Finite Group Databasebeta

← #13508919

Subgroups — #13508919

Order 1792.

558 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2951C211yesmaximal, characteristic, complemented
2#11736014C211yesmaximal, characteristic, complemented
3#11729836C211yesmaximal, characteristic, complemented
4#11729836C211yesmaximal, characteristic, complemented
5#11728668C211yesmaximal, characteristic, complemented
6#11728667C211yesmaximal, characteristic, complemented
7#11728666C211yesmaximal, characteristic, complemented
8C4×#6402C211yesmaximal, characteristic, complemented
9#11728643C211yesmaximal, characteristic, complemented
10#11728642C211yesmaximal, characteristic, complemented
11#11728639C211yesmaximal, characteristic, complemented
12C4×#6400C211yesmaximal, characteristic, complemented
13#11728354C211yesmaximal, characteristic, complemented
14#11728353C211yesmaximal, characteristic, complemented
15#11727637C211yesmaximal, characteristic, complemented
16C2×C7×#135C2211yescharacteristic
17C7×#520—21no—
18#90226—21no—
19C7×#440—21no—
20C7×#440—21no—
21C4×C7×Q16C2211yescharacteristic
22C4×C7×Q16C2211yescharacteristic
23C4×C7×D8C2211yescharacteristic
24C4×C7×D8C2211yescharacteristic
25C2×C7×#107C2211yescharacteristic
26C7×#417—21no—
27C7×#405C2211yescharacteristic
28C2×#6402C2211yescharacteristic
29C2×#6402C2211yescharacteristic
30C2×#6400C2211yescharacteristic
31C2×#6400C2211yescharacteristic
32#89907—21no—
33#89887C2211yescharacteristic
34#89887C2211yescharacteristic
35#89485—21no—
36#89483—21no—
37#89467C2211yescharacteristic
38C4×D56C2211yescharacteristic
39C4×D56C2211yescharacteristic
40#89455C2211yescharacteristic
41#89359C2211yescharacteristic
42#89359C2211yescharacteristic
43#89357C2211yescharacteristic
44#89357C2211yescharacteristic
45C22⋊(C7⋊C16)C2211yescharacteristic
46C22⋊(C7⋊C16)C2211yescharacteristic
47#89286C2211yescharacteristic
48#89286C2211yescharacteristic
49#89286C2211yescharacteristic
50#89286C2211yescharacteristic
51#89285C2211yescharacteristic
52#89285C2211yescharacteristic
53#89284C2211yescharacteristic
54#89284C2211yescharacteristic
55C4⋊(C7⋊C16)C2211yescharacteristic
56C4⋊(C7⋊C16)C2211yescharacteristic
57#89256C2211yescharacteristic
58C4×C7⋊C16C2211yescharacteristic
59#33679—71nomaximal, Sylow
60C2×C7×#41—21no—
61C7×#135—22no—
62C7×#135—22no—
63C2×C7×Q16C2311yescharacteristic
64C2×C7×SD16—21no—
65C2×C7×D8C2311yescharacteristic
66C2×C7×M16D411yescharacteristic
67C22×C7×C8C2311yescharacteristic
68C4×C7×Q8—21no—
69C4×C7×Q8—21no—
70C4×C7×D4—21no—
71C4×C7×D4—21no—
72C4×C7×D4—41no—
73C4×C7×D4—41no—
74C7×#117—41no—
75C7×#117D411yescharacteristic
76C2×C7×C4⋊C4—21no—
77C2×C7×C4⋊C4—41no—
78#6493—21no—
79#6493—21no—
80C2×D56—21no—
81C2×D56C2311yescharacteristic
82C2×D56C2311yescharacteristic
83#6457—41no—
84#6457—41no—
85#6447—41no—
86#6447—41no—
87#6437—21no—
88C4×D28—21no—
89C4×D28—21no—
90#6433—21no—
91C7×#107—21no—
92C7×#107C2311yescharacteristic
93C7×#107C2311yescharacteristic
94C7×#103—21no—
95C7×#103—21no—
96C7×#102—21no—
97C7×#102—21no—
98C7×#97D411yescharacteristic
99C7×#97D411yescharacteristic
100C4×C7×C8C2311yescharacteristic
101C4×C7×C8C2311yescharacteristic
102#6402—22no—
103#6402—22no—
104#6400—22no—
105#6400—22no—
106#6395—21no—
107#6395—21no—
108#6395—21no—
109#6395—21no—
110#6392C2311yescharacteristic
111#6392C2311yescharacteristic
112#6391—21no—
113C2×C7⋊C16C2311yescharacteristic
114C2×C7⋊C16C2311yescharacteristic
115C2×C7⋊C16C2311yescharacteristic
116C2×C7⋊C16C2311yescharacteristic
117#3176—71no—
118#2951D711yescharacteristic, complemented
119#2233—71no—
120#2232—71no—
121#2229—71no—
122#2228—71no—
123#2207—71no—
124#2206—71no—
125#2186—71no—
126#2186—71no—
127#2177—71no—
128#2175—71no—
129C4×SD32—71no—
130C4×D16—71no—
131#2164—71no—
132C7×#41—24no—
133C7×#41—42no—
134C2×C7×Q8—21no—
135C2×C7×D4—21no—
136C2×C7×D4—41no—
137C22×C4×C7—41no—
138C22×C4×C7—41no—
139C22×C4×C7—41no—
140C22×C4×C7—41no—
141C22×C4×C7C2×D411yescharacteristic
142C2×C22⋊D7—41no—
143C2×C22⋊D7—41no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28—41no—
147C2×D28—41no—
148C2×C4×D7—41no—
149C2×C4×D7—41no—
150C7×Q16—21no—
151C7×Q16#4112yes—
152C7×SD16—24no—
153C7×D8—21no—
154C7×D8#4112yes—
155C7×M16—41no—
156C2×C7×C8—21no—
157C2×C7×C8—21no—
158C2×C7×C8C2×D411yescharacteristic
159C2×C7×C8C2×D411yescharacteristic
160C2×C7×C8C2×D411yescharacteristic
161C2×C7×C8C2411yescharacteristic
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—41no—
165C7×C4⋊C4—41no—
166C7×C4⋊C4—41no—
167C7×C4⋊C4D811yescharacteristic
168C7×C4⋊C4D811yescharacteristic
169C7×C22⋊C4—41no—
170C7×C22⋊C4—41no—
171C7×C22⋊C4D811yescharacteristic
172C7×C22⋊C4D811yescharacteristic
173C42×C7—41no—
174C42×C7—41no—
175C42×C7C2×D411yescharacteristic
176C42×C7C2×D411yescharacteristic
177(C2×D7)⋊C4—41no—
178(C2×D7)⋊C4—41no—
179(C2×D7)⋊C4—41no—
180(C2×D7)⋊C4—41no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183Dic7⋊C4—41no—
184Dic7⋊C4—41no—
185D56—21no—
186D56—21no—
187D56—21no—
188D56—21no—
189D56—42no—
190C7⋊C16—21no—
191C7⋊C16—21no—
192C7⋊C16—21no—
193C7⋊C16—21no—
194C2×#135C2×D711yescharacteristic
195#540—141no—
196#520—21no—
197C2×SD32—71no—
198C2×SD32—71no—
199C2×D16—71no—
200C2×D16—71no—
201#499—71no—
202#493—71no—
203#490—141no—
204#469—141no—
205#466—71no—
206#466—71no—
207#442—141no—
208#440—21no—
209#440—21no—
210C4×Q16C2×D711yescharacteristic
211C4×Q16C2×D711yescharacteristic
212C4×D8—71no—
213C4×D8—71no—
214C4×D8C2×D711yescharacteristic
215C4×D8C2×D711yescharacteristic
216C2×#107C2×D711yescharacteristic
217#417—21no—
218#405C2×D711yescharacteristic
219#367—71no—
220#367—71no—
221#366—71no—
222#366—71no—
223C4⋊C16—71no—
224C4⋊C16—71no—
225#358—71no—
226#358—71no—
227#357—71no—
228#357—71no—
229#357—71no—
230#357—71no—
231#357—71no—
232#357—71no—
233C22⋊C16—71no—
234C22⋊C16—71no—
235#346—71no—
236C4×C16—71no—
237C23×C7—41no—
238C23×C7C2×D811yescharacteristic
239C22×D7—41no—
240C22×D7—41no—
241C7×Q8—22no—
242C7×Q8—41no—
243C7×D4—22no—
244C7×D4—44no—
245C7×D4—41no—
246C2×C4×C7—21no—
247C2×C4×C7—21no—
248C2×C4×C7—44no—
249C2×C4×C7—41no—
250C2×C4×C7—41no—
251C2×C4×C7—41no—
252C2×C4×C7—161no—
253C2×C4×C7—161no—
254C2×C4×C7—161no—
255C2×C4×C7C2×D811yescharacteristic
256C2×C4×C7C2×D811yescharacteristic
257C2×C4×C7C2×D811yescharacteristic
258C2×C4×C7C2×D811yescharacteristic
259C2×C4×C7C2×D811yescharacteristic
260C2×C4×C7C22×D411yescharacteristic
261C22⋊D7—161no—
262C22⋊D7—161no—
263C2×Dic7—41no—
264C2×Dic7—41no—
265D28—41no—
266D28—41no—
267D28—41no—
268D28—41no—
269D28—161no—
270D28—161no—
271C4×D7—161no—
272C4×D7—161no—
273C7×C8—21no—
274C7×C8—21no—
275C7×C8—21no—
276C7×C8C2×#4111yescharacteristic
277C7×C8C2×#4111yescharacteristic
278C2×#41—21no—
279#135—22no—
280#135—22no—
281C2×Q16C22×D711yescharacteristic
282C2×SD16—21no—
283C2×D8—71no—
284C2×D8—71no—
285C2×D8—141no—
286C2×D8C22×D711yescharacteristic
287C2×M16C22⋊D711yescharacteristic
288C22×C8C22×D711yescharacteristic
289#127—141no—
290#125—141no—
291#123—281no—
292#123—281no—
293#121—141no—
294#121—141no—
295#121—281no—
296#121—281no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4—41no—
302C4×D4—41no—
303C4×D4—141no—
304C4×D4—141no—
305#117—41no—
306#117C22⋊D711yescharacteristic
307C2×C4⋊C4—21no—
308C2×C4⋊C4—41no—
309SD32—142no—
310SD32—142no—
311D16—142no—
312D16—142no—
313C2×C16—71no—
314C2×C16—71no—
315C2×C16—71no—
316C2×C16—71no—
317#107—21no—
318#107—71no—
319#107—71no—
320#107C22×D711yescharacteristic
321#107C22×D711yescharacteristic
322#106—141no—
323#103—21no—
324#103—21no—
325#102—21no—
326#102—21no—
327#102—141no—
328#102—141no—
329#102—141no—
330#102—141no—
331#97C22⋊D711yescharacteristic
332#97C22⋊D711yescharacteristic
333C4×C8C22×D711yescharacteristic
334C4×C8C22×D711yescharacteristic
335C22×C7—21no—
336C22×C7—21no—
337C22×C7—21no—
338C22×C7—42no—
339C22×C7—81no—
340C22×C7—81no—
341C22×C7C22×D811yescharacteristic
342C2×D7—81no—
343C2×D7—81no—
344C2×D7—81no—
345C2×D7—81no—
346C2×D7—81no—
347C2×D7—81no—
348C4×C7—21no—
349C4×C7—21no—
350C4×C7—21no—
351C4×C7—21no—
352C4×C7—21no—
353C4×C7—42no—
354C4×C7—81no—
355C4×C7—81no—
356C4×C7#52511yescharacteristic
357C4×C7#57511yescharacteristic
358Dic7—81no—
359Dic7—81no—
360#41—24no—
361#41—42no—
362C2×Q8—21no—
363C2×D4—21no—
364C2×D4—41no—
365C2×D4—141no—
366C2×D4—141no—
367C2×D4—281no—
368C2×D4—281no—
369C2×D4—281no—
370C2×D4—281no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—281no—
376C22×C4—281no—
377C22×C4C2×C22⋊D711yescharacteristic
378Q16—21no—
379Q16#115712yes—
380SD16—24no—
381D8—21no—
382D8—141no—
383D8—141no—
384D8—141no—
385D8—141no—
386D8—282no—
387D8#115712yes—
388M16—41no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8C2×C22⋊D711yescharacteristic
392C2×C8C2×C22⋊D711yescharacteristic
393C2×C8C2×C22⋊D711yescharacteristic
394C2×C8C23×D711yescharacteristic
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—141no—
401C4⋊C4—141no—
402C4⋊C4—281no—
403C4⋊C4—281no—
404C4⋊C4#114111yescharacteristic
405C4⋊C4#114111yescharacteristic
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—281no—
409C22⋊C4—281no—
410C22⋊C4—281no—
411C22⋊C4—281no—
412C22⋊C4#114111yescharacteristic
413C22⋊C4#114111yescharacteristic
414C42—41no—
415C42—41no—
416C42C2×C22⋊D711yescharacteristic
417C42C2×C22⋊D711yescharacteristic
418C16—141no—
419C16—141no—
420C16—141no—
421C16—141no—
422C2×C7—41no—
423C2×C7—82no—
424C2×C7#310011yescharacteristic
425C2×C7#341511yescharacteristic
426C2×C7#341611yescharacteristic
427D7—161no—
428D7—161no—
429C23—41no—
430C23—281no—
431C23—281no—
432C23C2×#114111yescharacteristic
433Q8—22no—
434Q8—41no—
435D4—22no—
436D4—44no—
437D4—41no—
438D4—281no—
439D4—281no—
440D4—281no—
441D4—281no—
442D4—1121no—
443D4—1121no—
444D4—1121no—
445D4—1121no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—44no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—161no—
453C2×C4—161no—
454C2×C4—161no—
455C2×C4—281no—
456C2×C4—281no—
457C2×C4—1121no—
458C2×C4—1121no—
459C2×C4C2×#114111yescharacteristic
460C2×C4C2×#114111yescharacteristic
461C2×C4C2×#114111yescharacteristic
462C2×C4C2×#114111yescharacteristic
463C2×C4C2×#114111yescharacteristic
464C2×C4C22×C22⋊D711yescharacteristic
465C8—21no—
466C8—21no—
467C8—21no—
468C8C2×#115711yescharacteristic
469C8C2×#115711yescharacteristic
470C7#3367911yescharacteristic, complemented, Sylow
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—81no—
476C22—81no—
477C22—561no—
478C22—561no—
479C22—561no—
480C22—561no—
481C22—561no—
482C22—561no—
483C22C22×#114111yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—42no—
490C4—81no—
491C4—81no—
492C4—561no—
493C4—561no—
494C4#9048211yescharacteristic
495C4#9051311yescharacteristic
496C2—41no—
497C2—82no—
498C2—1121no—
499C2—1121no—
500C2#1173975311yescharacteristic
501C2#1174145211yescharacteristic
502C2#1174146111yescharacteristic
subgroupG/H
centerC22C22×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2951C2
socleC22×C7C22×D8
hypercenter#2951D7
hypocenterC7#33679
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. C7×#2951C2semidirect
2. #11736014C2semidirect
3. #11729836C2semidirect
4. #11729836C2semidirect
5. #11728668C2semidirect
6. #11728667C2semidirect
7. #11728666C2semidirect
8. C4×#6402C2semidirect
9. #11728643C2semidirect
10. #11728642C2semidirect
11. #11728639C2semidirect
12. C4×#6400C2semidirect
13. #11728354C2semidirect
14. #11728353C2semidirect
15. #11727637C2semidirect
118. #2951D7semidirect
470. C7#33679semidirect

Maximal subgroups

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

subgroupindex
C7×#29512
#117360142
#117298362
#117286682
#117286672
#117286662
C4×#64022
#117286432
#117286422
#117286392
C4×#64002
#117283542
#117283532
#117276372
#336797