Finite Group Databasebeta

← #13361998

Subgroups — #13361998

Order 1792.

730 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11741231C211yesmaximal, characteristic
2#11736271C211yesmaximal, characteristic
3#11735987C211yesmaximal, characteristic, complemented
4C7×#1765C211yesmaximal, characteristic, complemented
5#11729763C211yesmaximal, characteristic, complemented
6#11728125C211yesmaximal, characteristic, complemented
7#11728111C211yesmaximal, characteristic
8#11728041C211yesmaximal, characteristic, complemented
9#11727642C211yesmaximal, characteristic, complemented
10#11727593C211yesmaximal, characteristic, complemented
11#11727558C211yesmaximal, characteristic, complemented
12#11726683C211yesmaximal, characteristic, complemented
13#11726672C211yesmaximal, characteristic, complemented
14#11726499C211yesmaximal, characteristic, complemented
15#11726486C211yesmaximal, characteristic, complemented
16C2×C7×#131C2211yescharacteristic
17#90523C2211yescharacteristic
18C2×#6464C2211yescharacteristic
19#90208C2211yescharacteristic
20C2×C7×#104C2211yescharacteristic
21C7×#416C2211yescharacteristic
22C7×#405C2211yescharacteristic
23C2×C7×#97C2211yescharacteristic
24C2×#6411C2211yescharacteristic
25#90006C2211yescharacteristic
26#90005C2211yescharacteristic
27C2×#6399C2211yescharacteristic
28#89901C2211yescharacteristic
29#89900—22no—
30#89891C2211yescharacteristic
31#89890C2211yescharacteristic
32#89890C2211yescharacteristic
33#89885C2211yescharacteristic
34C8×C22⋊D7—22no—
35#89873C2211yescharacteristic
36C2×#6389C2211yescharacteristic
37C2×#6389C2211yescharacteristic
38C2×C8×Dic7C2211yescharacteristic
39C2×C8×Dic7C2211yescharacteristic
40#89774C2211yescharacteristic
41#89771C2211yescharacteristic
42#89704C2211yescharacteristic
43C2×#6379C2211yescharacteristic
44C2×C4×C7⋊C8C2211yescharacteristic
45#89569C412yescomplemented
46#89567C412yescomplemented
47#89537C412yescomplemented
48#89536C412yescomplemented
49#89473—22no—
50C8×Dic14—22no—
51C7×#341C2211yescharacteristic
52C7×#339C2211yescharacteristic
53#89353C2211yescharacteristic
54#89353C2211yescharacteristic
55#89350C2211yescharacteristic
56#89347C2211yescharacteristic
57#89327C2211yescharacteristic
58#89318C2211yescharacteristic
59#21663—71nomaximal, Sylow
60C2×C7×#41C2311yescharacteristic
61C2×#1157C2311yescharacteristic
62C7×#131—22no—
63C7×#131—22no—
64C2×C7×M16—21no—
65C2×C7×M16C2311yescharacteristic
66C2×C7×M16C2311yescharacteristic
67C22×C7×C8—21no—
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C7×#117C2311yescharacteristic
71C2×C42×C7C2311yescharacteristic
72#6510C2×C411yescharacteristic
73#6507C2×C411yescharacteristic
74#6503C2×C411yescharacteristic
75#6502—22no—
76#6499C2×C411yescharacteristic
77C4×C22⋊D7—22no—
78#6489C2311yescharacteristic
79C2×C4×Dic7C2311yescharacteristic
80C2×C4×Dic7C2311yescharacteristic
81C2×#1136C2311yescharacteristic
82C2×#1136C2311yescharacteristic
83C22×C7⋊C8C2311yescharacteristic
84C22×C7⋊C8C2311yescharacteristic
85#6464—22no—
86#6464—22no—
87C2×#1131—21no—
88C2×C8×D7—21no—
89#6453C2×C412yes—
90#6451C2×C412yes—
91#6441—22no—
92C4×Dic14—22no—
93C7×#104—22no—
94C7×#103C2×C412yes—
95C7×#102C2×C412yes—
96C7×#97—21no—
97C7×#97—22no—
98C4×C7×C8—21no—
99#6411—22no—
100#6409C2×C412yes—
101#6406C2×C412yes—
102C7⋊#95—22no—
103C22⋊(C7⋊C8)—22no—
104#6399—22no—
105#6395C2×C412yes—
106(C2×D7)⋊C8—22no—
107#6390C2×C412yes—
108#6389—21no—
109#6389—22no—
110#6389C2×C412yes—
111#6389C2×C412yes—
112Dic7⋊C8—22no—
113Dic7⋊C8—22no—
114Dic7⋊C8—22no—
115C8×Dic7—21no—
116C8×Dic7—22no—
117C8×Dic7D412yes—
118C8×Dic7D412yes—
119#6384C2×C412yes—
120#6383C2×C412yes—
121#6379—22no—
122C4⋊(C7⋊C8)—22no—
123C4×C7⋊C8—22no—
124#3135—71no—
125#2960—71no—
126#2931—71no—
127#1995—71no—
128#1963—71no—
129#1937—71no—
130#1937—71no—
131#1937—71no—
132#1937—71no—
133#1838—71no—
134#1765—71no—
135#1765—71no—
136#1765—71no—
137#1765D711yescharacteristic, complemented
138#1726—71no—
139C7×#41—21no—
140C7×#41—21no—
141C7×#41—21no—
142C7×#41—21no—
143C2×C7×Q8C22×C411yescharacteristic
144C2×C7×D4—21no—
145C2×C7×D4C22×C411yescharacteristic
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7C2411yescharacteristic
149C2×C22⋊D7—21no—
150C2×C22⋊D7—22no—
151C22×Dic7—21no—
152C22×Dic7#4112yes—
153#1157—21no—
154#1157—21no—
155#1157—21no—
156#1157—21no—
157C2×D28C22×C411yescharacteristic
158C2×C4×D7—21no—
159C2×Dic14—22no—
160C2×Dic14C22×C411yescharacteristic
161C7×M16—21no—
162C7×M16—21no—
163C7×M16—21no—
164C7×M16—21no—
165C7×M16—42no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8—21no—
169C2×C7×C8—21no—
170C2×C7×C8—42no—
171C2×C7×C8C22×C412yes—
172C2×C7×C8C2×D411yescharacteristic
173C2×C7×C8C2×D411yescharacteristic
174C2×C7×C8#4112yes—
175C2×C7×C8#4112yes—
176C7×C4⋊C4C22×C412yes—
177C7×C22⋊C4—21no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7#4112yes—
181C22⋊Dic7—21no—
182C22⋊Dic7—22no—
183(C2×D7)⋊C4—22no—
184C4⋊Dic7—22no—
185C4⋊Dic7C22×C412yes—
186Dic7⋊C4—22no—
187Dic7⋊C4—22no—
188Dic7⋊C4—22no—
189C4×Dic7—21no—
190C4×Dic7—21no—
191C4×Dic7C22×C411yescharacteristic
192C4×Dic7C22×C411yescharacteristic
193C4×Dic7C2×D412yes—
194C4×Dic7#4112yes—
195#1136—21no—
196#1136—21no—
197#1136—21no—
198#1136—21no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8C22×C412yes—
206C2×C7⋊C8C2×D412yes—
207#1131—42no—
208C8×D7—42no—
209C2×#131—71no—
210C2×#131C2×D711yescharacteristic
211#533—71no—
212#532—71no—
213#491—72no—
214#490—72no—
215#489—72no—
216#489—72no—
217#446—142no—
218C8×Q8—142no—
219#436—142no—
220C8×D4—142no—
221#424—71no—
222#424—71no—
223#423—71no—
224#423—71no—
225C2×#104—71no—
226C2×#104—71no—
227C2×#104—71no—
228C2×#104C2×D711yescharacteristic
229#416—71no—
230#416—71no—
231#416—71no—
232#416C2×D711yescharacteristic
233#408—71no—
234#408—71no—
235#405—71no—
236#405C2×D711yescharacteristic
237C2×#97—71no—
238C2×#97—71no—
239C2×#97C2×D711yescharacteristic
240C2×C4×C8—71no—
241C2×C4×C8—71no—
242C2×C4×C8—71no—
243#376—71no—
244#341—71no—
245#341—71no—
246#341—71no—
247#341C2×D711yescharacteristic
248#339—71no—
249#339—71no—
250#339—71no—
251#339C2×D711yescharacteristic
252C23×C7—21no—
253C23×C7C2×#4111yescharacteristic
254C22×D7—21no—
255C7×Q8—21no—
256C7×Q8—21no—
257C7×D4—21no—
258C7×D4—21no—
259C7×D4—41no—
260C7×D4—41no—
261C2×C4×C7—22no—
262C2×C4×C7—21no—
263C2×C4×C7—22no—
264C2×C4×C7—42no—
265C2×C4×C7—41no—
266C2×C4×C7—41no—
267C2×C4×C7C23×C411yescharacteristic
268C2×C4×C7C22×D411yescharacteristic
269C2×C4×C7C2×#4111yescharacteristic
270C2×C4×C7C2×#4111yescharacteristic
271C2×C4×C7C2×#4111yescharacteristic
272C2×C4×C7C2×#4111yescharacteristic
273C22⋊D7—44no—
274C22⋊D7—41no—
275C22⋊D7—41no—
276C2×Dic7—22no—
277C2×Dic7—21no—
278C2×Dic7—22no—
279C2×Dic7—42no—
280C2×Dic7—42no—
281C2×Dic7C4×D412yes—
282C2×Dic7C4×D412yes—
283D28—21no—
284D28—21no—
285C4×D7—41no—
286C4×D7—41no—
287Dic14—21no—
288Dic14—21no—
289Dic14—44no—
290C7×C8—22no—
291C7×C8—41no—
292C7×C8—41no—
293C7×C8#12412yes—
294C7×C8#12412yes—
295C7⋊C8—22no—
296C7⋊C8—22no—
297C7⋊C8—41no—
298C7⋊C8—41no—
299C2×#41—71no—
300C2×#41C22×D711yescharacteristic
301#131—22no—
302#131—22no—
303#131—142no—
304#131—142no—
305C2×M16—21no—
306C2×M16—71no—
307C2×M16—71no—
308C2×M16—141no—
309C2×M16C22×D711yescharacteristic
310C2×M16C22×D711yescharacteristic
311C22×C8—21no—
312C22×C8—71no—
313C22×C8—71no—
314C22×C8—141no—
315C22×C8C22×D711yescharacteristic
316C22×C8C22×D711yescharacteristic
317C4⋊Q8—72no—
318C4⋊Q8—71no—
319#127—71no—
320#125—72no—
321#124—71no—
322#124—71no—
323C22⋊Q8—142no—
324#121—142no—
325C4×Q8—142no—
326C4×D4—142no—
327#117—71no—
328#117C22×D711yescharacteristic
329C2×C42—71no—
330C2×C42—71no—
331C2×C42C22×D711yescharacteristic
332C4⋊C8—142no—
333C4⋊C8—142no—
334C4⋊C8—142no—
335C4⋊C8—142no—
336#104—22no—
337#104—142no—
338#104—142no—
339#104—142no—
340#103—72no—
341#103—72no—
342#103—72no—
343#103C4×D712yes—
344#102—72no—
345#102—72no—
346#102—72no—
347#102C4×D712yes—
348C22⋊C8—142no—
349C22⋊C8—142no—
350#97—21no—
351#97—22no—
352#97—72no—
353#97—72no—
354#97—142no—
355#97—141no—
356C4×C8—21no—
357C4×C8—72no—
358C4×C8—72no—
359C4×C8—142no—
360C4×C8—141no—
361C4×C8—142no—
362#95—142no—
363C22×C7—22no—
364C22×C7—42no—
365C22×C7—41no—
366C22×C7C2×C4×D411yescharacteristic
367C22×C7#52911yescharacteristic
368C22×C7#52911yescharacteristic
369C2×D7—42no—
370C2×D7—41no—
371C4×C7—42no—
372C4×C7—42no—
373C4×C7—41no—
374C4×C7C4×#4111yescharacteristic
375C4×C7C4×#4111yescharacteristic
376C4×C7C2×#12411yescharacteristic
377C4×C7#53211yescharacteristic
378Dic7—22no—
379Dic7—22no—
380Dic7—42no—
381Dic7—42no—
382Dic7—41no—
383#41—21no—
384#41—21no—
385#41—21no—
386#41—21no—
387#41—141no—
388#41—141no—
389#41—141no—
390#41—141no—
391C2×Q8—71no—
392C2×Q8—142no—
393C2×Q8C2×C4×D711yescharacteristic
394C2×D4—21no—
395C2×D4—71no—
396C2×D4—142no—
397C2×D4—141no—
398C2×D4C2×C4×D711yescharacteristic
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—72no—
402C22×C4—141no—
403C22×C4—141no—
404C22×C4C23×D711yescharacteristic
405M16—21no—
406M16—21no—
407M16—21no—
408M16—21no—
409M16—42no—
410M16—141no—
411M16—141no—
412M16—141no—
413M16—141no—
414M16—282no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—42no—
420C2×C8—72no—
421C2×C8—72no—
422C2×C8—141no—
423C2×C8—141no—
424C2×C8—141no—
425C2×C8—141no—
426C2×C8—141no—
427C2×C8—141no—
428C2×C8—282no—
429C2×C8C2×C4×D712yes—
430C2×C8#115712yes—
431C2×C8D4×D711yescharacteristic
432C2×C8D4×D711yescharacteristic
433C2×C8#115912yes—
434C4⋊C4—72no—
435C4⋊C4—142no—
436C4⋊C4—142no—
437C4⋊C4—142no—
438C4⋊C4—142no—
439C4⋊C4C2×C4×D712yes—
440C22⋊C4—21no—
441C22⋊C4—142no—
442C22⋊C4—142no—
443C22⋊C4—141no—
444C42—21no—
445C42—21no—
446C42—71no—
447C42—71no—
448C42—72no—
449C42—72no—
450C42—141no—
451C42—141no—
452C42#115712yes—
453C2×C7—21no—
454C2×C7—21no—
455C2×C7—81no—
456C2×C7C4×#12411yescharacteristic
457C2×C7#295512yes—
458D7—81no—
459C23—21no—
460C23—141no—
461C23D7×#4111yescharacteristic
462Q8—21no—
463Q8—21no—
464Q8—141no—
465Q8—141no—
466Q8—284no—
467D4—21no—
468D4—21no—
469D4—41no—
470D4—41no—
471D4—141no—
472D4—141no—
473D4—284no—
474D4—281no—
475D4—281no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—72no—
483C2×C4—72no—
484C2×C4—142no—
485C2×C4—142no—
486C2×C4—141no—
487C2×C4—282no—
488C2×C4—282no—
489C2×C4—281no—
490C2×C4—281no—
491C2×C4C22×C4×D711yescharacteristic
492C2×C4C2×#115711yescharacteristic
493C2×C4C2×#115711yescharacteristic
494C2×C4C2×D4×D711yescharacteristic
495C2×C4C2×#115911yescharacteristic
496C2×C4D7×#4111yescharacteristic
497C8—22no—
498C8—41no—
499C8—41no—
500C8—142no—
501C8—142no—
502C8—281no—
503C8—281no—
504C8#644812yes—
505C8#644812yes—
506C7#2166311yescharacteristic, complemented, Sylow
507C22—22no—
508C22—41no—
509C22—42no—
510C22—281no—
511C22—282no—
512C22#9021911yescharacteristic
513C22#9022911yescharacteristic
514C22C4×D4×D711yescharacteristic
515C4—42no—
516C4—41no—
517C4—42no—
518C4—142no—
519C4—142no—
520C4—282no—
521C4—281no—
522C4—282no—
523C4C4×#115711yescharacteristic
524C4C2×#644811yescharacteristic
525C4C4×#115911yescharacteristic
526C4#9023911yescharacteristic
527C2—21no—
528C2—21no—
529C2—81no—
530C2—561no—
531C2C4×#644811yescharacteristic
532C2#1173734212yes—
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1765C2
socleC22×C7C2×C4×D4
prime radical#90523C22
cosocleC22×C4×C7C24
hypercenter#1765D7
hypocenterC7#21663
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1765

Complements

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

NH ≅ G/Ntype
3. #11735987C2semidirect
4. C7×#1765C2semidirect
5. #11729763C2semidirect
6. #11728125C2semidirect
8. #11728041C2semidirect
9. #11727642C2semidirect
10. #11727593C2semidirect
11. #11727558C2semidirect
12. #11726683C2semidirect
13. #11726672C2semidirect
14. #11726499C2semidirect
15. #11726486C2semidirect
45. #89569C4semidirect
46. #89567C4semidirect
47. #89537C4semidirect
48. #89536C4semidirect
137. #1765D7semidirect
506. C7#21663semidirect

Maximal subgroups

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

subgroupindex
#117412312
#117362712
#117359872
C7×#17652
#117297632
#117281252
#117281112
#117280412
#117276422
#117275932
#117275582
#117266832
#117266722
#117264992
#117264862
#216637