Finite Group Databasebeta

← #13395345

Subgroups — #13395345

Order 1792.

1118 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2541C211yesmaximal, characteristic
2C2×#89847C212yesmaximal, complemented
3C2×#89847C211yesmaximal, characteristic, complemented
4C2×C4×#1142C211yesmaximal, characteristic, complemented
5C4⋊C4×Dic14C211yesmaximal, characteristic, complemented
6#11729518C211yesmaximal, characteristic, complemented
7#11729516C212yesmaximal, complemented
8#11729358C211yesmaximal, characteristic, complemented
9#11727414C212yesmaximal, complemented
10#11727406C212yesmaximal, complemented
11#11727238C211yesmaximal, characteristic, complemented
12C2×C7×#127C2211yescharacteristic
13C2×C7×#127C2212yes—
14C2×C7×#121—21no—
15C2×C7×#121—21no—
16C2×C4×C7×D4C2211yescharacteristic
17C22×#1142C2212yes—
18C22×#1142C2211yescharacteristic
19C2×C4×Dic14C2212yes—
20C2×C4×Dic14C2211yescharacteristic
21C7×#392—22no—
22C7×#390C2212yes—
23C4×C7×C4⋊C4C2211yescharacteristic
24C2×#6406C2212yes—
25C2×#6406C2212yes—
26C2×#6406C2211yescharacteristic
27C2×#6406C2212yes—
28C2×#6406C2212yes—
29#89847—28no—
30#89847—24no—
31C4×#1142—24no—
32#89748—21no—
33C4⋊C4×Dic7—21no—
34C2×#6384C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36#89707C2211yescharacteristic
37#89700C2212yes—
38C2×C4⋊(C7⋊C8)C2212yes—
39C2×C4⋊(C7⋊C8)C2211yescharacteristic
40C2×C4×C7⋊C8C2211yescharacteristic
41#89419—22no—
42#89324C2212yes—
43#89319C2212yes—
44#24728—71nomaximal, Sylow
45C22×C7×D4—21no—
46C22×C7×D4—22no—
47C22×C7×D4—22no—
48C22×C7×D4—21no—
49C22×C7×D4C2312yes—
50C22×C7×D4C2311yescharacteristic
51C23×C4×C7—21no—
52C22×Dic14C2311yescharacteristic
53C7×#127—24no—
54C7×#127—22no—
55C7×#127—22no—
56C7×#127—42no—
57C7×#121—44no—
58C7×#121—44no—
59C4×C7×D4—22no—
60C4×C7×D4—22no—
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C7×C4⋊C4C2311yescharacteristic
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4—22no—
66C2×C7×C22⋊C4—22no—
67C2×C42×C7C2312yes—
68C2×C42×C7C2311yescharacteristic
69C2×#1142—22no—
70C2×#1142—22no—
71C2×#1142—24no—
72C2×#1142—24no—
73C2×#1142—24no—
74C2×#1142D414yescomplemented
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7C2312yes—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×Dic7⋊C4—21no—
80C2×Dic7⋊C4—22no—
81C2×Dic7⋊C4—22no—
82C2×C4×Dic7—21no—
83C2×C4×Dic7—22no—
84C22×C7⋊C8C2311yescharacteristic
85C22×C7⋊C8C2312yes—
86C22×C7⋊C8C2311yescharacteristic
87C4×Dic14—22no—
88C4×Dic14—24no—
89C4×Dic14—42no—
90C4×Dic14D414yescomplemented
91C7×#95D412yes—
92#6406—24no—
93#6406—22no—
94#6406—24no—
95#6406—24no—
96#6406—42no—
97C7⋊#95—22no—
98#6384—22no—
99#6382—22no—
100C4⋊(C7⋊C8)—22no—
101C4⋊(C7⋊C8)—24no—
102C4×C7⋊C8—22no—
103C2×#460—71no—
104C2×#460—72no—
105C2×C4×SD16—71no—
106#2541D711yescharacteristic
107Q8×C4⋊C4—71no—
108#2065—72no—
109#2041—72no—
110#1957—71no—
111#1939—71no—
112#1880—72no—
113#1802—71no—
114C24×C7—21no—
115C24×C7—22no—
116C2×C7×D4—21no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4—21no—
120C2×C7×D4—21no—
121C2×C7×D4—22no—
122C2×C7×D4—22no—
123C2×C7×D4—21no—
124C2×C7×D4—22no—
125C2×C7×D4—21no—
126C2×C7×D4—22no—
127C2×C7×D4—44no—
128C2×C7×D4—44no—
129C2×C7×D4—44no—
130C2×C7×D4—42no—
131C2×C7×D4—42no—
132C2×C7×D4—44no—
133C2×C7×D4—42no—
134C2×C7×D4—42no—
135C2×C7×D4—81no—
136C2×C7×D4—82no—
137C2×C7×D4—81no—
138C2×C7×D4—82no—
139C2×C7×D4C2×D412yes—
140C22×C4×C7—21no—
141C22×C4×C7—42no—
142C22×C4×C7—42no—
143C22×C4×C7—42no—
144C22×C4×C7C2×D412yes—
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D412yes—
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2411yescharacteristic
149C22×Dic7—21no—
150C22×Dic7—21no—
151C22×Dic7—22no—
152C2×Dic14—21no—
153C2×Dic14—21no—
154C2×Dic14—21no—
155C2×Dic14C2×D412yes—
156C2×Dic14C2×D412yes—
157C2×Dic14#4112yes—
158C7×C4⋊C4—22no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4SD1614yescomplemented
162C7×C22⋊C4—44no—
163C7×C22⋊C4—42no—
164C7×C22⋊C4—44no—
165C42×C7—22no—
166C42×C7—22no—
167C42×C7C2×D412yes—
168C42×C7#4112yes—
169#1142—28no—
170#1142—48no—
171C4⋊Dic7—21no—
172C4⋊Dic7—22no—
173C4⋊Dic7—22no—
174C4⋊Dic7—42no—
175C4⋊Dic7—81no—
176C4⋊Dic7C2×D412yes—
177Dic7⋊C4—24no—
178Dic7⋊C4—44no—
179Dic7⋊C4—82no—
180C4×Dic7—24no—
181C4×Dic7—44no—
182C2×C7⋊C8—21no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8—22no—
185C2×C7⋊C8—22no—
186C2×C7⋊C8—21no—
187C2×C7⋊C8—22no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—21no—
190C2×C7⋊C8C2×D412yes—
191C22×SD16—71no—
192C22×SD16—72no—
193C2×#127C2×D711yescharacteristic
194C2×#127C2×D712yes—
195C2×#121—21no—
196C2×#121—21no—
197C2×C4×Q8—71no—
198C2×C4×Q8—72no—
199C2×C4×D4C2×D711yescharacteristic
200#460—144no—
201#460—148no—
202C4×SD16—144no—
203C2×#106—71no—
204C2×C4⋊C8—71no—
205C2×C4⋊C8—72no—
206C2×#103—71no—
207C2×#102—72no—
208C2×#102—72no—
209C2×#102—72no—
210C2×#102—71no—
211C2×#102—72no—
212C2×C4×C8—71no—
213#392—22no—
214#390C2×D712yes—
215#389—72no—
216#389—142no—
217#384—71no—
218#384—141no—
219C4×C4⋊C4—141no—
220C4×C4⋊C4C2×D711yescharacteristic
221#340—72no—
222#336—72no—
223C23×C7—22no—
224C23×C7—41no—
225C23×C7—41no—
226C23×C7—41no—
227C23×C7—41no—
228C23×C7—41no—
229C23×C7—41no—
230C23×C7—42no—
231C23×C7—42no—
232C23×C7—42no—
233C23×C7—42no—
234C23×C7—42no—
235C23×C7—42no—
236C23×C7—42no—
237C23×C7C22×D411yescharacteristic
238C7×D4—22no—
239C7×D4—22no—
240C7×D4—42no—
241C7×D4—41no—
242C7×D4—42no—
243C7×D4—42no—
244C7×D4—41no—
245C7×D4—42no—
246C7×D4—88no—
247C7×D4—84no—
248C7×D4—88no—
249C7×D4—84no—
250C2×C4×C7—22no—
251C2×C4×C7—21no—
252C2×C4×C7—22no—
253C2×C4×C7—21no—
254C2×C4×C7—22no—
255C2×C4×C7—22no—
256C2×C4×C7—22no—
257C2×C4×C7—41no—
258C2×C4×C7—41no—
259C2×C4×C7—41no—
260C2×C4×C7—84no—
261C2×C4×C7#12112yes—
262C2×C4×C7#12112yes—
263C2×C4×C7C2×SD1614yes—
264C2×C4×C7C2×SD1612yes—
265C2×C4×C7AGL(1,ℤ/8)12yes—
266C2×C4×C7C22×D411yescharacteristic
267C2×C4×C7C22×D411yescharacteristic
268C2×C4×C7C2×#4111yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7H(5,𝔽2)11yescharacteristic
271C2×C4×C72−1+411yescharacteristic
272C2×Dic7—22no—
273C2×Dic7—22no—
274C2×Dic7—22no—
275C2×Dic7—22no—
276C2×Dic7—41no—
277C2×Dic7—41no—
278C2×Dic7—42no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281Dic14—22no—
282Dic14—22no—
283Dic14—22no—
284Dic14#12114yescomplemented
285C7⋊C8—22no—
286C7⋊C8—42no—
287C7⋊C8—41no—
288C22×Q8—71no—
289C22×D4—21no—
290C22×D4—22no—
291C22×D4—22no—
292C22×D4—21no—
293C22×D4C22×D712yes—
294C22×D4C22×D711yescharacteristic
295C23×C4—21no—
296C2×SD16—74no—
297C2×SD16—142no—
298C2×SD16—142no—
299C2×SD16—144no—
300C2×SD16—144no—
301C2×SD16—144no—
302C22×C8—71no—
303C22×C8—72no—
304C22×C8—71no—
305#127—24no—
306#127—22no—
307#127—22no—
308#127—42no—
309#121—44no—
310#121—44no—
311C4×Q8—74no—
312C4×Q8—144no—
313C4×Q8—142no—
314C4×Q8—282no—
315C4×D4—22no—
316C4×D4—22no—
317C2×C4⋊C4—71no—
318C2×C4⋊C4—72no—
319C2×C4⋊C4—141no—
320C2×C4⋊C4—142no—
321C2×C4⋊C4—141no—
322C2×C4⋊C4—142no—
323C2×C4⋊C4—141no—
324C2×C4⋊C4C22⋊D711yescharacteristic
325C2×C4⋊C4C22⋊D711yescharacteristic
326C2×C4⋊C4C22×D711yescharacteristic
327C2×C22⋊C4—21no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C42—141no—
331C2×C42—142no—
332C2×C42C22×D712yes—
333C2×C42C22×D711yescharacteristic
334#106—142no—
335C4⋊C8—144no—
336C4⋊C8—142no—
337#103—142no—
338#102—142no—
339#102—144no—
340#102—144no—
341#102—144no—
342#102—282no—
343C4×C8—142no—
344#95—142no—
345#95C22⋊D712yes—
346C22×C7—42no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—81no—
350C22×C7—81no—
351C22×C7—81no—
352C22×C7—81no—
353C22×C7—82no—
354C22×C7—82no—
355C22×C7—82no—
356C22×C7—82no—
357C22×C7—82no—
358C22×C7—82no—
359C22×C7—82no—
360C22×C7C2×#12111yescharacteristic
361C22×C7D4211yescharacteristic
362C22×C7#54811yescharacteristic
363C22×C7C22×SD1611yescharacteristic
364C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
365C22×C7#57611yescharacteristic
366C22×C7#57711yescharacteristic
367C4×C7—24no—
368C4×C7—22no—
369C4×C7—42no—
370C4×C7—81no—
371C4×C7#46014yescomplemented
372C4×C7C2×#12111yescharacteristic
373C4×C7#53711yescharacteristic
374C4×C7#54711yescharacteristic
375C4×C7#55311yescharacteristic
376Dic7—24no—
377Dic7—42no—
378Dic7—82no—
379C24—21no—
380C24—22no—
381C2×Q8—72no—
382C2×Q8—72no—
383C2×Q8—72no—
384C2×Q8—141no—
385C2×Q8—141no—
386C2×Q8—141no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—21no—
393C2×D4—22no—
394C2×D4—21no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—42no—
402C2×D4—44no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—81no—
407C2×D4—82no—
408C2×D4—81no—
409C2×D4—82no—
410C2×D4D4×D712yes—
411C22×C4—21no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—142no—
418C22×C4C2×C22⋊D712yes—
419C22×C4C2×C22⋊D712yes—
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C23×D711yescharacteristic
423SD16—148no—
424SD16—288no—
425C2×C8—72no—
426C2×C8—141no—
427C2×C8—141no—
428C2×C8—141no—
429C2×C8—142no—
430C2×C8—142no—
431C2×C8—141no—
432C2×C8—142no—
433C2×C8—141no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—72no—
438C4⋊C4—144no—
439C4⋊C4—141no—
440C4⋊C4—142no—
441C4⋊C4—142no—
442C4⋊C4—284no—
443C4⋊C4—282no—
444C4⋊C4—562no—
445C4⋊C4—561no—
446C4⋊C4#114214yescomplemented
447C22⋊C4—44no—
448C22⋊C4—44no—
449C22⋊C4—42no—
450C42—22no—
451C42—22no—
452C42—144no—
453C42—284no—
454C42D4×D712yes—
455C42#115912yes—
456C2×C7—82no—
457C2×C7—162no—
458C2×C7#241611yescharacteristic
459C2×C7C2×#46011yescharacteristic
460C2×C7#304511yescharacteristic
461C2×C7#327411yescharacteristic
462C2×C7D4×SD1611yescharacteristic
463C2×C7#336211yescharacteristic
464C2×C7#336811yescharacteristic
465C23—22no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23C22×C22⋊D711yescharacteristic
480Q8—74no—
481Q8—142no—
482Q8—142no—
483Q8—142no—
484D4—22no—
485D4—22no—
486D4—42no—
487D4—41no—
488D4—42no—
489D4—42no—
490D4—41no—
491D4—42no—
492D4—88no—
493D4—84no—
494D4—88no—
495D4—84no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—84no—
507C2×C4—142no—
508C2×C4—142no—
509C2×C4—142no—
510C2×C4—142no—
511C2×C4—281no—
512C2×C4—281no—
513C2×C4—282no—
514C2×C4—282no—
515C2×C4—282no—
516C2×C4#649512yes—
517C2×C4C2×#114214yes—
518C2×C4C2×#114212yes—
519C2×C4#650112yes—
520C2×C4#650112yes—
521C2×C4C2×D4×D711yescharacteristic
522C2×C4C2×D4×D711yescharacteristic
523C2×C4C2×#115911yescharacteristic
524C2×C4#654811yescharacteristic
525C2×C4D7×#4111yescharacteristic
526C2×C4#655411yescharacteristic
527C8—142no—
528C8—282no—
529C8—281no—
530C7#2472811yescharacteristic, complemented, Sylow
531C22—42no—
532C22—42no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—82no—
545C22C2×#649511yescharacteristic
546C22C22×#114211yescharacteristic
547C22C2×#650111yescharacteristic
548C22D4×C22⋊D711yescharacteristic
549C22#9052211yescharacteristic
550C22#9052611yescharacteristic
551C22#9052711yescharacteristic
552C4—24no—
553C4—22no—
554C4—42no—
555C4—81no—
556C4—144no—
557C4—282no—
558C4—562no—
559C4#8984714yescomplemented
560C4D7×#12111yescharacteristic
561C4#9029911yescharacteristic
562C4#9030211yescharacteristic
563C4#9030511yescharacteristic
564C2—82no—
565C2—162no—
566C2#1174008611yescharacteristic
567C2C2×#8984711yescharacteristic
568C2#1174083011yescharacteristic
569C2#1174093711yescharacteristic
570C2D4×#114211yescharacteristic
571C2#1174103111yescharacteristic
572C2#1174103211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#2541C2
socleC23×C7C22×D4
hypercenter#2541D7
hypocenterC7#24728
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. C2×#89847C2semidirect
3. C2×#89847C2semidirect
4. C2×C4×#1142C2semidirect
5. C4⋊C4×Dic14C2semidirect
6. #11729518C2semidirect
7. #11729516C2semidirect
8. #11729358C2semidirect
9. #11727414C2semidirect
10. #11727406C2semidirect
11. #11727238C2semidirect
74. C2×#1142D4semidirect
90. C4×Dic14D4semidirect
161. C7×C4⋊C4SD16semidirect
284. Dic14#121semidirect
371. C4×C7#460semidirect
446. C4⋊C4#1142semidirect
530. C7#24728semidirect
559. C4#89847semidirect

Maximal subgroups

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

subgroupindex
C7×#25412
C2×#898472
C2×C4×#11422
C4⋊C4×Dic142
#117295182
#117295162
#117293582
#117274142
#117274062
#117272382
#247287