Finite Group Databasebeta

← #13382885

Subgroups — #13382885

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89653C211yesmaximal, characteristic
2C2×#89609C211yesmaximal, characteristic
3D7×#389C211yesmaximal, characteristic
4#11733408C211yesmaximal, characteristic
5C7×#2087C211yesmaximal, characteristic, complemented
6#11728375C211yesmaximal, characteristic, complemented
7#11728218C211yesmaximal, characteristic, complemented
8#11727377C211yesmaximal, characteristic, complemented
9#11727364C211yesmaximal, characteristic, complemented
10#11726621C211yesmaximal, characteristic
11#11726621C211yesmaximal, characteristic
12#11726621C211yesmaximal, characteristic
13#11726594C211yesmaximal, characteristic, complemented
14#11726563C211yesmaximal, characteristic, complemented
15#11726561C211yesmaximal, characteristic, complemented
16C22×#1131C2211yescharacteristic
17C22×#1131C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×#6455C2211yescharacteristic
20C2×D7×C4⋊C4—21no—
21C2×D7×C4⋊C4C2211yescharacteristic
22C2×D7×C4⋊C4C2211yescharacteristic
23C2×C42×D7C2211yescharacteristic
24C2×C7×#107C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C7×#389C2211yescharacteristic
27C7×#389C2211yescharacteristic
28C2×#6391C2211yescharacteristic
29C2×Dic7⋊C8C2211yescharacteristic
30C2×Dic7⋊C8C2211yescharacteristic
31#89761—21no—
32#89757C2211yescharacteristic
33#89757C2211yescharacteristic
34C4⋊C4×Dic7—21no—
35#89745C2211yescharacteristic
36#89745C2211yescharacteristic
37#89744—21no—
38C2×#6382C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40#89700C2211yescharacteristic
41#89700C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43#89653—24no—
44#89609—24no—
45#89440—21no—
46D7×#95—21no—
47#89421—21no—
48#89419—21no—
49C7×#340C2211yescharacteristic
50C7×#340C2211yescharacteristic
51C7×#340C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#89324C2211yescharacteristic
61#23670—71nomaximal, Sylow
62C22×C4×D7—21no—
63C22×C4×D7—21no—
64C22×C4×D7D411yescharacteristic
65C22×C4×D7D411yescharacteristic
66C22×C4×D7C2311yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C22×C7×C8C2311yescharacteristic
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C42×C7C2311yescharacteristic
76C2×(C2×D7)⋊C4—21no—
77C2×(C2×D7)⋊C4—21no—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2311yescharacteristic
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×Dic7⋊C4—21no—
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7D411yescharacteristic
93C2×C4×Dic7D411yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C2×#1131—22no—
98C2×#1131—22no—
99C2×#1131—22no—
100C2×#1131—22no—
101C2×#1131—22no—
102C2×#1131D414yes—
103#6455—22no—
104#6455—22no—
105#6455—22no—
106#6455Q814yes—
107D7×C4⋊C4—22no—
108D7×C4⋊C4—22no—
109D7×C4⋊C4—22no—
110D7×C4⋊C4—44no—
111D7×C4⋊C4Q814yes—
112C42×D7—22no—
113C42×D7—22no—
114C7×#107—22no—
115C7×C4⋊C8—22no—
116C7×#95—21no—
117C7×#95—21no—
118C7⋊#95—21no—
119C7⋊#95—21no—
120C7⋊#95—21no—
121C7⋊#95—21no—
122C7⋊#95—21no—
123C7⋊#95—21no—
124#6391—22no—
125Dic7⋊C8—22no—
126Dic7⋊C8—22no—
127#6382—22no—
128#6381—22no—
129C4⋊(C7⋊C8)—22no—
130C2×#428—71no—
131C2×#423—71no—
132#2296—71no—
133C2×#389—71no—
134#2088—71no—
135#2088—71no—
136#2088—71no—
137#2088—71no—
138#2087—71no—
139#2087D711yescharacteristic, complemented
140#2086—71no—
141#2086—71no—
142#1736—71no—
143#1736—71no—
144#1736—71no—
145C23×D7C2×D411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2411yescharacteristic
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7C2×D411yescharacteristic
158C22×Dic7C2×D411yescharacteristic
159C22×Dic7C2×D411yescharacteristic
160C2×C4×D7—22no—
161C2×C4×D7—22no—
162C2×C4×D7—24no—
163C2×C4×D7—22no—
164C2×C4×D7—22no—
165C2×C4×D7—41no—
166C2×C4×D7—41no—
167C2×C4×D7—42no—
168C2×C4×D7—42no—
169C2×C4×D7—42no—
170C2×C4×D7—42no—
171C2×C4×D7—42no—
172C2×C4×D7C2×D412yes—
173C2×C4×D7C2×Q812yes—
174C2×C4×D7C2×Q812yes—
175C2×C4×D7#4112yes—
176C2×C4×D7#4112yes—
177C2×C4×D7#4112yes—
178C2×C7×C8—21no—
179C2×C7×C8—21no—
180C2×C7×C8—21no—
181C2×C7×C8—21no—
182C2×C7×C8—21no—
183C2×C7×C8C2×D412yes—
184C7×C4⋊C4—21no—
185C7×C4⋊C4—21no—
186C7×C4⋊C4—21no—
187C7×C4⋊C4—21no—
188C7×C4⋊C4—21no—
189C7×C4⋊C4—21no—
190C7×C4⋊C4—42no—
191C7×C4⋊C4—42no—
192C7×C4⋊C4C2×Q812yes—
193C7×C4⋊C4C2×Q812yes—
194C42×C7—21no—
195C42×C7—21no—
196(C2×D7)⋊C4—24no—
197(C2×D7)⋊C4—42no—
198C4⋊Dic7—21no—
199C4⋊Dic7—21no—
200C4⋊Dic7—21no—
201C4⋊Dic7—21no—
202C4⋊Dic7—21no—
203C4⋊Dic7—21no—
204C4⋊Dic7—42no—
205C4⋊Dic7—42no—
206C4⋊Dic7C2×Q812yes—
207C4⋊Dic7C2×Q812yes—
208Dic7⋊C4—24no—
209Dic7⋊C4—42no—
210Dic7⋊C4—42no—
211Dic7⋊C4—42no—
212Dic7⋊C4—42no—
213Dic7⋊C4—42no—
214C4×Dic7—21no—
215C4×Dic7—21no—
216C4×Dic7—22no—
217C4×Dic7—22no—
218C4×Dic7—24no—
219C4×Dic7—42no—
220C2×C7⋊C8—21no—
221C2×C7⋊C8—21no—
222C2×C7⋊C8—21no—
223C2×C7⋊C8—21no—
224C2×C7⋊C8—21no—
225C2×C7⋊C8C2×D412yes—
226#1131—28no—
227#1131—44no—
228C22×M16—71no—
229C22×M16—71no—
230C2×#117—71no—
231C2×#117—71no—
232C22×C4⋊C4—71no—
233C22×C4⋊C4—71no—
234C22×C4⋊C4—141no—
235C22×C42—71no—
236#428—144no—
237C2×#107—71no—
238C2×#107C2×D711yescharacteristic
239C2×#106—71no—
240C2×#106—71no—
241#423—144no—
242C2×C4⋊C8—71no—
243C2×C4⋊C8—71no—
244C2×C4⋊C8—71no—
245C2×C4⋊C8C2×D711yescharacteristic
246#389—71no—
247#389—71no—
248#389—71no—
249#389—71no—
250#389—71no—
251#389—71no—
252#389—141no—
253#389—141no—
254#389C2×D711yescharacteristic
255#389C2×D711yescharacteristic
256#385—141no—
257#385—141no—
258C4×C4⋊C4—141no—
259C4×C4⋊C4—141no—
260C2×#95—141no—
261#340—71no—
262#340—71no—
263#340—71no—
264#340—71no—
265#340—71no—
266#340—71no—
267#340—71no—
268#340—71no—
269#340—71no—
270#340C2×D711yescharacteristic
271#340C2×D711yescharacteristic
272#340C2×D711yescharacteristic
273C23×C7C22×D411yescharacteristic
274C22×D7—21no—
275C22×D7—21no—
276C22×D7—21no—
277C22×D7—21no—
278C22×D7#12112yes—
279C22×D7C22⋊Q812yes—
280C22×D7C22⋊Q812yes—
281C2×C4×C7—21no—
282C2×C4×C7—21no—
283C2×C4×C7—21no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—21no—
287C2×C4×C7—21no—
288C2×C4×C7—41no—
289C2×C4×C7—41no—
290C2×C4×C7—41no—
291C2×C4×C7—41no—
292C2×C4×C7—41no—
293C2×C4×C7—41no—
294C2×C4×C7—41no—
295C2×C4×C7—41no—
296C2×C4×C7—41no—
297C2×C4×C7—41no—
298C2×C4×C7#13712yes—
299C2×C4×C7C22×D411yescharacteristic
300C2×C4×C7C22×Q811yescharacteristic
301C2×C4×C7C22×Q811yescharacteristic
302C2×C4×C7C2×#4111yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—21no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—21no—
311C2×Dic7—21no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C2×Dic7—41no—
320C2×Dic7—41no—
321C2×Dic7—41no—
322C2×Dic7#12112yes—
323C2×Dic7C22⋊Q812yes—
324C2×Dic7C22⋊Q812yes—
325C2×Dic7AGL(1,ℤ/8)12yes—
326C4×D7—44no—
327C4×D7—44no—
328C4×D7—48no—
329C4×D7—84no—
330C4×D7#12314yes—
331C4×D7#12514yes—
332C4×D7#12514yes—
333C4×D7C4⋊Q814yes—
334C7×C8—22no—
335C7×C8—41no—
336C7⋊C8—22no—
337C7⋊C8—41no—
338C23×C4—71no—
339C23×C4—71no—
340C23×C4—71no—
341C23×C4—141no—
342C23×C4—141no—
343C2×M16—74no—
344C2×M16—142no—
345C2×M16—142no—
346C2×M16—142no—
347C2×M16—142no—
348C2×M16—142no—
349C22×C8—71no—
350C22×C8—71no—
351C22×C8C22×D711yescharacteristic
352C22×C8C22×D711yescharacteristic
353#117—74no—
354#117—142no—
355#117—142no—
356#117—142no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—71no—
360C2×C4⋊C4—71no—
361C2×C4⋊C4—74no—
362C2×C4⋊C4—71no—
363C2×C4⋊C4—71no—
364C2×C4⋊C4—142no—
365C2×C4⋊C4—142no—
366C2×C4⋊C4—142no—
367C2×C4⋊C4—141no—
368C2×C4⋊C4—141no—
369C2×C4⋊C4—141no—
370C2×C4⋊C4—141no—
371C2×C4⋊C4—141no—
372C2×C4⋊C4—141no—
373C2×C4⋊C4—141no—
374C2×C4⋊C4—141no—
375C2×C4⋊C4—284no—
376C2×C4⋊C4C22×D711yescharacteristic
377C2×C4⋊C4C22×D711yescharacteristic
378C2×C4⋊C4C22×D711yescharacteristic
379C2×C4⋊C4C22×D711yescharacteristic
380C2×C22⋊C4—141no—
381C2×C22⋊C4—141no—
382C2×C42—71no—
383C2×C42—71no—
384C2×C42—71no—
385C2×C42—142no—
386C2×C42—142no—
387C2×C42—141no—
388C2×C42—141no—
389C2×C42C22×D711yescharacteristic
390#107—22no—
391#107—142no—
392#106—142no—
393#106—142no—
394C4⋊C8—22no—
395C4⋊C8—142no—
396C4⋊C8—142no—
397C4⋊C8—142no—
398#95—21no—
399#95—21no—
400#95—141no—
401#95—141no—
402#95—141no—
403#95—141no—
404#95—141no—
405#95—141no—
406C22×C7C2×#12111yescharacteristic
407C22×C7C2×C22⋊Q811yescharacteristic
408C22×C7C2×C22⋊Q811yescharacteristic
409C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
410C22×C7C2×#13711yescharacteristic
411C22×C7#57511yescharacteristic
412C22×C7#57511yescharacteristic
413C2×D7—22no—
414C2×D7—22no—
415C2×D7—22no—
416C2×D7—22no—
417C2×D7—22no—
418C2×D7—22no—
419C2×D7#39814yes—
420C4×C7—22no—
421C4×C7—42no—
422C4×C7—42no—
423C4×C7—41no—
424C4×C7—81no—
425C4×C7—81no—
426C4×C7C2×#12311yescharacteristic
427C4×C7C2×#12511yescharacteristic
428C4×C7C2×#12511yescharacteristic
429C4×C7C2×C4⋊Q811yescharacteristic
430Dic7—22no—
431Dic7—42no—
432Dic7—42no—
433Dic7—41no—
434Dic7—81no—
435Dic7—81no—
436Dic7#39814yes—
437C24—71no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—71no—
443C22×C4—72no—
444C22×C4—72no—
445C22×C4—72no—
446C22×C4—72no—
447C22×C4—72no—
448C22×C4—72no—
449C22×C4—71no—
450C22×C4—71no—
451C22×C4—142no—
452C22×C4—142no—
453C22×C4—142no—
454C22×C4—144no—
455C22×C4—141no—
456C22×C4—141no—
457C22×C4—142no—
458C22×C4—141no—
459C22×C4—141no—
460C22×C4—281no—
461C22×C4—281no—
462C22×C4—282no—
463C22×C4—282no—
464C22×C4—282no—
465C22×C4—282no—
466C22×C4—282no—
467C22×C4D4×D711yescharacteristic
468C22×C4D4×D711yescharacteristic
469C22×C4C23×D711yescharacteristic
470M16—148no—
471M16—284no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—72no—
478C2×C8—141no—
479C2×C8—141no—
480C2×C8—141no—
481C2×C8—141no—
482C2×C8—141no—
483C2×C8D4×D712yes—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—72no—
493C4⋊C4—72no—
494C4⋊C4—144no—
495C4⋊C4—141no—
496C4⋊C4—141no—
497C4⋊C4—141no—
498C4⋊C4—141no—
499C4⋊C4—141no—
500C4⋊C4—141no—
501C4⋊C4—282no—
502C4⋊C4—282no—
503C4⋊C4—282no—
504C4⋊C4—282no—
505C4⋊C4—282no—
506C4⋊C4—282no—
507C4⋊C4—282no—
508C4⋊C4Q8×D712yes—
509C4⋊C4Q8×D712yes—
510C22⋊C4—144no—
511C22⋊C4—282no—
512C42—21no—
513C42—21no—
514C42—142no—
515C42—142no—
516C42—144no—
517C42—141no—
518C42—141no—
519C42—282no—
520C2×C7C2×#39811yescharacteristic
521C2×C7#305511yescharacteristic
522C2×C7#305611yescharacteristic
523C2×C7#307711yescharacteristic
524C2×C7#307811yescharacteristic
525C2×C7#307911yescharacteristic
526C2×C7#308011yescharacteristic
527D7—24no—
528C23—72no—
529C23—72no—
530C23—72no—
531C23—141no—
532C23—141no—
533C23—141no—
534C23—141no—
535C23C2×D4×D711yescharacteristic
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—72no—
554C2×C4—72no—
555C2×C4—72no—
556C2×C4—74no—
557C2×C4—74no—
558C2×C4—74no—
559C2×C4—74no—
560C2×C4—72no—
561C2×C4—141no—
562C2×C4—141no—
563C2×C4—141no—
564C2×C4—141no—
565C2×C4—141no—
566C2×C4—142no—
567C2×C4—142no—
568C2×C4—284no—
569C2×C4—284no—
570C2×C4—288no—
571C2×C4—281no—
572C2×C4—281no—
573C2×C4—281no—
574C2×C4—281no—
575C2×C4—281no—
576C2×C4—281no—
577C2×C4—281no—
578C2×C4—281no—
579C2×C4—281no—
580C2×C4—281no—
581C2×C4—564no—
582C2×C4#648112yes—
583C2×C4C2×D4×D711yescharacteristic
584C2×C4C2×Q8×D711yescharacteristic
585C2×C4C2×Q8×D711yescharacteristic
586C2×C4D7×#4111yescharacteristic
587C2×C4D7×#4111yescharacteristic
588C2×C4D7×#4111yescharacteristic
589C8—22no—
590C8—41no—
591C8—142no—
592C8—281no—
593C7#2367011yescharacteristic, complemented, Sylow
594C22—74no—
595C22—142no—
596C22—142no—
597C22—142no—
598C22—142no—
599C22—142no—
600C22—142no—
601C22D7×#12111yescharacteristic
602C22D7×C22⋊Q811yescharacteristic
603C22D7×C22⋊Q811yescharacteristic
604C22C2×#648111yescharacteristic
605C22#9045911yescharacteristic
606C22D7×AGL(1,ℤ/8)11yescharacteristic
607C22#9046411yescharacteristic
608C4—22no—
609C4—41no—
610C4—42no—
611C4—42no—
612C4—74no—
613C4—81no—
614C4—81no—
615C4—142no—
616C4—281no—
617C4—282no—
618C4—282no—
619C4—561no—
620C4—561no—
621C4D7×#12311yescharacteristic
622C4D7×#12511yescharacteristic
623C4D7×#12511yescharacteristic
624C4D7×C4⋊Q811yescharacteristic
625C2—144no—
626C2D7×#39811yescharacteristic
627C2#1173820411yescharacteristic
628C2#1173830911yescharacteristic
629C2#1173855311yescharacteristic
630C2#1173855411yescharacteristic
631C2#1173861111yescharacteristic
632C2#1173861211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2087C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenter#2087D7
hypocenterC7#23670
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2087

Complements

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

NH ≅ G/Ntype
5. C7×#2087C2semidirect
6. #11728375C2semidirect
7. #11728218C2semidirect
8. #11727377C2semidirect
9. #11727364C2semidirect
13. #11726594C2semidirect
14. #11726563C2semidirect
15. #11726561C2semidirect
139. #2087D7semidirect
593. C7#23670semidirect

Maximal subgroups

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

subgroupindex
C2×#896532
C2×#896092
D7×#3892
#117334082
C7×#20872
#117283752
#117282182
#117273772
#117273642
#117266212
#117265942
#117265632
#117265612
#236707