Finite Group Databasebeta

← #13517273

Subgroups — #13517273

Order 1792.

2286 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90516C211yesmaximal, characteristic
2C7×#2311C211yesmaximal, characteristic
3#11741531C211yesmaximal, characteristic
4#11739688C212yesmaximal, complemented
5#11739607C212yesmaximal, complemented
6#11733942C212yesmaximal, complemented
7#11733939C212yesmaximal, complemented
8#11733745C214yesmaximal, complemented
9#11732127C212yesmaximal, complemented
10#11732126C212yesmaximal, complemented
11#11732116C214yesmaximal, complemented
12#11731327C212yesmaximal, complemented
13#11731321C212yesmaximal, complemented
14#11731316C212yesmaximal, complemented
15#11731311C212yesmaximal, complemented
16C22×C7×#41C2211yescharacteristic
17C23×Dic14C2211yescharacteristic
18C2×C7×#117C2211yescharacteristic
19C22×C7×C4⋊C4C2211yescharacteristic
20#90516C4116yescomplemented
21C2×#6507C2212yes—
22C2×#6499C2212yes—
23C2×#6498C2212yes—
24C2×#6498C2212yes—
25C2×#6489C2211yescharacteristic
26C2×#6489C2211yescharacteristic
27C2×#6487C2212yes—
28C2×#6487C2212yes—
29C22×C4×Dic7C2211yescharacteristic
30C2×#6450C2212yes—
31C2×#6450C2212yes—
32C7×#391C2212yes—
33C7×#390C2212yes—
34C7×#386C2212yes—
35C7×#386C2212yes—
36C7×#380C2212yes—
37C7×#380C2212yes—
38#90002C2211yescharacteristic
39#90002C2211yescharacteristic
40#89993C2211yescharacteristic
41#89993C2211yescharacteristic
42#89992C2212yes—
43#89992C2212yes—
44#89982C2214yes—
45C4×C22⋊Dic7C2212yes—
46C4×C22⋊Dic7C2212yes—
47C4×C22⋊Dic7C2214yes—
48#89750C2214yes—
49C4⋊C4×Dic7C2212yes—
50C4⋊C4×Dic7C2212yes—
51#89746C2212yes—
52#89746C2212yes—
53#89745C2212yes—
54#89745C2212yes—
55#89744C2214yes—
56#89718C2214yes—
57#89717C2214yes—
58#89717C2214yes—
59#89716C2214yes—
60#89716C2214yes—
61#89716C2214yes—
62#89716C2214yes—
63#89714C2214yes—
64C22⋊C4×Dic7C2214yes—
65C22⋊C4×Dic7C2214yes—
66#89705C2214yes—
67C4×Dic7⋊C4C2214yes—
68#89427C2214yes—
69#89424C2214yes—
70#89423C2214yes—
71#89421C2214yes—
72#89421C2214yes—
73#89419C2214yes—
74#89418C2214yes—
75#89418C2214yes—
76#34413—71nomaximal, Sylow
77C2×C7×#41—42no—
78C2×C7×#41—42no—
79C2×C7×#41C2×C418yes—
80C22×C7×Q8C2311yescharacteristic
81C22×C7×D4C2311yescharacteristic
82C22×C7×D4C2311yescharacteristic
83C22×C7×D4C2311yescharacteristic
84C23×C4×C7C2311yescharacteristic
85C23×C4×C7C2311yescharacteristic
86C23×C4×C7C2311yescharacteristic
87C23×C4×C7C2312yes—
88C23×Dic7C2312yes—
89C23×Dic7C2312yes—
90C22×Dic14—42no—
91C22×Dic14—42no—
92C22×Dic14C2×C418yescomplemented
93C22×Dic14C2312yes—
94C22×Dic14C2312yes—
95C7×#117—42no—
96C2×C7×C4⋊C4—42no—
97C2×C7×C4⋊C4C2312yes—
98C2×C7×C4⋊C4C2312yes—
99C2×C7×C4⋊C4C2312yes—
100C2×C7×C22⋊C4C2314yes—
101C2×C7×C22⋊C4C2314yes—
102C2×C7×C22⋊C4C2312yes—
103C2×C42×C7C2312yes—
104C2×C22⋊Dic7C2312yes—
105C2×C22⋊Dic7C2312yes—
106C2×C22⋊Dic7C2312yes—
107C2×C22⋊Dic7C2312yes—
108C2×C22⋊Dic7C2312yes—
109C2×C22⋊Dic7C2314yes—
110C2×C22⋊Dic7C2314yes—
111C2×C22⋊Dic7C2312yes—
112#6507C2×C4116yescomplemented
113#6499C2×C4116yes—
114#6498C2×C4116yescomplemented
115#6498C2×C4116yescomplemented
116#6489—42no—
117#6489C2×C418yescomplemented
118C2×C4⋊Dic7C2311yescharacteristic
119C2×C4⋊Dic7C2311yescharacteristic
120C2×C4⋊Dic7C2312yes—
121#6487C2×C4116yescomplemented
122#6487C2×C4116yescomplemented
123C2×Dic7⋊C4C2314yes—
124C2×Dic7⋊C4C2314yes—
125C2×Dic7⋊C4C2314yes—
126C2×Dic7⋊C4C2314yes—
127C2×Dic7⋊C4C2314yes—
128C2×Dic7⋊C4C2314yes—
129C2×C4×Dic7—42no—
130C2×C4×Dic7C2312yes—
131C2×C4×Dic7C2312yes—
132C2×C4×Dic7C2314yes—
133C2×C4×Dic7C2314yes—
134C2×C4×Dic7C2314yes—
135C2×C4×Dic7C2311yescharacteristic
136C2×C4×Dic7C2314yes—
137C2×C4×Dic7C2311yescharacteristic
138C2×C4×Dic7C2312yes—
139#6450—44no—
140#6450—44no—
141C7×#95C2314yes—
142C7×#95C2314yes—
143C7⋊#95C2312yes—
144C7⋊#95C2312yes—
145C7⋊#95C2314yes—
146C7⋊#95C2314yes—
147C7⋊#95C2314yes—
148C7⋊#95C2314yes—
149C7⋊#95C2314yes—
150C7⋊#95C2314yes—
151C7⋊#95C2314yes—
152C7⋊#95C2314yes—
153C7⋊#95C2312yes—
154C7⋊#95C2312yes—
155C2×#536—71no—
156#2381—72no—
157#2376—72no—
158#2374—72no—
159#2369—72no—
160#2362—72no—
161#2360—72no—
162#2347—72no—
163Q8×C22⋊C4—72no—
164#2337—74no—
165#2335—74no—
166#2323—72no—
167#2311D711yescharacteristic
168C4×C22⋊Q8—72no—
169#2297—71no—
170C24×C7C2411yescharacteristic
171C24×C7C2411yescharacteristic
172C24×C7C2411yescharacteristic
173C7×#41—416no—
174C2×C7×Q8—41no—
175C2×C7×Q8—41no—
176C2×C7×Q8C22×C414yes—
177C2×C7×D4—41no—
178C2×C7×D4—41no—
179C2×C7×D4—41no—
180C2×C7×D4—41no—
181C2×C7×D4—41no—
182C2×C7×D4—41no—
183C2×C7×D4C22×C414yes—
184C2×C7×D4C22×C414yes—
185C2×C7×D4C22×C414yes—
186C22×C4×C7—22no—
187C22×C4×C7—22no—
188C22×C4×C7—22no—
189C22×C4×C7—22no—
190C22×C4×C7—22no—
191C22×C4×C7—22no—
192C22×C4×C7—42no—
193C22×C4×C7—44no—
194C22×C4×C7C22×C414yes—
195C22×C4×C7C22×C414yes—
196C22×C4×C7C22×C414yes—
197C22×C4×C7C2414yes—
198C22×C4×C7C2414yes—
199C22×C4×C7C2411yescharacteristic
200C22×C4×C7C2411yescharacteristic
201C22×C4×C7C2411yescharacteristic
202C22×C4×C7C2411yescharacteristic
203C22×Dic7—24no—
204C22×Dic7—42no—
205C22×Dic7—42no—
206C22×Dic7—42no—
207C22×Dic7—42no—
208C22×Dic7C22×C418yes—
209C22×Dic7C2412yes—
210C22×Dic7C2412yes—
211C22×Dic7C2412yes—
212C22×Dic7C2414yes—
213C22×Dic7C2414yes—
214C22×Dic7C2412yes—
215C2×Dic14—24no—
216C2×Dic14—24no—
217C2×Dic14—416no—
218C2×Dic14—42no—
219C2×Dic14—42no—
220C2×Dic14C22×C418yes—
221C2×Dic14C22×C418yes—
222C7×C4⋊C4—24no—
223C7×C4⋊C4—24no—
224C7×C4⋊C4—42no—
225C7×C22⋊C4—44no—
226C7×C22⋊C4—44no—
227C7×C22⋊C4—42no—
228C42×C7—42no—
229C22⋊Dic7—24no—
230C22⋊Dic7—24no—
231C22⋊Dic7—42no—
232C22⋊Dic7—42no—
233C22⋊Dic7—42no—
234C22⋊Dic7C22×C4116yes—
235C22⋊Dic7C22×C4116yes—
236C22⋊Dic7C22×C418yes—
237C4⋊Dic7—41no—
238C4⋊Dic7—41no—
239C4⋊Dic7C22×C418yes—
240Dic7⋊C4—28no—
241Dic7⋊C4—44no—
242Dic7⋊C4—44no—
243Dic7⋊C4—44no—
244Dic7⋊C4C22×C4116yes—
245Dic7⋊C4C22×C4116yes—
246C4×Dic7—24no—
247C4×Dic7—24no—
248C4×Dic7—44no—
249C4×Dic7—44no—
250C4×Dic7—41no—
251C4×Dic7—41no—
252C4×Dic7—44no—
253C4×Dic7—44no—
254C4×Dic7C22×C418yes—
255C22×#41C2×D711yescharacteristic
256C23×Q8—71no—
257#536—716no—
258C2×C4⋊Q8—72no—
259C2×#124—72no—
260C2×#123—72no—
261C2×#123—72no—
262C2×C22⋊Q8—72no—
263C2×C22⋊Q8—72no—
264C2×C4×Q8—72no—
265C2×C4×Q8—72no—
266C2×#117—71no—
267C2×#117—71no—
268C2×#117C2×D711yescharacteristic
269C22×C4⋊C4C2×D711yescharacteristic
270C22×C42—71no—
271#391—71no—
272#391—71no—
273#391—72no—
274#391—72no—
275#391—74no—
276#391—74no—
277#391C2×D712yes—
278#390—71no—
279#390—71no—
280#390C2×D712yes—
281#389—72no—
282#389—74no—
283#389—72no—
284#389—74no—
285#388—74no—
286#388—74no—
287#388—74no—
288#388—74no—
289#388—74no—
290#388—74no—
291#387—74no—
292#387—74no—
293#387—74no—
294#387—74no—
295#386—72no—
296#386—72no—
297#386C2×D712yes—
298#386C2×D712yes—
299#385—74no—
300#385—74no—
301#383—74no—
302#380—74no—
303#380C2×D712yes—
304#380C2×D712yes—
305C4×C4⋊C4—72no—
306C4×C4⋊C4—72no—
307C4×C4⋊C4—74no—
308C4×C4⋊C4—74no—
309C4×C4⋊C4—74no—
310C4×C22⋊C4—74no—
311C4×C22⋊C4—72no—
312C4×C22⋊C4—72no—
313C4×C22⋊C4—74no—
314C4×C22⋊C4—74no—
315C23×C7—21no—
316C23×C7—21no—
317C23×C7—21no—
318C23×C7—21no—
319C23×C7—21no—
320C23×C7—21no—
321C23×C7—22no—
322C23×C7—22no—
323C23×C7—22no—
324C23×C7—22no—
325C23×C7—22no—
326C23×C7—22no—
327C23×C7C23×C412yes—
328C23×C7C23×C412yes—
329C23×C7C23×C412yes—
330C23×C7C2511yescharacteristic
331C7×Q8—44no—
332C7×D4—44no—
333C7×D4—44no—
334C7×D4—44no—
335C2×C4×C7—21no—
336C2×C4×C7—21no—
337C2×C4×C7—21no—
338C2×C4×C7—21no—
339C2×C4×C7—21no—
340C2×C4×C7—21no—
341C2×C4×C7—28no—
342C2×C4×C7—28no—
343C2×C4×C7—28no—
344C2×C4×C7—24no—
345C2×C4×C7—24no—
346C2×C4×C7—24no—
347C2×C4×C7—24no—
348C2×C4×C7—24no—
349C2×C4×C7—24no—
350C2×C4×C7—84no—
351C2×C4×C7C23×C412yes—
352C2×C4×C7C23×C412yes—
353C2×C4×C7C23×C412yes—
354C2×C4×C7C23×C412yes—
355C2×C4×C7H(5,𝔽2)12yes—
356C2×C4×C7H(5,𝔽2)12yes—
357C2×Dic7—22no—
358C2×Dic7—24no—
359C2×Dic7—24no—
360C2×Dic7—24no—
361C2×Dic7—24no—
362C2×Dic7—22no—
363C2×Dic7—22no—
364C2×Dic7—22no—
365C2×Dic7—22no—
366C2×Dic7—22no—
367C2×Dic7—22no—
368C2×Dic7—22no—
369C2×Dic7—22no—
370C2×Dic7—22no—
371C2×Dic7—22no—
372C2×Dic7—48no—
373C2×Dic7—48no—
374C2×Dic7C23×C418yes—
375C2×Dic7C23×C418yes—
376C2×Dic72−1+414yes—
377Dic14—216no—
378Dic14—48no—
379C2×#41—42no—
380C2×#41—42no—
381C2×#41C4×D718yes—
382C22×Q8—78no—
383C22×Q8—72no—
384C22×Q8—72no—
385C22×Q8—282no—
386C22×Q8—282no—
387C22×Q8C22×D711yescharacteristic
388C22×D4C22×D711yescharacteristic
389C22×D4C22×D711yescharacteristic
390C22×D4C22×D711yescharacteristic
391C23×C4—72no—
392C23×C4—72no—
393C23×C4C22×D711yescharacteristic
394C23×C4C22×D711yescharacteristic
395C23×C4C22×D711yescharacteristic
396C23×C4C22×D712yes—
397C4⋊Q8—716no—
398#124—716no—
399#123—716no—
400#123—716no—
401C22⋊Q8—716no—
402C22⋊Q8—716no—
403C4×Q8—284no—
404C4×Q8—284no—
405#117—42no—
406#117—78no—
407#117—282no—
408C2×C4⋊C4—42no—
409C2×C4⋊C4—72no—
410C2×C4⋊C4—74no—
411C2×C4⋊C4—74no—
412C2×C4⋊C4—74no—
413C2×C4⋊C4—71no—
414C2×C4⋊C4—71no—
415C2×C4⋊C4—74no—
416C2×C4⋊C4—74no—
417C2×C4⋊C4—74no—
418C2×C4⋊C4C22×D712yes—
419C2×C4⋊C4C22×D712yes—
420C2×C4⋊C4C22×D712yes—
421C2×C22⋊C4—72no—
422C2×C22⋊C4—74no—
423C2×C22⋊C4—74no—
424C2×C22⋊C4—72no—
425C2×C22⋊C4—72no—
426C2×C22⋊C4—72no—
427C2×C22⋊C4—72no—
428C2×C22⋊C4—72no—
429C2×C22⋊C4C22×D712yes—
430C2×C22⋊C4C22×D714yes—
431C2×C22⋊C4C22×D714yes—
432C2×C42—72no—
433C2×C42—72no—
434C2×C42—72no—
435C2×C42—74no—
436C2×C42—74no—
437C2×C42—74no—
438C2×C42—71no—
439C2×C42—74no—
440C2×C42—71no—
441C2×C42—282no—
442C2×C42C22×D712yes—
443#95—74no—
444#95—72no—
445#95—72no—
446#95—74no—
447#95—72no—
448#95—74no—
449#95—74no—
450#95—72no—
451#95—74no—
452#95—74no—
453#95—74no—
454#95—74no—
455#95C22×D714yes—
456#95C22×D714yes—
457C22×C7—22no—
458C22×C7—22no—
459C22×C7—22no—
460C22×C7—24no—
461C22×C7—24no—
462C22×C7—24no—
463C22×C7—42no—
464C22×C7—42no—
465C22×C7—42no—
466C22×C7—41no—
467C22×C7—41no—
468C22×C7—41no—
469C22×C7—41no—
470C22×C7—41no—
471C22×C7—41no—
472C22×C7C24×C411yescharacteristic
473C22×C7C2×H(5,𝔽2)11yescharacteristic
474C22×C7C2×H(5,𝔽2)11yescharacteristic
475C22×C7C2×2−1+412yes—
476C22×C7#58512yes—
477C4×C7—22no—
478C4×C7—22no—
479C4×C7—22no—
480C4×C7—44no—
481C4×C7—44no—
482C4×C7#51814yes—
483Dic7—28no—
484Dic7—28no—
485Dic7—24no—
486Dic7—42no—
487Dic7—42no—
488Dic7—42no—
489C24C23×D711yescharacteristic
490C24C23×D711yescharacteristic
491C24C23×D711yescharacteristic
492#41—416no—
493C2×Q8—41no—
494C2×Q8—41no—
495C2×Q8—78no—
496C2×Q8—78no—
497C2×Q8—144no—
498C2×Q8—144no—
499C2×Q8—282no—
500C2×Q8—282no—
501C2×Q8—2816no—
502C2×Q8C2×C4×D714yes—
503C2×D4—41no—
504C2×D4—41no—
505C2×D4—41no—
506C2×D4—41no—
507C2×D4—41no—
508C2×D4—41no—
509C2×D4C2×C4×D714yes—
510C2×D4C2×C4×D714yes—
511C2×D4C2×C4×D714yes—
512C22×C4—22no—
513C22×C4—22no—
514C22×C4—22no—
515C22×C4—22no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—42no—
519C22×C4—44no—
520C22×C4—72no—
521C22×C4—74no—
522C22×C4—74no—
523C22×C4—78no—
524C22×C4—72no—
525C22×C4—72no—
526C22×C4—72no—
527C22×C4—144no—
528C22×C4—282no—
529C22×C4—282no—
530C22×C4—282no—
531C22×C4—282no—
532C22×C4C2×C4×D714yes—
533C22×C4C2×C4×D714yes—
534C22×C4C2×C4×D714yes—
535C22×C4C23×D714yes—
536C22×C4C23×D714yes—
537C22×C4C23×D711yescharacteristic
538C22×C4C23×D711yescharacteristic
539C22×C4C23×D711yescharacteristic
540C22×C4C23×D711yescharacteristic
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—42no—
544C4⋊C4—716no—
545C4⋊C4—716no—
546C4⋊C4—78no—
547C4⋊C4—148no—
548C4⋊C4—284no—
549C4⋊C4—284no—
550C4⋊C4—284no—
551C4⋊C4—281no—
552C4⋊C4—281no—
553C22⋊C4—42no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—78no—
557C22⋊C4—716no—
558C22⋊C4—716no—
559C22⋊C4—144no—
560C22⋊C4—144no—
561C22⋊C4—282no—
562C22⋊C4—282no—
563C22⋊C4—282no—
564C42—42no—
565C42—78no—
566C42—144no—
567C42—144no—
568C42—284no—
569C42—284no—
570C42—281no—
571C42—281no—
572C42—284no—
573C42—284no—
574C2×C7—42no—
575C2×C7—42no—
576C2×C7—42no—
577C2×C7C2×#51811yescharacteristic
578C2×C7C4×2−1+412yes—
579C2×C7#354112yes—
580C2×C7#355011yescharacteristic
581C2×C7#355111yescharacteristic
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—22no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—22no—
593C23—22no—
594C23C22×C4×D712yes—
595C23C22×C4×D712yes—
596C23C22×C4×D712yes—
597C23C24×D711yescharacteristic
598Q8—44no—
599Q8—1416no—
600Q8—288no—
601D4—44no—
602D4—44no—
603D4—44no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—28no—
611C2×C4—28no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—78no—
620C2×C4—78no—
621C2×C4—74no—
622C2×C4—84no—
623C2×C4—142no—
624C2×C4—144no—
625C2×C4—144no—
626C2×C4—144no—
627C2×C4—142no—
628C2×C4—144no—
629C2×C4—142no—
630C2×C4—142no—
631C2×C4—142no—
632C2×C4—142no—
633C2×C4—142no—
634C2×C4—142no—
635C2×C4—142no—
636C2×C4—142no—
637C2×C4—142no—
638C2×C4—288no—
639C2×C4—288no—
640C2×C4C22×C4×D712yes—
641C2×C4C22×C4×D712yes—
642C2×C4C22×C4×D712yes—
643C2×C4C22×C4×D712yes—
644C2×C4#654812yes—
645C2×C4#654812yes—
646C7#3441311yescharacteristic, complemented, Sylow
647C22—22no—
648C22—22no—
649C22—22no—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22C23×C4×D711yescharacteristic
663C22C2×#654811yescharacteristic
664C22C2×#654811yescharacteristic
665C22#9061812yes—
666C22D7×2−1+412yes—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—44no—
671C4—44no—
672C4—148no—
673C4—144no—
674C4—148no—
675C4—282no—
676C4—282no—
677C4—282no—
678C4#9017714yes—
679C2—42no—
680C2—42no—
681C2—42no—
682C2C2×#9017711yescharacteristic
683C2#1174295512yes—
684C2#1174339511yescharacteristic
685C2#1174341711yescharacteristic
686C2#1174345312yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2311C2
socle, cosocleC23×C7C25
prime radicalC22×C7×#41C22
hypercenter#2311D7
hypocenterC7#34413
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2311

Complements

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

NH ≅ G/Ntype
4. #11739688C2semidirect
5. #11739607C2semidirect
6. #11733942C2semidirect
7. #11733939C2semidirect
8. #11733745C2semidirect
9. #11732127C2semidirect
10. #11732126C2semidirect
11. #11732116C2semidirect
12. #11731327C2semidirect
13. #11731321C2semidirect
14. #11731316C2semidirect
15. #11731311C2semidirect
20. #90516C4semidirect
92. C22×Dic14C2×C4semidirect
112. #6507C2×C4semidirect
114. #6498C2×C4semidirect
115. #6498C2×C4semidirect
117. #6489C2×C4semidirect
121. #6487C2×C4semidirect
122. #6487C2×C4semidirect
646. C7#34413semidirect

Maximal subgroups

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

subgroupindex
C2×#905162
C7×#23112
#117415312
#117396882
#117396072
#117339422
#117339392
#117337452
#117321272
#117321262
#117321162
#117313272
#117313212
#117313162
#117313112
#344137