Finite Group Databasebeta

← #13403909

Subgroups — #13403909

Order 1792.

978 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2537C211yesmaximal, characteristic
2#11739827C211yesmaximal, characteristic, complemented
3C2×D4×C7⋊C8C211yesmaximal, characteristic, complemented
4C2×#89779C211yesmaximal, characteristic, complemented
5#11729435C212yesmaximal, complemented
6#11729433C212yesmaximal, complemented
7C4×#6406C211yesmaximal, characteristic, complemented
8#11727336C211yesmaximal, characteristic, complemented
9#11727321C211yesmaximal, characteristic, complemented
10#11727272C212yesmaximal, complemented
11#11727266C212yesmaximal, complemented
12C2×C7×#121—21no—
13C2×C7×#121C2212yes—
14C2×C4×C7×D4C2211yescharacteristic
15C2×C4×C7×D4C2211yescharacteristic
16C2×#6499—21no—
17C23×C7⋊C8C2212yes—
18C2×#6432C2211yescharacteristic
19C7×#394—22no—
20C7×#384C2211yescharacteristic
21C7×#380—21no—
22C7×#380C2212yes—
23C2×#6406C2211yescharacteristic
24C2×#6406C2212yes—
25C2×#6406C2211yescharacteristic
26#89984C2212yes—
27#89984C2212yes—
28C2×C22⋊(C7⋊C8)C2212yes—
29D4×C7⋊C8—24no—
30#89779—24no—
31#89746—21no—
32C2×#6381C2212yes—
33C2×#6381C2212yes—
34#89725—22no—
35C4×C4⋊Dic7C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C2×C4×C7⋊C8C2211yescharacteristic
38#89324C2212yes—
39#89324C2212yes—
40#89324C2212yes—
41#89324C2211yescharacteristic
42#89324C2211yescharacteristic
43#25338—71nomaximal, Sylow
44C22×C7×D4—22no—
45C22×C7×D4D411yescharacteristic
46C22×C7×D4C2311yescharacteristic
47C23×C4×C7—21no—
48C23×C4×C7C2312yes—
49C22×Dic14—21no—
50C7×#121—24no—
51C7×#121—42no—
52C7×#121—44no—
53C4×C7×D4—22no—
54C4×C7×D4—22no—
55C4×C7×D4—22no—
56C4×C7×D4—22no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C4⋊C4C2312yes—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—22no—
65C2×C7×C22⋊C4D412yes—
66C2×C42×C7C2311yescharacteristic
67C2×C22⋊Dic7—22no—
68C2×C22⋊Dic7—22no—
69#6499—44no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7C2312yes—
72C2×C4⋊Dic7C2311yescharacteristic
73C2×C4⋊Dic7C2311yescharacteristic
74C2×C4×Dic7—21no—
75C22×C7⋊C8—24no—
76C22×C7⋊C8—24no—
77C22×C7⋊C8—42no—
78C22×C7⋊C8C2311yescharacteristic
79C22×C7⋊C8C2312yes—
80C22×C7⋊C8C2311yescharacteristic
81#6432—22no—
82#6432—22no—
83C7×#95—22no—
84#6406—22no—
85#6406—24no—
86#6406—22no—
87C7⋊#95—21no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90C22⋊(C7⋊C8)—24no—
91#6381—24no—
92#6381—42no—
93C4⋊(C7⋊C8)D414yescomplemented
94C4×C7⋊C8—22no—
95C2×#474—71no—
96C2×C8×D4—71no—
97#2547—71no—
98#2537D711yescharacteristic
99#2075—72no—
100#2033—72no—
101#1990—71no—
102#1947—71no—
103#1893—72no—
104#1808—72no—
105C4×#102—71no—
106C24×C7—21no—
107C24×C7C2×D412yes—
108C2×C7×D4—22no—
109C2×C7×D4—22no—
110C2×C7×D4—21no—
111C2×C7×D4—21no—
112C2×C7×D4—21no—
113C2×C7×D4—21no—
114C2×C7×D4—21no—
115C2×C7×D4—44no—
116C2×C7×D4—44no—
117C2×C7×D4—82no—
118C2×C7×D4D814yes—
119C2×C7×D4#4112yes—
120C22×C4×C7—22no—
121C22×C4×C7—22no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—22no—
125C22×C4×C7—22no—
126C22×C4×C7—21no—
127C22×C4×C7—22no—
128C22×C4×C7—42no—
129C22×C4×C7—42no—
130C22×C4×C7—42no—
131C22×C4×C7C2×D412yes—
132C22×C4×C7C2×D411yescharacteristic
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2411yescharacteristic
135C22×Dic7—21no—
136C22×Dic7—21no—
137C22×Dic7—22no—
138C2×Dic14—42no—
139C2×Dic14—42no—
140C2×Dic14—42no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—21no—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—42no—
147C7×C4⋊C4—81no—
148C7×C22⋊C4—24no—
149C7×C22⋊C4—44no—
150C7×C22⋊C4—44no—
151C7×C22⋊C4—42no—
152C42×C7C2×D412yes—
153C42×C7#4112yes—
154C22⋊Dic7—44no—
155C22⋊Dic7—44no—
156C4⋊Dic7—22no—
157C4⋊Dic7—21no—
158C4⋊Dic7—21no—
159C4⋊Dic7—22no—
160C4⋊Dic7—21no—
161C4⋊Dic7—21no—
162C4⋊Dic7—42no—
163C4×Dic7—42no—
164C2×C7⋊C8—21no—
165C2×C7⋊C8—22no—
166C2×C7⋊C8—21no—
167C2×C7⋊C8—22no—
168C2×C7⋊C8—21no—
169C2×C7⋊C8—21no—
170C2×C7⋊C8—21no—
171C2×C7⋊C8—48no—
172C2×C7⋊C8C2×D414yes—
173C2×C7⋊C8#4112yes—
174C23×C8—72no—
175C2×C4⋊Q8—71no—
176C2×#124—141no—
177C2×#121—21no—
178C2×#121C2×D712yes—
179C2×C4×D4C2×D711yescharacteristic
180C2×C4×D4C2×D711yescharacteristic
181#474—144no—
182C8×D4—144no—
183C2×#107—72no—
184C2×#107—72no—
185C2×C4⋊C8—71no—
186C2×#102—71no—
187C2×#102—72no—
188C2×#102—71no—
189C2×C22⋊C8—72no—
190C2×C4×C8—71no—
191#397—142no—
192#394—22no—
193#391—141no—
194#384C2×D711yescharacteristic
195#380—21no—
196#380—72no—
197#380—72no—
198#380C2×D712yes—
199C4×C4⋊C4—71no—
200#340—71no—
201#340—72no—
202#340—71no—
203#340—72no—
204#340—72no—
205C23×C7—22no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7—22no—
211C23×C7—22no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7—41no—
215C23×C7—41no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7C2×D814yes—
219C23×C7C22×D411yescharacteristic
220C7×D4—22no—
221C7×D4—22no—
222C7×D4—28no—
223C7×D4—41no—
224C7×D4—41no—
225C7×D4—88no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—22no—
229C2×C4×C7—22no—
230C2×C4×C7—22no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—42no—
234C2×C4×C7—42no—
235C2×C4×C7—41no—
236C2×C4×C7—42no—
237C2×C4×C7—41no—
238C2×C4×C7—42no—
239C2×C4×C7—41no—
240C2×C4×C7—42no—
241C2×C4×C7—84no—
242C2×C4×C7#12112yes—
243C2×C4×C7#12112yes—
244C2×C4×C7C2×D812yes—
245C2×C4×C7#13512yes—
246C2×C4×C7C22×D411yescharacteristic
247C2×C4×C7C2×#4111yescharacteristic
248C2×C4×C7C2×#4111yescharacteristic
249C2×C4×C7C2×#4111yescharacteristic
250C2×C4×C7H(5,𝔽2)11yescharacteristic
251C2×C4×C72−1+411yescharacteristic
252C2×Dic7—42no—
253C2×Dic7—41no—
254C2×Dic7—41no—
255C2×Dic7—41no—
256C2×Dic7—41no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261Dic14—84no—
262C7⋊C8—22no—
263C7⋊C8—24no—
264C7⋊C8—41no—
265C22×Q8—141no—
266C22×D4—22no—
267C22×D4C22⋊D711yescharacteristic
268C22×D4C22×D711yescharacteristic
269C23×C4—21no—
270C23×C4C22×D712yes—
271C22×C8—71no—
272C22×C8—72no—
273C22×C8—71no—
274C22×C8—144no—
275C22×C8—144no—
276C22×C8—282no—
277C4⋊Q8—142no—
278C4⋊Q8—142no—
279#124—284no—
280#121—24no—
281#121—42no—
282#121—44no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—72no—
290C2×C4⋊C4—71no—
291C2×C4⋊C4—71no—
292C2×C4⋊C4—141no—
293C2×C4⋊C4C22⋊D711yescharacteristic
294C2×C4⋊C4C22×D711yescharacteristic
295C2×C4⋊C4C22×D712yes—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—142no—
300C2×C22⋊C4—142no—
301C2×C22⋊C4C22⋊D712yes—
302C2×C42—141no—
303C2×C42C22×D711yescharacteristic
304#107—144no—
305#107—282no—
306C4⋊C8—74no—
307#102—142no—
308#102—144no—
309#102—142no—
310C22⋊C8—144no—
311C4×C8—142no—
312#95—22no—
313#95—141no—
314#95—142no—
315#95—142no—
316C22×C7—24no—
317C22×C7—24no—
318C22×C7—24no—
319C22×C7—42no—
320C22×C7—42no—
321C22×C7—42no—
322C22×C7—42no—
323C22×C7—42no—
324C22×C7—42no—
325C22×C7—42no—
326C22×C7—81no—
327C22×C7—81no—
328C22×C7—81no—
329C22×C7—81no—
330C22×C7C2×#12111yescharacteristic
331C22×C7#54611yescharacteristic
332C22×C7#54711yescharacteristic
333C22×C7C22×D811yescharacteristic
334C22×C7C2×#13511yescharacteristic
335C22×C7#57711yescharacteristic
336C22×C7#57811yescharacteristic
337C4×C7—22no—
338C4×C7—42no—
339C4×C7—82no—
340C4×C7—81no—
341C4×C7#46614yes—
342C4×C7#52911yescharacteristic
343C4×C7#54211yescharacteristic
344C4×C7#54611yescharacteristic
345C4×C7#54811yescharacteristic
346Dic7—81no—
347Dic7—81no—
348Dic7—82no—
349C24—21no—
350C24C2×C22⋊D712yes—
351C2×Q8—282no—
352C2×Q8—282no—
353C2×Q8—282no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—82no—
364C2×D4#114114yes—
365C2×D4#115912yes—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—141no—
378C22×C4—141no—
379C22×C4—142no—
380C22×C4C2×C22⋊D712yes—
381C22×C4C2×C22⋊D711yescharacteristic
382C22×C4C2×C22⋊D711yescharacteristic
383C22×C4C23×D711yescharacteristic
384C2×C8—74no—
385C2×C8—72no—
386C2×C8—141no—
387C2×C8—142no—
388C2×C8—141no—
389C2×C8—142no—
390C2×C8—141no—
391C2×C8—141no—
392C2×C8—141no—
393C2×C8—288no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—42no—
400C4⋊C4—81no—
401C4⋊C4—142no—
402C4⋊C4—141no—
403C4⋊C4—142no—
404C4⋊C4—141no—
405C4⋊C4—141no—
406C4⋊C4—141no—
407C4⋊C4—282no—
408C22⋊C4—24no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—284no—
413C22⋊C4—284no—
414C42—282no—
415C42D4×D712yes—
416C42#115912yes—
417C2×C7—44no—
418C2×C7—82no—
419C2×C7#242711yescharacteristic
420C2×C7C2×#46611yescharacteristic
421C2×C7#306311yescharacteristic
422C2×C7#329411yescharacteristic
423C2×C7#330211yescharacteristic
424C2×C7#333411yescharacteristic
425C2×C7#334011yescharacteristic
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23C2×#114114yes—
440C23C22×C22⋊D711yescharacteristic
441Q8—564no—
442D4—22no—
443D4—22no—
444D4—28no—
445D4—41no—
446D4—41no—
447D4—88no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—84no—
464C2×C4—281no—
465C2×C4—281no—
466C2×C4—282no—
467C2×C4—282no—
468C2×C4—281no—
469C2×C4—282no—
470C2×C4—281no—
471C2×C4—281no—
472C2×C4—281no—
473C2×C4C2×#114112yes—
474C2×C4#650212yes—
475C2×C4#650212yes—
476C2×C4#651312yes—
477C2×C4C2×D4×D711yescharacteristic
478C2×C4C2×#115911yescharacteristic
479C2×C4C2×#115911yescharacteristic
480C2×C4#654811yescharacteristic
481C2×C4D7×#4111yescharacteristic
482C2×C4#655411yescharacteristic
483C8—144no—
484C8—142no—
485C8—281no—
486C7#2533811yescharacteristic, complemented, Sylow
487C22—24no—
488C22—24no—
489C22—24no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22C22×#114111yescharacteristic
502C22C2×#650211yescharacteristic
503C22#9049711yescharacteristic
504C22C2×#651311yescharacteristic
505C22#9051511yescharacteristic
506C22#9052711yescharacteristic
507C22#9052911yescharacteristic
508C4—22no—
509C4—42no—
510C4—81no—
511C4—82no—
512C4—561no—
513C4—562no—
514C4—561no—
515C4#8981014yes—
516C4#9028911yescharacteristic
517C4#9029111yescharacteristic
518C4#9029411yescharacteristic
519C4#9030311yescharacteristic
520C2—44no—
521C2—82no—
522C2C2×#8981011yescharacteristic
523C2#1174009211yescharacteristic
524C2#1174086811yescharacteristic
525C2#1174092711yescharacteristic
526C2#1174095511yescharacteristic
527C2#1174096111yescharacteristic
528C2#1174109111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
Fitting, prime radicalC7×#2537C2
socleC23×C7C22×D4
hypercenter#2537D7
hypocenterC7#25338
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2537

Complements

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

NH ≅ G/Ntype
2. #11739827C2semidirect
3. C2×D4×C7⋊C8C2semidirect
4. C2×#89779C2semidirect
5. #11729435C2semidirect
6. #11729433C2semidirect
7. C4×#6406C2semidirect
8. #11727336C2semidirect
9. #11727321C2semidirect
10. #11727272C2semidirect
11. #11727266C2semidirect
93. C4⋊(C7⋊C8)D4semidirect
486. C7#25338semidirect

Maximal subgroups

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

subgroupindex
C7×#25372
#117398272
C2×D4×C7⋊C82
C2×#897792
#117294352
#117294332
C4×#64062
#117273362
#117273212
#117272722
#117272662
#253387