Finite Group Databasebeta

← #13367741

Subgroups — #13367741

Order 1792.

722 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11730925C211yesmaximal, characteristic
2C7×C8×C22⋊C4C211yesmaximal, characteristic
3C8×C22⋊Dic7C211yesmaximal, characteristic
4#11729719C211yesmaximal, characteristic
5#11727705C211yesmaximal, characteristic
6#11727663C211yesmaximal, characteristic
7#11727542C211yesmaximal, characteristic, complemented
8#11727540C211yesmaximal, characteristic, complemented
9#11727538C211yesmaximal, characteristic, complemented
10#11727538C211yesmaximal, characteristic, complemented
11#11727204C211yesmaximal, characteristic
12#11726373C211yesmaximal, characteristic, complemented
13#11726371C211yesmaximal, characteristic, complemented
14#11726365C211yesmaximal, characteristic, complemented
15#11726364C211yesmaximal, characteristic, complemented
16C23×C7×C8C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C2×C7×C22⋊C8C2211yescharacteristic
21C2×C4×C7×C8C2211yescharacteristic
22C2×C4×C7×C8C2211yescharacteristic
23C4×C7×C22⋊C4C2211yescharacteristic
24C4×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊(C7⋊C8)C2211yescharacteristic
26C2×Dic7⋊C8C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32#89721C411yescharacteristic
33#89720C411yescharacteristic
34#89720C411yescharacteristic
35#89719C411yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37C4×Dic7⋊C4C2211yescharacteristic
38C4×Dic7⋊C4C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41#89433C411yescharacteristic
42#89432C411yescharacteristic
43#89429C411yescharacteristic
44#89428C411yescharacteristic
45C7×#336C2211yescharacteristic
46C7×#336C2211yescharacteristic
47#89345C2211yescharacteristic
48#89345C2211yescharacteristic
49#89345C2211yescharacteristic
50#89345C2211yescharacteristic
51#89345C2211yescharacteristic
52#89345C2211yescharacteristic
53#89345C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89345C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#22288—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C22×Dic14C2×C411yescharacteristic
62C22×C7×C8—41no—
63C22×C7×C8—41no—
64C22×C7×C8Q814yescomplemented
65C22×C7×C8C2311yescharacteristic
66C22×C7×C8C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C2×C7×C22⋊C4C2×C411yescharacteristic
70C2×C7×C22⋊C4C2×C411yescharacteristic
71C2×C42×C7C2311yescharacteristic
72C2×C42×C7C2311yescharacteristic
73C2×C22⋊Dic7C2×C411yescharacteristic
74C2×C22⋊Dic7C2×C411yescharacteristic
75C2×C22⋊Dic7C2×C411yescharacteristic
76C2×C22⋊Dic7C2×C411yescharacteristic
77#6489—41no—
78#6489—41no—
79C2×C4⋊Dic7C2×C411yescharacteristic
80C2×C4⋊Dic7C2×C411yescharacteristic
81C2×C4⋊Dic7C2×C411yescharacteristic
82C2×C4⋊Dic7C2×C411yescharacteristic
83C2×C4⋊Dic7C2×C411yescharacteristic
84C2×Dic7⋊C4C2×C411yescharacteristic
85C2×Dic7⋊C4C2×C411yescharacteristic
86C2×Dic7⋊C4C2×C411yescharacteristic
87C2×Dic7⋊C4C2×C411yescharacteristic
88C2×Dic7⋊C4C2×C411yescharacteristic
89C2×Dic7⋊C4C2×C411yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91C2×C4×Dic7C2311yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C2×#1136—22no—
95C2×#1136—41no—
96C2×#1136—41no—
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C4×Dic14—41no—
102C4×Dic14—41no—
103C7×C22⋊C8—22no—
104C4×C7×C8—41no—
105C4×C7×C8—41no—
106C7×#95C2×C411yescharacteristic
107C7×#95C2×C411yescharacteristic
108C7⋊#95C2×C411yescharacteristic
109C7⋊#95C2×C411yescharacteristic
110C7⋊#95C2×C411yescharacteristic
111C7⋊#95C2×C411yescharacteristic
112C7⋊#95C2×C411yescharacteristic
113C7⋊#95C2×C411yescharacteristic
114C7⋊#95C2×C411yescharacteristic
115C7⋊#95C2×C411yescharacteristic
116C22⋊(C7⋊C8)—22no—
117Dic7⋊C8—41no—
118Dic7⋊C8—41no—
119Dic7⋊C8—41no—
120Dic7⋊C8—41no—
121C8×Dic7—41no—
122C8×Dic7—41no—
123C4⋊(C7⋊C8)—41no—
124C4⋊(C7⋊C8)—41no—
125#2396—71no—
126#1984—71no—
127#1984—71no—
128#1939—71no—
129#1939—71no—
130#1916—71no—
131#1916—71no—
132#1916—71no—
133#1916—71no—
134#1851—71no—
135#1851—71no—
136#1798—71no—
137C8×C22⋊C4—71no—
138C8×C22⋊C4D711yescharacteristic
139#1729—71no—
140C24×C7C22×C411yescharacteristic
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C22×C411yescharacteristic
146C22×C4×C7C22×C411yescharacteristic
147C22×C4×C7C22×C411yescharacteristic
148C22×C4×C7C22×C411yescharacteristic
149C22×C4×C7C22×C411yescharacteristic
150C22×C4×C7C2×Q812yes—
151C22×C4×C7#4112yes—
152C22×C4×C7C2411yescharacteristic
153C22×Dic7C22×C411yescharacteristic
154C22×Dic7C22×C411yescharacteristic
155C22×Dic7C22×C411yescharacteristic
156C22×Dic7C22×C411yescharacteristic
157C22×Dic7C22×C411yescharacteristic
158C22×Dic7C22×C411yescharacteristic
159C22×Dic7C22×C411yescharacteristic
160C22×Dic7C22×C411yescharacteristic
161C2×Dic14—22no—
162C2×Dic14—22no—
163C2×Dic14—41no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—21no—
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—44no—
175C2×C7×C8C2×Q812yes—
176C2×C7×C8C2×Q812yes—
177C7×C22⋊C4—22no—
178C7×C22⋊C4—22no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—21no—
182C42×C7—21no—
183C22⋊Dic7—22no—
184C22⋊Dic7—22no—
185C22⋊Dic7—41no—
186C22⋊Dic7—41no—
187C4⋊Dic7—22no—
188C4⋊Dic7—22no—
189C4⋊Dic7—41no—
190C4⋊Dic7—41no—
191C4⋊Dic7—41no—
192Dic7⋊C4—22no—
193Dic7⋊C4—22no—
194Dic7⋊C4—22no—
195Dic7⋊C4—22no—
196Dic7⋊C4—41no—
197Dic7⋊C4—41no—
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—21no—
201C4×Dic7—21no—
202C4×Dic7—21no—
203C4×Dic7—21no—
204C4×Dic7—21no—
205C4×Dic7—21no—
206#1136—44no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8—21no—
213C2×C7⋊C8—21no—
214C2×C7⋊C8—21no—
215C2×C7⋊C8—21no—
216C2×C7⋊C8—21no—
217C2×C7⋊C8—21no—
218C2×C7⋊C8—21no—
219C22×M16—71no—
220C23×C8C2×D711yescharacteristic
221C2×C4×Q8—71no—
222C2×#117—71no—
223C2×C4⋊C8—71no—
224C2×C4⋊C8—71no—
225C2×C4⋊C8—71no—
226C2×C4⋊C8—71no—
227C2×C4⋊C8—71no—
228C2×C4⋊C8—71no—
229C2×C22⋊C8—71no—
230C2×C22⋊C8C2×D711yescharacteristic
231C2×C4×C8—71no—
232C2×C4×C8—71no—
233C2×C4×C8C2×D711yescharacteristic
234C2×C4×C8C2×D711yescharacteristic
235#400—71no—
236#399—71no—
237#398—71no—
238#398—71no—
239#396—71no—
240#396—71no—
241#395—71no—
242#393—71no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C22⋊C4—71no—
247C4×C22⋊C4C2×D711yescharacteristic
248#336—71no—
249#336—71no—
250#336—71no—
251#336—71no—
252#336—71no—
253#336—71no—
254#336—71no—
255#336—71no—
256#336—71no—
257#336—71no—
258#336—71no—
259#336—71no—
260#336C2×D711yescharacteristic
261#336C2×D711yescharacteristic
262C23×C7—21no—
263C23×C7—21no—
264C23×C7—21no—
265C23×C7—21no—
266C23×C7—21no—
267C23×C7—21no—
268C23×C7C4×Q812yes—
269C23×C7C23×C411yescharacteristic
270C2×C4×C7—21no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—22no—
277C2×C4×C7—22no—
278C2×C4×C7—22no—
279C2×C4×C7—22no—
280C2×C4×C7—21no—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7C4×Q812yes—
287C2×C4×C7#13112yes—
288C2×C4×C7#13112yes—
289C2×C4×C7C22×Q811yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7H(5,𝔽2)11yescharacteristic
292C2×C4×C7H(5,𝔽2)11yescharacteristic
293C2×C4×C7H(5,𝔽2)11yescharacteristic
294C2×C4×C72−1+411yescharacteristic
295C2×Dic7—21no—
296C2×Dic7—21no—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—21no—
300C2×Dic7—21no—
301C2×Dic7—21no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—21no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—21no—
309C2×Dic7—21no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—21no—
313C2×Dic7—21no—
314C2×Dic7—21no—
315C2×Dic7—21no—
316C2×Dic7—21no—
317C2×Dic7#13112yes—
318C2×Dic7#13112yes—
319Dic14—44no—
320C7×C8—22no—
321C7×C8—22no—
322C7×C8—41no—
323C7×C8—41no—
324C7⋊C8—41no—
325C7⋊C8—41no—
326C7⋊C8—41no—
327C7⋊C8—41no—
328C22×Q8—71no—
329C23×C4C22×D711yescharacteristic
330C2×M16—142no—
331C2×M16—281no—
332C2×M16—281no—
333C22×C8—41no—
334C22×C8—41no—
335C22×C8—71no—
336C22×C8—71no—
337C22×C8—71no—
338C22×C8—71no—
339C22×C8Dic1414yescomplemented
340C22×C8C22×D711yescharacteristic
341C22×C8C22×D711yescharacteristic
342C22×C8C22×D711yescharacteristic
343C22×C8C22×D711yescharacteristic
344C4×Q8—281no—
345C4×Q8—281no—
346#117—281no—
347#117—281no—
348C2×C4⋊C4—71no—
349C2×C4⋊C4—71no—
350C2×C4⋊C4—71no—
351C2×C4⋊C4—71no—
352C2×C4⋊C4—71no—
353C2×C4⋊C4—71no—
354C2×C4⋊C4—71no—
355C2×C4⋊C4—71no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C22⋊C4—71no—
360C2×C22⋊C4—71no—
361C2×C22⋊C4—71no—
362C2×C22⋊C4—71no—
363C2×C22⋊C4C4×D711yescharacteristic
364C2×C22⋊C4C4×D711yescharacteristic
365C2×C42—71no—
366C2×C42—71no—
367C2×C42—71no—
368C2×C42—71no—
369C2×C42C22×D711yescharacteristic
370C2×C42C22×D711yescharacteristic
371C4⋊C8—281no—
372C4⋊C8—281no—
373C4⋊C8—281no—
374C4⋊C8—281no—
375C4⋊C8—281no—
376C4⋊C8—281no—
377C22⋊C8—22no—
378C22⋊C8—142no—
379C4×C8—41no—
380C4×C8—41no—
381C4×C8—281no—
382C4×C8—281no—
383#95—71no—
384#95—71no—
385#95—71no—
386#95—71no—
387#95—71no—
388#95—71no—
389#95—71no—
390#95—71no—
391#95C4×D711yescharacteristic
392#95C4×D711yescharacteristic
393C22×C7—22no—
394C22×C7—22no—
395C22×C7—22no—
396C22×C7—41no—
397C22×C7—41no—
398C22×C7—41no—
399C22×C7—41no—
400C22×C7C2×C4×Q811yescharacteristic
401C22×C7#51811yescharacteristic
402C22×C7#52011yescharacteristic
403C22×C7C2×#13111yescharacteristic
404C22×C7C2×#13111yescharacteristic
405C22×C7C2×#13111yescharacteristic
406C22×C7C2×#13111yescharacteristic
407C4×C7—22no—
408C4×C7—22no—
409C4×C7—22no—
410C4×C7—41no—
411C4×C7—41no—
412C4×C7#53811yescharacteristic
413C4×C7#53911yescharacteristic
414C4×C7#54311yescharacteristic
415C4×C7#54411yescharacteristic
416Dic7—22no—
417Dic7—22no—
418Dic7—41no—
419Dic7—41no—
420Dic7—41no—
421Dic7—41no—
422Dic7—41no—
423Dic7—41no—
424C24C2×C4×D711yescharacteristic
425C2×Q8—142no—
426C2×Q8—142no—
427C2×Q8—281no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—71no—
433C22×C4—71no—
434C22×C4—71no—
435C22×C4—71no—
436C22×C4—71no—
437C22×C4—71no—
438C22×C4—71no—
439C22×C4—71no—
440C22×C4C2×Dic1412yes—
441C22×C4C2×C4×D711yescharacteristic
442C22×C4C2×C4×D711yescharacteristic
443C22×C4C2×C4×D711yescharacteristic
444C22×C4C2×C4×D711yescharacteristic
445C22×C4C2×C4×D711yescharacteristic
446C22×C4#115712yes—
447C22×C4C23×D711yescharacteristic
448M16—284no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—44no—
460C2×C8—141no—
461C2×C8—141no—
462C2×C8—141no—
463C2×C8—141no—
464C2×C8—141no—
465C2×C8—141no—
466C2×C8—141no—
467C2×C8—141no—
468C2×C8—141no—
469C2×C8—141no—
470C2×C8—141no—
471C2×C8—141no—
472C2×C8C2×Dic1412yes—
473C2×C8C2×Dic1412yes—
474C4⋊C4—142no—
475C4⋊C4—142no—
476C4⋊C4—142no—
477C4⋊C4—142no—
478C4⋊C4—142no—
479C4⋊C4—142no—
480C4⋊C4—281no—
481C4⋊C4—281no—
482C4⋊C4—281no—
483C4⋊C4—281no—
484C4⋊C4—281no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—142no—
488C22⋊C4—142no—
489C22⋊C4—281no—
490C22⋊C4—281no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—141no—
496C42—141no—
497C42—141no—
498C42—141no—
499C42—141no—
500C42—141no—
501C42—141no—
502C42—141no—
503C2×C7—42no—
504C2×C7#232911yescharacteristic
505C2×C7#296011yescharacteristic
506C2×C7#296011yescharacteristic
507C2×C7#297711yescharacteristic
508C2×C7#297911yescharacteristic
509C2×C7#299411yescharacteristic
510C2×C7#299411yescharacteristic
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23C4×Dic1412yes—
518C23C22×C4×D711yescharacteristic
519Q8—284no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—72no—
537C2×C4—72no—
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—141no—
552C2×C4—141no—
553C2×C4—141no—
554C2×C4—141no—
555C2×C4—141no—
556C2×C4—141no—
557C2×C4—141no—
558C2×C4—141no—
559C2×C4—141no—
560C2×C4C4×Dic1412yes—
561C2×C4#646412yes—
562C2×C4#646412yes—
563C2×C4C22×Dic1411yescharacteristic
564C2×C4C2×#115711yescharacteristic
565C2×C4#654811yescharacteristic
566C2×C4#654811yescharacteristic
567C2×C4#655311yescharacteristic
568C2×C4#655411yescharacteristic
569C8—22no—
570C8—22no—
571C8—41no—
572C8—41no—
573C8—281no—
574C8—281no—
575C8—281no—
576C8—281no—
577C7#2228811yescharacteristic, complemented, Sylow
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22C2×C4×Dic1411yescharacteristic
586C22#9023611yescharacteristic
587C22#9023711yescharacteristic
588C22C2×#646411yescharacteristic
589C22C2×#646411yescharacteristic
590C22D7×#13111yescharacteristic
591C22D7×#13111yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—142no—
598C4—142no—
599C4—281no—
600C4—281no—
601C4—281no—
602C4—281no—
603C4—281no—
604C4—281no—
605C4#9017411yescharacteristic
606C4#9021011yescharacteristic
607C4#9025111yescharacteristic
608C4#9025511yescharacteristic
609C2—42no—
610C2#1173089211yescharacteristic
611C2#1173589011yescharacteristic
612C2#1173598711yescharacteristic
613C2#1173685411yescharacteristic
614C2#1173685511yescharacteristic
615C2#1173687411yescharacteristic
616C2#1173687511yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D7
commutator, socleC23×C7C23×C4
FittingC7×C8×C22⋊C4C2
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenterC8×C22⋊C4D7
hypocenterC7#22288
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ C8×C22⋊C4

Complements

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

NH ≅ G/Ntype
7. #11727542C2semidirect
8. #11727540C2semidirect
9. #11727538C2semidirect
10. #11727538C2semidirect
12. #11726373C2semidirect
13. #11726371C2semidirect
14. #11726365C2semidirect
15. #11726364C2semidirect
64. C22×C7×C8Q8semidirect
339. C22×C8Dic14semidirect
577. C7#22288semidirect

Maximal subgroups

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

subgroupindex
#117309252
C7×C8×C22⋊C42
C8×C22⋊Dic72
#117297192
#117277052
#117276632
#117275422
#117275402
#117275382
#117272042
#117263732
#117263712
#117263652
#117263642
#222887