharacteristic
448C4×C7#55411yescharacteristic
449Dic7—24no—
450Dic7—42no—
451Dic7—81no—
452Dic7—81no—
453C24—21no—
454C2×Q8—72no—
455C2×Q8—72no—
456C2×Q8—72no—
457C2×Q8—141no—
458C2×Q8—141no—
459C2×Q8—141no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4C2×C4×D712yes—
464C2×D4D4×D712yes—
465C2×D4Q8×D712yes—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4C2×C22⋊D711yescharacteristic
483C22×C4C2×C22⋊D711yescharacteristic
484C22×C4C2×C22⋊D711yescharacteristic
485C22×C4C2×C22⋊D711yescharacteristic
486C22×C4C2×C22⋊D711yescharacteristic
487C22×C4C2×C22⋊D711yescharacteristic
488C22×C4C23×D711yescharacteristic
489SD16—716no—
490C2×C8—72no—
491C2×C8—72no—
492C2×C8—72no—
493C2×C8—72no—
494C2×C8—141no—
495C2×C8—141no—
496C2×C8—141no—
497C2×C8—141no—
498C2×C8—141no—
499C2×C8—141no—
500C2×C8—141no—
501C2×C8—141no—
502C4⋊C4—22no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—41no—
509C4⋊C4—72no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4—144no—
513C4⋊C4—141no—
514C4⋊C4—141no—
515C4⋊C4—141no—
516C4⋊C4—141no—
517C4⋊C4—141no—
518C4⋊C4—282no—
519C4⋊C4—282no—
520C4⋊C4—282no—
521C4⋊C4—561no—
522C4⋊C4—561no—
523C4⋊C4—561no—
524C4⋊C4C2×C4×D712yes—
525C22⋊C4—24no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—144no—
533C42—282no—
534C42—282no—
535C42C2×C4×D712yes—
536C42#115712yes—
537C2×C7—44no—
538C2×C7D4×C4⋊C411yescharacteristic
539C2×C7C4×#13711yescharacteristic
540C2×C7#294611yescharacteristic
541C2×C7#326611yescharacteristic
542C2×C7#326811yescharacteristic
543C2×C7#338511yescharacteristic
544C2×C7#338811yescharacteristic
545C23—22no—
546C23—22no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23C22×C22⋊D711yescharacteristic
553Q8—74no—
554Q8—142no—
555Q8—142no—
556Q8—142no—
557D4—22no—
558D4—22no—
559D4—22no—
560D4D7×C4⋊C414yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—84no—
586C2×C4—84no—
587C2×C4—84no—
588C2×C4—142no—
589C2×C4—142no—
590C2×C4—142no—
591C2×C4—142no—
592C2×C4—281no—
593C2×C4—281no—
594C2×C4—281no—
595C2×C4—281no—
596C2×C4—281no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4C4×C22⋊D712yes—
601C2×C4C4×C22⋊D712yes—
602C2×C4#651412yes—
603C2×C4C22×C4×D711yescharacteristic
604C2×C4C2×#115711yescharacteristic
605C2×C4C2×D4×D711yescharacteristic
606C2×C4#654811yescharacteristic
607C2×C4C2×Q8×D711yescharacteristic
608C2×C4#655111yescharacteristic
609C8—74no—
610C8—142no—
611C8—281no—
612C8—281no—
613C7#2267811yescharacteristic, complemented, Sylow
614C22—24no—
615C22—42no—
616C22—42no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22C2×C4×C22⋊D711yescharacteristic
622C22D4×C22⋊D711yescharacteristic
623C22Q8×C22⋊D711yescharacteristic
624C22#9051311yescharacteristic
625C22C2×#651411yescharacteristic
626C22#9052611yescharacteristic
627C22#9052711yescharacteristic
628C4—22no—
629C4—22no—
630C4—22no—
631C4—41no—
632C4—41no—
633C4—41no—
634C4—42no—
635C4—81no—
636C4—81no—
637C4—144no—
638C4—282no—
639C4—561no—
640C4—561no—
641C4C2×D7×C4⋊C411yescharacteristic
642C4#9019511yescharacteristic
643C4#9019811yescharacteristic
644C4#9020211yescharacteristic
645C2—44no—
646C2C4⋊C4×C22⋊D711yescharacteristic
647C2C4×#651411yescharacteristic
648C2#1173998211yescharacteristic
649C2#1174095111yescharacteristic
650C2#1174095211yescharacteristic
651C2#1174107611yescharacteristic
652C2#1174107711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2349C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#2349D7
hypocenterC7#22678
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
5. C4⋊C4×Dic14C2semidirect
9. #11727386C2semidirect
10. #11727385C2semidirect
11. #11727374C2semidirect
12. #11727373C2semidirect
13. #11727313C2semidirect
14. #11727304C2semidirect
15. #11727232C2semidirect
35. #89790C4semidirect
112. C4×Dic14C2×C4semidirect
126. #6384C2×C4semidirect
131. #6381C2×C4semidirect
134. #6377C2×C4semidirect
206. #1142C4⋊C4semidirect
613. C7#22678semidirect

Maximal subgroups

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

subgroupindex
C7×#23492
C2×C4×#11422
C2×#897902
C4⋊C4×Dic142
#117293612
#117293602
#117293582
#117273862
#117273852
#117273742
#117273732
#117273132
#117273042
#117272322
#226787