Finite Group Databasebeta

← #13367732

Subgroups — #13367732

Order 1792.

814 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6441C211yesmaximal, characteristic
2C7×C8×C22⋊C4C211yesmaximal, characteristic
3C8×C22⋊Dic7C211yesmaximal, characteristic
4C2×#89345C211yesmaximal, characteristic
5#11727704C211yesmaximal, characteristic
6#11727662C211yesmaximal, characteristic
7#11727542C211yesmaximal, characteristic, complemented
8#11727540C211yesmaximal, characteristic, complemented
9#11727538C211yesmaximal, characteristic, complemented
10#11727538C211yesmaximal, characteristic, complemented
11#11727203C211yesmaximal, characteristic
12#11726373C211yesmaximal, characteristic, complemented
13#11726371C211yesmaximal, characteristic, complemented
14#11726365C211yesmaximal, characteristic, complemented
15#11726364C211yesmaximal, characteristic, complemented
16C23×C7×C8C2211yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19C2×#6441C411yescharacteristic
20C2×C4×Dic14C2211yescharacteristic
21C2×C7×C22⋊C8C2211yescharacteristic
22C2×C4×C7×C8C2211yescharacteristic
23C2×C4×C7×C8C2211yescharacteristic
24C4×C7×C22⋊C4C2211yescharacteristic
25C4×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊(C7⋊C8)C2211yescharacteristic
27C2×Dic7⋊C8C2211yescharacteristic
28C2×Dic7⋊C8C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×C8×Dic7C2211yescharacteristic
32C2×C8×Dic7C2211yescharacteristic
33#89722C411yescharacteristic
34#89718C411yescharacteristic
35#89714C411yescharacteristic
36C4×C4⋊Dic7C2211yescharacteristic
37C4×Dic7⋊C4C2211yescharacteristic
38C4×Dic7⋊C4C2211yescharacteristic
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41#89427C411yescharacteristic
42#89426C411yescharacteristic
43#89419C411yescharacteristic
44#89418C411yescharacteristic
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#22287—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C23×Dic7C2×C411yescharacteristic
62C23×Dic7C2×C411yescharacteristic
63C22×Dic14C2×C411yescharacteristic
64C22×C7×C8—41no—
65C22×C7×C8—41no—
66C22×C7×C8Q814yescomplemented
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C2×C7×C22⋊C4C2×C411yescharacteristic
72C2×C7×C22⋊C4C2×C411yescharacteristic
73C2×C42×C7C2311yescharacteristic
74C2×C42×C7C2311yescharacteristic
75C2×C22⋊Dic7C2×C411yescharacteristic
76C2×C22⋊Dic7C2×C411yescharacteristic
77C2×C4⋊Dic7C2×C411yescharacteristic
78C2×C4⋊Dic7C2×C411yescharacteristic
79C2×C4⋊Dic7C2×C411yescharacteristic
80C2×Dic7⋊C4C2×C411yescharacteristic
81C2×Dic7⋊C4C2×C411yescharacteristic
82C2×Dic7⋊C4C2×C411yescharacteristic
83C2×Dic7⋊C4C2×C411yescharacteristic
84C2×Dic7⋊C4C2×C411yescharacteristic
85C2×Dic7⋊C4C2×C411yescharacteristic
86C2×C4×Dic7—41no—
87C2×C4×Dic7—41no—
88C2×C4×Dic7C2×C411yescharacteristic
89C2×C4×Dic7C2×C411yescharacteristic
90C2×C4×Dic7C2311yescharacteristic
91C2×C4×Dic7C2311yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C22×C7⋊C8—22no—
95C22×C7⋊C8—41no—
96C22×C7⋊C8—41no—
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101#6441C818yescomplemented
102C4×Dic14—41no—
103C4×Dic14—41no—
104C7×C22⋊C8—22no—
105C4×C7×C8—41no—
106C4×C7×C8—41no—
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
116C7⋊#95C2×C411yescharacteristic
117C22⋊(C7⋊C8)—22no—
118Dic7⋊C8—41no—
119Dic7⋊C8—41no—
120Dic7⋊C8—41no—
121Dic7⋊C8—41no—
122C8×Dic7—41no—
123C8×Dic7—41no—
124C4⋊(C7⋊C8)—41no—
125C4⋊(C7⋊C8)—41no—
126C4×C22⋊Q8—71no—
127#1984—71no—
128#1984—71no—
129#1939—71no—
130#1939—71no—
131#1916—71no—
132#1916—71no—
133#1916—71no—
134#1916—71no—
135#1850—71no—
136#1850—71no—
137#1797—71no—
138C8×C22⋊C4—71no—
139C8×C22⋊C4D711yescharacteristic
140C2×#336—71no—
141C24×C7C22×C411yescharacteristic
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7C22×C411yescharacteristic
147C22×C4×C7C22×C411yescharacteristic
148C22×C4×C7C22×C411yescharacteristic
149C22×C4×C7C22×C411yescharacteristic
150C22×C4×C7C22×C411yescharacteristic
151C22×C4×C7C2×Q812yes—
152C22×C4×C7#4112yes—
153C22×C4×C7C2411yescharacteristic
154C22×Dic7—22no—
155C22×Dic7—41no—
156C22×Dic7—41no—
157C22×Dic7—41no—
158C22×Dic7—41no—
159C22×Dic7C2×C814yes—
160C22×Dic7C22×C411yescharacteristic
161C22×Dic7C22×C411yescharacteristic
162C22×Dic7C22×C411yescharacteristic
163C22×Dic7C22×C411yescharacteristic
164C22×Dic7C22×C411yescharacteristic
165C22×Dic7C22×C411yescharacteristic
166C22×Dic7C22×C411yescharacteristic
167C22×Dic7C22×C411yescharacteristic
168C2×Dic14—41no—
169C2×Dic14—41no—
170C2×Dic14C2×C814yescomplemented
171C2×C7×C8—21no—
172C2×C7×C8—21no—
173C2×C7×C8—21no—
174C2×C7×C8—21no—
175C2×C7×C8—21no—
176C2×C7×C8—21no—
177C2×C7×C8—21no—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—44no—
182C2×C7×C8C2×Q812yes—
183C2×C7×C8C2×Q812yes—
184C7×C22⋊C4—22no—
185C7×C22⋊C4C2×C814yes—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C42×C7—21no—
190C22⋊Dic7—22no—
191C22⋊Dic7C2×C814yes—
192C4⋊Dic7—41no—
193C4⋊Dic7—41no—
194C4⋊Dic7C2×C814yescomplemented
195Dic7⋊C4—41no—
196Dic7⋊C4—41no—
197Dic7⋊C4—41no—
198Dic7⋊C4—41no—
199Dic7⋊C4C2×C814yescomplemented
200Dic7⋊C4C2×C814yescomplemented
201C4×Dic7—21no—
202C4×Dic7—21no—
203C4×Dic7—21no—
204C4×Dic7—21no—
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—41no—
210C4×Dic7—41no—
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—
219C2×C7⋊C8—21no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—44no—
224C23×C8—71no—
225C23×C8C2×D711yescharacteristic
226C2×C22⋊Q8—71no—
227C2×C4×Q8—71no—
228C22×C42—71no—
229C2×C4⋊C8—71no—
230C2×C4⋊C8—71no—
231C2×C4⋊C8—71no—
232C2×C4⋊C8—71no—
233C2×C4⋊C8—71no—
234C2×C4⋊C8—71no—
235C2×C22⋊C8—71no—
236C2×C22⋊C8C2×D711yescharacteristic
237C2×C4×C8—71no—
238C2×C4×C8—71no—
239C2×C4×C8C2×D711yescharacteristic
240C2×C4×C8C2×D711yescharacteristic
241#391—71no—
242#389—71no—
243#387—71no—
244#387—71no—
245#385—71no—
246#385—71no—
247#380—71no—
248C4×C4⋊C4—71no—
249C4×C4⋊C4—71no—
250C4×C4⋊C4—71no—
251C4×C22⋊C4—71no—
252C4×C22⋊C4C2×D711yescharacteristic
253#336—71no—
254#336—71no—
255#336—71no—
256#336—71no—
257#336—71no—
258#336—71no—
259#336—71no—
260#336—71no—
261#336—71no—
262#336—71no—
263#336—71no—
264#336—71no—
265#336C2×D711yescharacteristic
266#336C2×D711yescharacteristic
267C23×C7—21no—
268C23×C7—21no—
269C23×C7—21no—
270C23×C7—21no—
271C23×C7C4×Q812yes—
272C23×C7C22×C812yes—
273C23×C7#13112yes—
274C23×C7C23×C411yescharacteristic
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—21no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—21no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—21no—
289C2×C4×C7—21no—
290C2×C4×C7—21no—
291C2×C4×C7C4×Q812yes—
292C2×C4×C7C22×C812yes—
293C2×C4×C7C22×C812yes—
294C2×C4×C7C22×Q811yescharacteristic
295C2×C4×C7C2×#4111yescharacteristic
296C2×C4×C7H(5,𝔽2)11yescharacteristic
297C2×C4×C7H(5,𝔽2)11yescharacteristic
298C2×C4×C7H(5,𝔽2)11yescharacteristic
299C2×C4×C72−1+411yescharacteristic
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—21no—
318C2×Dic7—21no—
319C2×Dic7—21no—
320C2×Dic7—44no—
321C2×Dic7—44no—
322C2×Dic7C22×C812yes—
323C2×Dic7C22×C812yes—
324C2×Dic7C22×C812yes—
325C2×Dic7C22×C812yes—
326Dic14—44no—
327C7×C8—22no—
328C7×C8—22no—
329C7×C8—41no—
330C7×C8—41no—
331C7⋊C8—41no—
332C7⋊C8—41no—
333C7⋊C8—41no—
334C7⋊C8—41no—
335C22×Q8—71no—
336C23×C4—71no—
337C23×C4—71no—
338C23×C4C22×D711yescharacteristic
339C22×C8—41no—
340C22×C8—41no—
341C22×C8—71no—
342C22×C8—71no—
343C22×C8—71no—
344C22×C8—71no—
345C22×C8—142no—
346C22×C8—281no—
347C22×C8—281no—
348C22×C8Dic1414yescomplemented
349C22×C8C22×D711yescharacteristic
350C22×C8C22×D711yescharacteristic
351C22×C8C22×D711yescharacteristic
352C22×C8C22×D711yescharacteristic
353C22⋊Q8—78no—
354C4×Q8—281no—
355C4×Q8—281no—
356C2×C4⋊C4—71no—
357C2×C4⋊C4—71no—
358C2×C4⋊C4—71no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—71no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—71no—
365C2×C22⋊C4—71no—
366C2×C22⋊C4—71no—
367C2×C22⋊C4C4×D711yescharacteristic
368C2×C22⋊C4C4×D711yescharacteristic
369C2×C42—71no—
370C2×C42—71no—
371C2×C42—71no—
372C2×C42—71no—
373C2×C42—71no—
374C2×C42—71no—
375C2×C42—281no—
376C2×C42—281no—
377C2×C42C22×D711yescharacteristic
378C2×C42C22×D711yescharacteristic
379C4⋊C8—281no—
380C4⋊C8—281no—
381C4⋊C8—281no—
382C4⋊C8—281no—
383C4⋊C8—281no—
384C4⋊C8—281no—
385C22⋊C8—22no—
386C22⋊C8—142no—
387C4×C8—41no—
388C4×C8—41no—
389C4×C8—281no—
390C4×C8—281no—
391#95—71no—
392#95—71no—
393#95—71no—
394#95—71no—
395#95—71no—
396#95—71no—
397#95—71no—
398#95—71no—
399#95C4×D711yescharacteristic
400#95C4×D711yescharacteristic
401C22×C7—22no—
402C22×C7—22no—
403C22×C7—22no—
404C22×C7—22no—
405C22×C7—22no—
406C22×C7—22no—
407C22×C7C8×Q814yes—
408C22×C7C2×C4×Q811yescharacteristic
409C22×C7#51811yescharacteristic
410C22×C7#52011yescharacteristic
411C22×C7C23×C811yescharacteristic
412C22×C7C2×#13111yescharacteristic
413C22×C7#56811yescharacteristic
414C22×C7#56811yescharacteristic
415C4×C7—22no—
416C4×C7—22no—
417C4×C7—22no—
418C4×C7—41no—
419C4×C7—41no—
420C4×C7#53811yescharacteristic
421C4×C7#53911yescharacteristic
422C4×C7#54311yescharacteristic
423C4×C7#54411yescharacteristic
424Dic7—22no—
425Dic7—22no—
426Dic7—22no—
427Dic7—22no—
428Dic7—41no—
429Dic7—41no—
430Dic7—41no—
431Dic7—41no—
432C24C2×C4×D711yescharacteristic
433C2×Q8—74no—
434C2×Q8—281no—
435C2×Q8—281no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—71no—
441C22×C4—71no—
442C22×C4—71no—
443C22×C4—71no—
444C22×C4—74no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—142no—
450C22×C4—281no—
451C22×C4—281no—
452C22×C4—281no—
453C22×C4—281no—
454C22×C4C2×Dic1412yes—
455C22×C4C2×C4×D711yescharacteristic
456C22×C4C2×C4×D711yescharacteristic
457C22×C4C2×C4×D711yescharacteristic
458C22×C4C2×C4×D711yescharacteristic
459C22×C4C2×C4×D711yescharacteristic
460C22×C4#115712yes—
461C22×C4C23×D711yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—44no—
473C2×C8—141no—
474C2×C8—141no—
475C2×C8—141no—
476C2×C8—141no—
477C2×C8—141no—
478C2×C8—141no—
479C2×C8—141no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8—141no—
484C2×C8—141no—
485C2×C8—284no—
486C2×C8C2×Dic1412yes—
487C2×C8C2×Dic1412yes—
488C4⋊C4—74no—
489C4⋊C4—74no—
490C4⋊C4—74no—
491C4⋊C4—281no—
492C4⋊C4—281no—
493C4⋊C4—281no—
494C4⋊C4—281no—
495C4⋊C4—281no—
496C4⋊C4—281no—
497C22⋊C4—22no—
498C22⋊C4—74no—
499C22⋊C4—142no—
500C22⋊C4C8×D714yes—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—141no—
506C42—141no—
507C42—141no—
508C42—141no—
509C42—141no—
510C42—141no—
511C42—141no—
512C42—141no—
513C42—281no—
514C42—281no—
515C2×C7—24no—
516C2×C7#232911yescharacteristic
517C2×C7C2×C8×Q811yescharacteristic
518C2×C7#296111yescharacteristic
519C2×C7#297211yescharacteristic
520C2×C7#297611yescharacteristic
521C2×C7#298811yescharacteristic
522C2×C7#299211yescharacteristic
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23C4×Dic1412yes—
528C23C2×C8×D712yes—
529C23#646412yes—
530C23C22×C4×D711yescharacteristic
531Q8—284no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—72no—
549C2×C4—72no—
550C2×C4—72no—
551C2×C4—72no—
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×C4—141no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—141no—
564C2×C4—141no—
565C2×C4—141no—
566C2×C4—141no—
567C2×C4—141no—
568C2×C4—141no—
569C2×C4—141no—
570C2×C4—141no—
571C2×C4—141no—
572C2×C4—284no—
573C2×C4—284no—
574C2×C4C4×Dic1412yes—
575C2×C4C2×C8×D712yes—
576C2×C4C2×C8×D712yes—
577C2×C4C22×Dic1411yescharacteristic
578C2×C4C2×#115711yescharacteristic
579C2×C4#654811yescharacteristic
580C2×C4#654811yescharacteristic
581C2×C4#655311yescharacteristic
582C2×C4#655411yescharacteristic
583C8—22no—
584C8—22no—
585C8—41no—
586C8—41no—
587C8—281no—
588C8—281no—
589C8—281no—
590C8—281no—
591C7#2228711yescharacteristic, complemented, Sylow
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22C8×Dic1414yes—
599C22C2×C4×Dic1411yescharacteristic
600C22#9023611yescharacteristic
601C22#9023711yescharacteristic
602C22C22×C8×D711yescharacteristic
603C22C2×#646411yescharacteristic
604C22#9044111yescharacteristic
605C22#9044111yescharacteristic
606C4—22no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—142no—
612C4—142no—
613C4—142no—
614C4—142no—
615C4—281no—
616C4—281no—
617C4—281no—
618C4—281no—
619C4#9017411yescharacteristic
620C4#9021011yescharacteristic
621C4#9025111yescharacteristic
622C4#9025511yescharacteristic
623C2—24no—
624C2#1173089211yescharacteristic
625C2C2×C8×Dic1411yescharacteristic
626C2#1173598811yescharacteristic
627C2#1173685011yescharacteristic
628C2#1173685211yescharacteristic
629C2#1173687111yescharacteristic
630C2#1173687311yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D7
commutatorC22×C7C23×C8
FittingC7×C8×C22⋊C4C2
socleC23×C7C23×C4
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenterC8×C22⋊C4D7
hypocenterC7#22287
LCSG ▷ C22×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
66. C22×C7×C8Q8semidirect
101. #6441C8semidirect
170. C2×Dic14C2×C8semidirect
194. C4⋊Dic7C2×C8semidirect
199. Dic7⋊C4C2×C8semidirect
200. Dic7⋊C4C2×C8semidirect
348. C22×C8Dic14semidirect
591. C7#22287semidirect

Maximal subgroups

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

subgroupindex
C4×#64412
C7×C8×C22⋊C42
C8×C22⋊Dic72
C2×#893452
#117277042
#117276622
#117275422
#117275402
#117275382
#117272032
#117263732
#117263712
#117263652
#117263642
#222877