Finite Group Databasebeta

← #13375583

Subgroups — #13375583

Order 1792.

788 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89874C211yesmaximal, characteristic
2C2×#89473C211yesmaximal, characteristic, complemented
3#11730925C211yesmaximal, characteristic
4C7×#1752C211yesmaximal, characteristic
5#11729728C211yesmaximal, characteristic
6#11727703C211yesmaximal, characteristic
7#11727661C211yesmaximal, characteristic
8#11727590C211yesmaximal, characteristic, complemented
9#11727588C211yesmaximal, characteristic, complemented
10#11727565C211yesmaximal, characteristic, complemented
11#11727204C211yesmaximal, characteristic
12#11726374C211yesmaximal, characteristic, complemented
13#11726371C211yesmaximal, characteristic, complemented
14#11726371C211yesmaximal, characteristic, complemented
15#11726367C211yesmaximal, characteristic, complemented
16C23×C7×C8C2211yescharacteristic
17C2×#6489C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×C4×Dic14C2211yescharacteristic
20C2×C7×C22⋊C8C2211yescharacteristic
21C2×C7×#97C2211yescharacteristic
22C2×C7×#97C2211yescharacteristic
23C4×C7×C22⋊C4C2211yescharacteristic
24C4×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊(C7⋊C8)C2211yescharacteristic
26#89874—24no—
27C2×#6389C2211yescharacteristic
28C2×#6389C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31C2×Dic7⋊C8C2211yescharacteristic
32C2×Dic7⋊C8C2211yescharacteristic
33C2×Dic7⋊C8C2211yescharacteristic
34C2×Dic7⋊C8C2211yescharacteristic
35C2×C8×Dic7C2211yescharacteristic
36C2×C8×Dic7C2211yescharacteristic
37#89721C411yescharacteristic
38#89720C411yescharacteristic
39#89720C411yescharacteristic
40#89719C411yescharacteristic
41C4×C4⋊Dic7C2211yescharacteristic
42C4×Dic7⋊C4C2211yescharacteristic
43C4×Dic7⋊C4C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×C4⋊(C7⋊C8)C2211yescharacteristic
46#89473—24no—
47#89433C411yescharacteristic
48#89432C411yescharacteristic
49#89429C411yescharacteristic
50#89428C411yescharacteristic
51C7×#336C2211yescharacteristic
52C7×#336C2211yescharacteristic
53#89345C2211yescharacteristic
54#89345C2211yescharacteristic
55#89345C2211yescharacteristic
56#89345C2211yescharacteristic
57#89345C2211yescharacteristic
58#89345C2211yescharacteristic
59#89319C2211yescharacteristic
60#89319C2211yescharacteristic
61#22961—71nomaximal, Sylow
62C23×C4×C7C2311yescharacteristic
63C22×Dic14C2×C411yescharacteristic
64C22×C7×C8—22no—
65C22×C7×C8—22no—
66C22×C7×C8—41no—
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×C22⋊Dic7C2×C411yescharacteristic
78C2×C22⋊Dic7C2×C411yescharacteristic
79#6489—22no—
80#6489—22no—
81C2×C4⋊Dic7C2×C411yescharacteristic
82C2×C4⋊Dic7C2×C411yescharacteristic
83C2×C4⋊Dic7C2×C411yescharacteristic
84C2×C4⋊Dic7C2×C411yescharacteristic
85C2×C4⋊Dic7C2×C411yescharacteristic
86C2×Dic7⋊C4C2×C411yescharacteristic
87C2×Dic7⋊C4C2×C411yescharacteristic
88C2×Dic7⋊C4C2×C411yescharacteristic
89C2×Dic7⋊C4C2×C411yescharacteristic
90C2×Dic7⋊C4C2×C411yescharacteristic
91C2×Dic7⋊C4C2×C411yescharacteristic
92C2×C4×Dic7C2311yescharacteristic
93C2×C4×Dic7C2311yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C2×C4×Dic7C2311yescharacteristic
96C2×#1136—22no—
97C2×#1136—22no—
98C2×#1136—41no—
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102C22×C7⋊C8C2311yescharacteristic
103C4×Dic14—22no—
104C4×Dic14—22no—
105C7×C22⋊C8Q814yescomplemented
106C7×#97—22no—
107C7×#97—41no—
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
117C7⋊#95C2×C411yescharacteristic
118C22⋊(C7⋊C8)—22no—
119#6389—22no—
120#6389—22no—
121Dic7⋊C8—22no—
122Dic7⋊C8—22no—
123Dic7⋊C8—22no—
124Dic7⋊C8—22no—
125Dic7⋊C8—41no—
126Dic7⋊C8D414yes—
127C8×Dic7—22no—
128C8×Dic7—41no—
129C4⋊(C7⋊C8)—22no—
130C4⋊(C7⋊C8)—41no—
131C2×#446—71no—
132C2×#424—71no—
133#2396—71no—
134#1985—71no—
135#1940—71no—
136#1939—71no—
137#1939—71no—
138#1917—71no—
139#1917—71no—
140#1851—71no—
141#1850—71no—
142#1771—71no—
143#1753—71no—
144#1752D711yescharacteristic
145#1750—71no—
146C24×C7C22×C411yescharacteristic
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—21no—
151C22×C4×C7C22×C411yescharacteristic
152C22×C4×C7C22×C411yescharacteristic
153C22×C4×C7C22×C411yescharacteristic
154C22×C4×C7C22×C411yescharacteristic
155C22×C4×C7C22×C411yescharacteristic
156C22×C4×C7C2×Q812yes—
157C22×C4×C7#4112yes—
158C22×C4×C7C2411yescharacteristic
159C22×Dic7C22×C411yescharacteristic
160C22×Dic7C22×C411yescharacteristic
161C22×Dic7C22×C411yescharacteristic
162C22×Dic7C22×C411yescharacteristic
163C22×Dic7C22×C411yescharacteristic
164C22×Dic7C22×C411yescharacteristic
165C22×Dic7C22×C411yescharacteristic
166C22×Dic7C22×C411yescharacteristic
167C2×Dic14—22no—
168C2×Dic14—41no—
169C2×Dic14—41no—
170C2×C7×C8—21no—
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—44no—
180C2×C7×C8C2×D412yes—
181C2×C7×C8C2×Q812yes—
182C2×C7×C8C2×Q812yes—
183C7×C22⋊C4—41no—
184C7×C22⋊C4—41no—
185C42×C7—21no—
186C42×C7—21no—
187C42×C7—21no—
188C42×C7—21no—
189C22⋊Dic7—22no—
190C22⋊Dic7—22no—
191C22⋊Dic7—22no—
192C22⋊Dic7—41no—
193C4⋊Dic7—22no—
194C4⋊Dic7—22no—
195C4⋊Dic7—22no—
196C4⋊Dic7—41no—
197C4⋊Dic7—41no—
198Dic7⋊C4—22no—
199Dic7⋊C4—22no—
200Dic7⋊C4—22no—
201Dic7⋊C4—22no—
202Dic7⋊C4—41no—
203Dic7⋊C4—41no—
204C4×Dic7—21no—
205C4×Dic7—21no—
206C4×Dic7—21no—
207C4×Dic7—21no—
208C4×Dic7—21no—
209C4×Dic7—21no—
210C4×Dic7C2×D412yes—
211C4×Dic7#4112yes—
212#1136—44no—
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—21no—
224C2×C7⋊C8C2×D412yes—
225C22×M16—71no—
226C23×C8C2×D711yescharacteristic
227C2×C4×Q8—71no—
228C2×#117—71no—
229#446—144no—
230#424—144no—
231C2×C4⋊C8—71no—
232C2×C4⋊C8—71no—
233C2×C4⋊C8—71no—
234C2×C4⋊C8—71no—
235C2×C4⋊C8—71no—
236C2×C4⋊C8—71no—
237C2×C4⋊C8—71no—
238C2×C4⋊C8—71no—
239C2×C22⋊C8—71no—
240C2×C22⋊C8C2×D711yescharacteristic
241C2×#97—71no—
242C2×#97—71no—
243C2×#97C2×D711yescharacteristic
244C2×#97C2×D711yescharacteristic
245C2×C4×C8—71no—
246C2×C4×C8—71no—
247#400—71no—
248#399—71no—
249#398—71no—
250#398—71no—
251#396—71no—
252#396—71no—
253#395—71no—
254#393—71no—
255C4×C4⋊C4—71no—
256C4×C4⋊C4—71no—
257C4×C4⋊C4—71no—
258C4×C22⋊C4—71no—
259C4×C22⋊C4C2×D711yescharacteristic
260#336—71no—
261#336—71no—
262#336—71no—
263#336—71no—
264#336—71no—
265#336—71no—
266#336—71no—
267#336—71no—
268#336C2×D711yescharacteristic
269#336C2×D711yescharacteristic
270C23×C7—21no—
271C23×C7—21no—
272C23×C7—21no—
273C23×C7—21no—
274C23×C7—21no—
275C23×C7—21no—
276C23×C7C4×Q812yes—
277C23×C7C23×C411yescharacteristic
278C2×C4×C7—21no—
279C2×C4×C7—21no—
280C2×C4×C7—22no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—21no—
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×C7—21no—
292C2×C4×C7—21no—
293C2×C4×C7—21no—
294C2×C4×C7—21no—
295C2×C4×C7—21no—
296C2×C4×C7C4×Q812yes—
297C2×C4×C7C22×D411yescharacteristic
298C2×C4×C7C22×Q811yescharacteristic
299C2×C4×C7C2×#4111yescharacteristic
300C2×C4×C7C2×#4111yescharacteristic
301C2×C4×C7H(5,𝔽2)11yescharacteristic
302C2×C4×C72−1+411yescharacteristic
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—21no—
321C2×Dic7—21no—
322C2×Dic7—21no—
323C2×Dic7—21no—
324C2×Dic7C4×D412yes—
325C2×Dic7C4×D412yes—
326C2×Dic7#13112yes—
327Dic14—44no—
328C7×C8—22no—
329C7×C8—22no—
330C7×C8—22no—
331C7×C8—41no—
332C7⋊C8—22no—
333C7⋊C8—41no—
334C7⋊C8—41no—
335C7⋊C8—41no—
336C22×Q8—71no—
337C23×C4C22×D711yescharacteristic
338C2×M16—142no—
339C2×M16—142no—
340C2×M16—281no—
341C22×C8—22no—
342C22×C8—22no—
343C22×C8—41no—
344C22×C8—71no—
345C22×C8—71no—
346C22×C8—71no—
347C22×C8—71no—
348C22×C8C22×D711yescharacteristic
349C22×C8C22×D711yescharacteristic
350C22×C8C22×D711yescharacteristic
351C22×C8C22×D711yescharacteristic
352C4×Q8—142no—
353C4×Q8—142no—
354#117—142no—
355#117—142no—
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×C4⋊C4—71no—
366C2×C4⋊C4—71no—
367C2×C22⋊C4—71no—
368C2×C22⋊C4—71no—
369C2×C22⋊C4—71no—
370C2×C22⋊C4—71no—
371C2×C22⋊C4C4×D711yescharacteristic
372C2×C22⋊C4C4×D711yescharacteristic
373C2×C42—71no—
374C2×C42—71no—
375C2×C42—71no—
376C2×C42—71no—
377C2×C42C22×D711yescharacteristic
378C2×C42C22×D711yescharacteristic
379C4⋊C8—74no—
380C4⋊C8—142no—
381C4⋊C8—142no—
382C4⋊C8—142no—
383C4⋊C8—142no—
384C4⋊C8—142no—
385C4⋊C8—281no—
386C4⋊C8—281no—
387C22⋊C8—142no—
388C22⋊C8Dic1414yescomplemented
389#97—22no—
390#97—41no—
391#97—142no—
392#97—142no—
393C4×C8—142no—
394C4×C8—281no—
395#95—71no—
396#95—71no—
397#95—71no—
398#95—71no—
399#95—71no—
400#95—71no—
401#95—71no—
402#95—71no—
403#95C4×D711yescharacteristic
404#95C4×D711yescharacteristic
405C22×C7—22no—
406C22×C7—22no—
407C22×C7—22no—
408C22×C7—41no—
409C22×C7—41no—
410C22×C7—41no—
411C22×C7—41no—
412C22×C7C2×C4×D411yescharacteristic
413C22×C7C2×C4×Q811yescharacteristic
414C22×C7#52011yescharacteristic
415C22×C7C2×#13111yescharacteristic
416C22×C7#56811yescharacteristic
417C22×C7#56811yescharacteristic
418C22×C7#56811yescharacteristic
419C4×C7—22no—
420C4×C7—41no—
421C4×C7—41no—
422C4×C7—41no—
423C4×C7—41no—
424C4×C7#54611yescharacteristic
425C4×C7D4×Q811yescharacteristic
426C4×C7#55211yescharacteristic
427C4×C7#55411yescharacteristic
428Dic7—22no—
429Dic7—22no—
430Dic7—22no—
431Dic7—41no—
432Dic7—41no—
433Dic7—41no—
434Dic7—41no—
435Dic7—41no—
436C24C2×C4×D711yescharacteristic
437C2×Q8—142no—
438C2×Q8—281no—
439C2×Q8—281no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—71no—
445C22×C4—71no—
446C22×C4—71no—
447C22×C4—71no—
448C22×C4—71no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—71no—
452C22×C4C2×Dic1412yes—
453C22×C4C2×C4×D711yescharacteristic
454C22×C4C2×C4×D711yescharacteristic
455C22×C4C2×C4×D711yescharacteristic
456C22×C4C2×C4×D711yescharacteristic
457C22×C4C2×C4×D711yescharacteristic
458C22×C4#115712yes—
459C22×C4C23×D711yescharacteristic
460M16—284no—
461C2×C8—21no—
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—44no—
471C2×C8—72no—
472C2×C8—141no—
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×C8C2×Dic1412yes—
484C2×C8C2×Dic1412yes—
485C2×C8D4×D712yes—
486C4⋊C4—142no—
487C4⋊C4—142no—
488C4⋊C4—142no—
489C4⋊C4—142no—
490C4⋊C4—142no—
491C4⋊C4—142no—
492C4⋊C4—142no—
493C4⋊C4—281no—
494C4⋊C4—281no—
495C4⋊C4—281no—
496C4⋊C4—281no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—142no—
500C22⋊C4—142no—
501C22⋊C4—142no—
502C22⋊C4—281no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—72no—
508C42—72no—
509C42—141no—
510C42—141no—
511C42—141no—
512C42—141no—
513C42—141no—
514C42—141no—
515C2×C7—42no—
516C2×C7#236011yescharacteristic
517C2×C7#293011yescharacteristic
518C2×C7#293511yescharacteristic
519C2×C7#296111yescharacteristic
520C2×C7#296211yescharacteristic
521C2×C7#299211yescharacteristic
522C2×C7#299511yescharacteristic
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23C4×Dic1412yes—
530C23C22×C4×D711yescharacteristic
531Q8—284no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—72no—
551C2×C4—72no—
552C2×C4—72no—
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—141no—
573C2×C4—141no—
574C2×C4C4×Dic1412yes—
575C2×C4C22×Dic1411yescharacteristic
576C2×C4C2×#115711yescharacteristic
577C2×C4C2×D4×D711yescharacteristic
578C2×C4#654811yescharacteristic
579C2×C4D7×#4111yescharacteristic
580C2×C4#655411yescharacteristic
581C8—22no—
582C8—22no—
583C8—22no—
584C8—41no—
585C8—142no—
586C8—281no—
587C8—281no—
588C8—281no—
589C7#2296111yescharacteristic, complemented, Sylow
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×C4×Dic1411yescharacteristic
598C22C4×D4×D711yescharacteristic
599C22#9023711yescharacteristic
600C22#9043611yescharacteristic
601C22#9043611yescharacteristic
602C22D7×#13111yescharacteristic
603C22#9044111yescharacteristic
604C4—22no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—142no—
610C4—142no—
611C4—142no—
612C4—281no—
613C4—281no—
614C4—281no—
615C4—281no—
616C4—281no—
617C4#9018111yescharacteristic
618C4#9022211yescharacteristic
619C4D4×Dic1411yescharacteristic
620C4#9023011yescharacteristic
621C2—42no—
622C2#1173091111yescharacteristic
623C2#1173592511yescharacteristic
624C2#1173598911yescharacteristic
625C2#1173690911yescharacteristic
626C2#1173691211yescharacteristic
627C2#1173692411yescharacteristic
628C2#1173692911yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D7
commutator, socleC23×C7C23×C4
FittingC7×#1752C2
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenter#1752D7
hypocenterC7#22961
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C22×C4 ◁ #1752

Complements

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

NH ≅ G/Ntype
2. C2×#89473C2semidirect
8. #11727590C2semidirect
9. #11727588C2semidirect
10. #11727565C2semidirect
12. #11726374C2semidirect
13. #11726371C2semidirect
14. #11726371C2semidirect
15. #11726367C2semidirect
105. C7×C22⋊C8Q8semidirect
388. C22⋊C8Dic14semidirect
589. C7#22961semidirect

Maximal subgroups

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

subgroupindex
C2×#898742
C2×#894732
#117309252
C7×#17522
#117297282
#117277032
#117276612
#117275902
#117275882
#117275652
#117272042
#117263742
#117263712
#117263672
#229617