Finite Group Databasebeta

← #13320477

Subgroups — #13320477

Order 1792.

984 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742906C211yesmaximal, characteristic
2C7×C8×#41C211yesmaximal, characteristic, complemented
3#11739966C211yesmaximal, characteristic, complemented
4#11728519C212yesmaximal, complemented
5#11728518C212yesmaximal, complemented
6#11728511C211yesmaximal, characteristic, complemented
7#11728509C211yesmaximal, characteristic, complemented
8#11727490C212yesmaximal, complemented
9#11727489C212yesmaximal, complemented
10#11727488C211yesmaximal, characteristic, complemented
11C2×#89254C211yesmaximal, characteristic, complemented
12C4×C7×#41C2211yescharacteristic
13#90524C411yescharacteristic
14#90521C411yescharacteristic
15#90273—21no—
16#90271—21no—
17#90253—21no—
18#90251—22no—
19#90239—21no—
20#90238—22no—
21#90226C412yes—
22#90225C412yes—
23#90216C411yescharacteristic
24#90210C411yescharacteristic
25#90170C2212yes—
26#90168C2212yes—
27C4×#1157C2211yescharacteristic
28C2×C4×D28C2211yescharacteristic
29C7×C8×Q8—21no—
30C7×C8×D4—21no—
31C7×C8×D4—22no—
32C7×#431C2211yescharacteristic
33C7×#431C2212yescomplemented
34C2×C4×C7×C8C2211yescharacteristic
35C2×C4×C7×C8C2212yescomplemented
36#89796—21no—
37#89784—21no—
38#89783—22no—
39#89698C2212yescomplemented
40#89697C2211yescharacteristic
41#89696C2212yescomplemented
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43#89254C2212yes—
44#89254C2212yes—
45#89254C2212yes—
46#89254C2212yes—
47#89254C2212yescomplemented
48#89254C2211yescharacteristic, complemented
49#89254C2211yescharacteristic, complemented
50#89251C2212yescomplemented
51#89251C2211yescharacteristic, complemented
52#89251C2211yescharacteristic, complemented
53#17483—71nomaximal, Sylow
54C2×C7×#41C2×C411yescharacteristic
55C2×#1157—22no—
56C2×#1157C2×C411yescharacteristic
57C22×D28C2×C411yescharacteristic
58C22×C4×D7—21no—
59C22×C7×C8—21no—
60C22×C7×C8—22no—
61C4×C7×Q8D411yescharacteristic
62C4×C7×Q8D411yescharacteristic
63C4×C7×D4D412yes—
64C4×C7×D4D411yescharacteristic
65C4×C7×D4D412yes—
66C4×C7×D4D411yescharacteristic
67C7×#117C2×C411yescharacteristic
68C7×#117C2×C412yes—
69C2×C42×C7C2311yescharacteristic
70C2×C42×C7C2312yes—
71#6510—21no—
72#6509—21no—
73#6503—21no—
74#6502—22no—
75#6501—21no—
76#6500—22no—
77#6493—22no—
78#6493C2×C412yes—
79#6493C2×C412yes—
80#6493C2×C412yes—
81#6492—22no—
82C4×C22⋊D7—21no—
83C4×C22⋊D7—21no—
84C4×C22⋊D7—22no—
85C2×(C2×D7)⋊C4—21no—
86#6489C2×C411yescharacteristic
87C2×C4⋊Dic7C2×C411yescharacteristic
88#6487C2×C412yes—
89C2×#1136—22no—
90C22×C7⋊C8—21no—
91#6461—22no—
92#6459—22no—
93#6458—21no—
94#6458—21no—
95#6457—22no—
96#6457—22no—
97#6453—21no—
98#6451—21no—
99#6448—22no—
100#6447—22no—
101#6447—22no—
102#6446—22no—
103#6445—22no—
104#6441—22no—
105#6439—22no—
106#6438—22no—
107#6438C2×C412yes—
108#6437C2×C412yes—
109C4×D28—21no—
110C4×D28—21no—
111C4×D28—22no—
112C4×D28C2×C412yes—
113C4×D28C2×C411yescharacteristic
114C4×D28C2312yes—
115C4×D28C2311yescharacteristic
116#6435—21no—
117#6435—22no—
118C42×D7—21no—
119#6433C2×C412yes—
120#6432C2×C412yes—
121C4×Dic14C2×C411yescharacteristic
122C4×Dic14C2311yescharacteristic
123C7×C4⋊C8—21no—
124C7×C4⋊C8—22no—
125C7×C22⋊C8—21no—
126C7×C22⋊C8—22no—
127C4×C7×C8—21no—
128C4×C7×C8—22no—
129C4×C7×C8C2311yescharacteristic
130C4×C7×C8C2311yescharacteristic
131C4×C7×C8C2312yes—
132C22⋊(C7⋊C8)—21no—
133C22⋊(C7⋊C8)—22no—
134C4⋊(C7⋊C8)—21no—
135C4⋊(C7⋊C8)—22no—
136C4⋊(C7⋊C8)C2312yes—
137C4⋊(C7⋊C8)C2311yescharacteristic
138C4⋊(C7⋊C8)C2311yescharacteristic
139#6377—22no—
140C4×C7⋊C8—21no—
141#3460—71no—
142#2989—71no—
143C8×#41D711yescharacteristic, complemented
144#1496—72no—
145#1495—72no—
146#1494—72no—
147#1492—72no—
148#1478—71no—
149#1477—71no—
150#1476—71no—
151C2×D4⋊C8—71no—
152C23×D7—21no—
153C7×#41—24no—
154C2×C7×Q8C22⋊C411yescharacteristic
155C2×C7×D4C22⋊C412yes—
156C2×C7×D4C22⋊C411yescharacteristic
157C22×C4×C7C22×C412yes—
158C22×C4×C7C22×C411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D412yes—
163C2×C22⋊D7—21no—
164C2×C22⋊D7—22no—
165C2×C22⋊D7—22no—
166C2×C22⋊D7—22no—
167C22×Dic7—21no—
168#1157—24no—
169#1157—82no—
170#1157—82no—
171C2×D28—24no—
172C2×D28—21no—
173C2×D28—22no—
174C2×D28—22no—
175C2×D28C22×C411yescharacteristic
176C2×D28C22×C412yes—
177C2×C4×D7—21no—
178C2×C4×D7—21no—
179C2×C4×D7—22no—
180C2×C4×D7—21no—
181C2×C4×D7—22no—
182C2×C4×D7—81no—
183C2×C4×D7—81no—
184C2×Dic14—22no—
185C2×Dic14C22×C411yescharacteristic
186C2×C7×C8—21no—
187C2×C7×C8—21no—
188C2×C7×C8—22no—
189C2×C7×C8—42no—
190C2×C7×C8—44no—
191C7×C4⋊C4C22⋊C411yescharacteristic
192C7×C4⋊C4C22⋊C411yescharacteristic
193C7×C4⋊C4C22⋊C412yes—
194C7×C4⋊C4C22⋊C412yes—
195C7×C22⋊C4C22⋊C412yes—
196C7×C22⋊C4C22⋊C412yes—
197C7×C22⋊C4C22⋊C412yes—
198C42×C7C22×C411yescharacteristic
199C42×C7C22×C412yes—
200C42×C7C2×D411yescharacteristic
201C42×C7C2×D412yes—
202C42×C7C2×D411yescharacteristic
203C42×C7C2×D412yes—
204C42×C7C2411yescharacteristic
205C22⋊Dic7—21no—
206C22⋊Dic7—22no—
207(C2×D7)⋊C4—21no—
208(C2×D7)⋊C4—22no—
209(C2×D7)⋊C4—22no—
210(C2×D7)⋊C4—22no—
211(C2×D7)⋊C4—22no—
212(C2×D7)⋊C4—22no—
213(C2×D7)⋊C4—22no—
214(C2×D7)⋊C4—21no—
215(C2×D7)⋊C4—22no—
216C4⋊Dic7—22no—
217C4⋊Dic7—21no—
218C4⋊Dic7C22×C411yescharacteristic
219C4⋊Dic7C22×C411yescharacteristic
220C4⋊Dic7C22×C412yes—
221Dic7⋊C4—21no—
222Dic7⋊C4—22no—
223Dic7⋊C4—22no—
224Dic7⋊C4—22no—
225Dic7⋊C4—21no—
226C4×Dic7—21no—
227C4×Dic7—21no—
228C4×Dic7—21no—
229#1136—82no—
230C2×C7⋊C8—21no—
231C2×C7⋊C8—22no—
232C2×C7⋊C8—21no—
233C2×C7⋊C8—81no—
234#544—71no—
235#542—72no—
236#542—141no—
237#540—72no—
238#540—141no—
239#538—142no—
240#537—71no—
241#535—71no—
242#535—71no—
243#532—72no—
244#532—141no—
245#529—72no—
246#529—141no—
247#525—142no—
248C4×#41—71no—
249C4×#41C2×D711yescharacteristic
250C2×C4×D4—71no—
251#446—141no—
252C8×Q8—21no—
253#436—141no—
254#435—142no—
255C8×D4—21no—
256C8×D4—22no—
257#432—72no—
258#431—71no—
259#431C2×D711yescharacteristic
260#431C2×D712yescomplemented
261#423—72no—
262C2×C4⋊C8—71no—
263C2×C4×C8C2×D711yescharacteristic
264C2×C4×C8C2×D712yescomplemented
265Q8⋊C8—71no—
266Q8⋊C8—71no—
267Q8⋊C8—72no—
268D4⋊C8—72no—
269D4⋊C8—72no—
270D4⋊C8—72no—
271D4⋊C8—72no—
272D4⋊C8—71no—
273D4⋊C8—71no—
274D4⋊C8—72no—
275C23×C7C2×C22⋊C412yes—
276C23×C7C2×C22⋊C411yescharacteristic
277C22×D7—22no—
278C22×D7—21no—
279C22×D7—42no—
280C22×D7—81no—
281C22×D7—81no—
282C7×Q8—22no—
283C7×D4—24no—
284C7×D4—22no—
285C2×C4×C7—24no—
286C2×C4×C7—21no—
287C2×C4×C7—22no—
288C2×C4×C7—21no—
289C2×C4×C7—22no—
290C2×C4×C7—41no—
291C2×C4×C7—42no—
292C2×C4×C7C2×C22⋊C411yescharacteristic
293C2×C4×C7C2×C22⋊C412yes—
294C2×C4×C7C2×C22⋊C412yes—
295C2×C4×C7C2×C22⋊C412yes—
296C2×C4×C7C2×C22⋊C411yescharacteristic
297C2×C4×C7C2×C22⋊C411yescharacteristic
298C2×C4×C7#13114yes—
299C2×C4×C7C23×C411yescharacteristic
300C2×C4×C7C22×D411yescharacteristic
301C2×C4×C7C22×D411yescharacteristic
302C22⋊D7—42no—
303C22⋊D7—82no—
304C22⋊D7—82no—
305C22⋊D7—82no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—22no—
310C2×Dic7—81no—
311D28—24no—
312D28—22no—
313D28—42no—
314D28—82no—
315D28—82no—
316C4×D7—42no—
317C4×D7—41no—
318C4×D7—42no—
319C4×D7—41no—
320C4×D7—42no—
321C4×D7—81no—
322C4×D7—82no—
323Dic14—22no—
324Dic14—82no—
325C7×C8—22no—
326C7×C8—41no—
327C7×C8—42no—
328C7⋊C8—42no—
329C7⋊C8—41no—
330C7⋊C8—41no—
331C2×#41—71no—
332C2×#41—142no—
333C2×#41C4×D711yescharacteristic
334C22×D4—71no—
335C23×C4—141no—
336C2×M16—142no—
337C22×C8—21no—
338C22×C8—22no—
339C22×C8—141no—
340C4⋊Q8—72no—
341C4⋊Q8—141no—
342#127—72no—
343#127—141no—
344#126—142no—
345#126—142no—
346#125—72no—
347#125—141no—
348#124—72no—
349#124—141no—
350#124—142no—
351#124—142no—
352#123—142no—
353#123—142no—
354#123—142no—
355#123—142no—
356C22⋊Q8—72no—
357C22⋊Q8—141no—
358C22⋊Q8—142no—
359C22⋊Q8—142no—
360#121—72no—
361#121—72no—
362#121—72no—
363#121—142no—
364#121—141no—
365#121—142no—
366#121—141no—
367#121—141no—
368#121—142no—
369#121—142no—
370#120—142no—
371#120—142no—
372C4×Q8—71no—
373C4×Q8—71no—
374C4×Q8D2811yescharacteristic
375C4×Q8C22⋊D711yescharacteristic
376C4×D4—72no—
377C4×D4—71no—
378C4×D4—72no—
379C4×D4—71no—
380C4×D4—141no—
381C4×D4—142no—
382C4×D4—141no—
383C4×D4—141no—
384C4×D4—142no—
385C4×D4—141no—
386C4×D4D2812yes—
387C4×D4D2811yescharacteristic
388C4×D4C22⋊D712yes—
389C4×D4C22⋊D711yescharacteristic
390#117—71no—
391#117—142no—
392#117—141no—
393#117C4×D711yescharacteristic
394#117C4×D712yes—
395C2×C4⋊C4—71no—
396C2×C22⋊C4—141no—
397C2×C42—141no—
398C2×C42C22×D711yescharacteristic
399C2×C42C22×D712yes—
400C4⋊C8—21no—
401C4⋊C8—22no—
402C4⋊C8—72no—
403C4⋊C8—71no—
404C4⋊C8—71no—
405C4⋊C8—141no—
406C4⋊C8—142no—
407C22⋊C8—21no—
408C22⋊C8—22no—
409C22⋊C8—141no—
410C22⋊C8—142no—
411#97—142no—
412C4×C8—21no—
413C4×C8—22no—
414C4×C8—141no—
415C4×C8C22×D711yescharacteristic
416C4×C8C22×D711yescharacteristic
417C4×C8C22×D712yes—
418C22×C7—21no—
419C22×C7—21no—
420C22×C7—22no—
421C22×C7—22no—
422C22×C7—22no—
423C22×C7#40812yes—
424C22×C7C22×C22⋊C411yescharacteristic
425C2×D7—41no—
426C2×D7—42no—
427C2×D7—42no—
428C2×D7—41no—
429C2×D7—41no—
430C2×D7—42no—
431C2×D7—84no—
432C4×C7—22no—
433C4×C7—22no—
434C4×C7—21no—
435C4×C7—21no—
436C4×C7—21no—
437C4×C7—22no—
438C4×C7#40812yes—
439C4×C7C2×#13112yes—
440C4×C7#57611yescharacteristic
441C4×C7#57711yescharacteristic
442Dic7—41no—
443Dic7—41no—
444Dic7—42no—
445Dic7—41no—
446C24—141no—
447#41—24no—
448#41—144no—
449#41—562no—
450#41—562no—
451C2×Q8—71no—
452C2×Q8—142no—
453C2×Q8(C2×D7)⋊C411yescharacteristic
454C2×D4—72no—
455C2×D4—71no—
456C2×D4—141no—
457C2×D4—142no—
458C2×D4—142no—
459C2×D4—142no—
460C2×D4—144no—
461C2×D4—141no—
462C2×D4—142no—
463C2×D4—142no—
464C2×D4(C2×D7)⋊C412yes—
465C2×D4(C2×D7)⋊C411yescharacteristic
466C22×C4—141no—
467C22×C4—141no—
468C22×C4—141no—
469C22×C4—142no—
470C22×C4—141no—
471C22×C4—142no—
472C22×C4—561no—
473C22×C4—561no—
474C22×C4C2×C4×D712yes—
475C22×C4C2×C4×D711yescharacteristic
476C22×C4C2×D2811yescharacteristic
477C22×C4C2×D2812yes—
478C22×C4C2×C22⋊D711yescharacteristic
479C22×C4C2×C22⋊D712yes—
480M16—562no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—22no—
484C2×C8—42no—
485C2×C8—44no—
486C2×C8—141no—
487C2×C8—141no—
488C2×C8—142no—
489C2×C8—561no—
490C4⋊C4—72no—
491C4⋊C4—71no—
492C4⋊C4—71no—
493C4⋊C4—141no—
494C4⋊C4—142no—
495C4⋊C4—142no—
496C4⋊C4—142no—
497C4⋊C4—141no—
498C4⋊C4—142no—
499C4⋊C4—141no—
500C4⋊C4(C2×D7)⋊C411yescharacteristic
501C4⋊C4(C2×D7)⋊C412yes—
502C4⋊C4(C2×D7)⋊C411yescharacteristic
503C4⋊C4(C2×D7)⋊C412yes—
504C22⋊C4—141no—
505C22⋊C4—142no—
506C22⋊C4—142no—
507C22⋊C4—141no—
508C22⋊C4—142no—
509C22⋊C4—142no—
510C22⋊C4—142no—
511C22⋊C4—142no—
512C22⋊C4—142no—
513C22⋊C4—142no—
514C22⋊C4—141no—
515C22⋊C4(C2×D7)⋊C412yes—
516C22⋊C4(C2×D7)⋊C412yes—
517C22⋊C4(C2×D7)⋊C412yes—
518C42—141no—
519C42—141no—
520C42—141no—
521C42C2×C4×D711yescharacteristic
522C42C2×C4×D712yes—
523C42C2×D2811yescharacteristic
524C42C2×D2812yes—
525C42C2×C22⋊D711yescharacteristic
526C42C2×C22⋊D712yes—
527C42C23×D711yescharacteristic
528C2×C7—22no—
529C2×C7—42no—
530C2×C7C2×#40811yescharacteristic
531C2×C7#289711yescharacteristic
532C2×C7#290111yescharacteristic
533D7—81no—
534D7—82no—
535C23—142no—
536C23—141no—
537C23—282no—
538C23—561no—
539C23—561no—
540C23C2×(C2×D7)⋊C412yes—
541C23C2×(C2×D7)⋊C411yescharacteristic
542Q8—22no—
543Q8—142no—
544Q8—562no—
545D4—24no—
546D4—22no—
547D4—144no—
548D4—142no—
549D4—282no—
550D4—282no—
551D4—562no—
552D4—562no—
553D4—562no—
554D4—562no—
555D4—562no—
556C2×C4—24no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—141no—
564C2×C4—141no—
565C2×C4—141no—
566C2×C4—142no—
567C2×C4—282no—
568C2×C4—282no—
569C2×C4—281no—
570C2×C4—282no—
571C2×C4—281no—
572C2×C4—561no—
573C2×C4—562no—
574C2×C4—561no—
575C2×C4#646414yes—
576C2×C4C2×(C2×D7)⋊C411yescharacteristic
577C2×C4C2×(C2×D7)⋊C412yes—
578C2×C4C2×(C2×D7)⋊C411yescharacteristic
579C2×C4C2×(C2×D7)⋊C412yes—
580C2×C4C2×(C2×D7)⋊C412yes—
581C2×C4C2×(C2×D7)⋊C411yescharacteristic
582C2×C4C22×C4×D711yescharacteristic
583C2×C4C22×D2811yescharacteristic
584C2×C4C22×C22⋊D711yescharacteristic
585C8—22no—
586C8—41no—
587C8—42no—
588C8—281no—
589C8—282no—
590C8—281no—
591C7#1748311yescharacteristic, complemented, Sylow
592C22—21no—
593C22—21no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—282no—
598C22—281no—
599C22—281no—
600C22—282no—
601C22—281no—
602C22—282no—
603C22—564no—
604C22#8988212yes—
605C22C22×(C2×D7)⋊C411yescharacteristic
606C4—21no—
607C4—22no—
608C4—21no—
609C4—21no—
610C4—22no—
611C4—22no—
612C4—282no—
613C4—281no—
614C4—281no—
615C4—281no—
616C4#8988212yes—
617C4C2×#646412yes—
618C4#9044211yescharacteristic
619C4#9044311yescharacteristic
620C2—22no—
621C2—42no—
622C2—562no—
623C2—561no—
624C2#1173998111yescharacteristic
625C2C2×#8988211yescharacteristic
626C2#1174124211yescharacteristic
subgroupG/H
centerC2×C4C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×C8×#41C2
socleC22×C7C22×C22⋊C4
prime radical#11742906C2
cosocleC42×C7C24
hypercenterC8×#41D7
hypocenterC7#17483
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
2. C7×C8×#41C2semidirect
3. #11739966C2semidirect
4. #11728519C2semidirect
5. #11728518C2semidirect
6. #11728511C2semidirect
7. #11728509C2semidirect
8. #11727490C2semidirect
9. #11727489C2semidirect
10. #11727488C2semidirect
11. C2×#89254C2semidirect
33. C7×#431C22semidirect
35. C2×C4×C7×C8C22semidirect
39. #89698C22semidirect
41. #89696C22semidirect
47. #89254C22semidirect
48. #89254C22semidirect
49. #89254C22semidirect
50. #89251C22semidirect
51. #89251C22semidirect
52. #89251C22semidirect
143. C8×#41D7semidirect
260. #431C2×D7semidirect
264. C2×C4×C8C2×D7semidirect
591. C7#17483semidirect

Maximal subgroups

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

subgroupindex
#117429062
C7×C8×#412
#117399662
#117285192
#117285182
#117285112
#117285092
#117274902
#117274892
#117274882
C2×#892542
#174837