Finite Group Databasebeta

← #13375641

Subgroups — #13375641

Order 1792.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2349C211yesmaximal, characteristic
2C2×#89781C211yesmaximal, characteristic
3#11731104C211yesmaximal, characteristic, complemented
4C4×#6406C211yesmaximal, characteristic
5C4×#6406C211yesmaximal, characteristic
6C4×#6406C211yesmaximal, characteristic
7#11729361C211yesmaximal, characteristic
8#11729360C211yesmaximal, characteristic
9#11727373C211yesmaximal, characteristic, complemented
10#11727372C211yesmaximal, characteristic, complemented
11#11727357C211yesmaximal, characteristic, complemented
12#11727356C211yesmaximal, characteristic, complemented
13#11727308C211yesmaximal, characteristic, complemented
14#11727294C211yesmaximal, characteristic, complemented
15C4⋊C4×C7⋊C8C211yesmaximal, characteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×C4×C7×D4C2211yescharacteristic
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6433C2211yescharacteristic
20C7×#389C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#385—21no—
23C7×#385—21no—
24C7×#382C2211yescharacteristic
25C7×#380—21no—
26C4×C7×C4⋊C4C2211yescharacteristic
27C2×C7×#95—21no—
28C2×#6406C2211yescharacteristic
29C2×#6406C2211yescharacteristic
30C2×#6406C2211yescharacteristic
31C2×#6406C2211yescharacteristic
32#89781C418yescomplemented
33#89750—21no—
34#89744—21no—
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40C4×C4⋊Dic7C2211yescharacteristic
41C4×C4⋊Dic7C2211yescharacteristic
42#89700C2211yescharacteristic
43#89700C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C2×C4×C7⋊C8C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47C2×C4×C7⋊C8C2211yescharacteristic
48#89424—21no—
49#89421—21no—
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89319C2211yescharacteristic
59#89319C2211yescharacteristic
60#22966—71nomaximal, Sylow
61C22×C7×D4C2311yescharacteristic
62C23×C4×C7—21no—
63C23×C4×C7—21no—
64C23×C4×C7—21no—
65C4×C7×D4—22no—
66C4×C7×D4—22no—
67C4×C7×D4—22no—
68C4×C7×D4—41no—
69C4×C7×D4—41no—
70C4×C7×D4C2×C414yes—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4D411yescharacteristic
74C2×C7×C4⋊C4D411yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C4⋊C4C2311yescharacteristic
77C2×C7×C4⋊C4C2311yescharacteristic
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—21no—
80C2×C7×C22⋊C4—21no—
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×Dic7⋊C4—21no—
92C2×C4×Dic7—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99#6433—22no—
100#6433C2×C414yescomplemented
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95D411yescharacteristic
106C7×#95D411yescharacteristic
107#6406—22no—
108#6406C2×C414yes—
109#6406C2×C414yes—
110#6406Q814yes—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C7⋊#95—21no—
114C7⋊#95—21no—
115C7⋊#95—21no—
116#6382—41no—
117#6382C2×C414yescomplemented
118#6381—41no—
119#6381C2×C414yescomplemented
120C4⋊(C7⋊C8)C2×C414yescomplemented
121C4×C7⋊C8—22no—
122C4×C7⋊C8—41no—
123C4×C7⋊C8—41no—
124C2×#478—71no—
125#2370—71no—
126#2349D711yescharacteristic
127#1991—71no—
128#1949—71no—
129#1942—71no—
130#1941—71no—
131#1921—71no—
132#1920—71no—
133#1864—71no—
134#1862—71no—
135C8×C4⋊C4—71no—
136C4×#102—71no—
137C4×#102—71no—
138C4×#102—71no—
139C24×C7—21no—
140C2×C7×D4—21no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4C22×C412yes—
144C2×C7×D4C2×Q812yes—
145C2×C7×D4#4112yes—
146C22×C4×C7—21no—
147C22×C4×C7—24no—
148C22×C4×C7—21no—
149C22×C4×C7—21no—
150C22×C4×C7—42no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7—42no—
154C22×C4×C7—81no—
155C22×C4×C7—81no—
156C22×C4×C7—81no—
157C22×C4×C7—81no—
158C22×C4×C7C2×D411yescharacteristic
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D411yescharacteristic
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D411yescharacteristic
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—21no—
166C22×Dic7—21no—
167C22×Dic7—21no—
168C22×Dic7—21no—
169C7×C4⋊C4—22no—
170C7×C4⋊C4—22no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—81no—
177C7×C4⋊C4—81no—
178C7×C4⋊C4C22×C412yes—
179C7×C22⋊C4—24no—
180C7×C22⋊C4—42no—
181C7×C22⋊C4—42no—
182C42×C7—21no—
183C42×C7—21no—
184C42×C7—21no—
185C42×C7—21no—
186C42×C7C22×C412yes—
187C42×C7#4112yes—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—21no—
193C4⋊Dic7C22×C412yes—
194C4⋊Dic7C22×C412yes—
195C4⋊Dic7C2×Q812yes—
196Dic7⋊C4—24no—
197Dic7⋊C4—24no—
198Dic7⋊C4—42no—
199Dic7⋊C4—81no—
200C4×Dic7—42no—
201C4×Dic7—81no—
202C4×Dic7—81no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8—21no—
209C2×C7⋊C8—21no—
210C2×C7⋊C8—21no—
211C2×C7⋊C8—21no—
212C2×C7⋊C8C22×C412yes—
213C2×C7⋊C8C22×C412yes—
214C2×C7⋊C8C2×Q812yes—
215C2×#125—71no—
216C2×C4×D4C2×D711yescharacteristic
217C2×C4×D4C2×D711yescharacteristic
218C2×C4×D4C2×D711yescharacteristic
219#478—78no—
220C2×#107—71no—
221C2×#107—71no—
222C2×#106—71no—
223C2×#106—71no—
224C2×C4⋊C8—71no—
225C2×#102—71no—
226C2×#102—71no—
227C2×#102—71no—
228C2×#102—71no—
229C2×C4×C8—71no—
230C2×C4×C8—71no—
231C2×C4×C8—71no—
232#389—71no—
233#389—71no—
234#389—141no—
235#389C2×D711yescharacteristic
236#389C2×D711yescharacteristic
237#387—141no—
238#387—141no—
239#385—21no—
240#385—21no—
241#382C2×D711yescharacteristic
242#380—21no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4—71no—
245C4×C4⋊C4—71no—
246C4×C4⋊C4—141no—
247C4×C4⋊C4C2×D711yescharacteristic
248C2×#95—21no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254#340—71no—
255#340—71no—
256#340—71no—
257#336—71no—
258#336—71no—
259C23×C7—22no—
260C23×C7—22no—
261C23×C7—22no—
262C23×C7—41no—
263C23×C7—41no—
264C23×C7—41no—
265C23×C7—41no—
266C23×C7C22×D411yescharacteristic
267C7×D4—22no—
268C7×D4—22no—
269C7×D4—22no—
270C7×D4C4×Q814yes—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—21no—
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×C7—41no—
287C2×C4×C7—41no—
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—84no—
295C2×C4×C7—84no—
296C2×C4×C7—84no—
297C2×C4×C7C4×D412yes—
298C2×C4×C7C4×D412yes—
299C2×C4×C7#13512yes—
300C2×C4×C7#13512yes—
301C2×C4×C7C23×C411yescharacteristic
302C2×C4×C7C22×Q811yescharacteristic
303C2×C4×C7C2×#4111yescharacteristic
304C2×C4×C7C2×#4111yescharacteristic
305C2×C4×C7H(5,𝔽2)11yescharacteristic
306C2×C4×C72−1+411yescharacteristic
307C2×Dic7—22no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—41no—
311C2×Dic7—41no—
312C2×Dic7—41no—
313C2×Dic7—41no—
314C2×Dic7—41no—
315C2×Dic7—41no—
316C2×Dic7—41no—
317C2×Dic7—41no—
318C2×Dic7—41no—
319C7⋊C8—22no—
320C7⋊C8—22no—
321C7⋊C8—22no—
322C7⋊C8—41no—
323C22×D4C22×D711yescharacteristic
324C23×C4—21no—
325C23×C4—21no—
326C23×C4—21no—
327C22×C8—71no—
328C22×C8—71no—
329C22×C8—71no—
330C22×C8—71no—
331#125—74no—
332#125—142no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—41no—
337C4×D4—41no—
338C4×D4C4×D714yes—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—71no—
342C2×C4⋊C4—71no—
343C2×C4⋊C4—71no—
344C2×C4⋊C4—71no—
345C2×C4⋊C4—141no—
346C2×C4⋊C4—141no—
347C2×C4⋊C4—141no—
348C2×C4⋊C4—141no—
349C2×C4⋊C4C22⋊D711yescharacteristic
350C2×C4⋊C4C22⋊D711yescharacteristic
351C2×C4⋊C4C22×D711yescharacteristic
352C2×C4⋊C4C22×D711yescharacteristic
353C2×C4⋊C4C22×D711yescharacteristic
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C42—141no—
358C2×C42—141no—
359C2×C42—141no—
360C2×C42C22×D711yescharacteristic
361C2×C42C22×D711yescharacteristic
362C2×C42C22×D711yescharacteristic
363#107—74no—
364#107—281no—
365#106—74no—
366#106—281no—
367C4⋊C8—74no—
368#102—74no—
369#102—74no—
370#102—74no—
371#102—142no—
372C4×C8—142no—
373C4×C8—281no—
374C4×C8—281no—
375#95—21no—
376#95—21no—
377#95—21no—
378#95—21no—
379#95—141no—
380#95—141no—
381#95—141no—
382#95—141no—
383#95—141no—
384#95C22⋊D711yescharacteristic
385#95C22⋊D711yescharacteristic
386C22×C7—24no—
387C22×C7—42no—
388C22×C7—42no—
389C22×C7—42no—
390C22×C7—42no—
391C22×C7—42no—
392C22×C7—42no—
393C22×C7C2×C4×D411yescharacteristic
394C22×C7#54611yescharacteristic
395C22×C7D4×Q811yescharacteristic
396C22×C7C2×#13511yescharacteristic
397C22×C7C2×#13511yescharacteristic
398C22×C7#57611yescharacteristic
399C22×C7#57711yescharacteristic
400C4×C7—22no—
401C4×C7—22no—
402C4×C7—22no—
403C4×C7—22no—
404C4×C7—42no—
405C4×C7—41no—
406C4×C7—41no—
407C4×C7—81no—
408C4×C7—81no—
409C4×C7C2×C4×Q811yescharacteristic
410C4×C7#52011yescharacteristic
411C4×C7#55211yescharacteristic
412C4×C7#55411yescharacteristic
413Dic7—42no—
414Dic7—42no—
415Dic7—42no—
416Dic7—81no—
417C24—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4C2×C4×D712yes—
422C2×D4#115912yes—
423C2×D4Q8×D712yes—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—81no—
433C22×C4—81no—
434C22×C4—81no—
435C22×C4—81no—
436C22×C4—141no—
437C22×C4—141no—
438C22×C4—141no—
439C22×C4—141no—
440C22×C4C2×C22⋊D711yescharacteristic
441C22×C4C2×C22⋊D711yescharacteristic
442C22×C4C2×C22⋊D711yescharacteristic
443C22×C4C2×C22⋊D711yescharacteristic
444C22×C4C2×C22⋊D711yescharacteristic
445C22×C4C2×C22⋊D711yescharacteristic
446C22×C4C23×D711yescharacteristic
447C2×C8—72no—
448C2×C8—72no—
449C2×C8—72no—
450C2×C8—141no—
451C2×C8—141no—
452C2×C8—141no—
453C2×C8—141no—
454C2×C8—141no—
455C2×C8—141no—
456C2×C8—141no—
457C2×C8—141no—
458C2×C8—141no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—72no—
467C4⋊C4—72no—
468C4⋊C4—72no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—144no—
472C4⋊C4—144no—
473C4⋊C4—141no—
474C4⋊C4—141no—
475C4⋊C4—141no—
476C4⋊C4—141no—
477C4⋊C4—141no—
478C4⋊C4—282no—
479C4⋊C4—561no—
480C4⋊C4C2×C4×D712yes—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—282no—
489C42—561no—
490C42—561no—
491C42C2×C4×D712yes—
492C42#115712yes—
493C2×C7—44no—
494C2×C7#236011yescharacteristic
495C2×C7C4×#13511yescharacteristic
496C2×C7#294611yescharacteristic
497C2×C7#329611yescharacteristic
498C2×C7#329711yescharacteristic
499C2×C7#338011yescharacteristic
500C2×C7#338111yescharacteristic
501C23—22no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C22×C22⋊D711yescharacteristic
509D4—22no—
510D4—22no—
511D4—22no—
512D4#645014yes—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—142no—
540C2×C4—142no—
541C2×C4—142no—
542C2×C4—281no—
543C2×C4—281no—
544C2×C4—281no—
545C2×C4—281no—
546C2×C4—281no—
547C2×C4—281no—
548C2×C4—281no—
549C2×C4—281no—
550C2×C4—281no—
551C2×C4C4×C22⋊D712yes—
552C2×C4C4×C22⋊D712yes—
553C2×C4#651312yes—
554C2×C4#651312yes—
555C2×C4C22×C4×D711yescharacteristic
556C2×C4C2×#115711yescharacteristic
557C2×C4C2×#115911yescharacteristic
558C2×C4#654811yescharacteristic
559C2×C4C2×Q8×D711yescharacteristic
560C2×C4#655111yescharacteristic
561C8—142no—
562C8—142no—
563C8—142no—
564C8—281no—
565C7#2296611yescharacteristic, complemented, Sylow
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22C2×C4×C22⋊D711yescharacteristic
574C22#9049711yescharacteristic
575C22Q8×C22⋊D711yescharacteristic
576C22C2×#651311yescharacteristic
577C22C2×#651311yescharacteristic
578C22#9052611yescharacteristic
579C22#9052711yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—81no—
588C4—81no—
589C4—282no—
590C4—282no—
591C4—282no—
592C4—561no—
593C4C2×#645011yescharacteristic
594C4#9019511yescharacteristic
595C4#9020211yescharacteristic
596C4#9020411yescharacteristic
597C2—44no—
598C2#1173960711yescharacteristic
599C2C4×#651311yescharacteristic
600C2#1173998211yescharacteristic
601C2#1174093911yescharacteristic
602C2#1174094011yescharacteristic
603C2#1174107311yescharacteristic
604C2#1174107411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2349C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
cosocleC22×C4×C7C24
hypercenter#2349D7
hypocenterC7#22966
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #11731104C2semidirect
9. #11727373C2semidirect
10. #11727372C2semidirect
11. #11727357C2semidirect
12. #11727356C2semidirect
13. #11727308C2semidirect
14. #11727294C2semidirect
15. C4⋊C4×C7⋊C8C2semidirect
32. #89781C4semidirect
100. #6433C2×C4semidirect
117. #6382C2×C4semidirect
119. #6381C2×C4semidirect
120. C4⋊(C7⋊C8)C2×C4semidirect
565. C7#22966semidirect

Maximal subgroups

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

subgroupindex
C7×#23492
C2×#897812
#117311042
C4×#64062
#117293612
#117293602
#117273732
#117273722
#117273572
#117273562
#117273082
#117272942
C4⋊C4×C7⋊C82
#229667