Finite Group Databasebeta

← #13355167

Subgroups — #13355167

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×C7×#137C211yesmaximal, characteristic
2C22×Q8×Dic7C211yesmaximal, characteristic, complemented
3#11741531C211yesmaximal, characteristic
4C2×#90004C211yesmaximal, characteristic
5C2×#89997C211yesmaximal, characteristic, complemented
6#11729819C214yesmaximal, complemented
7#11729802C214yesmaximal, complemented
8#11729739C211yesmaximal, characteristic, complemented
9#11729733C211yesmaximal, 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
18C2×Q8×Dic7—24no—
19C2×Q8×Dic7—24no—
20C2×Q8×Dic7—22no—
21C2×Q8×Dic7C2214yes—
22C2×#6489C2211yescharacteristic
23C22×C4⋊Dic7—21no—
24C22×C4⋊Dic7C2211yescharacteristic
25C22×C4×Dic7—21no—
26C22×C4×Dic7C2211yescharacteristic
27C22×#1136C2211yescharacteristic
28#90004—24no—
29#90002C2212yes—
30#89997—24no—
31C2×#6409C2214yes—
32C2×#6409C2212yes—
33#89993C2212yes—
34#89992—21no—
35#89992—21no—
36C2×#6406C2212yes—
37C4×C22⋊Dic7—22no—
38#89346C2214yes—
39#89345C2214yes—
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×Q8—22no—
48C22×C7×Q8—21no—
49C22×C7×Q8—22no—
50C22×C7×Q8—22no—
51C22×C7×Q8C2×C414yes—
52C22×C7×Q8D414yes—
53C22×C7×Q8D414yes—
54C22×C7×Q8C2311yescharacteristic
55C22×C7×Q8C2312yes—
56C22×C7×D4—21no—
57C22×C7×D4C2311yescharacteristic
58C23×C4×C7—21no—
59C23×C4×C7—21no—
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C23×Dic7D411yescharacteristic
63C23×Dic7D411yescharacteristic
64C7×#137—232no—
65C2×C7×Q16—24no—
66C2×C7×Q16—24no—
67C2×C7×Q16C2×C418yescomplemented
68C2×C7×SD16—24no—
69C2×C7×SD16—24no—
70C2×C7×SD16C2×C418yescomplemented
71C2×C7×M16—22no—
72C2×C7×M16—22no—
73C2×C7×M16C2×C414yes—
74C22×C7×C8C2312yes—
75C2×C22⋊Dic7—21no—
76C2×C22⋊Dic7—22no—
77C2×C22⋊Dic7—22no—
78Q8×Dic7—216no—
79Q8×Dic7—28no—
80Q8×Dic7—28no—
81#6489—22no—
82#6489—22no—
83C2×C4⋊Dic7—28no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7—24no—
87C2×C4⋊Dic7D414yes—
88C2×C4⋊Dic7C2312yes—
89C2×C4⋊Dic7C2312yes—
90C2×C4×Dic7—28no—
91C2×C4×Dic7—22no—
92C2×C4×Dic7—22no—
93C2×C4×Dic7—24no—
94C2×C4×Dic7—21no—
95C2×C4×Dic7D412yes—
96C2×C4×Dic7C2312yes—
97C2×#1136—22no—
98C2×#1136—22no—
99C2×#1136—22no—
100C22×C7⋊C8C2312yes—
101#6409—28no—
102#6409—24no—
103#6406—24no—
104C7⋊#95—22no—
105C7⋊#95—22no—
106C22×#137D711yescharacteristic
107C22×C4×Q8—71no—
108C2×#417—71no—
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—24no—
122C2×C7×Q8—28no—
123C2×C7×Q8—24no—
124C2×C7×Q8—24no—
125C2×C7×Q8—28no—
126C2×C7×Q8—28no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8—21no—
131C2×C7×Q8—21no—
132C2×C7×Q8—21no—
133C2×C7×Q8—21no—
134C2×C7×Q8—21no—
135C2×C7×Q8C22⋊C4116yescomplemented
136C2×C7×Q8C22×C412yes—
137C2×C7×Q8C22×C414yes—
138C2×C7×Q8C2×D414yes—
139C2×C7×Q8C2×D414yes—
140C2×C7×D4—24no—
141C2×C7×D4—21no—
142C2×C7×D4—21no—
143C2×C7×D4—21no—
144C2×C7×D4—21no—
145C2×C7×D4—21no—
146C2×C7×D4—42no—
147C2×C7×D4—42no—
148C2×C7×D4C22×C412yes—
149C22×C4×C7—24no—
150C22×C4×C7—24no—
151C22×C4×C7—24no—
152C22×C4×C7—22no—
153C22×C4×C7—21no—
154C22×C4×C7—24no—
155C22×C4×C7—42no—
156C22×C4×C7—42no—
157C22×C4×C7C22×C412yes—
158C22×C4×C7C2×D412yes—
159C22×C4×C7C2×D412yes—
160C22×C4×C7C2×D411yescharacteristic
161C22×C4×C7C2×D412yes—
162C22×C4×C7#4112yes—
163C22×C4×C7#4112yes—
164C22×C4×C7C2411yescharacteristic
165C22×Dic7—22no—
166C22×Dic7—24no—
167C22×Dic7—22no—
168C22×Dic7—24no—
169C22×Dic7—22no—
170C22×Dic7—22no—
171C22×Dic7—42no—
172C22×Dic7—44no—
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—44no—
182C22⋊Dic7—42no—
183C22⋊Dic7—44no—
184C4⋊Dic7—216no—
185C4⋊Dic7—24no—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7—22no—
189C4⋊Dic7C2×D414yes—
190C4×Dic7—216no—
191C4×Dic7—24no—
192C4×Dic7—22no—
193C4×Dic7—22no—
194C4×Dic7—42no—
195#1136—44no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—22no—
198C2×C7⋊C8—22no—
199C22×#41C2×D711yescharacteristic
200C23×Q8C2×D711yescharacteristic
201C2×#137—24no—
202C2×#137—24no—
203C2×#137Dic718yescomplemented
204C22×Q16C2×D712yes—
205C22×SD16C2×D712yes—
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—71no—
214C22×C4⋊C4—141no—
215C22×C42—71no—
216C22×C42—141no—
217#419—144no—
218#417—144no—
219C2×#103—74no—
220C2×#103—72no—
221C2×#102—72no—
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—24no—
241C7×Q8—24no—
242C7×Q8—24no—
243C7×Q8—216no—
244C7×Q8—22no—
245C7×Q8—22no—
246C7×Q8—41no—
247C7×Q8—41no—
248C7×Q8C2×C22⋊C418yes—
249C7×D4—22no—
250C7×D4—22no—
251C7×D4—48no—
252C7×D4—41no—
253C7×D4—41no—
254C2×C4×C7—24no—
255C2×C4×C7—24no—
256C2×C4×C7—24no—
257C2×C4×C7—216no—
258C2×C4×C7—22no—
259C2×C4×C7—41no—
260C2×C4×C7—41no—
261C2×C4×C7—48no—
262C2×C4×C7C2×C22⋊C414yes—
263C2×C4×C7C4×D414yes—
264C2×C4×C7C4×D412yes—
265C2×C4×C7#12012yes—
266C2×C4×C7#12114yes—
267C2×C4×C7#12112yes—
268C2×C4×C7#12314yes—
269C2×C4×C7C23×C411yescharacteristic
270C2×C4×C7C22×D411yescharacteristic
271C2×C4×C7C22×D411yescharacteristic
272C2×C4×C7C22×D411yescharacteristic
273C2×C4×C7C2×#4111yescharacteristic
274C2×C4×C7C2×#4111yescharacteristic
275C2×Dic7—28no—
276C2×Dic7—22no—
277C2×Dic7—22no—
278C2×Dic7—22no—
279C2×Dic7—24no—
280C2×Dic7—22no—
281C2×Dic7—22no—
282C2×Dic7—44no—
283C2×Dic7—48no—
284C2×Dic7—44no—
285C2×Dic7—42no—
286C2×Dic7—42no—
287C2×Dic7—42no—
288C2×Dic7#13714yescomplemented
289C7×C8—24no—
290C7⋊C8—42no—
291C2×#41—22no—
292C2×#41—22no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41C2×Dic714yes—
297C22×Q8—22no—
298C22×Q8—21no—
299C22×Q8—22no—
300C22×Q8—22no—
301C22×Q8C2×Dic714yes—
302C22×Q8C22⋊D714yes—
303C22×Q8C22⋊D714yes—
304C22×Q8C22×D711yescharacteristic
305C22×Q8C22×D712yes—
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—232no—
315C2×Q16—24no—
316C2×Q16—24no—
317C2×Q16C2×Dic718yescomplemented
318C2×SD16—24no—
319C2×SD16—24no—
320C2×SD16C2×Dic718yescomplemented
321C2×M16—22no—
322C2×M16—22no—
323C2×M16—142no—
324C2×M16—142no—
325C2×M16—142no—
326C2×M16C2×Dic714yes—
327C22×C8—72no—
328C22×C8C22×D712yes—
329C4×Q8—148no—
330C4×Q8—1416no—
331C4×Q8—148no—
332#117—142no—
333#117—142no—
334C2×C4⋊C4—74no—
335C2×C4⋊C4—72no—
336C2×C4⋊C4—72no—
337C2×C4⋊C4—144no—
338C2×C4⋊C4—148no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—142no—
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—144no—
352#103—148no—
353#102—144no—
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—28no—
372C4×C7—42no—
373C4×C7#38614yes—
374C4×C7C22×C22⋊C411yescharacteristic
375C4×C7C2×C4×D411yescharacteristic
376C4×C7C2×#12111yescharacteristic
377C4×C7C2×#12311yescharacteristic
378Dic7—24no—
379Dic7—42no—
380Dic7—44no—
381Dic7—82no—
382C24—21no—
383C24D4×D711yescharacteristic
384#41—28no—
385#41—28no—
386#41—44no—
387#41—44no—
388C2×Q8—24no—
389C2×Q8—28no—
390C2×Q8—28no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—28no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—21no—
402C2×Q8C22⋊Dic7116yescomplemented
403C2×Q8C22×Dic712yes—
404C2×Q8C22×Dic714yes—
405C2×Q8C2×C22⋊D714yes—
406C2×Q8C2×C22⋊D714yes—
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×D4C22×Dic712yes—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—72no—
425C22×C4—72no—
426C22×C4—144no—
427C22×C4—142no—
428C22×C4—144no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—282no—
433C22×C4—284no—
434C22×C4D4×D711yescharacteristic
435C22×C4D4×D712yes—
436C22×C4#115912yes—
437C22×C4#115912yes—
438C22×C4C22×Dic712yes—
439C22×C4C2×C22⋊D712yes—
440C22×C4C2×C22⋊D712yes—
441C22×C4C23×D711yescharacteristic
442Q16—216no—
443SD16—216no—
444M16—28no—
445M16—284no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—142no—
449C2×C8—142no—
450C2×C8—142no—
451C2×C8C22×Dic714yes—
452C4⋊C4—74no—
453C4⋊C4—1416no—
454C4⋊C4—144no—
455C4⋊C4—142no—
456C4⋊C4—142no—
457C4⋊C4—142no—
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—42no—
477C23—42no—
478C23—41no—
479C23—41no—
480C23D4×Dic712yes—
481C23#650012yes—
482C23#650212yes—
483C23C2×D4×D711yescharacteristic
484Q8—24no—
485Q8—24no—
486Q8—24no—
487Q8—216no—
488Q8—22no—
489Q8—22no—
490Q8—41no—
491Q8—41no—
492Q8C2×C22⋊Dic718yes—
493D4—22no—
494D4—22no—
495D4—48no—
496D4—41no—
497D4—41no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—216no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—48no—
506C2×C4—74no—
507C2×C4—148no—
508C2×C4—142no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—142no—
512C2×C4—144no—
513C2×C4—142no—
514C2×C4—284no—
515C2×C4—288no—
516C2×C4—284no—
517C2×C4—282no—
518C2×C4—282no—
519C2×C4—282no—
520C2×C4D4×Dic714yes—
521C2×C4D4×Dic712yes—
522C2×C4#649814yes—
523C2×C4#650012yes—
524C2×C4#650214yes—
525C2×C4#650212yes—
526C2×C4C2×C22⋊Dic714yes—
527C2×C4C2×D4×D711yescharacteristic
528C2×C4C2×#115911yescharacteristic
529C2×C4C2×#115911yescharacteristic
530C2×C4C23×Dic711yescharacteristic
531C2×C4C22×C22⋊D711yescharacteristic
532C2×C4C22×C22⋊D711yescharacteristic
533C8—24no—
534C8—282no—
535C7#2092611yescharacteristic, complemented, Sylow
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—44no—
542C22—82no—
543C22—81no—
544C22—81no—
545C22#8999214yes—
546C22D7×#13712yes—
547C22#9046812yes—
548C22C2×D4×Dic711yescharacteristic
549C22C2×#650011yescharacteristic
550C22C2×#650211yescharacteristic
551C4—28no—
552C4—42no—
553C4—144no—
554C4—282no—
555C4—284no—
556C4—562no—
557C4#8999214yes—
558C4C2×D4×Dic711yescharacteristic
559C4C2×#649811yescharacteristic
560C4C2×#650211yescharacteristic
561C4C22×C22⋊Dic711yescharacteristic
562C2—24no—
563C2—82no—
564C2Dic7×#13712yes—
565C2#1174141112yes—
566C2#1174142411yescharacteristic
567C2#1174142511yescharacteristic
568C2C2×#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#20926
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
2. C22×Q8×Dic7C2semidirect
5. C2×#89997C2semidirect
6. #11729819C2semidirect
7. #11729802C2semidirect
8. #11729739C2semidirect
9. #11729733C2semidirect
14. C2×C7×#137C4semidirect
67. C2×C7×Q16C2×C4semidirect
70. C2×C7×SD16C2×C4semidirect
135. C2×C7×Q8C22⋊C4semidirect
203. C2×#137Dic7semidirect
288. C2×Dic7#137semidirect
317. C2×Q16C2×Dic7semidirect
320. C2×SD16C2×Dic7semidirect
402. C2×Q8C22⋊Dic7semidirect
535. C7#20926semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1372
C22×Q8×Dic72
#117415312
C2×#900042
C2×#899972
#117298192
#117298022
#117297392
#117297332
#209267