Finite Group Databasebeta

← #13284490

Subgroups — #13284490

Order 1792.

1194 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89701C211yesmaximal, characteristic
2C2×#89705C211yesmaximal, characteristic
3C2×#89428C211yesmaximal, characteristic
4C7×#1434C211yesmaximal, characteristic
5#11727178C211yesmaximal, characteristic
6#11727176C212yesmaximal
7#11727172C212yesmaximal
8#11727172C212yesmaximal
9#11727166C211yesmaximal, characteristic
10#11727165C212yesmaximal
11#11727164C211yesmaximal, characteristic
12C23×Dic14C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2211yescharacteristic
15C22×C4⋊Dic7C2212yes—
16C22×Dic7⋊C4C2211yescharacteristic
17C22×Dic7⋊C4C2211yescharacteristic
18C2×C7×#95C2211yescharacteristic
19C2×C7×#95C2212yes—
20C2×C7×#95C2212yes—
21C2×C7×#95C2211yescharacteristic
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2211yescharacteristic
24C2×C7⋊#95C2211yescharacteristic
25C2×C7⋊#95C2211yescharacteristic
26C2×C7⋊#95C2212yes—
27C2×C7⋊#95C2211yescharacteristic
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32C2×C7⋊#95C2212yes—
33C2×C7⋊#95C2212yes—
34C2×C7⋊#95C2212yes—
35#89705—24no—
36#89701—24no—
37#89428C418yescomplemented
38#14626—71nomaximal, Sylow
39C23×C4×C7C2312yes—
40C23×C4×C7C2312yes—
41C23×C4×C7C2311yescharacteristic
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2311yescharacteristic
44C23×Dic7C2311yescharacteristic
45C23×Dic7C2312yes—
46C23×Dic7C2311yescharacteristic
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2311yescharacteristic
49C23×Dic7C2312yes—
50C22×Dic14—22no—
51C22×Dic14—22no—
52C22×Dic14—42no—
53C22×Dic14—42no—
54C22×Dic14C2×C414yes—
55C2×C42×C7—42no—
56C2×C42×C7D414yes—
57C2×C4⋊Dic7—22no—
58C2×C4⋊Dic7—42no—
59C2×C4⋊Dic7—42no—
60C2×C4⋊Dic7—42no—
61C2×C4⋊Dic7C2×C418yes—
62C2×Dic7⋊C4—22no—
63C2×Dic7⋊C4—22no—
64C2×Dic7⋊C4—22no—
65C2×Dic7⋊C4—42no—
66C2×Dic7⋊C4C2×C414yes—
67C7×#95—22no—
68C7×#95—42no—
69C7×#95—42no—
70C7×#95C2×C414yes—
71C7⋊#95—24no—
72C7⋊#95—24no—
73C7⋊#95—22no—
74C7⋊#95—22no—
75C7⋊#95—22no—
76C7⋊#95—22no—
77C7⋊#95—42no—
78C7⋊#95—42no—
79C7⋊#95—42no—
80C7⋊#95—42no—
81C7⋊#95—42no—
82C7⋊#95—42no—
83C7⋊#95C2×C418yes—
84C2×#395—71no—
85C2×#391—71no—
86C2×#387—71no—
87#1446—71no—
88#1445—72no—
89#1443—72no—
90#1443—72no—
91#1442—71no—
92#1441—71no—
93#1441—72no—
94#1434D711yescharacteristic
95C24×C7C2411yescharacteristic
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—21no—
101C22×C4×C7—22no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—21no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—22no—
118C22×C4×C7—22no—
119C22×C4×C7C22×C414yes—
120C22×C4×C7C22×C412yes—
121C22×C4×C7C2×D412yes—
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7#4112yes—
125C22×Dic7—21no—
126C22×Dic7—22no—
127C22×Dic7—21no—
128C22×Dic7—21no—
129C22×Dic7—21no—
130C22×Dic7—22no—
131C22×Dic7—21no—
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—22no—
138C22×Dic7—21no—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C22×Dic7—21no—
142C22×Dic7—21no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7—22no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—22no—
158C22×Dic7C22×C412yes—
159C22×Dic7C22×C414yes—
160C22×Dic7C22×C412yes—
161C2×Dic14—28no—
162C2×Dic14—48no—
163C2×Dic14—44no—
164C2×Dic14—44no—
165C2×Dic14—48no—
166C42×C7—44no—
167C4⋊Dic7—48no—
168C4⋊Dic7—44no—
169Dic7⋊C4—28no—
170Dic7⋊C4—44no—
171C23×Q8—71no—
172C22×C4⋊C4—71no—
173C22×C4⋊C4—72no—
174C22×C4⋊C4—71no—
175C22×C4⋊C4—71no—
176C22×C42C2×D711yescharacteristic
177#395—78no—
178#391—144no—
179#387—144no—
180C2×#95—72no—
181C2×#95—72no—
182C2×#95—71no—
183C2×#95—71no—
184C2×#95—71no—
185C2×#95—72no—
186C2×#95—71no—
187C2×#95—72no—
188C2×#95—72no—
189C2×#95—72no—
190C2×#95—72no—
191C2×#95—72no—
192C2×#95—72no—
193C2×#95C2×D711yescharacteristic
194C2×#95C2×D712yes—
195C2×#95C2×D712yes—
196C2×#95C2×D711yescharacteristic
197C23×C7C23×C411yescharacteristic
198C23×C7C22×D411yescharacteristic
199C23×C7C2×#4111yescharacteristic
200C23×C7H(5,𝔽2)12yes—
201C23×C7H(5,𝔽2)11yescharacteristic
202C23×C7H(5,𝔽2)12yes—
203C23×C72−1+411yescharacteristic
204C23×C72−1+412yes—
205C23×C72−1+412yes—
206C23×C72−1+411yescharacteristic
207C23×C72−1+411yescharacteristic
208C2×C4×C7—24no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—24no—
212C2×C4×C7—24no—
213C2×C4×C7—24no—
214C2×C4×C7—24no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—22no—
218C2×C4×C7—22no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7C4×D414yes—
237C2×Dic7—22no—
238C2×Dic7—22no—
239C2×Dic7—24no—
240C2×Dic7—24no—
241C2×Dic7—24no—
242C2×Dic7—24no—
243C2×Dic7—22no—
244C2×Dic7—22no—
245C2×Dic7—41no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—41no—
249C2×Dic7—41no—
250C2×Dic7—41no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7—41no—
255C2×Dic7—41no—
256C2×Dic7—41no—
257C2×Dic7—42no—
258C2×Dic7—41no—
259C2×Dic7—42no—
260C2×Dic7—41no—
261C2×Dic7—41no—
262C2×Dic7—41no—
263C2×Dic7—42no—
264C2×Dic7—41no—
265C2×Dic7—42no—
266C2×Dic7—42no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—42no—
270C2×Dic7—42no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273Dic14—416no—
274C22×Q8—74no—
275C22×Q8—142no—
276C22×Q8—142no—
277C22×Q8—282no—
278C22×Q8—282no—
279C23×C4—71no—
280C23×C4—72no—
281C23×C4—71no—
282C23×C4—71no—
283C23×C4—72no—
284C23×C4—71no—
285C23×C4C22×D712yes—
286C23×C4C22×D712yes—
287C23×C4C22×D711yescharacteristic
288C23×C4C22×D711yescharacteristic
289C23×C4C22×D711yescharacteristic
290C2×C4⋊C4—74no—
291C2×C4⋊C4—78no—
292C2×C4⋊C4—142no—
293C2×C4⋊C4—142no—
294C2×C4⋊C4—142no—
295C2×C4⋊C4—142no—
296C2×C4⋊C4—282no—
297C2×C4⋊C4—282no—
298C2×C4⋊C4—282no—
299C2×C4⋊C4—282no—
300C2×C42—42no—
301C2×C42D2814yes—
302#95—22no—
303#95—42no—
304#95—42no—
305#95—78no—
306#95—142no—
307#95—142no—
308#95—142no—
309#95—142no—
310#95—144no—
311#95—144no—
312#95—282no—
313#95—282no—
314#95—282no—
315#95—282no—
316#95—282no—
317#95—282no—
318#95C4×D714yes—
319C22×C7C2×C4×D411yescharacteristic
320C22×C7#51812yes—
321C22×C7#51912yes—
322C22×C7#51911yescharacteristic
323C22×C7#52011yescharacteristic
324C22×C7#53411yescharacteristic
325C22×C7#53511yescharacteristic
326C22×C7#53611yescharacteristic
327C22×C7#53611yescharacteristic
328C22×C7#53611yescharacteristic
329C22×C7#53711yescharacteristic
330C22×C7#53812yes—
331C22×C7#54112yes—
332C22×C7#54111yescharacteristic
333C22×C7#54211yescharacteristic
334C22×C7#56212yes—
335C22×C7#56212yes—
336C22×C7#56312yes—
337C22×C7#56312yes—
338C22×C7#56312yes—
339C22×C7#56312yes—
340C22×C7#56312yes—
341C22×C7#56312yes—
342C4×C7—24no—
343C4×C7—42no—
344C4×C7—44no—
345C4×C7—42no—
346C4×C7—82no—
347Dic7—44no—
348Dic7—42no—
349Dic7—42no—
350Dic7—82no—
351Dic7—81no—
352Dic7—81no—
353C24C23×D711yescharacteristic
354C2×Q8—148no—
355C2×Q8—288no—
356C2×Q8—284no—
357C2×Q8—284no—
358C2×Q8—288no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—72no—
383C22×C4—74no—
384C22×C4—72no—
385C22×C4—141no—
386C22×C4—142no—
387C22×C4—141no—
388C22×C4—141no—
389C22×C4—141no—
390C22×C4—142no—
391C22×C4—141no—
392C22×C4—142no—
393C22×C4—142no—
394C22×C4—142no—
395C22×C4—141no—
396C22×C4—142no—
397C22×C4—142no—
398C22×C4—142no—
399C22×C4—141no—
400C22×C4—141no—
401C22×C4—141no—
402C22×C4—141no—
403C22×C4—142no—
404C22×C4—142no—
405C22×C4—141no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—141no—
409C22×C4—142no—
410C22×C4—142no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—142no—
414C22×C4—142no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—142no—
418C22×C4C2×C4×D714yes—
419C22×C4C2×C4×D712yes—
420C22×C4C2×D2812yes—
421C22×C4C2×D2812yes—
422C22×C4C2×D2812yes—
423C22×C4#115712yes—
424C4⋊C4—148no—
425C4⋊C4—288no—
426C4⋊C4—284no—
427C4⋊C4—284no—
428C42—44no—
429C2×C7#232111yescharacteristic
430C2×C7#232311yescharacteristic
431C2×C7#232711yescharacteristic
432C2×C7#237812yes—
433C2×C7#238112yes—
434C2×C7#281411yescharacteristic
435C2×C7#281711yescharacteristic
436C2×C7#282011yescharacteristic
437C2×C7#282111yescharacteristic
438C2×C7#283112yes—
439C2×C7#283312yes—
440C23C22×C4×D711yescharacteristic
441C23C22×D2811yescharacteristic
442C23C2×#115711yescharacteristic
443C23#654811yescharacteristic
444C23#654812yes—
445C23#655111yescharacteristic
446C23#655312yes—
447C23#655412yes—
448C23#655412yes—
449C23#655411yescharacteristic
450C23#655411yescharacteristic
451Q8—2816no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—142no—
481C2×C4—142no—
482C2×C4—144no—
483C2×C4—144no—
484C2×C4—144no—
485C2×C4—144no—
486C2×C4—142no—
487C2×C4—142no—
488C2×C4—281no—
489C2×C4—282no—
490C2×C4—281no—
491C2×C4—281no—
492C2×C4—282no—
493C2×C4—281no—
494C2×C4—282no—
495C2×C4—282no—
496C2×C4—282no—
497C2×C4—281no—
498C2×C4—281no—
499C2×C4—282no—
500C2×C4—282no—
501C2×C4—282no—
502C2×C4—282no—
503C2×C4—282no—
504C2×C4—282no—
505C2×C4—281no—
506C2×C4—281no—
507C2×C4—282no—
508C2×C4—282no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—281no—
512C2×C4—281no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4C4×D2814yes—
517C7#1462611yescharacteristic, complemented, Sylow
518C22C2×C4×D2811yescharacteristic
519C22#9018411yescharacteristic
520C22#9020111yescharacteristic
521C22#9020712yes—
522C22#9020912yes—
523C22#9021611yescharacteristic
524C22#9021711yescharacteristic
525C22#9021711yescharacteristic
526C22#9021711yescharacteristic
527C22#9023612yes—
528C22#9023711yescharacteristic
529C22#9025112yes—
530C22#9025211yescharacteristic
531C22#9026311yescharacteristic
532C22#9027411yescharacteristic
533C22#9031212yes—
534C22#9033712yes—
535C22#9034212yes—
536C22#9034212yes—
537C22#9037512yes—
538C22#9039012yes—
539C22#9040312yes—
540C22#9040312yes—
541C4—24no—
542C4—42no—
543C4—42no—
544C4—44no—
545C4—82no—
546C4—282no—
547C4—284no—
548C4—282no—
549C4—561no—
550C4—561no—
551C4—562no—
552C2#1173083311yescharacteristic
553C2#1173101411yescharacteristic
554C2#1173120711yescharacteristic
555C2#1173149211yescharacteristic
556C2#1173155911yescharacteristic
557C2#1173170011yescharacteristic
558C2#1173173611yescharacteristic
559C2#1173242212yes—
560C2#1173273112yes—
561C2#1173293712yes—
562C2#1173325912yes—
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1434C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1434D7
hypocenterC7#14626
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1434

Complements

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

NH ≅ G/Ntype
37. #89428C4semidirect
517. C7#14626semidirect

Maximal subgroups

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

subgroupindex
C2×#897012
C2×#897052
C2×#894282
C7×#14342
#117271782
#117271762
#117271722
#117271662
#117271652
#117271642
#146267