Finite Group Databasebeta

← #13359473

Subgroups — #13359473

Order 1792.

1436 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic7C211yesmaximal, characteristic
2C2×C7×#451C211yesmaximal, characteristic
3C2×#89956C212yesmaximal, complemented
4Dic7×C22⋊Q8C211yesmaximal, characteristic
5#11729886C211yesmaximal, characteristic
6#11729819C212yesmaximal, complemented
7#11729446C211yesmaximal, characteristic
8#11727992C212yesmaximal, complemented
9#11727974C212yesmaximal, complemented
10#11727682C211yesmaximal, characteristic
11#11727660C211yesmaximal, characteristic
12C23×C7×Q8C2211yescharacteristic
13C22×C7×Q16C2212yes—
14C2×C7×C22⋊Q8C2211yescharacteristic
15C2×Q8×Dic7—28no—
16C2×Q8×Dic7—21no—
17C2×Q8×Dic7—21no—
18C2×Q8×Dic7C2211yescharacteristic
19C2×Q8×Dic7C2212yes—
20C2×Q8×Dic7C2212yes—
21C22×C4⋊Dic7—21no—
22C22×C4⋊Dic7C2211yescharacteristic
23C22×C4×Dic7—21no—
24C22×C4×Dic7C2211yescharacteristic
25C7×#451C418yescomplemented
26C2×C7×#103C2212yes—
27C2×C7×C22⋊C8C2211yescharacteristic
28#90002C2211yescharacteristic
29C2×#6409C2212yes—
30C2×#6409C2212yes—
31C2×#6409C2212yes—
32#89984C2211yescharacteristic
33C2×C22⋊(C7⋊C8)C2211yescharacteristic
34#89956—28no—
35C2×#6391C2212yes—
36C2×#6389C2212yes—
37#89757C2211yescharacteristic
38#89753—21no—
39C4⋊C4×Dic7—21no—
40C4⋊C4×Dic7C2211yescharacteristic
41#89722—21no—
42C22⋊C4×Dic7—21no—
43#89346C2212yes—
44#89345C2212yes—
45#89324C2212yes—
46#21455—71nomaximal, Sylow
47C22×C7×Q8—24no—
48C22×C7×Q8—24no—
49C22×C7×Q8—21no—
50C22×C7×Q8—22no—
51C22×C7×Q8C2×C414yes—
52C22×C7×Q8C2311yescharacteristic
53C22×C7×Q8C2312yes—
54C23×C4×C7—21no—
55C23×C4×C7C2311yescharacteristic
56C23×Dic7—21no—
57C23×Dic7D411yescharacteristic
58C23×Dic7D411yescharacteristic
59C2×C7×Q16—24no—
60C2×C7×Q16—24no—
61C2×C7×Q16C2×C418yescomplemented
62C22×C7×C8C2312yes—
63C7×C22⋊Q8—22no—
64C7×C22⋊Q8C2×C414yes—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C22⋊C4—21no—
68C2×C22⋊Dic7—21no—
69Q8×Dic7—28no—
70Q8×Dic7—28no—
71Q8×Dic7—24no—
72Q8×Dic7—24no—
73Q8×Dic7—24no—
74Q8×Dic7—24no—
75Q8×Dic7D418yes—
76C2×C4⋊Dic7—28no—
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7—22no—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—24no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2312yes—
86C2×C4×Dic7—28no—
87C2×C4×Dic7—22no—
88C2×C4×Dic7—22no—
89C2×C4×Dic7—24no—
90C2×C4×Dic7—21no—
91C2×C4×Dic7—21no—
92C2×C4×Dic7D411yescharacteristic
93C2×C4×Dic7D411yescharacteristic
94C2×C4×Dic7C2311yescharacteristic
95C2×C4×Dic7C2311yescharacteristic
96C22×C7⋊C8C2312yes—
97C7×#103C2×C418yescomplemented
98C7×C22⋊C8C2×C414yes—
99#6409—24no—
100#6409—24no—
101#6409—24no—
102C7⋊#95—21no—
103C7⋊#95—21no—
104C7⋊#95—21no—
105C7⋊#95—21no—
106C7⋊#95—21no—
107C22⋊(C7⋊C8)—22no—
108#6391—24no—
109#6389—24no—
110C22×C4×Q8—71no—
111C2×#451D711yescharacteristic
112C2×#441—72no—
113C4×C22⋊Q8—71no—
114#1878—72no—
115#1872—71no—
116#1865—72no—
117#1853—71no—
118#1787—71no—
119#1763—72no—
120#1749—71no—
121C24×C7C2×D411yescharacteristic
122C2×C7×Q8—24no—
123C2×C7×Q8—216no—
124C2×C7×Q8—24no—
125C2×C7×Q8—216no—
126C2×C7×Q8—24no—
127C2×C7×Q8—22no—
128C2×C7×Q8—22no—
129C2×C7×Q8—22no—
130C2×C7×Q8—21no—
131C2×C7×Q8—22no—
132C2×C7×Q8—22no—
133C2×C7×Q8C22×C412yes—
134C2×C7×Q8C22×C414yes—
135C2×C7×Q8C2×D414yes—
136C2×C7×Q8#4114yes—
137C22×C4×C7—24no—
138C22×C4×C7—24no—
139C22×C4×C7—24no—
140C22×C4×C7—22no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7C22×C412yes—
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7#4112yes—
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—22no—
150C22×Dic7—22no—
151C22×Dic7—22no—
152C22×Dic7—24no—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—22no—
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—41no—
159C22×Dic7—42no—
160C22×Dic7—42no—
161C22×Dic7C2×D411yescharacteristic
162C22×Dic7C2×D411yescharacteristic
163C22×Dic7C2×D411yescharacteristic
164C22×Dic7C2×D411yescharacteristic
165C7×Q16—216no—
166C2×C7×C8—22no—
167C2×C7×C8—22no—
168C2×C7×C8C22×C414yes—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4C22×C412yes—
172C7×C22⋊C4—24no—
173C22⋊Dic7—42no—
174C4⋊Dic7—28no—
175C4⋊Dic7—24no—
176C4⋊Dic7—22no—
177C4⋊Dic7—28no—
178C4⋊Dic7—22no—
179C4⋊Dic7—22no—
180C4⋊Dic7—42no—
181C4⋊Dic7—42no—
182C4⋊Dic7C2×D414yes—
183C4×Dic7—28no—
184C4×Dic7—22no—
185C4×Dic7—28no—
186C4×Dic7—22no—
187C4×Dic7—22no—
188C4×Dic7—42no—
189C4×Dic7—42no—
190C4×Dic7C2×D414yes—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8—22no—
194C23×Q8C2×D711yescharacteristic
195C22×Q16C2×D712yes—
196C2×C22⋊Q8C2×D711yescharacteristic
197C2×C4×Q8—72no—
198C2×C4×Q8—72no—
199C2×C4×Q8—71no—
200C2×C4×Q8—141no—
201C2×C4×Q8—141no—
202C2×C4×Q8—148no—
203C22×C4⋊C4—71no—
204C22×C4⋊C4—141no—
205C22×C42—71no—
206C22×C42—141no—
207#451Dic718yescomplemented
208#441—148no—
209C2×#106—72no—
210C2×#103—72no—
211C2×#103—72no—
212C2×#103—72no—
213C2×#103C2×D712yes—
214C2×C22⋊C8—71no—
215C2×C22⋊C8C2×D711yescharacteristic
216C2×#97—72no—
217#391—71no—
218#389—71no—
219#387—141no—
220#385—141no—
221#380—71no—
222C4×C4⋊C4—71no—
223C4×C4⋊C4—141no—
224C4×C22⋊C4—141no—
225#340—72no—
226#340—72no—
227#336—72no—
228C23×C7—21no—
229C23×C7—21no—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7C4×D412yes—
233C23×C7#13712yes—
234C23×C7#13712yes—
235C23×C7C22×D411yescharacteristic
236C7×Q8—216no—
237C7×Q8—24no—
238C7×Q8—24no—
239C7×Q8—24no—
240C7×Q8—24no—
241C7×Q8—42no—
242C7×Q8C4×D418yes—
243C2×C4×C7—216no—
244C2×C4×C7—24no—
245C2×C4×C7—24no—
246C2×C4×C7—21no—
247C2×C4×C7—24no—
248C2×C4×C7—21no—
249C2×C4×C7—24no—
250C2×C4×C7—24no—
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—41no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7C4×D412yes—
258C2×C4×C7C23×C411yescharacteristic
259C2×C4×C7C22×D412yes—
260C2×C4×C7C2×#4112yes—
261C2×C4×C7C2×#4111yescharacteristic
262C2×Dic7—28no—
263C2×Dic7—21no—
264C2×Dic7—21no—
265C2×Dic7—21no—
266C2×Dic7—21no—
267C2×Dic7—21no—
268C2×Dic7—21no—
269C2×Dic7—21no—
270C2×Dic7—24no—
271C2×Dic7—21no—
272C2×Dic7—21no—
273C2×Dic7—44no—
274C2×Dic7—48no—
275C2×Dic7—42no—
276C2×Dic7—42no—
277C2×Dic7—41no—
278C2×Dic7—41no—
279C2×Dic7—41no—
280C2×Dic7—41no—
281C2×Dic7—41no—
282C2×Dic7—41no—
283C2×Dic7#12012yes—
284C2×Dic7#12012yes—
285C2×Dic7#13712yes—
286C7×C8—24no—
287C7⋊C8—42no—
288C22×Q8—24no—
289C22×Q8—24no—
290C22×Q8—21no—
291C22×Q8—22no—
292C22×Q8C2×Dic714yes—
293C22×Q8C22×D711yescharacteristic
294C22×Q8C22×D712yes—
295C23×C4—21no—
296C23×C4—71no—
297C23×C4—71no—
298C23×C4—141no—
299C23×C4C22×D711yescharacteristic
300C2×Q16—24no—
301C2×Q16—24no—
302C2×Q16C2×Dic718yescomplemented
303C22×C8—72no—
304C22×C8C22×D712yes—
305C22⋊Q8—22no—
306C22⋊Q8C2×Dic714yes—
307C4×Q8—78no—
308C4×Q8—144no—
309C4×Q8—144no—
310C4×Q8—144no—
311C4×Q8—148no—
312C4×Q8—148no—
313C4×Q8—144no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—72no—
316C2×C4⋊C4—71no—
317C2×C4⋊C4—71no—
318C2×C4⋊C4—144no—
319C2×C4⋊C4—142no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—148no—
322C2×C4⋊C4—141no—
323C2×C4⋊C4—141no—
324C2×C4⋊C4—141no—
325C2×C4⋊C4C22×D711yescharacteristic
326C2×C22⋊C4—21no—
327C2×C22⋊C4—141no—
328C2×C42—71no—
329C2×C42—71no—
330C2×C42—71no—
331C2×C42—71no—
332C2×C42—144no—
333C2×C42—142no—
334C2×C42—142no—
335C2×C42—148no—
336C2×C42—141no—
337C2×C42—141no—
338#106—144no—
339#103—144no—
340#103—144no—
341#103—144no—
342#103C2×Dic718yescomplemented
343C22⋊C8—142no—
344C22⋊C8C2×Dic714yes—
345#97—144no—
346#95—141no—
347#95—141no—
348#95—141no—
349#95—141no—
350#95—141no—
351C22×C7—22no—
352C22×C7—22no—
353C22×C7—22no—
354C22×C7—22no—
355C22×C7—22no—
356C22×C7—22no—
357C22×C7#44114yescomplemented
358C22×C7C2×C4×D411yescharacteristic
359C22×C7C2×#12011yescharacteristic
360C22×C7#52511yescharacteristic
361C22×C7C2×#13711yescharacteristic
362C22×C7C2×#13711yescharacteristic
363C22×C7C2×#13711yescharacteristic
364C22×C7#57511yescharacteristic
365C4×C7—28no—
366C4×C7—22no—
367C4×C7—42no—
368C4×C7—42no—
369C4×C7C2×C4×D412yes—
370C4×C7#52512yes—
371Dic7—22no—
372Dic7—22no—
373Dic7—22no—
374Dic7—41no—
375Dic7—44no—
376Dic7—81no—
377Dic7—81no—
378C24D4×D711yescharacteristic
379C2×Q8—24no—
380C2×Q8—216no—
381C2×Q8—24no—
382C2×Q8—216no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8D4×D714yes—
391C2×Q8#115914yes—
392C2×Q8C22×Dic712yes—
393C2×Q8C22×Dic714yes—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—71no—
403C22×C4—71no—
404C22×C4—71no—
405C22×C4—71no—
406C22×C4—142no—
407C22×C4—142no—
408C22×C4—142no—
409C22×C4—142no—
410C22×C4—144no—
411C22×C4—142no—
412C22×C4—142no—
413C22×C4—141no—
414C22×C4—141no—
415C22×C4—281no—
416C22×C4—282no—
417C22×C4—282no—
418C22×C4D4×D711yescharacteristic
419C22×C4#115912yes—
420C22×C4C22×Dic712yes—
421C22×C4C23×D711yescharacteristic
422Q16—216no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—142no—
426C2×C8—142no—
427C2×C8—142no—
428C2×C8C22×Dic714yes—
429C4⋊C4—24no—
430C4⋊C4—21no—
431C4⋊C4—74no—
432C4⋊C4—148no—
433C4⋊C4—148no—
434C4⋊C4—144no—
435C4⋊C4—142no—
436C4⋊C4—142no—
437C4⋊C4—142no—
438C4⋊C4—282no—
439C4⋊C4—282no—
440C4⋊C4C22×Dic712yes—
441C22⋊C4—24no—
442C22⋊C4—282no—
443C42—74no—
444C42—142no—
445C42—142no—
446C42—148no—
447C42—148no—
448C42—142no—
449C42—282no—
450C42—282no—
451C2×C7—24no—
452C2×C7C4×#12011yescharacteristic
453C2×C7C2×#44111yescharacteristic
454C2×C7C4×#13711yescharacteristic
455C2×C7#300411yescharacteristic
456C2×C7#300711yescharacteristic
457C2×C7#311311yescharacteristic
458C2×C7#311711yescharacteristic
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23#648112yes—
464C23#648112yes—
465C23D4×Dic712yes—
466C23C2×D4×D711yescharacteristic
467Q8—216no—
468Q8—24no—
469Q8—24no—
470Q8—24no—
471Q8—24no—
472Q8—42no—
473Q8D4×Dic718yes—
474C2×C4—216no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—21no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—72no—
489C2×C4—72no—
490C2×C4—72no—
491C2×C4—148no—
492C2×C4—141no—
493C2×C4—141no—
494C2×C4—141no—
495C2×C4—141no—
496C2×C4—141no—
497C2×C4—141no—
498C2×C4—141no—
499C2×C4—141no—
500C2×C4—144no—
501C2×C4—141no—
502C2×C4—284no—
503C2×C4—288no—
504C2×C4—282no—
505C2×C4—282no—
506C2×C4—281no—
507C2×C4—281no—
508C2×C4—281no—
509C2×C4—281no—
510C2×C4—281no—
511C2×C4—281no—
512C2×C4D4×Dic712yes—
513C2×C4C2×D4×D712yes—
514C2×C4C2×#115912yes—
515C2×C4C2×#115911yescharacteristic
516C2×C4C23×Dic711yescharacteristic
517C8—24no—
518C8—282no—
519C7#2145511yescharacteristic, complemented, Sylow
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22#8995614yescomplemented
527C22#9027711yescharacteristic
528C22D7×#12011yescharacteristic
529C22C2×#648111yescharacteristic
530C22C2×#648111yescharacteristic
531C22D7×#13711yescharacteristic
532C22#9046811yescharacteristic
533C22C2×D4×Dic711yescharacteristic
534C4—28no—
535C4—22no—
536C4—42no—
537C4—42no—
538C4—142no—
539C4—142no—
540C4—142no—
541C4—281no—
542C4—284no—
543C4—561no—
544C4—561no—
545C4#9027712yes—
546C4C2×D4×Dic712yes—
547C2—24no—
548C2#1173761911yescharacteristic
549C2#1173762011yescharacteristic
550C2#1173762811yescharacteristic
551C2#1173762911yescharacteristic
552C2Dic7×#12011yescharacteristic
553C2C2×#8995611yescharacteristic
554C2Dic7×#13711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC2×C7×#451C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterC2×#451D7
hypocenterC7#21455
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
3. C2×#89956C2semidirect
6. #11729819C2semidirect
8. #11727992C2semidirect
9. #11727974C2semidirect
25. C7×#451C4semidirect
61. C2×C7×Q16C2×C4semidirect
97. C7×#103C2×C4semidirect
207. #451Dic7semidirect
302. C2×Q16C2×Dic7semidirect
342. #103C2×Dic7semidirect
357. C22×C7#441semidirect
519. C7#21455semidirect
526. C22#89956semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic72
C2×C7×#4512
C2×#899562
Dic7×C22⋊Q82
#117298862
#117298192
#117294462
#117279922
#117279742
#117276822
#117276602
#214557