Finite Group Databasebeta

← #13355034

Subgroups — #13355034

Order 1792.

1934 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6509C211yesmaximal, characteristic
2C2×D7×#117C211yesmaximal, characteristic
3C2×C7×#419C211yesmaximal, characteristic, complemented
4C2×#89997C211yesmaximal, characteristic, complemented
5C2×#89898C211yesmaximal, characteristic
6#11729739C211yesmaximal, characteristic, complemented
7#11729335C211yesmaximal, characteristic, complemented
8#11728023C214yesmaximal, complemented
9#11728022C214yesmaximal, complemented
10C23×C7×Q8C2211yescharacteristic
11C23×C4×D7C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C2×#6509—24no—
15C2×#6509—21no—
16C2×#6509—21no—
17C2×#6509C418yescomplemented
18C2×#6509C2212yes—
19C2×#6509C2212yes—
20C22×(C2×D7)⋊C4—21no—
21C2×#6489C2211yescharacteristic
22C22×C4⋊Dic7C2211yescharacteristic
23C22×Dic7⋊C4—21no—
24C22×#1136C2211yescharacteristic
25D7×#117—24no—
26D7×#117—24no—
27C2×#6455C2212yes—
28C2×D7×C4⋊C4C2212yes—
29C2×D7×C22⋊C4—21no—
30C2×#6440—21no—
31C2×#6435—21no—
32C2×C42×D7—21no—
33C7×#419—24no—
34C2×C7×#103C2214yescomplemented
35#89997—24no—
36C2×#6409C2214yescomplemented
37#89898—24no—
38C2×(C2×D7)⋊C8C2212yes—
39C2×(C2×D7)⋊C8C2212yes—
40#89346C2214yescomplemented
41#89324C2214yescomplemented
42#20911—71nomaximal, Sylow
43C24×D7D411yescharacteristic
44C22×C7×Q8—22no—
45C22×C7×Q8—21no—
46C22×C7×Q8—22no—
47C22×C7×Q8—22no—
48C22×C7×Q8—22no—
49C22×C7×Q8—22no—
50C22×C7×Q8C2×C414yescomplemented
51C22×C7×Q8C2312yes—
52C23×C4×C7—21no—
53C23×C4×C7C2311yescharacteristic
54C23×Dic7—21no—
55C23×Dic7D411yescharacteristic
56C22×C4×D7—22no—
57C22×C4×D7—22no—
58C22×C4×D7—22no—
59C22×C4×D7—22no—
60C22×C4×D7—22no—
61C22×C4×D7C2×C414yes—
62C22×C4×D7D411yescharacteristic
63C22×C4×D7D411yescharacteristic
64C22×C4×D7D414yes—
65C22×C4×D7C2311yescharacteristic
66C22×C4×D7C2311yescharacteristic
67C2×C7×M16—22no—
68C2×C7×M16—22no—
69C2×C7×M16—22no—
70C22×C7×C8C2312yes—
71C7×#117—22no—
72C7×#117D414yes—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C22⋊C4—21no—
75C2×C42×C7—21no—
76C2×C22⋊Dic7—21no—
77#6509—28no—
78#6509—28no—
79#6509—24no—
80#6509—24no—
81#6509C2×C418yes—
82#6509C2×C418yescomplemented
83C2×(C2×D7)⋊C4—28no—
84C2×(C2×D7)⋊C4—22no—
85C2×(C2×D7)⋊C4—22no—
86C2×(C2×D7)⋊C4—22no—
87#6489—22no—
88#6489D414yes—
89C2×C4⋊Dic7—21no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7C2×C414yescomplemented
92C2×C4⋊Dic7C2312yes—
93C2×C4⋊Dic7C2312yes—
94C2×Dic7⋊C4—28no—
95C2×Dic7⋊C4—22no—
96C2×Dic7⋊C4—22no—
97C2×Dic7⋊C4—22no—
98C2×C4×Dic7—21no—
99C2×C4×Dic7—22no—
100C2×#1136—22no—
101C2×#1136—22no—
102C2×#1136—22no—
103C22×C7⋊C8C2312yes—
104#6455—24no—
105#6455—24no—
106D7×C4⋊C4—24no—
107D7×C4⋊C4—24no—
108D7×C22⋊C4—28no—
109#6440—28no—
110#6435—28no—
111C42×D7—28no—
112C7×#103—28no—
113#6409—28no—
114(C2×D7)⋊C8—24no—
115(C2×D7)⋊C8—24no—
116C22×C22⋊Q8—71no—
117C22×#117—71no—
118C2×#419—71no—
119C2×#419D711yescharacteristic, complemented
120C2×#407—71no—
121#1873—74no—
122#1872—74no—
123#1736—71no—
124#1736—71no—
125C24×C7C2×D411yescharacteristic
126C23×D7—22no—
127C23×D7—22no—
128C23×D7—24no—
129C23×D7—42no—
130C23×D7C22⋊C414yes—
131C23×D7C2×D411yescharacteristic
132C23×D7C2×D411yescharacteristic
133C2×C7×Q8—24no—
134C2×C7×Q8—22no—
135C2×C7×Q8—28no—
136C2×C7×Q8—28no—
137C2×C7×Q8—22no—
138C2×C7×Q8—22no—
139C2×C7×Q8—22no—
140C2×C7×Q8—22no—
141C2×C7×Q8—42no—
142C2×C7×Q8—44no—
143C2×C7×Q8—44no—
144C2×C7×Q8—42no—
145C2×C7×Q8C22×C414yes—
146C22×C4×C7—24no—
147C22×C4×C7—22no—
148C22×C4×C7—22no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7C22×C412yes—
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D412yes—
155C22×C4×C7C2×D412yes—
156C22×C4×C7#4112yes—
157C22×C4×C7#4112yes—
158C22×C4×C7C2411yescharacteristic
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7—24no—
162C22×Dic7—22no—
163C22×Dic7—22no—
164C22×Dic7—44no—
165C22×Dic7C22⋊C414yes—
166C22×Dic7C2×D411yescharacteristic
167C22×Dic7C2×D411yescharacteristic
168C2×C4×D7—28no—
169C2×C4×D7—28no—
170C2×C4×D7—22no—
171C2×C4×D7—22no—
172C2×C4×D7—28no—
173C2×C4×D7—28no—
174C2×C4×D7—22no—
175C2×C4×D7—22no—
176C2×C4×D7—21no—
177C2×C4×D7—21no—
178C2×C4×D7—21no—
179C2×C4×D7—21no—
180C2×C4×D7—21no—
181C2×C4×D7—21no—
182C2×C4×D7—21no—
183C2×C4×D7—21no—
184C2×C4×D7—28no—
185C2×C4×D7—44no—
186C2×C4×D7—44no—
187C2×C4×D7C22⋊C414yes—
188C2×C4×D7C22⋊C414yes—
189C2×C4×D7C22×C412yes—
190C2×C4×D7C22×C412yes—
191C2×C4×D7C2×D412yes—
192C2×C4×D7C2×D412yes—
193C7×M16—44no—
194C2×C7×C8—22no—
195C2×C7×C8—22no—
196C2×C7×C8—22no—
197C7×C4⋊C4—22no—
198C7×C4⋊C4C2×D414yes—
199C7×C22⋊C4—24no—
200C42×C7—24no—
201C22⋊Dic7—24no—
202(C2×D7)⋊C4—28no—
203(C2×D7)⋊C4—28no—
204(C2×D7)⋊C4—28no—
205C4⋊Dic7—24no—
206C4⋊Dic7—22no—
207C4⋊Dic7—22no—
208C4⋊Dic7C22×C414yes—
209C4⋊Dic7C2×D414yes—
210Dic7⋊C4—28no—
211Dic7⋊C4—28no—
212Dic7⋊C4—28no—
213C4×Dic7—28no—
214C4×Dic7—24no—
215#1136—44no—
216C2×C7⋊C8—22no—
217C2×C7⋊C8—22no—
218C2×C7⋊C8—22no—
219C23×Q8C2×D711yescharacteristic
220C24×C4—71no—
221C22×M16—71no—
222C22×M16C2×D711yescharacteristic
223C2×C22⋊Q8—78no—
224C2×C22⋊Q8—72no—
225C2×C22⋊Q8—72no—
226C2×C22⋊Q8—141no—
227C2×C22⋊Q8—144no—
228C2×C22⋊Q8—141no—
229C2×#117—71no—
230C2×#117—72no—
231C2×#117—141no—
232C2×#117—141no—
233C2×#117—144no—
234C2×#117—144no—
235C2×#117C2×D711yescharacteristic
236C22×C4⋊C4—71no—
237C22×C4⋊C4—72no—
238C22×C4⋊C4—141no—
239C22×C22⋊C4—141no—
240C22×C22⋊C4—141no—
241C22×C42—141no—
242#419—24no—
243#419—144no—
244C2×#103—74no—
245C2×#103C2×D714yescomplemented
246#407—144no—
247C2×C22⋊C8—72no—
248C2×C22⋊C8—72no—
249#340—74no—
250#340—74no—
251C23×C7—22no—
252C23×C7—22no—
253C23×C7C2×C22⋊C412yes—
254C23×C7#12012yes—
255C23×C7#12312yes—
256C23×C7C22×D411yescharacteristic
257C22×D7—21no—
258C22×D7—21no—
259C22×D7—22no—
260C22×D7—21no—
261C22×D7—21no—
262C22×D7—22no—
263C22×D7—22no—
264C22×D7—28no—
265C22×D7—28no—
266C22×D7—44no—
267C22×D7—48no—
268C22×D7—48no—
269C22×D7C2×C22⋊C412yes—
270C22×D7C2×C22⋊C412yes—
271C22×D7#13714yes—
272C7×Q8—24no—
273C7×Q8—24no—
274C7×Q8—42no—
275C7×Q8—42no—
276C7×Q8—48no—
277C2×C4×C7—24no—
278C2×C4×C7—24no—
279C2×C4×C7—42no—
280C2×C4×C7—42no—
281C2×C4×C7—42no—
282C2×C4×C7—42no—
283C2×C4×C7—48no—
284C2×C4×C7C2×C22⋊C412yes—
285C2×C4×C7C4×D414yes—
286C2×C4×C7C4×D414yes—
287C2×C4×C7#12012yes—
288C2×C4×C7#12114yes—
289C2×C4×C7#12114yes—
290C2×C4×C7#12312yes—
291C2×C4×C7C23×C411yescharacteristic
292C2×C4×C7C22×D411yescharacteristic
293C2×C4×C7C22×D411yescharacteristic
294C2×C4×C7C22×D411yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7C2×#4111yescharacteristic
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—28no—
302C2×Dic7—24no—
303C2×Dic7—24no—
304C2×Dic7—48no—
305C2×Dic7—42no—
306C2×Dic7—42no—
307C2×Dic7—44no—
308C2×Dic7C2×C22⋊C412yes—
309C2×Dic7C2×C22⋊C412yes—
310C4×D7—22no—
311C4×D7—22no—
312C4×D7—22no—
313C4×D7—22no—
314C4×D7—22no—
315C4×D7—22no—
316C4×D7—22no—
317C4×D7—22no—
318C4×D7—28no—
319C4×D7—28no—
320C4×D7—416no—
321C7×C8—42no—
322C7⋊C8—42no—
323C25—71no—
324C22×Q8—22no—
325C22×Q8—21no—
326C22×Q8—22no—
327C22×Q8—22no—
328C22×Q8—22no—
329C22×Q8—22no—
330C22×Q8C4×D714yescomplemented
331C22×Q8C22×D712yes—
332C23×C4—21no—
333C23×C4—71no—
334C23×C4—74no—
335C23×C4—71no—
336C23×C4—74no—
337C23×C4—71no—
338C23×C4—71no—
339C23×C4—71no—
340C23×C4—142no—
341C23×C4—142no—
342C23×C4—142no—
343C23×C4—142no—
344C23×C4—142no—
345C23×C4—141no—
346C23×C4C22×D711yescharacteristic
347C2×M16—22no—
348C2×M16—22no—
349C2×M16—22no—
350C2×M16—142no—
351C2×M16—142no—
352C2×M16—142no—
353C22×C8—72no—
354C22×C8C22×D712yes—
355C22⋊Q8—78no—
356C22⋊Q8—78no—
357C22⋊Q8—148no—
358C22⋊Q8—148no—
359C22⋊Q8—144no—
360C22⋊Q8—144no—
361#117—22no—
362#117—74no—
363#117—148no—
364#117—148no—
365#117—144no—
366#117—142no—
367#117—144no—
368#117C22⋊D714yes—
369C2×C4⋊C4—74no—
370C2×C4⋊C4—72no—
371C2×C4⋊C4—72no—
372C2×C4⋊C4—144no—
373C2×C4⋊C4—142no—
374C2×C4⋊C4—148no—
375C2×C4⋊C4—142no—
376C2×C4⋊C4—142no—
377C2×C4⋊C4—141no—
378C2×C4⋊C4—142no—
379C2×C4⋊C4—144no—
380C2×C4⋊C4C22×D712yes—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—148no—
383C2×C22⋊C4—142no—
384C2×C22⋊C4—148no—
385C2×C22⋊C4—142no—
386C2×C22⋊C4—142no—
387C2×C22⋊C4—141no—
388C2×C42—21no—
389C2×C42—148no—
390C2×C42—142no—
391C2×C42—141no—
392#103—28no—
393#103—148no—
394C22⋊C8—144no—
395C22⋊C8—144no—
396C22×C7—24no—
397C22×C7—22no—
398C22×C7—22no—
399C22×C7—24no—
400C22×C7#38614yes—
401C22×C7C22×C22⋊C411yescharacteristic
402C22×C7C2×#12011yescharacteristic
403C22×C7C2×#12311yescharacteristic
404C22×C7C2×#13712yes—
405C22×C7#57512yes—
406C2×D7—24no—
407C2×D7—22no—
408C2×D7—24no—
409C2×D7—22no—
410C2×D7—22no—
411C2×D7—22no—
412C2×D7—22no—
413C2×D7—416no—
414C2×D7—42no—
415C2×D7—42no—
416C2×D7#41914yescomplemented
417C4×C7—44no—
418C4×C7—44no—
419C4×C7#38614yes—
420C4×C7C2×C4×D411yescharacteristic
421C4×C7C2×C4×D411yescharacteristic
422C4×C7C2×#12111yescharacteristic
423C4×C7C2×#12111yescharacteristic
424Dic7—22no—
425Dic7—22no—
426Dic7—44no—
427Dic7—44no—
428C24—71no—
429C24—71no—
430C24—74no—
431C24—144no—
432C24—142no—
433C24—142no—
434C24—282no—
435C24D4×D711yescharacteristic
436C2×Q8—24no—
437C2×Q8—22no—
438C2×Q8—28no—
439C2×Q8—28no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—42no—
445C2×Q8—44no—
446C2×Q8—42no—
447C2×Q8—44no—
448C2×Q8C2×C4×D714yes—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—74no—
455C22×C4—71no—
456C22×C4—71no—
457C22×C4—74no—
458C22×C4—74no—
459C22×C4—72no—
460C22×C4—72no—
461C22×C4—72no—
462C22×C4—72no—
463C22×C4—148no—
464C22×C4—142no—
465C22×C4—142no—
466C22×C4—148no—
467C22×C4—142no—
468C22×C4—142no—
469C22×C4—148no—
470C22×C4—148no—
471C22×C4—142no—
472C22×C4—142no—
473C22×C4—148no—
474C22×C4—141no—
475C22×C4—141no—
476C22×C4—141no—
477C22×C4—141no—
478C22×C4—141no—
479C22×C4—141no—
480C22×C4—141no—
481C22×C4—141no—
482C22×C4—144no—
483C22×C4—142no—
484C22×C4—142no—
485C22×C4—284no—
486C22×C4—284no—
487C22×C4—284no—
488C22×C4C2×C4×D712yes—
489C22×C4#115712yes—
490C22×C4D4×D711yescharacteristic
491C22×C4D4×D712yes—
492C22×C4D4×D712yes—
493C22×C4#115912yes—
494C22×C4C2×C22⋊D712yes—
495C22×C4C23×D711yescharacteristic
496M16—44no—
497M16—284no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—142no—
502C2×C8—142no—
503C2×C8—142no—
504C4⋊C4—22no—
505C4⋊C4—74no—
506C4⋊C4—74no—
507C4⋊C4—148no—
508C4⋊C4—148no—
509C4⋊C4—148no—
510C4⋊C4—144no—
511C4⋊C4—142no—
512C4⋊C4—142no—
513C4⋊C4C2×C22⋊D714yes—
514C22⋊C4—24no—
515C22⋊C4—148no—
516C22⋊C4—148no—
517C22⋊C4—148no—
518C22⋊C4—144no—
519C42—24no—
520C42—148no—
521C42—144no—
522C2×C7—24no—
523C2×C7C2×#38611yescharacteristic
524C2×C7C2×#41911yescharacteristic
525C2×C7#289611yescharacteristic
526C2×C7#300712yes—
527C2×C7#309512yes—
528D7—24no—
529D7—42no—
530C23—22no—
531C23—22no—
532C23—72no—
533C23—74no—
534C23—72no—
535C23—148no—
536C23—141no—
537C23—142no—
538C23—141no—
539C23—141no—
540C23—142no—
541C23—142no—
542C23—141no—
543C23—148no—
544C23—284no—
545C23—288no—
546C23—288no—
547C23D7×C22⋊C412yes—
548C23#644612yes—
549C23#650012yes—
550C23C2×D4×D711yescharacteristic
551Q8—24no—
552Q8—24no—
553Q8—42no—
554Q8—42no—
555Q8—48no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—48no—
563C2×C4—72no—
564C2×C4—72no—
565C2×C4—141no—
566C2×C4—141no—
567C2×C4—141no—
568C2×C4—141no—
569C2×C4—148no—
570C2×C4—142no—
571C2×C4—142no—
572C2×C4—142no—
573C2×C4—142no—
574C2×C4—142no—
575C2×C4—142no—
576C2×C4—142no—
577C2×C4—142no—
578C2×C4—148no—
579C2×C4—148no—
580C2×C4—144no—
581C2×C4—144no—
582C2×C4—2816no—
583C2×C4—288no—
584C2×C4—282no—
585C2×C4—284no—
586C2×C4—282no—
587C2×C4D7×C22⋊C412yes—
588C2×C4(C22⋊D7)⋊C414yes—
589C2×C4#644612yes—
590C2×C4#644714yes—
591C2×C4C4×C22⋊D714yes—
592C2×C4#650012yes—
593C2×C4#650114yes—
594C2×C4C22×C4×D711yescharacteristic
595C2×C4C2×#115711yescharacteristic
596C2×C4C2×D4×D711yescharacteristic
597C2×C4C2×D4×D711yescharacteristic
598C2×C4C2×#115911yescharacteristic
599C2×C4C22×C22⋊D711yescharacteristic
600C8—42no—
601C8—282no—
602C7#2091111yescharacteristic, complemented, Sylow
603C22—22no—
604C22—22no—
605C22—24no—
606C22—24no—
607C22—74no—
608C22—142no—
609C22—144no—
610C22—142no—
611C22—144no—
612C22—142no—
613C22—142no—
614C22—142no—
615C22—2816no—
616C22—282no—
617C22—282no—
618C22#8972814yes—
619C22C2×D7×C22⋊C411yescharacteristic
620C22C2×#644611yescharacteristic
621C22D7×#13712yes—
622C22#9046812yes—
623C22C2×#650011yescharacteristic
624C4—44no—
625C4—44no—
626C4—142no—
627C4—142no—
628C4—284no—
629C4—284no—
630C4#8972814yes—
631C4C2×(C22⋊D7)⋊C411yescharacteristic
632C4C2×#644711yescharacteristic
633C4C2×C4×C22⋊D711yescharacteristic
634C4C2×#650111yescharacteristic
635C2—24no—
636C2—144no—
637C2—282no—
638C2D7×#41911yescharacteristic
639C2#1173640111yescharacteristic
640C2#1173641012yes—
641C2C2×#8972811yescharacteristic
642C2#1174142612yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#419C2
socleC23×C7C22×D4
prime radicalC24×D7D4
cosocleC22×C4×C7C24
hypercenterC2×#419D7
hypocenterC7#20911
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#419

Complements

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

NH ≅ G/Ntype
3. C2×C7×#419C2semidirect
4. C2×#89997C2semidirect
6. #11729739C2semidirect
7. #11729335C2semidirect
8. #11728023C2semidirect
9. #11728022C2semidirect
17. C2×#6509C4semidirect
34. C2×C7×#103C22semidirect
36. C2×#6409C22semidirect
40. #89346C22semidirect
41. #89324C22semidirect
50. C22×C7×Q8C2×C4semidirect
82. #6509C2×C4semidirect
91. C2×C4⋊Dic7C2×C4semidirect
119. C2×#419D7semidirect
245. C2×#103C2×D7semidirect
330. C22×Q8C4×D7semidirect
416. C2×D7#419semidirect
602. C7#20911semidirect

Maximal subgroups

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

subgroupindex
C22×#65092
C2×D7×#1172
C2×C7×#4192
C2×#899972
C2×#898982
#117297392
#117293352
#117280232
#117280222
#209117