Finite Group Databasebeta

← #13355161

Subgroups — #13355161

Order 1792.

1802 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6514C211yesmaximal, characteristic
2C22×#6450C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic
4#11741531C211yesmaximal, characteristic
5C2×#89892C211yesmaximal, characteristic, complemented
6#11729733C211yesmaximal, characteristic, complemented
7#11729312C211yesmaximal, characteristic, complemented
8#11728035C214yesmaximal, complemented
9#11727967C214yesmaximal, complemented
10C22×C7×#41C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C2×#6514—24no—
15C2×#6514—24no—
16C2×#6514C418yescomplemented
17C22×#1144C2212yes—
18C22×#1142C2212yes—
19C2×#6489C2211yescharacteristic
20C22×Dic7⋊C4—21no—
21C22×C4×Dic7—21no—
22C22×C4×Dic7C2211yescharacteristic
23C22×#1136C2211yescharacteristic
24C2×#6450—24no—
25C2×#6450—24no—
26C2×#6450—22no—
27C2×#6450C2214yes—
28C7×#417—24no—
29C2×C7×#103C2212yes—
30C2×C7×#102C2212yes—
31#90002C2212yes—
32#89993C2212yes—
33#89992—21no—
34#89992—21no—
35C4×C22⋊Dic7—22no—
36#89892—24no—
37C2×#6390C2214yes—
38#89345C2214yes—
39#89324C2214yes—
40#20926—71nomaximal, Sylow
41C2×C7×#41—22no—
42C2×C7×#41—22no—
43C2×C7×#41—22no—
44C2×C7×#41—22no—
45C2×C7×#41—22no—
46C2×C7×#41C2×C414yes—
47C22×C7×Q8C2311yescharacteristic
48C22×C7×D4—21no—
49C22×C7×D4C2311yescharacteristic
50C23×C4×C7—21no—
51C23×C4×C7—21no—
52C23×C4×C7C2311yescharacteristic
53C23×Dic7—21no—
54C23×Dic7D411yescharacteristic
55C23×Dic7D411yescharacteristic
56C22×Dic14—21no—
57C22×Dic14—22no—
58C22×Dic14—22no—
59C22×Dic14—22no—
60C22×Dic14C2×C414yes—
61C22×Dic14D414yes—
62C22×Dic14D414yes—
63C22×Dic14C2312yes—
64C2×C7×M16—22no—
65C2×C7×M16—22no—
66C2×C7×M16—22no—
67C22×C7×C8C2312yes—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4D414yes—
71C2×C7×C4⋊C4C2312yes—
72C2×C22⋊Dic7—21no—
73C2×C22⋊Dic7—22no—
74C2×C22⋊Dic7—22no—
75#6514—232no—
76C2×#1144—24no—
77C2×#1144—24no—
78C2×#1144C2×C418yescomplemented
79C2×#1142—24no—
80C2×#1142—24no—
81C2×#1142C2×C418yescomplemented
82#6489—22no—
83#6489—22no—
84C2×C4⋊Dic7C2312yes—
85C2×Dic7⋊C4—28no—
86C2×Dic7⋊C4—24no—
87C2×C4×Dic7—28no—
88C2×C4×Dic7—22no—
89C2×C4×Dic7—22no—
90C2×C4×Dic7—24no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7D412yes—
93C2×C4×Dic7C2312yes—
94C2×#1136—22no—
95C2×#1136—22no—
96C2×#1136C2×C414yes—
97C22×C7⋊C8C2312yes—
98#6450—216no—
99#6450—28no—
100#6450—28no—
101C7×#103—24no—
102C7×#102—24no—
103C7⋊#95—22no—
104C7⋊#95—22no—
105#6390—28no—
106C22×#137—71no—
107C22×C4×Q8—71no—
108C2×#417D711yescharacteristic
109C2×#419—71no—
110#2297—71no—
111#1880—74no—
112#1878—74no—
113#1736—71no—
114#1728—71no—
115C24×C7—21no—
116C24×C7C2×D411yescharacteristic
117C7×#41—28no—
118C7×#41—28no—
119C7×#41—44no—
120C7×#41—44no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×Q8—21no—
124C2×C7×Q8—21no—
125C2×C7×Q8—21no—
126C2×C7×Q8C22×C412yes—
127C2×C7×D4—24no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—42no—
134C2×C7×D4—42no—
135C2×C7×D4C22×C412yes—
136C22×C4×C7—24no—
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—24no—
140C22×C4×C7—44no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7C22×C412yes—
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D412yes—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D412yes—
148C22×C4×C7#4112yes—
149C22×C4×C7#4112yes—
150C22×C4×C7C2411yescharacteristic
151C22×Dic7—24no—
152C22×Dic7—22no—
153C22×Dic7—24no—
154C22×Dic7—24no—
155C22×Dic7—22no—
156C22×Dic7—24no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—42no—
160C22×Dic7C2×D412yes—
161C22×Dic7C2×D412yes—
162C2×Dic14—28no—
163C2×Dic14—24no—
164C2×Dic14—24no—
165C2×Dic14—28no—
166C2×Dic14—24no—
167C2×Dic14—28no—
168C2×Dic14—22no—
169C2×Dic14—22no—
170C2×Dic14—22no—
171C2×Dic14C22⋊C4116yescomplemented
172C2×Dic14C22×C414yes—
173C2×Dic14C2×D414yes—
174C2×Dic14C2×D414yes—
175C7×M16—44no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8—22no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4C2×D414yes—
182C22⋊Dic7—44no—
183C22⋊Dic7—42no—
184C22⋊Dic7—44no—
185#1144—216no—
186#1142—216no—
187C4⋊Dic7—22no—
188C4⋊Dic7—22no—
189Dic7⋊C4—216no—
190C4×Dic7—216no—
191C4×Dic7—24no—
192C4×Dic7—22no—
193C4×Dic7—22no—
194C4×Dic7—42no—
195#1136—28no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8C22×C414yes—
199C22×#41C2×D711yescharacteristic
200C23×Q8—71no—
201C2×#137—78no—
202C2×#137—144no—
203C2×#137—144no—
204C22×Q16—72no—
205C22×SD16—72no—
206C22×M16—71no—
207C22×M16C2×D711yescharacteristic
208C2×C4×Q8—74no—
209C2×C4×Q8—142no—
210C2×C4×Q8—144no—
211C2×C4×Q8—144no—
212C2×#117—71no—
213C22×C4⋊C4—141no—
214C22×C4⋊C4C2×D711yescharacteristic
215C22×C42—71no—
216C22×C42—141no—
217#419—144no—
218#417—24no—
219C2×#103—74no—
220C2×#103C2×D712yes—
221C2×#102C2×D712yes—
222#391—72no—
223#390—72no—
224#386—141no—
225#386—141no—
226C4×C22⋊C4—142no—
227#340—74no—
228#336—74no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7—22no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7C4×D412yes—
237C23×C7#12012yes—
238C23×C7#12112yes—
239C23×C7C22×D411yescharacteristic
240C7×Q8—22no—
241C7×Q8—22no—
242C7×Q8—41no—
243C7×Q8—41no—
244C7×D4—22no—
245C7×D4—22no—
246C7×D4—48no—
247C7×D4—41no—
248C7×D4—41no—
249C2×C4×C7—22no—
250C2×C4×C7—24no—
251C2×C4×C7—48no—
252C2×C4×C7—41no—
253C2×C4×C7—41no—
254C2×C4×C7—44no—
255C2×C4×C7—48no—
256C2×C4×C7C2×C22⋊C414yes—
257C2×C4×C7C4×D412yes—
258C2×C4×C7C4×D414yes—
259C2×C4×C7#12012yes—
260C2×C4×C7#12112yes—
261C2×C4×C7#12114yes—
262C2×C4×C7#12314yes—
263C2×C4×C7C23×C411yescharacteristic
264C2×C4×C7C22×D411yescharacteristic
265C2×C4×C7C22×D411yescharacteristic
266C2×C4×C7C22×D411yescharacteristic
267C2×C4×C7C2×#4111yescharacteristic
268C2×C4×C7C2×#4111yescharacteristic
269C2×Dic7—28no—
270C2×Dic7—24no—
271C2×Dic7—22no—
272C2×Dic7—24no—
273C2×Dic7—24no—
274C2×Dic7—22no—
275C2×Dic7—216no—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—44no—
280C2×Dic7—42no—
281C2×Dic7—42no—
282C2×Dic7—42no—
283C2×Dic7#13714yescomplemented
284Dic14—24no—
285Dic14—24no—
286Dic14—24no—
287Dic14—216no—
288Dic14C2×C22⋊C418yes—
289C7×C8—42no—
290C7⋊C8—24no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41C4×D714yes—
297C22×Q8—74no—
298C22×Q8—74no—
299C22×Q8—74no—
300C22×Q8—72no—
301C22×Q8—142no—
302C22×Q8—141no—
303C22×Q8—142no—
304C22×Q8—142no—
305C22×Q8C22×D711yescharacteristic
306C22×D4—21no—
307C22×D4C22×D711yescharacteristic
308C23×C4—21no—
309C23×C4—21no—
310C23×C4—71no—
311C23×C4—71no—
312C23×C4—141no—
313C23×C4C22×D711yescharacteristic
314#137—1432no—
315C2×Q16—78no—
316C2×Q16—144no—
317C2×Q16—144no—
318C2×SD16—78no—
319C2×SD16—144no—
320C2×SD16—144no—
321C2×M16—22no—
322C2×M16—22no—
323C2×M16—22no—
324C2×M16—74no—
325C2×M16—142no—
326C2×M16—142no—
327C22×C8—72no—
328C22×C8C22×D712yes—
329C4×Q8—148no—
330C4×Q8—1416no—
331C4×Q8—148no—
332#117—142no—
333#117—142no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—72no—
337C2×C4⋊C4—144no—
338C2×C4⋊C4—148no—
339C2×C4⋊C4C22⋊D714yes—
340C2×C4⋊C4C22×D712yes—
341C2×C22⋊C4—142no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4—141no—
344C2×C42—72no—
345C2×C42—72no—
346C2×C42—144no—
347C2×C42—148no—
348C2×C42—142no—
349C2×C42—142no—
350C2×C42—141no—
351#103—24no—
352#103—148no—
353#102—24no—
354#95—142no—
355#95—142no—
356C22×C7—24no—
357C22×C7—22no—
358C22×C7—22no—
359C22×C7—24no—
360C22×C7—44no—
361C22×C7—42no—
362C22×C7—81no—
363C22×C7—81no—
364C22×C7—82no—
365C22×C7#38614yes—
366C22×C7C2×C4×D411yescharacteristic
367C22×C7C2×#12011yescharacteristic
368C22×C7C2×#12111yescharacteristic
369C22×C7C2×#13712yes—
370C22×C7#57512yes—
371C4×C7—44no—
372C4×C7—42no—
373C4×C7#38614yes—
374C4×C7C22×C22⋊C411yescharacteristic
375C4×C7C2×C4×D411yescharacteristic
376C4×C7C2×#12111yescharacteristic
377C4×C7C2×#12311yescharacteristic
378Dic7—24no—
379Dic7—28no—
380Dic7—42no—
381Dic7—82no—
382C24—21no—
383C24D4×D711yescharacteristic
384#41—28no—
385#41—28no—
386#41—44no—
387#41—44no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—716no—
394C2×Q8—74no—
395C2×Q8—74no—
396C2×Q8—74no—
397C2×Q8—144no—
398C2×Q8—148no—
399C2×Q8—144no—
400C2×Q8—148no—
401C2×Q8—144no—
402C2×Q8—148no—
403C2×Q8—142no—
404C2×Q8—142no—
405C2×Q8—142no—
406C2×Q8C2×C4×D712yes—
407C2×D4—24no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4C2×C4×D712yes—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—44no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—72no—
424C22×C4—72no—
425C22×C4—144no—
426C22×C4—144no—
427C22×C4—144no—
428C22×C4—142no—
429C22×C4—144no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—282no—
434C22×C4C2×C4×D712yes—
435C22×C4#115712yes—
436C22×C4D4×D712yes—
437C22×C4D4×D712yes—
438C22×C4D4×D711yescharacteristic
439C22×C4#115912yes—
440C22×C4C2×C22⋊D712yes—
441C22×C4C23×D711yescharacteristic
442Q16—1416no—
443SD16—1416no—
444M16—44no—
445M16—148no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—74no—
450C2×C8—142no—
451C2×C8—142no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—1416no—
455C4⋊C4—142no—
456C4⋊C4—142no—
457C4⋊C4C2×C22⋊D714yes—
458C22⋊C4—284no—
459C22⋊C4—284no—
460C22⋊C4—282no—
461C42—1416no—
462C42—144no—
463C42—142no—
464C42—142no—
465C42—282no—
466C2×C7—24no—
467C2×C7—82no—
468C2×C7C2×#38611yescharacteristic
469C2×C7C4×#13712yes—
470C2×C7#300411yescharacteristic
471C2×C7#300511yescharacteristic
472C2×C7#303712yes—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23(C22⋊D7)⋊C412yes—
481C23#644712yes—
482C23#650012yes—
483C23C2×D4×D711yescharacteristic
484Q8—22no—
485Q8—22no—
486Q8—41no—
487Q8—41no—
488Q8—78no—
489Q8—144no—
490Q8—144no—
491Q8—144no—
492Q8—1416no—
493D4—22no—
494D4—22no—
495D4—48no—
496D4—41no—
497D4—41no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—48no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—48no—
505C2×C4—74no—
506C2×C4—148no—
507C2×C4—144no—
508C2×C4—144no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—144no—
512C2×C4—1416no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—142no—
516C2×C4—284no—
517C2×C4—282no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4D7×C22⋊C414yes—
521C2×C4(C22⋊D7)⋊C412yes—
522C2×C4#644614yes—
523C2×C4#644712yes—
524C2×C4C4×C22⋊D714yes—
525C2×C4#650012yes—
526C2×C4#650114yes—
527C2×C4C22×C4×D711yescharacteristic
528C2×C4C2×#115711yescharacteristic
529C2×C4C2×D4×D711yescharacteristic
530C2×C4C2×D4×D711yescharacteristic
531C2×C4C2×#115911yescharacteristic
532C2×C4C22×C22⋊D711yescharacteristic
533C8—42no—
534C8—144no—
535C7#2092611yescharacteristic, complemented, Sylow
536C22—24no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—42no—
541C22—44no—
542C22—82no—
543C22—81no—
544C22—81no—
545C22#8972814yes—
546C22C2×(C22⋊D7)⋊C411yescharacteristic
547C22C2×#644711yescharacteristic
548C22#9046412yes—
549C22D7×#13712yes—
550C22C2×#650011yescharacteristic
551C4—42no—
552C4—44no—
553C4—148no—
554C4—144no—
555C4—282no—
556C4—562no—
557C4#8972814yes—
558C4C2×D7×C22⋊C411yescharacteristic
559C4C2×#644611yescharacteristic
560C4C2×C4×C22⋊D711yescharacteristic
561C4C2×#650111yescharacteristic
562C2—24no—
563C2—82no—
564C2#1173631212yes—
565C2#1173635812yes—
566C2C2×#8972811yescharacteristic
567C2#1174139011yescharacteristic
568C2#1174142411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC22×C7×#41C22
cosocleC22×C4×C7C24
hypercenterC2×#417D7
hypocenterC7#20926
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#6450C2semidirect
5. C2×#89892C2semidirect
6. #11729733C2semidirect
7. #11729312C2semidirect
8. #11728035C2semidirect
9. #11727967C2semidirect
16. C2×#6514C4semidirect
78. C2×#1144C2×C4semidirect
81. C2×#1142C2×C4semidirect
171. C2×Dic14C22⋊C4semidirect
283. C2×Dic7#137semidirect
535. C7#20926semidirect

Maximal subgroups

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

subgroupindex
C22×#65142
C22×#64502
C2×C7×#4172
#117415312
C2×#898922
#117297332
#117293122
#117280352
#117279672
#209267