Finite Group Databasebeta

← #13355212

Subgroups — #13355212

Order 1792.

1766 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#137C211yesmaximal, characteristic
2C2×#90518C211yesmaximal, characteristic
3C2×#90005C211yesmaximal, characteristic
4#11741507C211yesmaximal, characteristic, complemented
5C22×#6409C211yesmaximal, characteristic, complemented
6#11729819C214yesmaximal, complemented
7#11729801C214yesmaximal, complemented
8#11729739C211yesmaximal, characteristic, complemented
9#11729734C211yesmaximal, characteristic, complemented
10C22×C7×#41C2211yescharacteristic
11C23×C7×Q8C2211yescharacteristic
12C2×C7×#137—24no—
13C2×C7×#137—24no—
14C2×C7×#137C418yescomplemented
15C22×C7×Q16C2212yes—
16C22×C7×SD16C2212yes—
17C22×C7×M16C2211yescharacteristic
18#90518—24no—
19#90518—24no—
20C2×Q8×Dic7C2212yes—
21C2×D4×Dic7—22no—
22C2×D4×Dic7C2212yes—
23C2×#6489—21no—
24C2×#6489C2211yescharacteristic
25C2×#6489C2211yescharacteristic
26C22×C4⋊Dic7—21no—
27C22×C4⋊Dic7C2211yescharacteristic
28C23×C7⋊C8C2211yescharacteristic
29#90005—24no—
30#90002—21no—
31#90002—21no—
32#90002C2214yes—
33C2×#6409—24no—
34C2×#6409C2214yes—
35C2×#6409C2212yes—
36C2×#6406C2212yes—
37#89983—22no—
38#89346C2214yes—
39#89345C2214yes—
40#20931—71nomaximal, Sylow
41C2×C7×#41—22no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41C2×C414yes—
45C2×C7×#41D414yes—
46C2×C7×#41D414yes—
47C22×C7×Q8—21no—
48C22×C7×Q8—22no—
49C22×C7×Q8—22no—
50C22×C7×Q8—22no—
51C22×C7×Q8—22no—
52C22×C7×Q8—22no—
53C22×C7×Q8C2×C414yescomplemented
54C22×C7×Q8C2312yes—
55C22×C7×Q8C2311yescharacteristic
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C23×Dic7—22no—
63C7×#137—232no—
64C2×C7×Q16—24no—
65C2×C7×Q16—24no—
66C2×C7×Q16C2×C418yescomplemented
67C2×C7×SD16—24no—
68C2×C7×SD16—24no—
69C2×C7×SD16C2×C418yescomplemented
70C2×C7×M16—22no—
71C2×C7×M16—22no—
72C2×C7×M16C2×C414yescomplemented
73C22×C7×C8C2312yes—
74C2×C22⋊Dic7—22no—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7D412yes—
77Q8×Dic7—24no—
78Q8×Dic7—24no—
79D4×Dic7—216no—
80D4×Dic7—24no—
81D4×Dic7—24no—
82#6489—28no—
83#6489—22no—
84#6489—22no—
85#6489—22no—
86#6489—22no—
87C2×C4⋊Dic7—28no—
88C2×C4⋊Dic7—21no—
89C2×C4⋊Dic7—22no—
90C2×C4⋊Dic7—22no—
91C2×C4⋊Dic7D411yescharacteristic
92C2×C4⋊Dic7D411yescharacteristic
93C2×C4⋊Dic7D414yes—
94C2×C4⋊Dic7C2312yes—
95C2×C4⋊Dic7C2312yes—
96C2×C4×Dic7—22no—
97C2×C4×Dic7—21no—
98C2×C4×Dic7C2312yes—
99C22×C7⋊C8—22no—
100C22×C7⋊C8—22no—
101C22×C7⋊C8—22no—
102C22×C7⋊C8C2312yes—
103#6409—24no—
104#6409—28no—
105#6406—24no—
106C7⋊#95—24no—
107C7⋊#95—24no—
108C22×#137D711yescharacteristic
109C2×#520—71no—
110C2×#416—71no—
111C22×#103—71no—
112#2310—71no—
113#1878—74no—
114#1877—74no—
115#1736—71no—
116#1729—71no—
117C24×C7—21no—
118C24×C7C2×D411yescharacteristic
119C7×#41—28no—
120C7×#41—28no—
121C7×#41—28no—
122C7×#41C22⋊C4116yes—
123C2×C7×Q8—24no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—22no—
128C2×C7×Q8—28no—
129C2×C7×Q8—28no—
130C2×C7×Q8—22no—
131C2×C7×Q8—22no—
132C2×C7×Q8—22no—
133C2×C7×Q8—22no—
134C2×C7×Q8—42no—
135C2×C7×Q8—42no—
136C2×C7×Q8—44no—
137C2×C7×Q8—44no—
138C2×C7×Q8C22×C414yes—
139C2×C7×Q8C22×C412yes—
140C2×C7×Q8C2×D412yes—
141C2×C7×Q8C2×D412yes—
142C2×C7×D4—24no—
143C2×C7×D4—24no—
144C2×C7×D4—24no—
145C2×C7×D4—21no—
146C2×C7×D4—21no—
147C2×C7×D4—21no—
148C2×C7×D4C22×C412yes—
149C2×C7×D4C2×D412yes—
150C2×C7×D4C2×D412yes—
151C22×C4×C7—21no—
152C22×C4×C7—24no—
153C22×C4×C7—24no—
154C22×C4×C7—24no—
155C22×C4×C7—24no—
156C22×C4×C7—22no—
157C22×C4×C7—42no—
158C22×C4×C7—42no—
159C22×C4×C7C22×C412yes—
160C22×C4×C7C2×D412yes—
161C22×C4×C7C2×D412yes—
162C22×C4×C7C2×D411yescharacteristic
163C22×C4×C7C2×D412yes—
164C22×C4×C7#4112yes—
165C22×C4×C7#4112yes—
166C22×C4×C7C2411yescharacteristic
167C22×Dic7—22no—
168C22×Dic7—28no—
169C22×Dic7—24no—
170C22×Dic7—22no—
171C22×Dic7—84no—
172C22×Dic7—82no—
173C22×Dic7C2×D412yes—
174C22×Dic7C2×D412yes—
175C7×Q16—216no—
176C7×SD16—216no—
177C7×M16—28no—
178C2×C7×C8—22no—
179C2×C7×C8—22no—
180C2×C7×C8C22×C414yes—
181C22⋊Dic7—28no—
182C22⋊Dic7—24no—
183C22⋊Dic7—42no—
184C4⋊Dic7—28no—
185C4⋊Dic7—22no—
186C4⋊Dic7—24no—
187C4⋊Dic7—22no—
188C4⋊Dic7—22no—
189C4⋊Dic7—22no—
190C4⋊Dic7—22no—
191C4⋊Dic7C2×D414yes—
192C4×Dic7—28no—
193C4×Dic7—22no—
194C4×Dic7—22no—
195C4×Dic7—42no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C2×C7⋊C8—44no—
200C22×#41C2×D711yescharacteristic
201C23×Q8C2×D711yescharacteristic
202C2×#137—24no—
203C2×#137—24no—
204C2×#137Dic718yescomplemented
205C22×Q16C2×D712yes—
206C22×SD16C2×D712yes—
207C22×M16C2×D711yescharacteristic
208C23×C8—71no—
209#520—144no—
210#520—144no—
211C2×C4×Q8—72no—
212C2×C4×D4—72no—
213C2×C4×D4—142no—
214C2×#117—71no—
215C2×#117—71no—
216C2×#117—141no—
217C22×C4⋊C4—71no—
218C22×C4⋊C4—141no—
219#416—144no—
220C2×#103—74no—
221C2×#103—72no—
222C2×#103—144no—
223C2×#102—72no—
224#391—74no—
225#391—141no—
226#391—141no—
227#381—142no—
228#340—74no—
229#336—74no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—22no—
233C23×C7—22no—
234C23×C7—22no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7C4×D412yes—
238C23×C7#12012yes—
239C23×C7#12112yes—
240C23×C7C22×D411yescharacteristic
241C7×Q8—22no—
242C7×Q8—22no—
243C7×Q8—22no—
244C7×Q8—24no—
245C7×Q8—24no—
246C7×Q8—42no—
247C7×Q8—42no—
248C7×Q8—48no—
249C7×Q8C2×C22⋊C414yes—
250C7×D4—216no—
251C7×D4—22no—
252C7×D4—22no—
253C7×D4—22no—
254C7×D4C2×C22⋊C414yes—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—22no—
258C2×C4×C7—216no—
259C2×C4×C7—24no—
260C2×C4×C7—42no—
261C2×C4×C7—42no—
262C2×C4×C7—48no—
263C2×C4×C7C2×C22⋊C414yes—
264C2×C4×C7C4×D414yes—
265C2×C4×C7C4×D412yes—
266C2×C4×C7#12012yes—
267C2×C4×C7#12114yes—
268C2×C4×C7#12112yes—
269C2×C4×C7#12314yes—
270C2×C4×C7C23×C411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C22×D411yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×C4×C7C2×#4111yescharacteristic
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—24no—
279C2×Dic7—24no—
280C2×Dic7—24no—
281C2×Dic7—44no—
282C2×Dic7—42no—
283C2×Dic7—42no—
284C2×Dic7—42no—
285C2×Dic7—84no—
286C2×Dic7—88no—
287C7×C8—24no—
288C7⋊C8—42no—
289C2×#41—22no—
290C2×#41—22no—
291C2×#41—22no—
292C2×#41C2×Dic714yes—
293C2×#41C22⋊D714yes—
294C2×#41C22⋊D714yes—
295C22×Q8—21no—
296C22×Q8—22no—
297C22×Q8—22no—
298C22×Q8—22no—
299C22×Q8—22no—
300C22×Q8—22no—
301C22×Q8C2×Dic714yescomplemented
302C22×Q8C22×D712yes—
303C22×Q8C22×D711yescharacteristic
304C22×D4—21no—
305C22×D4C22×D711yescharacteristic
306C23×C4—21no—
307C23×C4—21no—
308C23×C4—142no—
309C23×C4—141no—
310C23×C4C22×D711yescharacteristic
311#137—232no—
312C2×Q16—24no—
313C2×Q16—24no—
314C2×Q16C2×Dic718yescomplemented
315C2×SD16—24no—
316C2×SD16—24no—
317C2×SD16C2×Dic718yescomplemented
318C2×M16—22no—
319C2×M16—22no—
320C2×M16C2×Dic714yescomplemented
321C22×C8—72no—
322C22×C8—142no—
323C22×C8—142no—
324C22×C8—142no—
325C22×C8C22×D712yes—
326C4×Q8—144no—
327C4×Q8—144no—
328C4×D4—144no—
329C4×D4—1416no—
330C4×D4—144no—
331#117—148no—
332#117—142no—
333#117—142no—
334#117—142no—
335#117—142no—
336C2×C4⋊C4—74no—
337C2×C4⋊C4—72no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—71no—
340C2×C4⋊C4—72no—
341C2×C4⋊C4—142no—
342C2×C4⋊C4—148no—
343C2×C4⋊C4—141no—
344C2×C4⋊C4—142no—
345C2×C22⋊C4—72no—
346C2×C22⋊C4—142no—
347C2×C22⋊C4—141no—
348C2×C42—72no—
349C2×C42—142no—
350C2×C42—141no—
351#103—148no—
352#103—144no—
353#102—144no—
354#95—144no—
355#95—144no—
356C22×C7—24no—
357C22×C7—22no—
358C22×C7—22no—
359C22×C7—24no—
360C22×C7—24no—
361C22×C7—44no—
362C22×C7—42no—
363C22×C7—42no—
364C22×C7—44no—
365C22×C7#38614yes—
366C22×C7C2×C4×D411yescharacteristic
367C22×C7C2×#12011yescharacteristic
368C22×C7C2×#12111yescharacteristic
369C22×C7#57712yes—
370C22×C7#57812yes—
371C4×C7—24no—
372C4×C7—44no—
373C4×C7#38614yes—
374C4×C7C22×C22⋊C411yescharacteristic
375C4×C7C2×C4×D411yescharacteristic
376C4×C7C2×#12111yescharacteristic
377C4×C7C2×#12311yescharacteristic
378Dic7—42no—
379Dic7—44no—
380Dic7—42no—
381Dic7—82no—
382C24—21no—
383C24D4×D711yescharacteristic
384#41—28no—
385#41—28no—
386#41—28no—
387#41C22⋊Dic7116yes—
388C2×Q8—24no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—22no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—44no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8C22×Dic714yes—
404C2×Q8C22×Dic712yes—
405C2×Q8C2×C22⋊D712yes—
406C2×Q8C2×C22⋊D712yes—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4C22×Dic712yes—
414C2×D4C2×C22⋊D712yes—
415C2×D4C2×C22⋊D712yes—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—72no—
425C22×C4—72no—
426C22×C4—142no—
427C22×C4—148no—
428C22×C4—144no—
429C22×C4—142no—
430C22×C4—564no—
431C22×C4—562no—
432C22×C4D4×D711yescharacteristic
433C22×C4D4×D712yes—
434C22×C4#115912yes—
435C22×C4#115912yes—
436C22×C4C22×Dic712yes—
437C22×C4C2×C22⋊D712yes—
438C22×C4C2×C22⋊D712yes—
439C22×C4C23×D711yescharacteristic
440Q16—216no—
441SD16—216no—
442M16—28no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—142no—
446C2×C8—142no—
447C2×C8—142no—
448C2×C8—284no—
449C2×C8C22×Dic714yes—
450C4⋊C4—74no—
451C4⋊C4—148no—
452C4⋊C4—144no—
453C4⋊C4—142no—
454C4⋊C4—142no—
455C4⋊C4—142no—
456C4⋊C4—142no—
457C4⋊C4—142no—
458C22⋊C4—148no—
459C22⋊C4—144no—
460C22⋊C4—282no—
461C42—148no—
462C42—142no—
463C42—142no—
464C42—282no—
465C2×C7—24no—
466C2×C7—44no—
467C2×C7C2×#38611yescharacteristic
468C2×C7#294712yes—
469C2×C7#300911yescharacteristic
470C2×C7#301111yescharacteristic
471C2×C7#304612yes—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—42no—
478C23—42no—
479C23D4×Dic712yes—
480C23#650012yes—
481C23#650212yes—
482C23C2×D4×D711yescharacteristic
483Q8—22no—
484Q8—22no—
485Q8—22no—
486Q8—24no—
487Q8—24no—
488Q8—42no—
489Q8—42no—
490Q8—48no—
491Q8C2×C22⋊Dic714yes—
492D4—216no—
493D4—22no—
494D4—22no—
495D4—22no—
496D4C2×C22⋊Dic714yes—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—216no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—48no—
505C2×C4—144no—
506C2×C4—144no—
507C2×C4—142no—
508C2×C4—144no—
509C2×C4—142no—
510C2×C4—284no—
511C2×C4—282no—
512C2×C4—282no—
513C2×C4—282no—
514C2×C4—564no—
515C2×C4—568no—
516C2×C4D4×Dic714yes—
517C2×C4D4×Dic712yes—
518C2×C4#649814yes—
519C2×C4#650012yes—
520C2×C4#650214yes—
521C2×C4#650212yes—
522C2×C4C2×C22⋊Dic714yes—
523C2×C4C2×D4×D711yescharacteristic
524C2×C4C2×#115911yescharacteristic
525C2×C4C2×#115911yescharacteristic
526C2×C4C23×Dic711yescharacteristic
527C2×C4C22×C22⋊D711yescharacteristic
528C2×C4C22×C22⋊D711yescharacteristic
529C8—24no—
530C8—282no—
531C7#2093111yescharacteristic, complemented, Sylow
532C22—24no—
533C22—24no—
534C22—22no—
535C22—22no—
536C22—24no—
537C22—44no—
538C22—44no—
539C22—42no—
540C22—42no—
541C22#8999214yes—
542C22#9046912yes—
543C22#9047012yes—
544C22C2×D4×Dic711yescharacteristic
545C22C2×#650011yescharacteristic
546C22C2×#650211yescharacteristic
547C4—24no—
548C4—44no—
549C4—282no—
550C4—284no—
551C4—282no—
552C4—562no—
553C4#8999214yes—
554C4C2×D4×Dic711yescharacteristic
555C4C2×#649811yescharacteristic
556C4C2×#650211yescharacteristic
557C4C22×C22⋊Dic711yescharacteristic
558C2—24no—
559C2—44no—
560C2#1174141012yes—
561C2#1174141612yes—
562C2#1174142711yescharacteristic
563C2#1174142811yescharacteristic
564C2C2×#8999211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC22×C7×#137C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC22×#137D7
hypocenterC7#20931
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#137

Complements

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

NH ≅ G/Ntype
4. #11741507C2semidirect
5. C22×#6409C2semidirect
6. #11729819C2semidirect
7. #11729801C2semidirect
8. #11729739C2semidirect
9. #11729734C2semidirect
14. C2×C7×#137C4semidirect
53. C22×C7×Q8C2×C4semidirect
66. C2×C7×Q16C2×C4semidirect
69. C2×C7×SD16C2×C4semidirect
72. C2×C7×M16C2×C4semidirect
204. C2×#137Dic7semidirect
301. C22×Q8C2×Dic7semidirect
314. C2×Q16C2×Dic7semidirect
317. C2×SD16C2×Dic7semidirect
320. C2×M16C2×Dic7semidirect
531. C7#20931semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1372
C2×#905182
C2×#900052
#117415072
C22×#64092
#117298192
#117298012
#117297392
#117297342
#209317