Finite Group Databasebeta

← #13321605

Subgroups — #13321605

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742855C211yesmaximal, characteristic
2C7×#2357C211yesmaximal, characteristic, complemented
3#11739747C211yesmaximal, characteristic, complemented
4#11727298C212yesmaximal, complemented
5#11727287C211yesmaximal, characteristic, complemented
6#11727286C212yesmaximal, complemented
7#11727285C211yesmaximal, characteristic, complemented
8#11727224C212yesmaximal, complemented
9#11727199C211yesmaximal, characteristic, complemented
10#11727198C211yesmaximal, characteristic, complemented
11#11727197C212yesmaximal, complemented
12C2×C4×C7×D4C2211yescharacteristic
13#90495—21no—
14#90482C2212yes—
15C2×#6493C2211yescharacteristic
16C2×#6487C2211yescharacteristic
17#90256C412yescomplemented
18#90247—22no—
19#90243C412yescomplemented
20#90241C412yescomplemented
21#90232C412yescomplemented
22#90231—22no—
23#90206—21no—
24#90185—21no—
25#90177—21no—
26#90170C2212yes—
27C7×#390—21no—
28C7×#388—21no—
29C7×#388—21no—
30C7×#382C2211yescharacteristic, complemented
31C7×#381C2211yescharacteristic
32C7×#381C2211yescharacteristic, complemented
33C4×C7×C4⋊C4—21no—
34C4×C7×C4⋊C4C2211yescharacteristic, complemented
35C4×C7×C22⋊C4C2211yescharacteristic
36C4×C7×C22⋊C4C2211yescharacteristic, complemented
37#89979C2211yescharacteristic
38#89979C2211yescharacteristic
39C2×C22⋊(C7⋊C8)C2212yescomplemented
40#89783—22no—
41#89783—22no—
42#89698C2212yescomplemented
43#89247C2212yescomplemented
44#89247C2212yescomplemented
45#89247C2212yescomplemented
46#89247C2212yescomplemented
47#89246C2212yes—
48#89246C2212yes—
49#89246C2212yes—
50#89246C2212yes—
51#17625—71nomaximal, Sylow
52C22×C7×D4D411yescharacteristic
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C22×C22⋊D7—21no—
56C2×#1157C2×C412yes—
57C22×D28C2×C411yescharacteristic
58C22×Dic14C2×C411yescharacteristic
59C4×C7×D4C2×C412yes—
60C4×C7×D4C2×C412yes—
61C4×C7×D4D412yes—
62C4×C7×D4D412yes—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4D411yescharacteristic
66C2×C7×C22⋊C4—21no—
67C2×C7×C22⋊C4—21no—
68C2×C7×C22⋊C4—21no—
69C2×C7×C22⋊C4—21no—
70C2×C7×C22⋊C4D411yescharacteristic
71C2×C7×C22⋊C4D411yescharacteristic
72C2×C42×C7—21no—
73C2×C42×C7—21no—
74C2×C42×C7—21no—
75C2×C42×C7C2311yescharacteristic
76#6516—22no—
77C2×C22⋊Dic7—21no—
78#6502—22no—
79#6502—22no—
80#6500—22no—
81#6498—22no—
82#6493—21no—
83#6493—21no—
84#6493C2×C412yes—
85#6493C2×C412yes—
86#6493C2312yes—
87#6492—22no—
88#6492—22no—
89C4×C22⋊D7—22no—
90C4×C22⋊D7—22no—
91C4×C22⋊D7—22no—
92C2×(C2×D7)⋊C4—21no—
93#6489C2×C412yes—
94C2×C4⋊Dic7C2×C411yescharacteristic
95C2×C4⋊Dic7C2×C411yescharacteristic
96#6487—21no—
97#6487—21no—
98#6487C2×C412yes—
99#6487C2×C412yes—
100#6487C2312yes—
101C2×Dic7⋊C4—21no—
102C2×#1136—21no—
103C2×#1136—21no—
104C22×C7⋊C8—22no—
105#6461—22no—
106#6459—22no—
107#6457—22no—
108#6452—22no—
109#6448—22no—
110#6447—22no—
111#6446—22no—
112(C22⋊D7)⋊C4—22no—
113(C22⋊D7)⋊C4—22no—
114D7×C22⋊C4—22no—
115#6442—22no—
116#6440—22no—
117#6439—22no—
118#6438C2×C412yes—
119C4×D28C2×C412yes—
120#6435—22no—
121#6433C2×C412yes—
122C4×Dic14C2×C412yes—
123C7×#95—21no—
124C7×#95—21no—
125C7×#95—21no—
126C7×#95C2311yescharacteristic
127C7×#95C2311yescharacteristic
128C7×#95C2311yescharacteristic
129C7×#95C2311yescharacteristic
130C22⋊(C7⋊C8)—22no—
131C22⋊(C7⋊C8)—21no—
132C22⋊(C7⋊C8)—21no—
133C22⋊(C7⋊C8)C2312yes—
134C22⋊(C7⋊C8)C2312yes—
135C4⋊(C7⋊C8)—22no—
136#6377—22no—
137#3505—71no—
138#2973—71no—
139#2357D711yescharacteristic, complemented
140#1516—72no—
141#1512—72no—
142#1511—71no—
143#1510—71no—
144#1509—72no—
145#1508—71no—
146#1507—71no—
147#1501—72no—
148C24×C7C2×D411yescharacteristic
149C24×C7C2×D411yescharacteristic
150C23×D7—21no—
151C2×C7×D4—41no—
152C2×C7×D4—41no—
153C2×C7×D4C22⋊C412yes—
154C2×C7×D4C22⋊C412yes—
155C22×C4×C7—21no—
156C22×C4×C7—21no—
157C22×C4×C7—21no—
158C22×C4×C7—21no—
159C22×C4×C7—81no—
160C22×C4×C7—81no—
161C22×C4×C7C22×C411yescharacteristic
162C22×C4×C7C22×C412yes—
163C22×C4×C7C22×C411yescharacteristic
164C22×C4×C7C2×D411yescharacteristic
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D411yescharacteristic
167C22×C4×C7C2×D411yescharacteristic
168C22×C4×C7C2×D412yes—
169C22×C4×C7C2×D411yescharacteristic
170C22×C4×C7C2×D411yescharacteristic
171C22×C4×C7C2411yescharacteristic
172C2×C22⋊D7—22no—
173C2×C22⋊D7—22no—
174C2×C22⋊D7—22no—
175C2×C22⋊D7—22no—
176C2×C22⋊D7—42no—
177C2×C22⋊D7—42no—
178C22×Dic7—21no—
179C22×Dic7—21no—
180C22×Dic7—21no—
181#1157—82no—
182C2×D28—21no—
183C2×D28—81no—
184C2×D28C22×C412yes—
185C2×C4×D7—22no—
186C2×C4×D7—22no—
187C2×Dic14—21no—
188C2×Dic14—81no—
189C2×Dic14C22×C412yes—
190C7×C4⋊C4—42no—
191C7×C4⋊C4—42no—
192C7×C4⋊C4C22⋊C412yes—
193C7×C4⋊C4C22⋊C412yes—
194C7×C22⋊C4—42no—
195C7×C22⋊C4—42no—
196C7×C22⋊C4—81no—
197C7×C22⋊C4—81no—
198C7×C22⋊C4C22⋊C412yes—
199C7×C22⋊C4C22⋊C412yes—
200C7×C22⋊C4C22⋊C412yes—
201C7×C22⋊C4C22⋊C412yes—
202C42×C7—42no—
203C42×C7—81no—
204C42×C7—81no—
205C42×C7C22×C412yes—
206C42×C7C2×D412yes—
207C22⋊Dic7—22no—
208C22⋊Dic7—22no—
209C22⋊Dic7—22no—
210C22⋊Dic7—22no—
211(C2×D7)⋊C4—22no—
212(C2×D7)⋊C4—22no—
213(C2×D7)⋊C4—22no—
214(C2×D7)⋊C4—22no—
215(C2×D7)⋊C4—22no—
216(C2×D7)⋊C4—22no—
217C4⋊Dic7—21no—
218C4⋊Dic7—21no—
219C4⋊Dic7C22×C412yes—
220C4⋊Dic7C22×C412yes—
221Dic7⋊C4—22no—
222Dic7⋊C4—22no—
223Dic7⋊C4—22no—
224Dic7⋊C4—22no—
225Dic7⋊C4—22no—
226Dic7⋊C4—22no—
227C4×Dic7—22no—
228C4×Dic7—22no—
229#1136—81no—
230#1136—81no—
231C2×C7⋊C8—22no—
232C2×C7⋊C8—22no—
233C2×C7⋊C8—82no—
234#553—72no—
235#553—142no—
236#552—72no—
237#551—142no—
238#548—72no—
239#546—72no—
240#539—141no—
241#538—141no—
242#534—141no—
243#529—72no—
244#525—72no—
245C2×C22⋊Q8—71no—
246C2×#121—71no—
247#518—141no—
248C2×C4×D4C2×D711yescharacteristic
249#435—142no—
250#435—142no—
251#432—72no—
252#407—71no—
253#407—71no—
254C2×C22⋊C8—72no—
255#390—21no—
256#388—21no—
257#388—21no—
258#382C2×D711yescharacteristic, complemented
259#381C2×D711yescharacteristic
260#381C2×D711yescharacteristic, complemented
261C4×C4⋊C4—21no—
262C4×C4⋊C4C2×D711yescharacteristic, complemented
263C4×C22⋊C4C2×D711yescharacteristic
264C4×C22⋊C4C2×D711yescharacteristic, complemented
265#328—72no—
266#328—72no—
267#328—72no—
268#328—72no—
269#327—72no—
270#327—72no—
271#327—72no—
272#327—72no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7—21no—
277C23×C7—41no—
278C23×C7—41no—
279C23×C7—41no—
280C23×C7—41no—
281C23×C7C2×C22⋊C411yescharacteristic
282C23×C7C2×C22⋊C412yes—
283C23×C7C2×C22⋊C411yescharacteristic
284C23×C7C22×D411yescharacteristic
285C22×D7—22no—
286C22×D7—42no—
287C22×D7—81no—
288C22×D7—81no—
289C7×D4—42no—
290C7×D4—42no—
291C2×C4×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—22no—
294C2×C4×C7—42no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7—41no—
299C2×C4×C7—41no—
300C2×C4×C7—41no—
301C2×C4×C7—41no—
302C2×C4×C7—42no—
303C2×C4×C7—41no—
304C2×C4×C7—41no—
305C2×C4×C7—41no—
306C2×C4×C7—41no—
307C2×C4×C7—41no—
308C2×C4×C7—41no—
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×C7C2×C22⋊C412yes—
316C2×C4×C7C2×C22⋊C412yes—
317C2×C4×C7C2×C22⋊C411yescharacteristic
318C2×C4×C7C2×C22⋊C412yes—
319C2×C4×C7C2×C22⋊C411yescharacteristic
320C2×C4×C7C23×C411yescharacteristic
321C2×C4×C7C22×D411yescharacteristic
322C22⋊D7—44no—
323C22⋊D7—42no—
324C22⋊D7—42no—
325C22⋊D7—82no—
326C22⋊D7—82no—
327C2×Dic7—22no—
328C2×Dic7—22no—
329C2×Dic7—22no—
330C2×Dic7—42no—
331C2×Dic7—81no—
332C2×Dic7—81no—
333D28—42no—
334D28—81no—
335C4×D7—82no—
336C4×D7—82no—
337Dic14—42no—
338Dic14—81no—
339C7⋊C8—42no—
340C7⋊C8—42no—
341C2×#41—72no—
342C22×Q8—71no—
343C22×D4—71no—
344C22×D4—141no—
345C22×D4C22⋊D711yescharacteristic
346C23×C4C22×D711yescharacteristic
347C23×C4C22×D711yescharacteristic
348C2×M16—141no—
349C2×M16—141no—
350C22×C8—142no—
351#126—142no—
352#126—142no—
353#126—142no—
354#125—72no—
355#125—142no—
356#124—72no—
357#124—142no—
358#123—142no—
359#123—142no—
360#123—142no—
361#123—142no—
362#123—142no—
363C22⋊Q8—72no—
364C22⋊Q8—72no—
365C22⋊Q8—72no—
366C22⋊Q8—141no—
367C22⋊Q8—141no—
368C22⋊Q8—142no—
369#121—72no—
370#121—72no—
371#121—72no—
372#121—142no—
373#121—142no—
374#121—141no—
375#121—141no—
376#121—142no—
377#120—142no—
378#120—142no—
379C4×Q8—72no—
380C4×D4—72no—
381C4×D4—142no—
382C4×D4—142no—
383C4×D4—142no—
384C4×D4—142no—
385C4×D4—142no—
386C4×D4C4×D712yes—
387C4×D4C4×D712yes—
388C4×D4D2812yes—
389C4×D4D2812yes—
390#117—72no—
391#117—142no—
392#117—142no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—71no—
396C2×C4⋊C4—71no—
397C2×C4⋊C4—141no—
398C2×C4⋊C4C22⋊D711yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—141no—
404C2×C22⋊C4—141no—
405C2×C22⋊C4—142no—
406C2×C22⋊C4C22⋊D711yescharacteristic
407C2×C22⋊C4C22⋊D711yescharacteristic
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42C22×D711yescharacteristic
412C4⋊C8—142no—
413C22⋊C8—72no—
414C22⋊C8—72no—
415C22⋊C8—142no—
416C22⋊C8—141no—
417C22⋊C8—141no—
418#97—142no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95C22×D711yescharacteristic
423#95C22×D711yescharacteristic
424#95C22×D711yescharacteristic
425#95C22×D711yescharacteristic
426C22×C7—21no—
427C22×C7—21no—
428C22×C7—21no—
429C22×C7—22no—
430C22×C7—21no—
431C22×C7—21no—
432C22×C7—21no—
433C22×C7—22no—
434C22×C7—21no—
435C22×C7—22no—
436C22×C7—21no—
437C22×C7—81no—
438C22×C7—81no—
439C22×C7—81no—
440C22×C7—81no—
441C22×C7C22×C22⋊C411yescharacteristic
442C22×C7#57511yescharacteristic
443C22×C7#57511yescharacteristic
444C2×D7—42no—
445C2×D7—42no—
446C2×D7—42no—
447C2×D7—162no—
448C4×C7—21no—
449C4×C7—21no—
450C4×C7—21no—
451C4×C7—22no—
452C4×C7—21no—
453C4×C7—22no—
454C4×C7—22no—
455C4×C7—42no—
456C4×C7—81no—
457C4×C7—81no—
458C4×C7—81no—
459Dic7—42no—
460Dic7—42no—
461Dic7—42no—
462C24—141no—
463C24C2×C22⋊D711yescharacteristic
464C24C2×C22⋊D711yescharacteristic
465#41—562no—
466C2×Q8—72no—
467C2×Q8—141no—
468C2×Q8—561no—
469C2×D4—41no—
470C2×D4—41no—
471C2×D4—72no—
472C2×D4—142no—
473C2×D4—142no—
474C2×D4—142no—
475C2×D4—142no—
476C2×D4—141no—
477C2×D4—282no—
478C2×D4—282no—
479C2×D4—561no—
480C2×D4(C2×D7)⋊C412yes—
481C2×D4(C2×D7)⋊C412yes—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—81no—
487C22×C4—81no—
488C22×C4—141no—
489C22×C4—141no—
490C22×C4—141no—
491C22×C4—142no—
492C22×C4—142no—
493C22×C4C2×C4×D711yescharacteristic
494C22×C4C2×C4×D711yescharacteristic
495C22×C4C2×C4×D712yes—
496C22×C4C2×D2811yescharacteristic
497C22×C4C2×D2811yescharacteristic
498C22×C4C2×D2812yes—
499C22×C4C2×C22⋊D711yescharacteristic
500C22×C4C2×C22⋊D711yescharacteristic
501C22×C4C2×C22⋊D711yescharacteristic
502C22×C4C2×C22⋊D711yescharacteristic
503C22×C4C23×D711yescharacteristic
504M16—561no—
505M16—561no—
506C2×C8—142no—
507C2×C8—142no—
508C2×C8—562no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—72no—
512C4⋊C4—72no—
513C4⋊C4—142no—
514C4⋊C4—142no—
515C4⋊C4—142no—
516C4⋊C4—142no—
517C4⋊C4—142no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—142no—
521C4⋊C4(C2×D7)⋊C412yes—
522C4⋊C4(C2×D7)⋊C412yes—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—81no—
526C22⋊C4—81no—
527C22⋊C4—142no—
528C22⋊C4—142no—
529C22⋊C4—142no—
530C22⋊C4—142no—
531C22⋊C4—142no—
532C22⋊C4—142no—
533C22⋊C4—142no—
534C22⋊C4—142no—
535C22⋊C4—142no—
536C22⋊C4—142no—
537C22⋊C4(C2×D7)⋊C412yes—
538C22⋊C4(C2×D7)⋊C412yes—
539C22⋊C4(C2×D7)⋊C412yes—
540C22⋊C4(C2×D7)⋊C412yes—
541C42—42no—
542C42—81no—
543C42—81no—
544C42—142no—
545C42—142no—
546C42C2×C4×D712yes—
547C42C2×D2812yes—
548C2×C7—21no—
549C2×C7—21no—
550C2×C7—41no—
551C2×C7—41no—
552C2×C7—42no—
553C2×C7#288311yescharacteristic
554C2×C7#289611yescharacteristic
555C2×C7#290111yescharacteristic
556D7—82no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—142no—
566C23—282no—
567C23—561no—
568C23—561no—
569C23C2×(C2×D7)⋊C411yescharacteristic
570C23C2×(C2×D7)⋊C412yes—
571C23C2×(C2×D7)⋊C411yescharacteristic
572C23C22×C22⋊D711yescharacteristic
573Q8—282no—
574Q8—561no—
575D4—42no—
576D4—42no—
577D4—282no—
578D4—284no—
579D4—282no—
580D4—282no—
581D4—561no—
582D4—562no—
583D4—562no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—142no—
609C2×C4—142no—
610C2×C4—142no—
611C2×C4—282no—
612C2×C4—562no—
613C2×C4—562no—
614C2×C4—561no—
615C2×C4—561no—
616C2×C4C2×(C2×D7)⋊C412yes—
617C2×C4C2×(C2×D7)⋊C412yes—
618C2×C4C2×(C2×D7)⋊C412yes—
619C2×C4C2×(C2×D7)⋊C411yescharacteristic
620C2×C4C2×(C2×D7)⋊C411yescharacteristic
621C2×C4C22×C4×D711yescharacteristic
622C2×C4C22×D2811yescharacteristic
623C8—282no—
624C8—282no—
625C7#1762511yescharacteristic, complemented, Sylow
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—22no—
631C22—21no—
632C22—22no—
633C22—21no—
634C22—21no—
635C22—22no—
636C22—21no—
637C22—81no—
638C22—81no—
639C22—81no—
640C22—81no—
641C22—282no—
642C22—282no—
643C22—282no—
644C22—1122no—
645C22C22×(C2×D7)⋊C411yescharacteristic
646C22#9051311yescharacteristic
647C22#9051311yescharacteristic
648C4—22no—
649C4—22no—
650C4—21no—
651C4—21no—
652C4—22no—
653C4—21no—
654C4—21no—
655C4—42no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—282no—
660C4—282no—
661C4—282no—
662C2—21no—
663C2—21no—
664C2—41no—
665C2—41no—
666C2—42no—
667C2—562no—
668C2#1173965811yescharacteristic
669C2#1173998111yescharacteristic
670C2#1173999111yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2357C2
socleC22×C7C22×C22⋊C4
prime radical#11742855C2
cosocleC22×C4×C7C24
hypercenter#2357D7
hypocenterC7#17625
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2357

Complements

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

NH ≅ G/Ntype
2. C7×#2357C2semidirect
3. #11739747C2semidirect
4. #11727298C2semidirect
5. #11727287C2semidirect
6. #11727286C2semidirect
7. #11727285C2semidirect
8. #11727224C2semidirect
9. #11727199C2semidirect
10. #11727198C2semidirect
11. #11727197C2semidirect
17. #90256C4semidirect
19. #90243C4semidirect
20. #90241C4semidirect
21. #90232C4semidirect
30. C7×#382C22semidirect
32. C7×#381C22semidirect
34. C4×C7×C4⋊C4C22semidirect
36. C4×C7×C22⋊C4C22semidirect
39. C2×C22⋊(C7⋊C8)C22semidirect
42. #89698C22semidirect
43. #89247C22semidirect
44. #89247C22semidirect
45. #89247C22semidirect
46. #89247C22semidirect
139. #2357D7semidirect
258. #382C2×D7semidirect
260. #381C2×D7semidirect
262. C4×C4⋊C4C2×D7semidirect
264. C4×C22⋊C4C2×D7semidirect
625. C7#17625semidirect

Maximal subgroups

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

subgroupindex
#117428552
C7×#23572
#117397472
#117272982
#117272872
#117272862
#117272852
#117272242
#117271992
#117271982
#117271972
#176257