Finite Group Databasebeta

← #13380614

Subgroups — #13380614

Order 1792.

2142 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×C7×#535C211yesmaximal, characteristic
2C2×#90516C211yesmaximal, characteristic, complemented
3C23×#1142C211yesmaximal, characteristic, complemented
4C2×#90006C211yesmaximal, characteristic, complemented
5C2×#89987C211yesmaximal, characteristic, complemented
6C2×#89776C211yesmaximal, characteristic, complemented
7#11729439C218yesmaximal, complemented
8#11729334C211yesmaximal, characteristic, complemented
9C22×C7×#41C2211yescharacteristic
10C23×C7×D4C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C7×#535—24no—
13C7×#535—24no—
14C2×C7×#127—21no—
15C2×C7×#124—21no—
16C2×C7×#121C2214yes—
17C2×C7×#120—22no—
18C2×C7×#117C2211yescharacteristic
19#90516—24no—
20#90516—24no—
21C2×#6507—21no—
22C2×#6499—21no—
23C2×#6498—22no—
24C22×#1142—24no—
25C22×#1142—24no—
26C22×#1142—24no—
27C22×#1142C2214yes—
28C2×#6489C2211yescharacteristic
29C2×#6487C2214yescomplemented
30C22×#1136C2211yescharacteristic
31C23×C7⋊C8C2211yescharacteristic
32#90006—24no—
33#89987—24no—
34C2×#6406C2214yes—
35C2×C22⋊(C7⋊C8)C2214yescomplemented
36#89776—24no—
37C2×#6384C2214yescomplemented
38#89324C2214yescomplemented
39#23388—71nomaximal, Sylow
40C25×C7—21no—
41C2×C7×#41—22no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41—22no—
47C22×C7×Q8D411yescharacteristic
48C22×C7×D4—24no—
49C22×C7×D4—22no—
50C22×C7×D4—22no—
51C22×C7×D4—22no—
52C22×C7×D4—21no—
53C22×C7×D4—22no—
54C22×C7×D4—22no—
55C22×C7×D4D412yes—
56C22×C7×D4D411yescharacteristic
57C22×C7×D4D414yes—
58C22×C7×D4C2312yes—
59C23×C4×C7C2312yes—
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C22×Dic14—22no—
63C22×Dic14—22no—
64C22×Dic14—22no—
65C22×Dic14—22no—
66C22×Dic14—21no—
67C22×Dic14—22no—
68C22×Dic14D414yescomplemented
69C22×Dic14C2312yes—
70C7×#127—28no—
71C7×#124—28no—
72C7×#121—28no—
73C7×#121—28no—
74C7×#120—216no—
75C7×#117—22no—
76C7×#117D414yes—
77C2×C7×C4⋊C4C2312yes—
78C2×C7×C22⋊C4—21no—
79C2×C7×C22⋊C4—24no—
80C2×C42×C7—21no—
81C2×C22⋊Dic7—24no—
82C2×C22⋊Dic7—21no—
83#6507—28no—
84#6499—28no—
85#6498—216no—
86C2×#1142—232no—
87C2×#1142—28no—
88C2×#1142—28no—
89C2×#1142—28no—
90#6489—22no—
91#6489D414yescomplemented
92C2×C4⋊Dic7C2312yes—
93#6487—28no—
94#6487—28no—
95C2×Dic7⋊C4—24no—
96C2×C4×Dic7—21no—
97C2×#1136—22no—
98C2×#1136—22no—
99C2×#1136—22no—
100C22×C7⋊C8—22no—
101C22×C7⋊C8—22no—
102C22×C7⋊C8D414yescomplemented
103C22×C7⋊C8C2312yes—
104C22×C7⋊C8C2312yes—
105#6406—28no—
106C22⋊(C7⋊C8)—28no—
107#6384—28no—
108C23×SD16—71no—
109C2×#536—71no—
110C2×#535D711yescharacteristic
111C2×#419—71no—
112C2×#418—71no—
113C2×#408—71no—
114#2000—78no—
115#1735—71no—
116C24×C7—22no—
117C24×C7—24no—
118C24×C7—42no—
119C24×C7—42no—
120C24×C7—84no—
121C24×C7C2×D412yes—
122C24×C7C2×D411yescharacteristic
123C7×#41—44no—
124C7×#41—44no—
125C7×#41—44no—
126C7×#41—44no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×D4—24no—
131C2×C7×D4—24no—
132C2×C7×D4—24no—
133C2×C7×D4—216no—
134C2×C7×D4—28no—
135C2×C7×D4—22no—
136C2×C7×D4—24no—
137C2×C7×D4—22no—
138C2×C7×D4—28no—
139C2×C7×D4—22no—
140C2×C7×D4—22no—
141C2×C7×D4—22no—
142C2×C7×D4—22no—
143C2×C7×D4—22no—
144C2×C7×D4—22no—
145C2×C7×D4—42no—
146C2×C7×D4—44no—
147C2×C7×D4—42no—
148C2×C7×D4—44no—
149C2×C7×D4—88no—
150C2×C7×D4C2×D414yes—
151C22×C4×C7—22no—
152C22×C4×C7—22no—
153C22×C4×C7—22no—
154C22×C4×C7—22no—
155C22×C4×C7—22no—
156C22×C4×C7—22no—
157C22×C4×C7—22no—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D411yescharacteristic
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D412yes—
163C22×C4×C7C2×D412yes—
164C22×C4×C7C2×D412yes—
165C22×C4×C7#4112yes—
166C22×C4×C7C2411yescharacteristic
167C22×Dic7—24no—
168C22×Dic7—22no—
169C22×Dic7—22no—
170C22×Dic7—42no—
171C22×Dic7—42no—
172C2×Dic14—28no—
173C2×Dic14—28no—
174C2×Dic14—22no—
175C2×Dic14—22no—
176C2×Dic14—22no—
177C2×Dic14—22no—
178C2×Dic14—24no—
179C2×Dic14—22no—
180C2×Dic14—42no—
181C2×Dic14—44no—
182C2×Dic14—42no—
183C2×Dic14—44no—
184C2×Dic14C2×D414yes—
185C7×C4⋊C4—22no—
186C7×C4⋊C4C2×D414yes—
187C7×C22⋊C4—216no—
188C7×C22⋊C4—24no—
189C42×C7—24no—
190C22⋊Dic7—216no—
191C22⋊Dic7—24no—
192#1142—232no—
193C4⋊Dic7—22no—
194C4⋊Dic7C2×D414yes—
195Dic7⋊C4—216no—
196C4×Dic7—24no—
197#1136—44no—
198C2×C7⋊C8—28no—
199C2×C7⋊C8—22no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8—22no—
202C2×C7⋊C8—22no—
203C2×C7⋊C8—22no—
204C2×C7⋊C8C2×D414yes—
205C22×#41C2×D711yescharacteristic
206C23×Q8—71no—
207C23×D4C2×D711yescharacteristic
208C22×SD16—74no—
209C22×SD16—144no—
210C22×SD16—144no—
211C22×SD16—144no—
212C22×M16—71no—
213C23×C8—71no—
214#536—144no—
215#536—144no—
216#535—24no—
217#535—24no—
218C2×C4⋊Q8—141no—
219C2×#127—21no—
220C2×#124—21no—
221C2×#124—141no—
222C2×#123—142no—
223C2×C22⋊Q8—74no—
224C2×#121C2×D714yes—
225C2×#120—22no—
226C2×#117—71no—
227C2×#117C2×D711yescharacteristic
228#419—144no—
229#418—144no—
230C2×#103—74no—
231C2×#102—74no—
232#408—144no—
233C2×C22⋊C8—74no—
234#340—74no—
235C23×C7—24no—
236C23×C7—22no—
237C23×C7—24no—
238C23×C7—22no—
239C23×C7—22no—
240C23×C7—28no—
241C23×C7—42no—
242C23×C7—42no—
243C23×C7—48no—
244C23×C7—48no—
245C23×C7—88no—
246C23×C7—88no—
247C23×C7—88no—
248C23×C7#12012yes—
249C23×C7#12012yes—
250C23×C7#12112yes—
251C23×C7C22×D411yescharacteristic
252C7×Q8—44no—
253C7×D4—24no—
254C7×D4—24no—
255C7×D4—48no—
256C7×D4—42no—
257C7×D4—42no—
258C7×D4—48no—
259C7×D4—44no—
260C7×D4—816no—
261C2×C4×C7—22no—
262C2×C4×C7—24no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—44no—
266C2×C4×C7—42no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—42no—
270C2×C4×C7#12014yes—
271C2×C4×C7#12012yes—
272C2×C4×C7#12012yes—
273C2×C4×C7#12114yes—
274C2×C4×C7#12114yes—
275C2×C4×C7#12112yes—
276C2×C4×C7#12714yes—
277C2×C4×C7C22×D411yescharacteristic
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×D411yescharacteristic
281C2×C4×C7C22×D411yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—44no—
288C2×Dic7—48no—
289Dic14—24no—
290Dic14—24no—
291Dic14—42no—
292Dic14—42no—
293Dic14—48no—
294C7⋊C8—24no—
295C7⋊C8—42no—
296C25—21no—
297C2×#41—22no—
298C2×#41—22no—
299C2×#41—22no—
300C2×#41—22no—
301C2×#41—22no—
302C2×#41—22no—
303C22×Q8—74no—
304C22×Q8—72no—
305C22×Q8—142no—
306C22×Q8—141no—
307C22×Q8—142no—
308C22×Q8—142no—
309C22×Q8—142no—
310C22×Q8—142no—
311C22×Q8C22⋊D711yescharacteristic
312C22×D4—24no—
313C22×D4—22no—
314C22×D4—22no—
315C22×D4—22no—
316C22×D4—22no—
317C22×D4—21no—
318C22×D4—22no—
319C22×D4C22⋊D712yes—
320C22×D4C22⋊D711yescharacteristic
321C22×D4C22⋊D714yes—
322C22×D4C22×D712yes—
323C23×C4—141no—
324C23×C4C22×D712yes—
325C23×C4C22×D711yescharacteristic
326C2×SD16—148no—
327C2×SD16—148no—
328C2×SD16—1432no—
329C2×SD16—148no—
330C2×M16—142no—
331C2×M16—142no—
332C2×M16—142no—
333C22×C8—74no—
334C22×C8—72no—
335C22×C8—72no—
336C22×C8—142no—
337C22×C8—142no—
338C4⋊Q8—148no—
339#127—28no—
340#124—28no—
341#124—148no—
342#123—1416no—
343C22⋊Q8—148no—
344C22⋊Q8—148no—
345#121—28no—
346#121—28no—
347#120—216no—
348#117—22no—
349#117—74no—
350#117—142no—
351#117C22⋊D714yes—
352C2×C4⋊C4—72no—
353C2×C4⋊C4—144no—
354C2×C4⋊C4C22×D712yes—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—141no—
358C2×C22⋊C4—144no—
359C2×C42—21no—
360C2×C42—141no—
361#103—148no—
362#102—148no—
363C22⋊C8—148no—
364C22×C7—24no—
365C22×C7—24no—
366C22×C7—24no—
367C22×C7—42no—
368C22×C7—44no—
369C22×C7—44no—
370C22×C7—48no—
371C22×C7—44no—
372C22×C7—44no—
373C22×C7—816no—
374C22×C7—84no—
375C22×C7—84no—
376C22×C7#39214yes—
377C22×C7C2×#12011yescharacteristic
378C22×C7C2×#12011yescharacteristic
379C22×C7C2×#12111yescharacteristic
380C22×C7#57714yes—
381C4×C7—42no—
382C4×C7—44no—
383C4×C7#39214yes—
384C4×C7C2×#12011yescharacteristic
385C4×C7C2×#12111yescharacteristic
386C4×C7C2×#12111yescharacteristic
387C4×C7C2×#12711yescharacteristic
388Dic7—44no—
389Dic7—44no—
390C24—22no—
391C24—24no—
392C24—42no—
393C24—42no—
394C24—84no—
395C24C2×C22⋊D712yes—
396C24C2×C22⋊D711yescharacteristic
397#41—44no—
398#41—44no—
399#41—44no—
400#41—44no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—74no—
405C2×Q8—144no—
406C2×Q8—142no—
407C2×Q8—148no—
408C2×Q8—148no—
409C2×Q8—142no—
410C2×Q8—142no—
411C2×Q8—142no—
412C2×Q8—142no—
413C2×Q8—282no—
414C2×Q8—284no—
415C2×Q8—284no—
416C2×Q8—282no—
417C2×D4—24no—
418C2×D4—24no—
419C2×D4—24no—
420C2×D4—216no—
421C2×D4—28no—
422C2×D4—22no—
423C2×D4—28no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—42no—
433C2×D4—44no—
434C2×D4—42no—
435C2×D4—44no—
436C2×D4—88no—
437C2×D4C2×C22⋊D714yes—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—144no—
446C22×C4—142no—
447C22×C4—142no—
448C22×C4—282no—
449C22×C4—282no—
450C22×C4D4×D712yes—
451C22×C4D4×D712yes—
452C22×C4D4×D712yes—
453C22×C4#115912yes—
454C22×C4C2×C22⋊D712yes—
455C22×C4C2×C22⋊D711yescharacteristic
456C22×C4C2×C22⋊D712yes—
457C22×C4C2×C22⋊D712yes—
458C22×C4C23×D711yescharacteristic
459SD16—1432no—
460M16—284no—
461C2×C8—74no—
462C2×C8—148no—
463C2×C8—142no—
464C2×C8—142no—
465C2×C8—142no—
466C2×C8—142no—
467C2×C8—142no—
468C4⋊C4—22no—
469C4⋊C4—74no—
470C4⋊C4—1416no—
471C4⋊C4—142no—
472C4⋊C4C2×C22⋊D714yes—
473C22⋊C4—216no—
474C22⋊C4—24no—
475C22⋊C4—1416no—
476C22⋊C4—144no—
477C42—24no—
478C42—144no—
479C2×C7—24no—
480C2×C7—82no—
481C2×C7—84no—
482C2×C7C2×#39211yescharacteristic
483C2×C7#300912yes—
484C2×C7#301012yes—
485C2×C7#305712yes—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—28no—
490C23—24no—
491C23—22no—
492C23—42no—
493C23—42no—
494C23—48no—
495C23—48no—
496C23—88no—
497C23—88no—
498C23—88no—
499C23#650012yes—
500C23#650212yes—
501C23#651612yes—
502C23C22×C22⋊D711yescharacteristic
503Q8—44no—
504Q8—144no—
505Q8—144no—
506Q8—282no—
507Q8—282no—
508Q8—288no—
509D4—24no—
510D4—24no—
511D4—48no—
512D4—42no—
513D4—42no—
514D4—48no—
515D4—44no—
516D4—816no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—144no—
527C2×C4—144no—
528C2×C4—282no—
529C2×C4—288no—
530C2×C4—284no—
531C2×C4—282no—
532C2×C4#650014yes—
533C2×C4#650012yes—
534C2×C4#650114yes—
535C2×C4#650214yes—
536C2×C4#650212yes—
537C2×C4#650314yes—
538C2×C4#651612yes—
539C2×C4C2×D4×D711yescharacteristic
540C2×C4C2×D4×D711yescharacteristic
541C2×C4C2×D4×D711yescharacteristic
542C2×C4C2×#115911yescharacteristic
543C2×C4C22×C22⋊D711yescharacteristic
544C2×C4C22×C22⋊D711yescharacteristic
545C8—144no—
546C8—282no—
547C7#2338811yescharacteristic, complemented, Sylow
548C22—24no—
549C22—24no—
550C22—24no—
551C22—42no—
552C22—44no—
553C22—44no—
554C22—48no—
555C22—44no—
556C22—44no—
557C22—816no—
558C22—84no—
559C22—84no—
560C22#8999514yes—
561C22C2×#650011yescharacteristic
562C22C2×#650211yescharacteristic
563C22#9052714yes—
564C22C2×#651611yescharacteristic
565C4—42no—
566C4—44no—
567C4—284no—
568C4—284no—
569C4#8999514yes—
570C4C2×#650011yescharacteristic
571C4C2×#650111yescharacteristic
572C4C2×#650211yescharacteristic
573C4C2×#650311yescharacteristic
574C2—24no—
575C2—82no—
576C2—84no—
577C2#1174082212yes—
578C2#1174083512yes—
579C2C2×#8999511yescharacteristic
580C2#1174155012yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC2×C7×#535C2
socleC23×C7C22×D4
hypercenterC2×#535D7
hypocenterC7#23388
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#535

Complements

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

NH ≅ G/Ntype
2. C2×#90516C2semidirect
3. C23×#1142C2semidirect
4. C2×#90006C2semidirect
5. C2×#89987C2semidirect
6. C2×#89776C2semidirect
7. #11729439C2semidirect
8. #11729334C2semidirect
29. C2×#6487C22semidirect
35. C2×C22⋊(C7⋊C8)C22semidirect
37. C2×#6384C22semidirect
38. #89324C22semidirect
68. C22×Dic14D4semidirect
91. #6489D4semidirect
102. C22×C7⋊C8D4semidirect
547. C7#23388semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5352
C2×#905162
C23×#11422
C2×#900062
C2×#899872
C2×#897762
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#233887