Finite Group Databasebeta

← #13370327

Subgroups — #13370327

Order 1792.

1292 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Dic14C211yesmaximal, characteristic
2C2×#89473C212yesmaximal, complemented
3C7×#1749C211yesmaximal, characteristic
4#11729284C211yesmaximal, characteristic
5#11727662C211yesmaximal, characteristic
6#11727662C211yesmaximal, characteristic
7#11727660C211yesmaximal, characteristic
8Dic7×C22⋊C8C211yesmaximal, characteristic
9#11727592C212yesmaximal, complemented
10#11726374C212yesmaximal, complemented
11#11726373C212yesmaximal, complemented
12C23×Dic14C411yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C411yescharacteristic
15C22×Dic7⋊C4C411yescharacteristic
16C22×Dic7⋊C4C411yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C22×C4×Dic7C2211yescharacteristic
19C2×C4×Dic14—28no—
20C2×C4×Dic14C411yescharacteristic
21C2×C4×Dic14C411yescharacteristic
22C2×C4×Dic14C411yescharacteristic
23C2×C4×Dic14C411yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C2×C4×Dic14C2211yescharacteristic
26C2×C4×Dic14C2211yescharacteristic
27C2×C4×Dic14C2211yescharacteristic
28C2×C7×C22⋊C8C2211yescharacteristic
29C2×C7×C22⋊C8C2211yescharacteristic
30C2×C7×#97C2212yes—
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×C22⋊(C7⋊C8)C2211yescharacteristic
33C2×#6389C2212yes—
34C2×Dic7⋊C8C2212yes—
35C2×Dic7⋊C8C2212yes—
36C2×C8×Dic7C2212yes—
37C2×C4⋊(C7⋊C8)C2212yes—
38#89473—28no—
39C7×#336C2212yes—
40#89345C2212yes—
41#89345C2212yes—
42#89345C2212yes—
43#89345C2212yes—
44#89319C2212yes—
45#22510—71nomaximal, Sylow
46C23×C4×C7C2×C411yescharacteristic
47C23×C4×C7C2×C411yescharacteristic
48C23×C4×C7C2311yescharacteristic
49C23×Dic7C2×C411yescharacteristic
50C23×Dic7C2×C411yescharacteristic
51C23×Dic7C2×C411yescharacteristic
52C23×Dic7C2×C411yescharacteristic
53C22×Dic14—24no—
54C22×Dic14—24no—
55C22×Dic14—24no—
56C22×Dic14C2×C411yescharacteristic
57C22×Dic14C2×C411yescharacteristic
58C22×Dic14C2×C411yescharacteristic
59C22×Dic14C2×C411yescharacteristic
60C22×C7×C8C2312yes—
61C22×C7×C8C2312yes—
62C2×C42×C7—24no—
63C2×C42×C7C2×C411yescharacteristic
64C2×C42×C7C2×C411yescharacteristic
65C2×C42×C7C2311yescharacteristic
66C2×C42×C7C2311yescharacteristic
67C2×C4⋊Dic7—24no—
68C2×C4⋊Dic7C2×C411yescharacteristic
69C2×C4⋊Dic7C2×C411yescharacteristic
70C2×C4⋊Dic7C2×C411yescharacteristic
71C2×C4⋊Dic7C2×C411yescharacteristic
72C2×Dic7⋊C4—24no—
73C2×Dic7⋊C4—24no—
74C2×Dic7⋊C4C2×C411yescharacteristic
75C2×Dic7⋊C4C2×C411yescharacteristic
76C2×Dic7⋊C4C2×C411yescharacteristic
77C2×Dic7⋊C4C2×C411yescharacteristic
78C2×Dic7⋊C4C2×C411yescharacteristic
79C2×Dic7⋊C4C2×C411yescharacteristic
80C2×Dic7⋊C4C2×C411yescharacteristic
81C2×Dic7⋊C4C2×C411yescharacteristic
82C2×C4×Dic7—24no—
83C2×C4×Dic7—24no—
84C2×C4×Dic7C2×C411yescharacteristic
85C2×C4×Dic7C2×C411yescharacteristic
86C2×C4×Dic7C2×C411yescharacteristic
87C2×C4×Dic7C2×C411yescharacteristic
88C2×C4×Dic7C2311yescharacteristic
89C2×C4×Dic7C2311yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91C2×C4×Dic7C2311yescharacteristic
92C22×C7⋊C8C2312yes—
93C22×C7⋊C8C2312yes—
94C4×Dic14—24no—
95C4×Dic14—24no—
96C4×Dic14—24no—
97C4×Dic14—24no—
98C4×Dic14—24no—
99C4×Dic14—24no—
100C4×Dic14—24no—
101C4×Dic14D418yes—
102C7×C22⋊C8—22no—
103C7×C22⋊C8Q814yescomplemented
104C7×#97—24no—
105C22⋊(C7⋊C8)—22no—
106C22⋊(C7⋊C8)—22no—
107#6389—24no—
108Dic7⋊C8—24no—
109Dic7⋊C8—24no—
110C8×Dic7—24no—
111C4⋊(C7⋊C8)—24no—
112C22×C4×Q8—71no—
113C2×#446—72no—
114#1985—72no—
115#1985—72no—
116#1984—72no—
117#1797—71no—
118#1797—71no—
119#1797—71no—
120#1749—71no—
121#1749D711yescharacteristic
122C4×C22⋊C8—71no—
123C24×C7C22×C411yescharacteristic
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—41no—
132C22×C4×C7C22×C411yescharacteristic
133C22×C4×C7C22×C411yescharacteristic
134C22×C4×C7C22×C411yescharacteristic
135C22×C4×C7C22×C411yescharacteristic
136C22×C4×C7C22×C411yescharacteristic
137C22×C4×C7C2×Q812yes—
138C22×C4×C7#4112yes—
139C22×C4×C7C2411yescharacteristic
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—41no—
150C22×Dic7—41no—
151C22×Dic7M1614yes—
152C22×Dic7C22×C411yescharacteristic
153C22×Dic7C22×C411yescharacteristic
154C22×Dic7C22×C411yescharacteristic
155C22×Dic7C22×C411yescharacteristic
156C22×Dic7C22×C411yescharacteristic
157C22×Dic7C22×C411yescharacteristic
158C22×Dic7C22×C411yescharacteristic
159C22×Dic7C22×C411yescharacteristic
160C2×Dic14—232no—
161C2×Dic14—22no—
162C2×Dic14—22no—
163C2×Dic14—22no—
164C2×Dic14—22no—
165C2×Dic14—22no—
166C2×Dic14—22no—
167C2×Dic14—22no—
168C2×Dic14—22no—
169C2×Dic14—22no—
170C2×Dic14—22no—
171C2×Dic14C22⋊C414yes—
172C2×Dic14M1614yes—
173C2×C7×C8—22no—
174C2×C7×C8—22no—
175C2×C7×C8—22no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8C2×Q814yescomplemented
179C42×C7—22no—
180C42×C7—22no—
181C42×C7—22no—
182C42×C7C2×D414yes—
183C4⋊Dic7—22no—
184C4⋊Dic7—22no—
185C4⋊Dic7—22no—
186C4⋊Dic7C22⋊C414yes—
187Dic7⋊C4—22no—
188Dic7⋊C4—22no—
189Dic7⋊C4—22no—
190Dic7⋊C4—22no—
191Dic7⋊C4—22no—
192Dic7⋊C4—22no—
193Dic7⋊C4C22⋊C414yes—
194Dic7⋊C4C22⋊C414yes—
195C4×Dic7—22no—
196C4×Dic7—22no—
197C4×Dic7—22no—
198C4×Dic7—22no—
199C4×Dic7—22no—
200C4×Dic7—22no—
201C4×Dic7C2×D414yes—
202C4×Dic7C2×D414yes—
203C2×C7⋊C8—22no—
204C2×C7⋊C8—22no—
205C2×C7⋊C8—22no—
206C2×C7⋊C8—22no—
207C2×C7⋊C8—22no—
208C2×C7⋊C8—22no—
209C23×Q8—71no—
210C2×C4×Q8—71no—
211C2×C4×Q8—71no—
212C2×C4×Q8—71no—
213C2×C4×Q8—71no—
214C2×C4×Q8—71no—
215C2×C4×Q8—71no—
216C2×C4×Q8—71no—
217C2×C4×Q8—71no—
218C2×C4×Q8—148no—
219C22×C4⋊C4—71no—
220C22×C4⋊C4—71no—
221C22×C4⋊C4—71no—
222C22×C42—71no—
223C22×C42—71no—
224C22×C42C2×D711yescharacteristic
225#446—148no—
226C2×C4⋊C8—72no—
227C2×C4⋊C8—72no—
228C2×C4⋊C8—72no—
229C2×C22⋊C8—71no—
230C2×C22⋊C8—71no—
231C2×C22⋊C8C2×D711yescharacteristic
232C2×C22⋊C8C2×D711yescharacteristic
233C2×#97—72no—
234C2×#97C2×D712yes—
235C2×C4×C8—72no—
236#336—72no—
237#336—72no—
238#336—72no—
239#336—72no—
240#336—72no—
241#336C2×D712yes—
242C23×C7—21no—
243C23×C7—21no—
244C23×C7—21no—
245C23×C7—21no—
246C23×C7C4×Q812yes—
247C23×C7C2×M1612yes—
248C23×C7#13112yes—
249C23×C7C23×C411yescharacteristic
250C2×C4×C7—28no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—24no—
260C2×C4×C7—24no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—44no—
265C2×C4×C7C2×C22⋊C412yes—
266C2×C4×C7C2×C22⋊C412yes—
267C2×C4×C7C4×Q812yes—
268C2×C4×C7C2×M1612yes—
269C2×C4×C7C22×D412yes—
270C2×C4×C7C22×Q811yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C72−1+412yes—
273C2×Dic7—28no—
274C2×Dic7—21no—
275C2×Dic7—21no—
276C2×Dic7—21no—
277C2×Dic7—21no—
278C2×Dic7—21no—
279C2×Dic7—21no—
280C2×Dic7—21no—
281C2×Dic7—21no—
282C2×Dic7—21no—
283C2×Dic7—21no—
284C2×Dic7—28no—
285C2×Dic7—21no—
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—21no—
289C2×Dic7—21no—
290C2×Dic7—21no—
291C2×Dic7—44no—
292C2×Dic7—44no—
293C2×Dic7C2×C22⋊C412yes—
294C2×Dic7C2×C22⋊C412yes—
295C2×Dic7C2×C22⋊C412yes—
296C2×Dic7C2×C22⋊C412yes—
297C2×Dic7C2×M1612yes—
298C2×Dic7C2×M1612yes—
299C2×Dic7C2×M1612yes—
300C2×Dic7C2×M1612yes—
301Dic14—28no—
302Dic14—28no—
303Dic14—28no—
304Dic14—28no—
305C7×C8—24no—
306C7×C8—42no—
307C7⋊C8—42no—
308C7⋊C8—42no—
309C22×Q8—71no—
310C22×Q8—71no—
311C22×Q8—71no—
312C22×Q8—71no—
313C22×Q8—144no—
314C22×Q8—144no—
315C22×Q8—144no—
316C23×C4—71no—
317C23×C4—71no—
318C23×C4—71no—
319C23×C4—71no—
320C23×C4C4×D711yescharacteristic
321C23×C4C4×D711yescharacteristic
322C23×C4C22×D711yescharacteristic
323C22×C8—72no—
324C22×C8—72no—
325C22×C8C22×D712yes—
326C22×C8C22×D712yes—
327C4×Q8—78no—
328C4×Q8—144no—
329C4×Q8—144no—
330C4×Q8—144no—
331C4×Q8—144no—
332C4×Q8—144no—
333C4×Q8—144no—
334C4×Q8—144no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—71no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—71no—
346C2×C4⋊C4—71no—
347C2×C4⋊C4—144no—
348C2×C4⋊C4—144no—
349C2×C4⋊C4—144no—
350C2×C42—24no—
351C2×C42—71no—
352C2×C42—71no—
353C2×C42—71no—
354C2×C42—71no—
355C2×C42—71no—
356C2×C42—71no—
357C2×C42—71no—
358C2×C42—71no—
359C2×C42—144no—
360C2×C42—144no—
361C2×C42C4×D711yescharacteristic
362C2×C42C4×D711yescharacteristic
363C2×C42C22×D711yescharacteristic
364C2×C42C22×D711yescharacteristic
365C4⋊C8—144no—
366C4⋊C8—144no—
367C4⋊C8—144no—
368C22⋊C8—22no—
369C22⋊C8—142no—
370C22⋊C8—142no—
371C22⋊C8Dic1414yescomplemented
372#97—24no—
373#97—144no—
374C4×C8—144no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—22no—
378C22×C7—22no—
379C22×C7—22no—
380C22×C7—22no—
381C22×C7#44614yescomplemented
382C22×C7C22×C22⋊C411yescharacteristic
383C22×C7C2×C4×Q811yescharacteristic
384C22×C7#51911yescharacteristic
385C22×C7C22×M1611yescharacteristic
386C22×C7C22×M1611yescharacteristic
387C22×C7C2×#13111yescharacteristic
388C22×C7#56811yescharacteristic
389C4×C7—22no—
390C4×C7—22no—
391C4×C7—22no—
392C4×C7—22no—
393C4×C7—41no—
394C4×C7#54812yes—
395C4×C7D4×Q812yes—
396Dic7—22no—
397Dic7—22no—
398Dic7—22no—
399Dic7—22no—
400Dic7—22no—
401Dic7—41no—
402Dic7—41no—
403Dic7#40714yes—
404C24C2×C4×D711yescharacteristic
405C2×Q8—74no—
406C2×Q8—74no—
407C2×Q8—142no—
408C2×Q8—142no—
409C2×Q8—142no—
410C2×Q8—142no—
411C2×Q8—142no—
412C2×Q8—142no—
413C2×Q8—142no—
414C2×Q8—142no—
415C2×Q8—1432no—
416C2×Q8—142no—
417C2×Q8—142no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—71no—
427C22×C4—74no—
428C22×C4—71no—
429C22×C4—71no—
430C22×C4—71no—
431C22×C4—71no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—71no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—142no—
442C22×C4—142no—
443C22×C4—142no—
444C22×C4—281no—
445C22×C4—281no—
446C22×C4C2×Dic1412yes—
447C22×C4C2×C4×D711yescharacteristic
448C22×C4C2×C4×D711yescharacteristic
449C22×C4C2×C4×D711yescharacteristic
450C22×C4C2×C4×D711yescharacteristic
451C22×C4C2×C4×D711yescharacteristic
452C22×C4#115712yes—
453C22×C4C23×D711yescharacteristic
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—142no—
460C2×C8—142no—
461C2×C8—142no—
462C2×C8—142no—
463C2×C8—142no—
464C2×C8—142no—
465C2×C8C2×Dic1414yescomplemented
466C4⋊C4—74no—
467C4⋊C4—74no—
468C4⋊C4—74no—
469C4⋊C4—142no—
470C4⋊C4—142no—
471C4⋊C4—142no—
472C4⋊C4—142no—
473C4⋊C4—142no—
474C4⋊C4—142no—
475C4⋊C4—142no—
476C4⋊C4—142no—
477C4⋊C4—142no—
478C42—22no—
479C42—22no—
480C42—22no—
481C42—74no—
482C42—74no—
483C42—142no—
484C42—142no—
485C42—142no—
486C42—142no—
487C42—142no—
488C42—142no—
489C42D4×D714yes—
490C2×C7—24no—
491C2×C7Q8×C22⋊C411yescharacteristic
492C2×C7C2×#40711yescharacteristic
493C2×C7#288011yescharacteristic
494C2×C7C2×#44611yescharacteristic
495C2×C7Q8×M1611yescharacteristic
496C2×C7#298311yescharacteristic
497C2×C7#298411yescharacteristic
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C4×Dic1412yes—
503C23D7×M1612yes—
504C23#647012yes—
505C23C22×C4×D711yescharacteristic
506Q8—148no—
507Q8—148no—
508Q8—148no—
509Q8—148no—
510C2×C4—28no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—44no—
525C2×C4—72no—
526C2×C4—72no—
527C2×C4—72no—
528C2×C4—72no—
529C2×C4—72no—
530C2×C4—72no—
531C2×C4—72no—
532C2×C4—72no—
533C2×C4—148no—
534C2×C4—148no—
535C2×C4—141no—
536C2×C4—141no—
537C2×C4—141no—
538C2×C4—141no—
539C2×C4—141no—
540C2×C4—141no—
541C2×C4—141no—
542C2×C4—141no—
543C2×C4—141no—
544C2×C4—141no—
545C2×C4—141no—
546C2×C4—141no—
547C2×C4—141no—
548C2×C4—141no—
549C2×C4—141no—
550C2×C4—141no—
551C2×C4—284no—
552C2×C4—284no—
553C2×C4C4×Dic1412yes—
554C2×C4D7×C22⋊C412yes—
555C2×C4D7×C22⋊C412yes—
556C2×C4D7×M1612yes—
557C2×C4C22×Dic1411yescharacteristic
558C2×C4C2×#115711yescharacteristic
559C2×C4C2×D4×D712yes—
560C2×C4#655412yes—
561C8—24no—
562C8—42no—
563C8—282no—
564C8—282no—
565C7#2251011yescharacteristic, complemented, Sylow
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22#8947314yescomplemented
573C22C2×C4×Dic1411yescharacteristic
574C22C2×D7×C22⋊C411yescharacteristic
575C22#9020711yescharacteristic
576C22C2×D7×M1611yescharacteristic
577C22C2×D7×M1611yescharacteristic
578C22C2×#647011yescharacteristic
579C22#9043611yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—74no—
586C4—142no—
587C4—142no—
588C4—142no—
589C4—142no—
590C4—142no—
591C4—281no—
592C4—281no—
593C4D4×Dic1412yes—
594C4#9024312yes—
595C2—24no—
596C2C22⋊C4×Dic1411yescharacteristic
597C2C2×#8947311yescharacteristic
598C2M16×Dic1411yescharacteristic
599C2D7×#40711yescharacteristic
600C2#1173607611yescharacteristic
601C2#1173675911yescharacteristic
602C2#1173676011yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D7
commutator, socleC23×C7C23×C4
FittingC7×#1749C2
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenter#1749D7
hypocenterC7#22510
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. C2×#89473C2semidirect
9. #11727592C2semidirect
10. #11726374C2semidirect
11. #11726373C2semidirect
103. C7×C22⋊C8Q8semidirect
178. C2×C7×C8C2×Q8semidirect
371. C22⋊C8Dic14semidirect
381. C22×C7#446semidirect
465. C2×C8C2×Dic14semidirect
565. C7#22510semidirect
572. C22#89473semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic142
C2×#894732
C7×#17492
#117292842
#117276622
#117276602
Dic7×C22⋊C82
#117275922
#117263742
#117263732
#225107