Finite Group Databasebeta

← #13354775

Subgroups — #13354775

Order 1792.

1550 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×Dic7C211yesmaximal, characteristic
2C2×C7×#417C211yesmaximal, characteristic
3C2×#90004C211yesmaximal, characteristic
4C2×#89893C211yesmaximal, characteristic, complemented
5C2×#89757C211yesmaximal, characteristic, complemented
6#11729733C211yesmaximal, characteristic, complemented
7#11729312C211yesmaximal, characteristic, complemented
8#11727992C212yesmaximal, complemented
9#11727990C212yesmaximal, complemented
10#11727924C212yesmaximal, complemented
11#11727922C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C22×C7×M16C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15#41×Dic7—24no—
16#41×Dic7—24no—
17C2×Q8×Dic7C2211yescharacteristic
18C2×Q8×Dic7C2211yescharacteristic
19C2×D4×Dic7—21no—
20C2×D4×Dic7—21no—
21C2×D4×Dic7C2211yescharacteristic
22C2×D4×Dic7C2211yescharacteristic
23C2×#6489—21no—
24C2×#6489C2211yescharacteristic
25C22×C4⋊Dic7—21no—
26C22×C4⋊Dic7C2211yescharacteristic
27C22×C4×Dic7—21no—
28C22×C4×Dic7C2211yescharacteristic
29C22×#1136C2211yescharacteristic
30C7×#417C418yescomplemented
31C2×C7×#103C2212yes—
32C2×C7×#102C2212yes—
33#90004—24no—
34C2×#6409C2212yes—
35C2×#6406C2212yes—
36C2×C7⋊#95—21no—
37#89893—24no—
38C2×#6392C2212yes—
39C2×#6391C2212yes—
40#89757—21no—
41#89757—21no—
42#89757C2212yes—
43#89757C2212yes—
44#89345C2212yes—
45#89345C2212yes—
46#89324C2212yes—
47#89324C2212yes—
48#20875—71nomaximal, Sylow
49C2×C7×#41—22no—
50C2×C7×#41—22no—
51C2×C7×#41—22no—
52C2×C7×#41C2×C414yes—
53C2×C7×#41D414yes—
54C2×C7×#41Q814yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C23×Dic7—21no—
63C23×Dic7—21no—
64C23×Dic7D411yescharacteristic
65C23×Dic7D411yescharacteristic
66C2×C7×M16—22no—
67C2×C7×M16—22no—
68C2×C7×M16C2×C414yescomplemented
69C22×C7×C8C2312yes—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4C2×C414yescomplemented
73C2×C7×C4⋊C4C2312yes—
74C2×C22⋊Dic7—21no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—21no—
77Q8×Dic7—22no—
78Q8×Dic7—22no—
79Q8×Dic7—22no—
80Q8×Dic7—22no—
81D4×Dic7—28no—
82D4×Dic7—28no—
83D4×Dic7—22no—
84D4×Dic7—22no—
85D4×Dic7—22no—
86D4×Dic7—22no—
87#6489—28no—
88#6489—22no—
89#6489D414yes—
90C2×C4⋊Dic7—21no—
91C2×C4⋊Dic7—21no—
92C2×C4⋊Dic7—22no—
93C2×C4⋊Dic7—22no—
94C2×C4⋊Dic7—21no—
95C2×C4⋊Dic7—22no—
96C2×C4⋊Dic7—22no—
97C2×C4⋊Dic7—44no—
98C2×C4⋊Dic7C2312yes—
99C2×C4⋊Dic7C2311yescharacteristic
100C2×C4⋊Dic7C2311yescharacteristic
101C2×C4×Dic7—28no—
102C2×C4×Dic7—21no—
103C2×C4×Dic7—22no—
104C2×C4×Dic7—22no—
105C2×C4×Dic7—21no—
106C2×C4×Dic7—21no—
107C2×C4×Dic7D411yescharacteristic
108C2×C4×Dic7D411yescharacteristic
109C2×C4×Dic7C2311yescharacteristic
110C2×C4×Dic7C2311yescharacteristic
111C2×#1136—22no—
112C2×#1136—22no—
113C2×#1136—22no—
114C22×C7⋊C8C2312yes—
115C7×#103C2×C418yescomplemented
116C7×#102C2×C418yescomplemented
117#6409—24no—
118#6406—24no—
119C7⋊#95—22no—
120C7⋊#95—22no—
121#6392—24no—
122#6391—24no—
123C2×C4×#41—71no—
124C2×#428—71no—
125C2×#417—71no—
126C2×#417D711yescharacteristic
127C2×#389—71no—
128#1865—72no—
129#1864—72no—
130#1863—72no—
131#1862—72no—
132#1736—71no—
133#1728—71no—
134C24×C7—21no—
135C24×C7C2×D411yescharacteristic
136C7×#41—28no—
137C7×#41—28no—
138C7×#41—28no—
139C7×#41C4⋊C4116yescomplemented
140C2×C7×Q8—21no—
141C2×C7×Q8—21no—
142C2×C7×Q8—21no—
143C2×C7×Q8C22×C412yes—
144C2×C7×Q8C2×D412yes—
145C2×C7×Q8C2×Q812yes—
146C2×C7×D4—24no—
147C2×C7×D4—24no—
148C2×C7×D4—24no—
149C2×C7×D4—21no—
150C2×C7×D4—21no—
151C2×C7×D4—21no—
152C2×C7×D4C22×C412yes—
153C2×C7×D4C2×D412yes—
154C2×C7×D4C2×Q812yes—
155C22×C4×C7—21no—
156C22×C4×C7—24no—
157C22×C4×C7—24no—
158C22×C4×C7—24no—
159C22×C4×C7—24no—
160C22×C4×C7—22no—
161C22×C4×C7—42no—
162C22×C4×C7—42no—
163C22×C4×C7C22×C412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7C2×D411yescharacteristic
166C22×C4×C7C2×D412yes—
167C22×C4×C7C2×Q812yes—
168C22×C4×C7#4112yes—
169C22×C4×C7#4112yes—
170C22×C4×C7C2411yescharacteristic
171C22×Dic7—24no—
172C22×Dic7—24no—
173C22×Dic7—22no—
174C22×Dic7—22no—
175C22×Dic7—21no—
176C22×Dic7—21no—
177C22×Dic7—22no—
178C22×Dic7—22no—
179C22×Dic7—22no—
180C22×Dic7—41no—
181C22×Dic7—41no—
182C22×Dic7—42no—
183C22×Dic7—42no—
184C22×Dic7—42no—
185C22×Dic7—42no—
186C22×Dic7—42no—
187C22×Dic7—42no—
188C22×Dic7—42no—
189C22×Dic7C2×D411yescharacteristic
190C22×Dic7C2×D411yescharacteristic
191C22×Dic7C2×D411yescharacteristic
192C22×Dic7C2×D411yescharacteristic
193C7×M16—28no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C2×C7×C8C22×C414yes—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—22no—
199C7×C4⋊C4C22×C414yes—
200C22⋊Dic7—24no—
201C22⋊Dic7—24no—
202C22⋊Dic7—24no—
203C4⋊Dic7—24no—
204C4⋊Dic7—24no—
205C4⋊Dic7—21no—
206C4⋊Dic7—21no—
207C4⋊Dic7—22no—
208C4⋊Dic7—22no—
209C4⋊Dic7—44no—
210C4⋊Dic7—42no—
211C4⋊Dic7—44no—
212C4⋊Dic7C2×D412yes—
213C4⋊Dic7C2×D412yes—
214C4×Dic7—24no—
215C4×Dic7—24no—
216C4×Dic7—22no—
217C4×Dic7—24no—
218C4×Dic7—22no—
219C4×Dic7—21no—
220C4×Dic7—21no—
221C4×Dic7—21no—
222C4×Dic7—21no—
223#1136—44no—
224C2×C7⋊C8—22no—
225C2×C7⋊C8—22no—
226C2×C7⋊C8—22no—
227C22×#41C2×D711yescharacteristic
228C22×M16—71no—
229C22×M16C2×D711yescharacteristic
230C4×#41—144no—
231C4×#41—144no—
232C2×C4×Q8—71no—
233C2×C4×Q8—71no—
234C2×C4×D4—71no—
235C2×C4×D4—71no—
236C2×C4×D4—141no—
237C2×C4×D4—141no—
238C2×#117—71no—
239C2×#117—141no—
240C22×C4⋊C4—71no—
241C22×C4⋊C4—141no—
242C22×C4⋊C4C2×D711yescharacteristic
243C22×C42—71no—
244C22×C42—141no—
245#428—144no—
246C2×#107—72no—
247C2×#106—72no—
248#417—144no—
249#417Dic718yescomplemented
250C2×#103—72no—
251C2×#103C2×D712yes—
252C2×#102—72no—
253C2×#102C2×D712yes—
254#389—72no—
255#389—72no—
256#389—141no—
257#389—141no—
258C2×#95—141no—
259#340—72no—
260#340—72no—
261#336—72no—
262#336—72no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—22no—
268C23×C7—22no—
269C23×C7—22no—
270C23×C7—41no—
271C23×C7—41no—
272C23×C7—41no—
273C23×C7—41no—
274C23×C7C4×D412yes—
275C23×C7#12012yes—
276C23×C7C22⋊Q812yes—
277C23×C7C22×D411yescharacteristic
278C7×Q8—22no—
279C7×Q8—22no—
280C7×Q8—22no—
281C7×Q8C2×C4⋊C414yescomplemented
282C7×D4—216no—
283C7×D4—22no—
284C7×D4—22no—
285C7×D4—22no—
286C7×D4C2×C4⋊C414yescomplemented
287C2×C4×C7—22no—
288C2×C4×C7—22no—
289C2×C4×C7—22no—
290C2×C4×C7—216no—
291C2×C4×C7—24no—
292C2×C4×C7—48no—
293C2×C4×C7—42no—
294C2×C4×C7—42no—
295C2×C4×C7C2×C4⋊C414yes—
296C2×C4×C7C4×D414yes—
297C2×C4×C7C4×D412yes—
298C2×C4×C7#12012yes—
299C2×C4×C7C22⋊Q814yes—
300C2×C4×C7C22⋊Q812yes—
301C2×C4×C7#12314yes—
302C2×C4×C7C23×C411yescharacteristic
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×D411yescharacteristic
305C2×C4×C7C22×Q811yescharacteristic
306C2×C4×C7C2×#4111yescharacteristic
307C2×C4×C7C2×#4111yescharacteristic
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—22no—
315C2×Dic7—22no—
316C2×Dic7—21no—
317C2×Dic7—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—44no—
321C2×Dic7—44no—
322C2×Dic7—48no—
323C2×Dic7—48no—
324C2×Dic7—41no—
325C2×Dic7—41no—
326C2×Dic7—41no—
327C2×Dic7—41no—
328C2×Dic7—42no—
329C2×Dic7—42no—
330C2×Dic7—42no—
331C2×Dic7—84no—
332C2×Dic7AGL(1,ℤ/8)12yes—
333C2×Dic7#13712yes—
334C7×C8—24no—
335C7⋊C8—42no—
336C2×#41—22no—
337C2×#41—22no—
338C2×#41—22no—
339C2×#41Dic1414yes—
340C2×#41D2814yes—
341C2×#41C2×Dic714yes—
342C22×Q8C22×D711yescharacteristic
343C22×D4—21no—
344C22×D4C22×D711yescharacteristic
345C23×C4—21no—
346C23×C4—21no—
347C23×C4—71no—
348C23×C4—71no—
349C23×C4—141no—
350C23×C4—141no—
351C23×C4—141no—
352C23×C4C22×D711yescharacteristic
353C2×M16—22no—
354C2×M16—22no—
355C2×M16—142no—
356C2×M16—142no—
357C2×M16—142no—
358C2×M16C2×Dic714yescomplemented
359C22×C8—72no—
360C22×C8C22×D712yes—
361C4×Q8—142no—
362C4×Q8—142no—
363C4×Q8—142no—
364C4×Q8—142no—
365C4×D4—142no—
366C4×D4—142no—
367C4×D4—148no—
368C4×D4—148no—
369C4×D4—142no—
370C4×D4—142no—
371#117—74no—
372#117—148no—
373#117—142no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—71no—
377C2×C4⋊C4—71no—
378C2×C4⋊C4—72no—
379C2×C4⋊C4—141no—
380C2×C4⋊C4—141no—
381C2×C4⋊C4—142no—
382C2×C4⋊C4—142no—
383C2×C4⋊C4—142no—
384C2×C4⋊C4—142no—
385C2×C4⋊C4—141no—
386C2×C4⋊C4—284no—
387C2×C4⋊C4C2×Dic714yescomplemented
388C2×C4⋊C4C22×D712yes—
389C2×C22⋊C4—141no—
390C2×C22⋊C4—141no—
391C2×C22⋊C4—141no—
392C2×C42—71no—
393C2×C42—71no—
394C2×C42—71no—
395C2×C42—71no—
396C2×C42—141no—
397C2×C42—141no—
398C2×C42—148no—
399C2×C42—142no—
400C2×C42—141no—
401C2×C42—142no—
402#107—144no—
403#106—144no—
404#103—144no—
405#103C2×Dic718yescomplemented
406#102—144no—
407#102C2×Dic718yescomplemented
408#95—142no—
409#95—142no—
410C22×C7—22no—
411C22×C7—22no—
412C22×C7—22no—
413C22×C7—22no—
414C22×C7—22no—
415C22×C7—22no—
416C22×C7—24no—
417C22×C7—42no—
418C22×C7—42no—
419C22×C7—42no—
420C22×C7—42no—
421C22×C7—42no—
422C22×C7—42no—
423C22×C7#38514yes—
424C22×C7C2×C4×D411yescharacteristic
425C22×C7C2×#12011yescharacteristic
426C22×C7C2×C22⋊Q811yescharacteristic
427C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
428C22×C7C2×#13711yescharacteristic
429C22×C7#57511yescharacteristic
430C22×C7#57511yescharacteristic
431C4×C7—24no—
432C4×C7—44no—
433C4×C7#38514yes—
434C4×C7C22×C4⋊C411yescharacteristic
435C4×C7C2×C4×D411yescharacteristic
436C4×C7C2×C22⋊Q811yescharacteristic
437C4×C7C2×#12311yescharacteristic
438Dic7—22no—
439Dic7—22no—
440Dic7—41no—
441Dic7—42no—
442Dic7—41no—
443Dic7—42no—
444Dic7—82no—
445C24—21no—
446C24D4×D711yescharacteristic
447#41—28no—
448#41—28no—
449#41—28no—
450#41C4⋊Dic7116yescomplemented
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8C2×Dic1412yes—
455C2×Q8C2×D2812yes—
456C2×Q8C22×Dic712yes—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—24no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4C2×Dic1412yes—
464C2×D4C2×D2812yes—
465C2×D4C22×Dic712yes—
466C22×C4—21no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—71no—
475C22×C4—71no—
476C22×C4—71no—
477C22×C4—71no—
478C22×C4—142no—
479C22×C4—142no—
480C22×C4—144no—
481C22×C4—144no—
482C22×C4—142no—
483C22×C4—142no—
484C22×C4—141no—
485C22×C4—141no—
486C22×C4—142no—
487C22×C4—282no—
488C22×C4—282no—
489C22×C4—281no—
490C22×C4—281no—
491C22×C4—282no—
492C22×C4—282no—
493C22×C4—282no—
494C22×C4—282no—
495C22×C4—282no—
496C22×C4C2×Dic1412yes—
497C22×C4C2×D2812yes—
498C22×C4D4×D711yescharacteristic
499C22×C4D4×D712yes—
500C22×C4#115912yes—
501C22×C4#115912yes—
502C22×C4C22×Dic712yes—
503C22×C4C23×D711yescharacteristic
504M16—28no—
505M16—284no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—142no—
509C2×C8—142no—
510C2×C8—142no—
511C2×C8C22×Dic714yes—
512C4⋊C4—24no—
513C4⋊C4—22no—
514C4⋊C4—72no—
515C4⋊C4—72no—
516C4⋊C4—144no—
517C4⋊C4—144no—
518C4⋊C4—141no—
519C4⋊C4—141no—
520C4⋊C4—142no—
521C4⋊C4—142no—
522C4⋊C4—284no—
523C4⋊C4—284no—
524C4⋊C4—282no—
525C4⋊C4C22×Dic714yes—
526C22⋊C4—144no—
527C22⋊C4—144no—
528C22⋊C4—144no—
529C42—144no—
530C42—144no—
531C42—142no—
532C42—142no—
533C42—144no—
534C42—141no—
535C42—141no—
536C42—141no—
537C42—141no—
538C2×C7—24no—
539C2×C7—44no—
540C2×C7C2×#38511yescharacteristic
541C2×C7C4×AGL(1,ℤ/8)11yescharacteristic
542C2×C7C4×#13711yescharacteristic
543C2×C7#300611yescharacteristic
544C2×C7#300711yescharacteristic
545C2×C7#307911yescharacteristic
546C2×C7#308011yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23#644112yes—
559C23#644512yes—
560C23D4×Dic712yes—
561C23C2×D4×D711yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—22no—
565Q8C2×C4⋊Dic714yescomplemented
566D4—216no—
567D4—22no—
568D4—22no—
569D4—22no—
570D4C2×C4⋊Dic714yescomplemented
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—216no—
575C2×C4—24no—
576C2×C4—48no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—72no—
580C2×C4—72no—
581C2×C4—141no—
582C2×C4—141no—
583C2×C4—141no—
584C2×C4—141no—
585C2×C4—142no—
586C2×C4—142no—
587C2×C4—141no—
588C2×C4—141no—
589C2×C4—141no—
590C2×C4—141no—
591C2×C4—141no—
592C2×C4—141no—
593C2×C4—284no—
594C2×C4—284no—
595C2×C4—288no—
596C2×C4—288no—
597C2×C4—281no—
598C2×C4—281no—
599C2×C4—281no—
600C2×C4—281no—
601C2×C4—282no—
602C2×C4—282no—
603C2×C4—282no—
604C2×C4—564no—
605C2×C4#644114yes—
606C2×C4#644112yes—
607C2×C4#644512yes—
608C2×C4#644914yes—
609C2×C4C2×C4⋊Dic714yes—
610C2×C4D4×Dic714yes—
611C2×C4D4×Dic712yes—
612C2×C4C22×Dic1411yescharacteristic
613C2×C4C22×D2811yescharacteristic
614C2×C4C2×D4×D711yescharacteristic
615C2×C4C2×#115911yescharacteristic
616C2×C4C2×#115911yescharacteristic
617C2×C4C23×Dic711yescharacteristic
618C8—24no—
619C8—282no—
620C7#2087511yescharacteristic, complemented, Sylow
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22#8972214yes—
635C22C2×#644111yescharacteristic
636C22C2×#644511yescharacteristic
637C22D7×AGL(1,ℤ/8)11yescharacteristic
638C22#9046411yescharacteristic
639C22D7×#13711yescharacteristic
640C22#9046811yescharacteristic
641C22C2×D4×Dic711yescharacteristic
642C4—24no—
643C4—44no—
644C4—142no—
645C4—142no—
646C4—281no—
647C4—282no—
648C4—281no—
649C4—282no—
650C4—562no—
651C4#8972214yes—
652C4C2×#644111yescharacteristic
653C4C2×#644911yescharacteristic
654C4C22×C4⋊Dic711yescharacteristic
655C4C2×D4×Dic711yescharacteristic
656C2—24no—
657C2—44no—
658C2#1173631511yescharacteristic
659C2#1173631611yescharacteristic
660C2#1173632911yescharacteristic
661C2#1173633011yescharacteristic
662C2C2×#8972211yescharacteristic
663C2Dic7×AGL(1,ℤ/8)11yescharacteristic
664C2Dic7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#417D7
hypocenterC7#20875
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
4. C2×#89893C2semidirect
5. C2×#89757C2semidirect
6. #11729733C2semidirect
7. #11729312C2semidirect
8. #11727992C2semidirect
9. #11727990C2semidirect
10. #11727924C2semidirect
11. #11727922C2semidirect
30. C7×#417C4semidirect
68. C2×C7×M16C2×C4semidirect
72. C2×C7×C4⋊C4C2×C4semidirect
115. C7×#103C2×C4semidirect
116. C7×#102C2×C4semidirect
139. C7×#41C4⋊C4semidirect
249. #417Dic7semidirect
281. C7×Q8C2×C4⋊C4semidirect
286. C7×D4C2×C4⋊C4semidirect
358. C2×M16C2×Dic7semidirect
387. C2×C4⋊C4C2×Dic7semidirect
405. #103C2×Dic7semidirect
407. #102C2×Dic7semidirect
450. #41C4⋊Dic7semidirect
565. Q8C2×C4⋊Dic7semidirect
570. D4C2×C4⋊Dic7semidirect
620. C7#20875semidirect

Maximal subgroups

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

subgroupindex
C2×#41×Dic72
C2×C7×#4172
C2×#900042
C2×#898932
C2×#897572
#117297332
#117293122
#117279922
#117279902
#117279242
#117279222
#208757