Finite Group Databasebeta

← #13276203

Subgroups — #13276203

Order 1792.

1710 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6450C211yesmaximal, characteristic
2C2×C4×Dic7⋊C4C211yesmaximal, characteristic
3C2×#89422C211yesmaximal, characteristic
4C7×#1436C211yesmaximal, characteristic
5#11729318C211yesmaximal, characteristic
6#11729290C211yesmaximal, characteristic
7#11727178C212yesmaximal
8#11727175C212yesmaximal
9#11727159C211yesmaximal, characteristic
10Dic7⋊#95C212yesmaximal
11Dic7⋊#95C212yesmaximal
12C23×Dic14C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C42×C7C2211yescharacteristic
15C22×Dic7⋊C4C2212yes—
16C22×Dic7⋊C4C2211yescharacteristic
17C22×Dic7⋊C4C2211yescharacteristic
18C22×Dic7⋊C4C2212yes—
19C22×C4×Dic7C2211yescharacteristic
20C22×C4×Dic7C2211yescharacteristic
21C22×C4×Dic7C2211yescharacteristic
22C22×C4×Dic7C2211yescharacteristic
23C2×#6450—24no—
24C2×#6450—24no—
25C2×#6450C418yescomplemented
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⋊#95C2211yescharacteristic
35C2×C7⋊#95C2212yes—
36C2×C7⋊#95C2212yes—
37C2×C7⋊#95C2212yes—
38C4×Dic7⋊C4—24no—
39#89422—24no—
40#13925—71nomaximal, Sylow
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C23×C4×C7C2312yes—
44C23×C4×C7C2311yescharacteristic
45C23×C4×C7C2311yescharacteristic
46C23×Dic7C2312yes—
47C23×Dic7C2311yescharacteristic
48C23×Dic7C2312yes—
49C23×Dic7C2311yescharacteristic
50C23×Dic7C2311yescharacteristic
51C23×Dic7C2311yescharacteristic
52C22×Dic14—24no—
53C22×Dic14—22no—
54C22×Dic14—22no—
55C22×Dic14—24no—
56C22×Dic14C2×C414yes—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4—22no—
59C2×C7×C4⋊C4C2×C414yes—
60C2×C42×C7—22no—
61C2×C42×C7—42no—
62C2×Dic7⋊C4—22no—
63C2×Dic7⋊C4—22no—
64C2×Dic7⋊C4—22no—
65C2×Dic7⋊C4—22no—
66C2×Dic7⋊C4—24no—
67C2×Dic7⋊C4—24no—
68C2×Dic7⋊C4—24no—
69C2×Dic7⋊C4—24no—
70C2×Dic7⋊C4—24no—
71C2×Dic7⋊C4C2×C414yes—
72C2×Dic7⋊C4C2×C414yes—
73C2×Dic7⋊C4Q818yescomplemented
74C2×C4×Dic7—22no—
75C2×C4×Dic7—22no—
76C2×C4×Dic7—22no—
77C2×C4×Dic7—22no—
78C2×C4×Dic7—22no—
79C2×C4×Dic7—42no—
80C2×C4×Dic7C2×C414yes—
81C2×C4×Dic7C2×C414yes—
82C2×C4×Dic7C2×C414yes—
83C2×C4×Dic7D414yes—
84C2×C4×Dic7D414yes—
85#6450—232no—
86C7×#95—22no—
87C7×#95—24no—
88C7×#95—42no—
89C7⋊#95—22no—
90C7⋊#95—24no—
91C7⋊#95—24no—
92C7⋊#95—24no—
93C7⋊#95—24no—
94C7⋊#95—24no—
95C7⋊#95—42no—
96C7⋊#95—42no—
97C7⋊#95—42no—
98C22×C4×Q8—71no—
99C2×#382—71no—
100C2×C4×C4⋊C4—71no—
101#1446—72no—
102#1445—72no—
103#1436—71no—
104#1436—71no—
105#1436—71no—
106#1436D711yescharacteristic
107C4⋊#95—72no—
108C4⋊#95—72no—
109C24×C7C2411yescharacteristic
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—21no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—22no—
128C22×C4×C7—22no—
129C22×C4×C7—22no—
130C22×C4×C7—22no—
131C22×C4×C7—22no—
132C22×C4×C7C22×C412yes—
133C22×C4×C7C22×C412yes—
134C22×C4×C7C22×C412yes—
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×Q814yes—
138C22×C4×C7#4114yes—
139C22×Dic7—21no—
140C22×Dic7—21no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—22no—
148C22×Dic7—22no—
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—22no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—22no—
161C22×Dic7C22×C412yes—
162C22×Dic7C22×C412yes—
163C22×Dic7C22×C412yes—
164C22×Dic7C22×C412yes—
165C22×Dic7C2×D412yes—
166C22×Dic7C2×D412yes—
167C22×Dic7C2×D412yes—
168C22×Dic7C2×D412yes—
169C22×Dic7C2×Q814yes—
170C22×Dic7C2×Q814yes—
171C22×Dic7#4114yes—
172C22×Dic7#4114yes—
173C2×Dic14—216no—
174C2×Dic14—216no—
175C2×Dic14—28no—
176C2×Dic14—28no—
177C2×Dic14—28no—
178C7×C4⋊C4—28no—
179C42×C7—44no—
180Dic7⋊C4—28no—
181Dic7⋊C4—28no—
182Dic7⋊C4—216no—
183Dic7⋊C4—48no—
184C4×Dic7—28no—
185C4×Dic7—28no—
186C4×Dic7—44no—
187C4×Dic7C22⋊C4116yes—
188C23×Q8—71no—
189C2×C4×Q8—78no—
190C2×C4×Q8—144no—
191C2×C4×Q8—144no—
192C22×C4⋊C4—71no—
193C22×C4⋊C4—71no—
194C22×C4⋊C4—72no—
195C22×C4⋊C4—72no—
196C22×C4⋊C4C2×D711yescharacteristic
197C22×C42—71no—
198C22×C42—71no—
199C22×C42—71no—
200C22×C42—71no—
201C22×C42C2×D711yescharacteristic
202#382—144no—
203C4×C4⋊C4—144no—
204C2×#95—72no—
205C2×#95—72no—
206C2×#95—72no—
207C2×#95—71no—
208C2×#95—72no—
209C2×#95—72no—
210C2×#95—72no—
211C2×#95—72no—
212C2×#95—72no—
213C2×#95C2×D712yes—
214C2×#95C2×D711yescharacteristic
215C2×#95C2×D712yes—
216C23×C7C23×C411yescharacteristic
217C23×C7C22×D411yescharacteristic
218C23×C7C22×D411yescharacteristic
219C23×C7C22×Q812yes—
220C23×C7C2×#4112yes—
221C23×C7C2×#4112yes—
222C23×C7C2×#4112yes—
223C23×C7H(5,𝔽2)12yes—
224C23×C72−1+412yes—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—24no—
228C2×C4×C7—24no—
229C2×C4×C7—24no—
230C2×C4×C7—22no—
231C2×C4×C7—22no—
232C2×C4×C7—24no—
233C2×C4×C7—22no—
234C2×C4×C7—24no—
235C2×C4×C7—24no—
236C2×C4×C7—24no—
237C2×C4×C7—24no—
238C2×C4×C7—42no—
239C2×C4×C7—42no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—42no—
247C2×C4×C7—42no—
248C2×C4×C7—42no—
249C2×C4×C7—42no—
250C2×C4×C7—42no—
251C2×C4×C7—42no—
252C2×C4×C7C2×C22⋊C414yes—
253C2×C4×C7#11714yes—
254C2×Dic7—22no—
255C2×Dic7—22no—
256C2×Dic7—22no—
257C2×Dic7—22no—
258C2×Dic7—24no—
259C2×Dic7—24no—
260C2×Dic7—24no—
261C2×Dic7—24no—
262C2×Dic7—22no—
263C2×Dic7—24no—
264C2×Dic7—24no—
265C2×Dic7—24no—
266C2×Dic7—24no—
267C2×Dic7—42no—
268C2×Dic7—42no—
269C2×Dic7—41no—
270C2×Dic7—41no—
271C2×Dic7—42no—
272C2×Dic7—42no—
273C2×Dic7—42no—
274C2×Dic7—42no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—42no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7C2×C22⋊C414yes—
282C2×Dic7C2×C22⋊C414yes—
283C2×Dic7#11714yes—
284C2×Dic7C4×Q818yes—
285C2×Dic7C22⋊Q818yes—
286C2×Dic7#12414yes—
287C2×Dic7C4⋊Q814yes—
288Dic14—232no—
289C22×Q8—74no—
290C22×Q8—142no—
291C22×Q8—142no—
292C22×Q8—144no—
293C22×Q8—144no—
294C23×C4—71no—
295C23×C4—71no—
296C23×C4—71no—
297C23×C4—71no—
298C23×C4—72no—
299C23×C4—72no—
300C23×C4C22×D712yes—
301C23×C4C22×D711yescharacteristic
302C23×C4C22×D712yes—
303C23×C4C22×D711yescharacteristic
304C23×C4C22×D711yescharacteristic
305C4×Q8—1432no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—74no—
309C2×C4⋊C4—74no—
310C2×C4⋊C4—78no—
311C2×C4⋊C4—142no—
312C2×C4⋊C4—142no—
313C2×C4⋊C4—144no—
314C2×C4⋊C4—144no—
315C2×C4⋊C4—142no—
316C2×C4⋊C4—142no—
317C2×C4⋊C4—144no—
318C2×C4⋊C4—144no—
319C2×C4⋊C4—144no—
320C2×C4⋊C4C4×D714yes—
321C2×C42—22no—
322C2×C42—42no—
323C2×C42—74no—
324C2×C42—74no—
325C2×C42—74no—
326C2×C42—74no—
327C2×C42—74no—
328C2×C42—142no—
329C2×C42—142no—
330C2×C42—142no—
331C2×C42—142no—
332C2×C42—142no—
333C2×C42—282no—
334#95—24no—
335#95—22no—
336#95—42no—
337#95—144no—
338#95—144no—
339#95—144no—
340#95—144no—
341#95—144no—
342#95—142no—
343#95—282no—
344#95—282no—
345#95—282no—
346C22×C7C22×C22⋊C411yescharacteristic
347C22×C7C2×#11711yescharacteristic
348C22×C7C2×#11711yescharacteristic
349C22×C7C2×C4×Q812yes—
350C22×C7#52012yes—
351C22×C7C2×C22⋊Q812yes—
352C22×C7#52512yes—
353C22×C7#52512yes—
354C22×C7C2×#12411yescharacteristic
355C22×C7C2×C4⋊Q811yescharacteristic
356C22×C7#53511yescharacteristic
357C22×C7#53611yescharacteristic
358C22×C7#53712yes—
359C22×C7#55412yes—
360C22×C7#55412yes—
361C22×C7#55612yes—
362C22×C7#55612yes—
363C22×C7#55712yes—
364C22×C7#55712yes—
365C22×C7#55912yes—
366C22×C7#55912yes—
367C4×C7—24no—
368C4×C7—24no—
369C4×C7—42no—
370C4×C7—44no—
371C4×C7—82no—
372Dic7—24no—
373Dic7—24no—
374Dic7—42no—
375Dic7—44no—
376Dic7—82no—
377Dic7#39118yes—
378C24C23×D711yescharacteristic
379C2×Q8—148no—
380C2×Q8—148no—
381C2×Q8—1416no—
382C2×Q8—1416no—
383C2×Q8—148no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—72no—
407C22×C4—72no—
408C22×C4—72no—
409C22×C4—74no—
410C22×C4—74no—
411C22×C4—72no—
412C22×C4—72no—
413C22×C4—72no—
414C22×C4—72no—
415C22×C4—74no—
416C22×C4—72no—
417C22×C4—74no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—142no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4C2×C4×D712yes—
441C22×C4C2×C4×D712yes—
442C22×C4C2×C4×D712yes—
443C22×C4#115714yes—
444C22×C4D4×D712yes—
445C22×C4D4×D712yes—
446C22×C4Q8×D714yes—
447C4⋊C4—28no—
448C4⋊C4—1416no—
449C4⋊C4—148no—
450C4⋊C4—148no—
451C4⋊C4—288no—
452C42—44no—
453C42—716no—
454C42—148no—
455C42—148no—
456C42—284no—
457C2×C7#228011yescharacteristic
458C2×C7C2×#39111yescharacteristic
459C2×C7#231111yescharacteristic
460C2×C7#235512yes—
461C2×C7#235512yes—
462C2×C7#258012yes—
463C2×C7#258112yes—
464C2×C7#258412yes—
465C2×C7#259212yes—
466C23C22×C4×D711yescharacteristic
467C23C2×#115712yes—
468C23C2×D4×D711yescharacteristic
469C23C2×D4×D711yescharacteristic
470C23#654812yes—
471C23C2×Q8×D712yes—
472C23D7×#4112yes—
473C23D7×#4112yes—
474C23#655412yes—
475Q8—1432no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—74no—
504C2×C4—74no—
505C2×C4—78no—
506C2×C4—78no—
507C2×C4—74no—
508C2×C4—74no—
509C2×C4—74no—
510C2×C4—142no—
511C2×C4—142no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—144no—
515C2×C4—144no—
516C2×C4—144no—
517C2×C4—144no—
518C2×C4—142no—
519C2×C4—144no—
520C2×C4—144no—
521C2×C4—144no—
522C2×C4—144no—
523C2×C4—282no—
524C2×C4—282no—
525C2×C4—282no—
526C2×C4—281no—
527C2×C4—282no—
528C2×C4—281no—
529C2×C4—282no—
530C2×C4—282no—
531C2×C4—282no—
532C2×C4—282no—
533C2×C4—282no—
534C2×C4—282no—
535C2×C4—282no—
536C2×C4—282no—
537C2×C4#643514yes—
538C2×C4D7×C22⋊C414yes—
539C7#1392511yescharacteristic, complemented, Sylow
540C22C2×#643511yescharacteristic
541C22C2×D7×C22⋊C411yescharacteristic
542C22#9020212yes—
543C22D7×#11711yescharacteristic
544C22#9022412yes—
545C22#9023712yes—
546C22#9023812yes—
547C22#9023812yes—
548C22#9025012yes—
549C22#9025812yes—
550C22C4×Q8×D712yes—
551C22#9028311yescharacteristic
552C22#9028812yes—
553C22D7×C22⋊Q812yes—
554C22#9032712yes—
555C22#9033911yescharacteristic
556C22#9035512yes—
557C22D7×#12411yescharacteristic
558C22#9037612yes—
559C22#9039312yes—
560C22D7×C4⋊Q811yescharacteristic
561C4—24no—
562C4—24no—
563C4—42no—
564C4—44no—
565C4—78no—
566C4—82no—
567C4—144no—
568C4—144no—
569C4—282no—
570C4—284no—
571C4—562no—
572C2#1173100611yescharacteristic
573C2#1173108612yes—
574C2#1173193212yes—
575C2#1173210711yescharacteristic
576C2#1173240812yes—
577C2#1173253512yes—
578C2#1173306412yes—
579C2#1173352512yes—
580C2D7×#39111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC23×C7C23×C4
FittingC7×#1436C2
socle, prime radical, cosocleC24×C7C24
hypercenter#1436D7
hypocenterC7#13925
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ #1436

Complements

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

NH ≅ G/Ntype
25. C2×#6450C4semidirect
73. C2×Dic7⋊C4Q8semidirect
539. C7#13925semidirect

Maximal subgroups

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

subgroupindex
C22×#64502
C2×C4×Dic7⋊C42
C2×#894222
C7×#14362
#117293182
#117292902
#117271782
#117271752
#117271592
Dic7⋊#952
#139257