Finite Group Databasebeta

← #13387711

Subgroups — #13387711

Order 1792.

1218 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
1C2×#90270C211yesmaximal, characteristic, complemented
2C2×C4×C7×SD16C211yesmaximal, characteristic, complemented
3#11739862C211yesmaximal, characteristic
4C2×#89780C211yesmaximal, characteristic, complemented
5#11729808C212yesmaximal, complemented
6#11729380C211yesmaximal, characteristic, complemented
7#11728333C212yesmaximal, complemented
8#11728028C212yesmaximal, complemented
9#11727954C212yesmaximal, complemented
10#11727541C211yesmaximal, characteristic, complemented
11#11727531C211yesmaximal, characteristic, complemented
12C22×C7×SD16C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14C2×C4×C7×D4C2211yescharacteristic
15C2×#6509C2212yes—
16C2×#6501C2212yes—
17C2×#6493—21no—
18#90270—24no—
19#90270—24no—
20C2×#6459—22no—
21C2×D7×C4⋊C4C2212yes—
22C2×C4×D28—21no—
23C2×C4×D28C2211yescharacteristic
24C2×#6433—21no—
25C2×#6432C2211yescharacteristic
26C4×C7×SD16—24no—
27C2×C7×#106C2211yescharacteristic
28C2×C7×#103C2211yescharacteristic
29C2×C7×#102C2211yescharacteristic
30C2×C4×C7×C8C2211yescharacteristic
31C2×#6409C2212yes—
32C2×#6406C2212yes—
33#89984—21no—
34C2×(C2×D7)⋊C8C2212yes—
35C2×(C2×D7)⋊C8C2212yes—
36#89780—24no—
37#89759C2212yes—
38C2×#6382C2212yes—
39#89733—22no—
40#89707C2211yescharacteristic
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42#89346C2212yes—
43#89346C2212yes—
44#89324C2212yes—
45#24118—71nomaximal, Sylow
46C22×C7×Q8C2311yescharacteristic
47C22×C7×D4C2311yescharacteristic
48C23×C4×C7—21no—
49C22×C22⋊D7—22no—
50C22×D28D411yescharacteristic
51C22×C4×D7—22no—
52C22×C4×D7C2312yes—
53C22×Dic14—21no—
54C2×C7×SD16—22no—
55C2×C7×SD16—22no—
56C2×C7×SD16—22no—
57C22×C7×C8C2311yescharacteristic
58C22×C7×C8C2311yescharacteristic
59C4×C7×Q8—22no—
60C4×C7×Q8D414yescomplemented
61C4×C7×D4—22no—
62C4×C7×D4—22no—
63C2×C7×C4⋊C4—21no—
64C2×C7×C4⋊C4C2311yescharacteristic
65C2×C7×C4⋊C4C2311yescharacteristic
66C2×C7×C22⋊C4—21no—
67C2×C42×C7—21no—
68C2×C42×C7C2311yescharacteristic
69C2×C22⋊Dic7—22no—
70#6509—24no—
71#6509—24no—
72#6501—24no—
73#6501—42no—
74#6493—44no—
75C2×(C2×D7)⋊C4—22no—
76C2×(C2×D7)⋊C4—22no—
77C2×(C2×D7)⋊C4D412yes—
78C2×C4⋊Dic7—21no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×C4⋊Dic7D411yescharacteristic
82C2×C4⋊Dic7C2312yes—
83C2×C4⋊Dic7C2312yes—
84C2×Dic7⋊C4—22no—
85C2×Dic7⋊C4—22no—
86C22×C7⋊C8C2312yes—
87#6459—216no—
88D7×C4⋊C4—24no—
89D7×C4⋊C4—24no—
90C4×D28—28no—
91C4×D28—22no—
92C4×D28—22no—
93#6433—28no—
94#6432—22no—
95#6432D414yescomplemented
96C7×#106—22no—
97C7×#103—22no—
98C7×#102—22no—
99C4×C7×C8—22no—
100#6409—42no—
101#6406—24no—
102C7⋊#95—22no—
103(C2×D7)⋊C8—24no—
104(C2×D7)⋊C8—42no—
105#6382—24no—
106C4⋊(C7⋊C8)—22no—
107C2×#554—71no—
108C2×#476—71no—
109C2×C4×SD16D711yescharacteristic, complemented
110#2537—71no—
111#2035—72no—
112#2000—72no—
113#1990—71no—
114#1954—71no—
115#1874—72no—
116#1853—72no—
117#1803—71no—
118C24×C7—21no—
119C23×D7C2×D412yes—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8C2×D412yes—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4#4112yes—
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—42no—
136C22×C4×C7—42no—
137C22×C4×C7—42no—
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2411yescharacteristic
141C2×C22⋊D7—44no—
142C2×C22⋊D7—44no—
143C2×C22⋊D7—82no—
144C22×Dic7—22no—
145C22×Dic7—22no—
146C22×Dic7C2×D412yes—
147C2×D28—22no—
148C2×D28—22no—
149C2×D28SD1614yescomplemented
150C2×C4×D7—22no—
151C2×C4×D7—22no—
152C2×C4×D7—22no—
153C2×C4×D7—22no—
154C2×C4×D7—22no—
155C2×C4×D7—22no—
156C2×C4×D7—28no—
157C2×C4×D7—82no—
158C2×C4×D7—82no—
159C2×Dic14—24no—
160C2×Dic14—81no—
161C2×Dic14—81no—
162C7×SD16—28no—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8#4112yes—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4C2×D412yes—
174C7×C22⋊C4—42no—
175C42×C7—24no—
176C42×C7C2×D412yes—
177C42×C7C2×D412yes—
178C22⋊Dic7—44no—
179(C2×D7)⋊C4—28no—
180(C2×D7)⋊C4—24no—
181(C2×D7)⋊C4—44no—
182C4⋊Dic7—24no—
183C4⋊Dic7—22no—
184C4⋊Dic7—22no—
185C4⋊Dic7—22no—
186C4⋊Dic7—22no—
187C4⋊Dic7—42no—
188C4⋊Dic7—81no—
189C4⋊Dic7C2×D414yes—
190Dic7⋊C4—28no—
191Dic7⋊C4—28no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C2×C7⋊C8—22no—
195C22×SD16C2×D711yescharacteristic
196#554—144no—
197#554—144no—
198C2×C4⋊Q8—71no—
199C2×#125—141no—
200C2×C22⋊Q8—72no—
201C2×C22⋊Q8—142no—
202C2×#121—72no—
203C2×#121—141no—
204C2×C4×Q8C2×D711yescharacteristic
205C2×C4×D4—71no—
206C2×C4×D4—141no—
207C2×C4×D4C2×D711yescharacteristic
208C22×C4⋊C4—72no—
209#476—144no—
210C4×SD16—24no—
211C2×#106—72no—
212C2×#106C2×D711yescharacteristic
213C2×C4⋊C8—71no—
214C2×#103—72no—
215C2×#103C2×D711yescharacteristic
216C2×#102—72no—
217C2×#102C2×D711yescharacteristic
218C2×C22⋊C8—72no—
219C2×C22⋊C8—72no—
220C2×C4×C8C2×D711yescharacteristic
221#394—142no—
222#384—71no—
223#380—72no—
224#380—141no—
225#340—72no—
226#340—72no—
227#340—72no—
228C23×C7—22no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7C22×D411yescharacteristic
236C22×D7—22no—
237C22×D7—22no—
238C22×D7—22no—
239C22×D7—22no—
240C22×D7—22no—
241C22×D7—22no—
242C22×D7C2×SD1614yes—
243C7×Q8—22no—
244C7×Q8—22no—
245C7×Q8—41no—
246C7×Q8—41no—
247C7×D4—22no—
248C7×D4—22no—
249C7×D4—41no—
250C7×D4—41no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—41no—
262C2×C4×C7—84no—
263C2×C4×C7#12012yes—
264C2×C4×C7#12012yes—
265C2×C4×C7C2×SD1612yes—
266C2×C4×C7#13712yes—
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7H(5,𝔽2)11yescharacteristic
272C2×C4×C72−1+411yescharacteristic
273C22⋊D7—88no—
274C2×Dic7—22no—
275C2×Dic7—24no—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283D28—28no—
284C4×D7—42no—
285C4×D7—42no—
286C4×D7—42no—
287C4×D7—42no—
288C4×D7—88no—
289Dic14—84no—
290C7×C8—22no—
291C7×C8—41no—
292C7⋊C8—42no—
293C22×Q8—141no—
294C22×Q8C22×D711yescharacteristic
295C22×D4—71no—
296C22×D4—142no—
297C22×D4C22×D711yescharacteristic
298C23×C4—21no—
299C23×C4—72no—
300C23×C4—142no—
301C2×SD16—22no—
302C2×SD16—22no—
303C2×SD16—22no—
304C22×C8—72no—
305C22×C8C22×D711yescharacteristic
306C22×C8C22×D711yescharacteristic
307C4⋊Q8—74no—
308C4⋊Q8—142no—
309#125—148no—
310C22⋊Q8—1416no—
311C22⋊Q8—144no—
312C22⋊Q8—144no—
313#121—144no—
314#121—282no—
315#121—284no—
316C4×Q8—22no—
317C4×Q8C22⋊D714yescomplemented
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—148no—
321C4×D4—142no—
322C4×D4—142no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—72no—
325C2×C4⋊C4—71no—
326C2×C4⋊C4—72no—
327C2×C4⋊C4—144no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—142no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4—144no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4C22×D711yescharacteristic
335C2×C4⋊C4C22×D711yescharacteristic
336C2×C22⋊C4—21no—
337C2×C22⋊C4—72no—
338C2×C22⋊C4—142no—
339C2×C22⋊C4—142no—
340C2×C22⋊C4—142no—
341C2×C42—21no—
342C2×C42C22×D711yescharacteristic
343#106—22no—
344#106—144no—
345C4⋊C8—142no—
346#103—22no—
347#103—282no—
348#102—22no—
349#102—144no—
350C22⋊C8—144no—
351C22⋊C8—282no—
352C4×C8—22no—
353#95—142no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—42no—
357C22×C7—81no—
358C22×C7—81no—
359C22×C7—81no—
360C22×C7—81no—
361C22×C7C2×#12011yescharacteristic
362C22×C7#54611yescharacteristic
363C22×C7#54711yescharacteristic
364C22×C7C22×SD1611yescharacteristic
365C22×C7C2×#13711yescharacteristic
366C22×C7#57611yescharacteristic
367C22×C7#57711yescharacteristic
368C2×D7—24no—
369C2×D7—24no—
370C2×D7—24no—
371C2×D7—42no—
372C2×D7—42no—
373C2×D7—42no—
374C2×D7—42no—
375C4×C7—22no—
376C4×C7—42no—
377C4×C7—42no—
378C4×C7—81no—
379C4×C7#44814yes—
380C4×C7#52511yescharacteristic
381C4×C7#53711yescharacteristic
382C4×C7#54811yescharacteristic
383C4×C7#55011yescharacteristic
384Dic7—42no—
385Dic7—44no—
386Dic7—82no—
387C24—21no—
388C24—72no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—144no—
395C2×Q8—561no—
396C2×Q8—561no—
397C2×Q8C2×C22⋊D712yes—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—74no—
404C2×D4—142no—
405C2×D4—142no—
406C2×D4—284no—
407C2×D4—284no—
408C2×D4—562no—
409C2×D4#115712yes—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—72no—
417C22×C4—148no—
418C22×C4—142no—
419C22×C4—142no—
420C22×C4—142no—
421C22×C4—142no—
422C22×C4—142no—
423C22×C4—142no—
424C22×C4—142no—
425C22×C4—142no—
426C22×C4—562no—
427C22×C4—562no—
428C22×C4D4×D711yescharacteristic
429C22×C4D4×D711yescharacteristic
430C22×C4C23×D711yescharacteristic
431SD16—28no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—142no—
438C2×C8—142no—
439C2×C8—142no—
440C2×C8#115712yes—
441C4⋊C4—24no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—74no—
446C4⋊C4—148no—
447C4⋊C4—148no—
448C4⋊C4—142no—
449C4⋊C4—144no—
450C4⋊C4—142no—
451C4⋊C4—142no—
452C4⋊C4—142no—
453C4⋊C4—282no—
454C4⋊C4—561no—
455C4⋊C4C2×C22⋊D712yes—
456C22⋊C4—42no—
457C22⋊C4—148no—
458C22⋊C4—144no—
459C22⋊C4—284no—
460C22⋊C4—284no—
461C42—24no—
462C42D4×D712yes—
463C42C2×C22⋊D712yes—
464C2×C7—82no—
465C2×C7#240711yescharacteristic
466C2×C7C2×#44811yescharacteristic
467C2×C7#301211yescharacteristic
468C2×C7#329511yescharacteristic
469C2×C7#333111yescharacteristic
470C2×C7#333511yescharacteristic
471C2×C7#335711yescharacteristic
472D7—44no—
473C23—22no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—74no—
481C23—142no—
482C23—142no—
483C23—142no—
484C23—142no—
485C23—142no—
486C23—142no—
487C23C2×D4×D711yescharacteristic
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492Q8—564no—
493D4—22no—
494D4—22no—
495D4—41no—
496D4—41no—
497D4—148no—
498D4—568no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—84no—
511C2×C4—144no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—282no—
516C2×C4—282no—
517C2×C4—282no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4—282no—
521C2×C4—282no—
522C2×C4—282no—
523C2×C4—282no—
524C2×C4—568no—
525C2×C4D7×SD1612yes—
526C2×C4#647812yes—
527C2×C4#650012yes—
528C2×C4#650012yes—
529C2×C4C2×#115711yescharacteristic
530C2×C4C2×#115711yescharacteristic
531C2×C4C2×D4×D711yescharacteristic
532C2×C4#655311yescharacteristic
533C2×C4#655411yescharacteristic
534C2×C4C22×C22⋊D711yescharacteristic
535C8—22no—
536C8—41no—
537C8—282no—
538C7#2411811yescharacteristic, complemented, Sylow
539C22—42no—
540C22—42no—
541C22—42no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22—81no—
546C22—144no—
547C22—144no—
548C22—144no—
549C22—282no—
550C22—282no—
551C22—282no—
552C22—282no—
553C22#9024111yescharacteristic
554C22#9024211yescharacteristic
555C22C2×D7×SD1611yescharacteristic
556C22C2×#647811yescharacteristic
557C22#9046011yescharacteristic
558C22#9046111yescharacteristic
559C22C2×#650011yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—81no—
564C4—284no—
565C4—282no—
566C4—562no—
567C4#8994214yes—
568C4#9024311yescharacteristic
569C4#9024411yescharacteristic
570C4#9048211yescharacteristic
571C4#9052411yescharacteristic
572C2—82no—
573C2—284no—
574C2#1173712111yescharacteristic
575C2#1173713111yescharacteristic
576C2#1173713311yescharacteristic
577C2#1173713811yescharacteristic
578C2#1173984711yescharacteristic
579C2C2×#8994211yescharacteristic
580C2#1174135711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C4×C7×SD16C2
socleC23×C7C22×D4
prime radical#11739862C2
hypercenterC2×C4×SD16D7
hypocenterC7#24118
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#90270C2semidirect
2. C2×C4×C7×SD16C2semidirect
4. C2×#89780C2semidirect
5. #11729808C2semidirect
6. #11729380C2semidirect
7. #11728333C2semidirect
8. #11728028C2semidirect
9. #11727954C2semidirect
10. #11727541C2semidirect
11. #11727531C2semidirect
60. C4×C7×Q8D4semidirect
95. #6432D4semidirect
109. C2×C4×SD16D7semidirect
149. C2×D28SD16semidirect
317. C4×Q8C22⋊D7semidirect
538. C7#24118semidirect

Maximal subgroups

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

subgroupindex
C2×#902702
C2×C4×C7×SD162
#117398622
C2×#897802
#117298082
#117293802
#117283332
#117280282
#117279542
#117275412
#117275312
#241187