Finite Group Databasebeta

← #33239307

Subgroups — #33239307

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989022C211yesmaximal, characteristic
2#135×Dic73C211yesmaximal, characteristic, complemented
3C2×#17988669C211yesmaximal, characteristic, complemented
4#22611804C212yesmaximal, complemented
5#22611798C212yesmaximal, complemented
6#22611780C212yesmaximal, complemented
7#22611775C212yesmaximal, complemented
8#22611717C211yesmaximal, characteristic, complemented
9#22611648C211yesmaximal, characteristic, complemented
10M32×Dic73C211yesmaximal, characteristic, complemented
11C73×#2139C211yesmaximal, characteristic, complemented
12C22×D584C2211yescharacteristic
13#41×Dic73—21no—
14C2×#14556553—21no—
15#17989022C418yescomplemented
16C2×#14556534C2212yes—
17C2×#14556532C2212yes—
18C2×#14556531C2211yescharacteristic
19Q16×Dic73C2211yescharacteristic, complemented
20Q16×Dic73C2211yescharacteristic, complemented
21SD16×Dic73—21no—
22SD16×Dic73—21no—
23D8×Dic73C2211yescharacteristic, complemented
24D8×Dic73C2211yescharacteristic, complemented
25#17988669—21no—
26#17988669—21no—
27#17988669C2212yes—
28#17988669C2212yes—
29#17988625C2211yescharacteristic
30#17988583—21no—
31C2×#14556511—21no—
32C2×C8×Dic73C2211yescharacteristic
33#17988414C2212yescomplemented
34#17988412C2212yescomplemented
35#17988389C2212yes—
36#17988389C2212yes—
37#17988362C2212yescomplemented
38C16×Dic73C2212yescomplemented
39C2×#135×C73C2211yescharacteristic
40C2×M32×C73C2211yescharacteristic
41C2×#107×C73C2211yescharacteristic
42#358×C73C2212yescomplemented
43#357×C73C2212yescomplemented
44C22×D292—21no—
45C22×C4×D73—41no—
46C2×D584—21no—
47C2×D584—22no—
48C2×D584C2×C414yescomplemented
49C2×D584C2312yes—
50C22×#10596831D411yescharacteristic
51Q8×Dic73—21no—
52Q8×Dic73—21no—
53D4×Dic73—21no—
54D4×Dic73—21no—
55D4×Dic73—41no—
56D4×Dic73—41no—
57#14556553—22no—
58#14556553—22no—
59#14556553—41no—
60#14556553—41no—
61#14556550—21no—
62#14556550—41no—
63C2×#11924005—41no—
64C2×C4×Dic73—41no—
65C2×C4×Dic73D411yescharacteristic
66#14556534C2×C418yescomplemented
67#14556532C2×C418yescomplemented
68#14556531C2×C414yescomplemented
69C2×#11924003C2312yes—
70#14556525C2311yescharacteristic
71#14556525C2311yescharacteristic
72#14556523—21no—
73#14556515—21no—
74#14556515—21no—
75#14556512—21no—
76#14556512—21no—
77#14556511—22no—
78#14556511—22no—
79C8×Dic73D411yescharacteristic
80C8×Dic73D411yescharacteristic
81C8×Dic73C2311yescharacteristic
82C8×Dic73C2311yescharacteristic
83C2×#41×C73—21no—
84#135×C73—22no—
85#135×C73C2×C414yescomplemented
86C2×Q16×C73C2311yescharacteristic
87C2×SD16×C73—21no—
88C2×D8×C73C2311yescharacteristic
89C22×C8×C73C2311yescharacteristic
90C2×C4⋊C4×C73—21no—
91M32×C73—22no—
92C2×C16×C73C2312yes—
93#107×C73—21no—
94#107×C73C2312yes—
95C23×D73—41no—
96C2×D292—24no—
97C2×D292—22no—
98C2×D292—41no—
99C2×D292—42no—
100C22×Dic73—41no—
101C22×Dic73—41no—
102C22×Dic73D811yescharacteristic
103C22×Dic73D811yescharacteristic
104C2×C4×D73—42no—
105C2×C4×D73—42no—
106C2×C4×D73—84no—
107D584—24no—
108D584—22no—
109D584C22×C414yes—
110C2×#10596831—22no—
111C2×#10596831C2×D411yescharacteristic
112C2×#10596831C2×D411yescharacteristic
113C4⋊Dic73—21no—
114C4⋊Dic73—21no—
115C4⋊Dic73—41no—
116C4⋊Dic73—41no—
117C22⋊Dic73—41no—
118C22⋊Dic73—41no—
119C22⋊Dic73—41no—
120#11924005—42no—
121#11924005—42no—
122C4×Dic73—41no—
123C4×Dic73—41no—
124C4×Dic73—41no—
125C4×Dic73D811yescharacteristic
126C4×Dic73D811yescharacteristic
127C4×Dic73C2×D411yescharacteristic
128C4×Dic73C2×D411yescharacteristic
129#11924003C22×C414yescomplemented
130#41×C73—24no—
131#41×C73—42no—
132C2×Q8×C73—21no—
133C2×D4×C73—21no—
134C2×D4×C73—41no—
135C22×C4×C73—41no—
136C22×C4×C73—41no—
137C22×C4×C73C2×D411yescharacteristic
138Q16×C73—21no—
139Q16×C73C22×C412yescomplemented
140SD16×C73—24no—
141D8×C73—21no—
142D8×C73C22×C412yescomplemented
143C2×C8×C73C22×C412yes—
144C2×C8×C73C2×D411yescharacteristic
145C2×C8×C73#4112yes—
146C2×C8×C73C2411yescharacteristic
147C4⋊C4×C73—22no—
148C4⋊C4×C73—41no—
149C16×C73—22no—
150C23×C73—41no—
151C23×C73C2×D811yescharacteristic
152C22×D73—42no—
153C22×D73—44no—
154C22×D73—82no—
155C22×D73—82no—
156Q8×C73—22no—
157Q8×C73—41no—
158D4×C73—22no—
159D4×C73—44no—
160D4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—44no—
163C2×C4×C73—42no—
164C2×C4×C73—82no—
165C2×C4×C73C4×D412yes—
166C2×C4×C73C2×D811yescharacteristic
167C2×C4×C73#13512yes—
168C2×C4×C73C22×D411yescharacteristic
169C2×Dic73—22no—
170C2×Dic73—41no—
171C2×Dic73—41no—
172C2×Dic73—41no—
173C2×Dic73—82no—
174C2×Dic73—82no—
175C2×Dic73C2×D811yescharacteristic
176C2×Dic73C2×D811yescharacteristic
177C2×Dic73C2×D811yescharacteristic
178C2×Dic73C2×D811yescharacteristic
179D292—24no—
180D292—42no—
181D292—44no—
182C4×D73—84no—
183C4×D73—84no—
184C8×C73C4×D412yes—
185C8×C73C23×C411yescharacteristic
186C8×C73C2×#4111yescharacteristic
187#10596831—21no—
188#10596831—21no—
189C22×C73—21no—
190C22×C73—21no—
191C22×C73—42no—
192C22×C73—81no—
193C22×C73—81no—
194C22×C73C4×D812yes—
195C22×C73C22×D811yescharacteristic
196C2×D73—44no—
197C2×D73—88no—
198C2×D73—82no—
199C2×D73—82no—
200C4×C73—42no—
201C4×C73—82no—
202C4×C73C4×D812yes—
203C4×C73C2×C4×D411yescharacteristic
204C4×C73C2×#13511yescharacteristic
205Dic73—21no—
206Dic73—21no—
207Dic73—21no—
208Dic73—81no—
209Dic73—81no—
210Dic73#50912yescomplemented
211C4×#509—731nomaximal, Sylow
212C2×C73—22no—
213C2×C73—82no—
214C2×C73C2×C4×D811yescharacteristic
215C2×C73C2×#50911yescharacteristic
216C2×C73#341411yescharacteristic
217D73—84no—
218C2×#509—731no—
219C4×#135—731no—
220C2×C4×D8—731no—
221#2177—732no—
222#2175—732no—
223C4×SD32—732no—
224C4×D16—732no—
225#2159—731no—
226#2140—731no—
227#2139D7311yescharacteristic, complemented
228C4×M32—731no—
229C73C4×#50911yescharacteristic, complemented, Sylow
230C2×#135C2×D7311yescharacteristic
231C22×D8—731no—
232C4×#41—1461no—
233C2×C4×D4—1461no—
234#509—738no—
235C2×SD32—732no—
236C2×D16—732no—
237C2×M32—731no—
238C2×M32C2×D7311yescharacteristic
239C4×Q16—731no—
240C4×Q16—731no—
241C4×SD16—1461no—
242C4×SD16—1461no—
243C4×D8—731no—
244C4×D8—731no—
245C4×D8—732no—
246C4×D8—732no—
247C4×D8—1461no—
248C4×D8—1461no—
249#427—731no—
250C2×#107C2×D7311yescharacteristic
251#416—1461no—
252C2×#102—1461no—
253C2×C4×C8—731no—
254#367—732no—
255#366—732no—
256#358C2×D7312yescomplemented
257#357—732no—
258#357—732no—
259#357C2×D7312yescomplemented
260#346—732no—
261C4×C16—732no—
262C2×#41—21no—
263C22×D4—1461no—
264C23×C4—2921no—
265#135—22no—
266#135C4×D7314yescomplemented
267C2×Q16C22×D7311yescharacteristic
268C2×SD16—21no—
269C2×D8—734no—
270C2×D8—732no—
271C2×D8—1461no—
272C2×D8—1462no—
273C2×D8C22×D7311yescharacteristic
274C22×C8—731no—
275C22×C8C22×D7311yescharacteristic
276C4×Q8—1461no—
277C4×Q8—1461no—
278C4×D4—2921no—
279C4×D4—2921no—
280C4×D4—2921no—
281C4×D4—2921no—
282C4×D4—1462no—
283C4×D4—1462no—
284C4×D4—1461no—
285C4×D4—1461no—
286#117—2921no—
287#117—1461no—
288C2×C4⋊C4—21no—
289C2×C22⋊C4—2921no—
290C2×C42—2921no—
291C2×C42—731no—
292SD32—738no—
293D16—738no—
294M32—22no—
295M32—734no—
296C2×C16—732no—
297C2×C16C22×D7312yes—
298#107—21no—
299#107—731no—
300#107—731no—
301#107C22×D7312yes—
302#106—1461no—
303#103—1461no—
304#103—1461no—
305#102—1462no—
306#102—1462no—
307#102—1461no—
308#102—1461no—
309C4×C8—731no—
310C4×C8—731no—
311C4×C8—731no—
312C4×C8—731no—
313C24—2921no—
314#41—24no—
315#41—42no—
316C2×Q8—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—2921no—
320C2×D4—2922no—
321C2×D4—1464no—
322C2×D4—1462no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—2922no—
326C22×C4—2922no—
327C22×C4—2921no—
328C22×C4—2921no—
329C22×C4—5844no—
330C22×C4—731no—
331C22×C4—731no—
332C22×C4D4×D7311yescharacteristic
333Q16—21no—
334Q16C2×C4×D7312yescomplemented
335SD16—24no—
336D8—21no—
337D8—734no—
338D8—1464no—
339D8—1462no—
340D8C2×C4×D7312yescomplemented
341C2×C8—731no—
342C2×C8—731no—
343C2×C8—1462no—
344C2×C8C2×C4×D7312yes—
345C2×C8D4×D7311yescharacteristic
346C2×C8#1192402712yes—
347C2×C8C23×D7311yescharacteristic
348C4⋊C4—22no—
349C4⋊C4—41no—
350C4⋊C4—2921no—
351C4⋊C4—2921no—
352C4⋊C4—1461no—
353C4⋊C4—1461no—
354C22⋊C4—2922no—
355C22⋊C4—2922no—
356C22⋊C4—2921no—
357C22⋊C4—2921no—
358C22⋊C4—2921no—
359C42—2921no—
360C42—2921no—
361C42—2921no—
362C42—731no—
363C42—731no—
364C42—731no—
365C42—731no—
366C16—22no—
367C16—734no—
368C23—41no—
369C23—2924no—
370C23—2922no—
371C23—5842no—
372C23—5842no—
373C23D8×D7311yescharacteristic
374Q8—22no—
375Q8—41no—
376D4—22no—
377D4—44no—
378D4—41no—
379D4—2924no—
380D4—2922no—
381D4—1464no—
382C2×C4—41no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—2921no—
387C2×C4—2921no—
388C2×C4—2921no—
389C2×C4—5844no—
390C2×C4—5844no—
391C2×C4—5842no—
392C2×C4—5842no—
393C2×C4—731no—
394C2×C4—731no—
395C2×C4—731no—
396C2×C4—731no—
397C2×C4—1462no—
398C2×C4#1455655112yes—
399C2×C4D8×D7311yescharacteristic
400C2×C4#1455661212yes—
401C2×C4C2×D4×D7311yescharacteristic
402C8—1461no—
403C8—1461no—
404C8#1455655112yes—
405C8C22×C4×D7311yescharacteristic
406C8C2×#1192402711yescharacteristic
407C22—21no—
408C22—21no—
409C22—42no—
410C22—81no—
411C22—81no—
412C22—2924no—
413C22—5848no—
414C22—5842no—
415C22—5842no—
416C22#1798866712yes—
417C22C2×D8×D7311yescharacteristic
418C4—42no—
419C4—82no—
420C4—5841no—
421C4—5841no—
422C4—732no—
423C4—1461no—
424C4—1461no—
425C4—1461no—
426C4#1798866712yes—
427C4C2×#1455655111yescharacteristic
428C4C2×#1455661211yescharacteristic
429C2—22no—
430C2—82no—
431C2—5844no—
432C2C2×#1798866711yescharacteristic
433C2#509×D7311yescharacteristic
434C2#2262306611yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2139C2
socleC22×C73C22×D8
prime radicalC2×#17989022C2
cosocleC2×C8×C73C24
hypercenter#2139D73
hypocenterC73C4×#509
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
2. #135×Dic73C2semidirect
3. C2×#17988669C2semidirect
4. #22611804C2semidirect
5. #22611798C2semidirect
6. #22611780C2semidirect
7. #22611775C2semidirect
8. #22611717C2semidirect
9. #22611648C2semidirect
10. M32×Dic73C2semidirect
11. C73×#2139C2semidirect
15. #17989022C4semidirect
19. Q16×Dic73C22semidirect
20. Q16×Dic73C22semidirect
23. D8×Dic73C22semidirect
24. D8×Dic73C22semidirect
33. #17988414C22semidirect
34. #17988412C22semidirect
37. #17988362C22semidirect
38. C16×Dic73C22semidirect
42. #358×C73C22semidirect
43. #357×C73C22semidirect
48. C2×D584C2×C4semidirect
66. #14556534C2×C4semidirect
67. #14556532C2×C4semidirect
68. #14556531C2×C4semidirect
85. #135×C73C2×C4semidirect
129. #11924003C22×C4semidirect
139. Q16×C73C22×C4semidirect
142. D8×C73C22×C4semidirect
210. Dic73#509semidirect
227. #2139D73semidirect
229. C73C4×#509semidirect
256. #358C2×D73semidirect
259. #357C2×D73semidirect
266. #135C4×D73semidirect
334. Q16C2×C4×D73semidirect
340. D8C2×C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890222
#135×Dic732
C2×#179886692
#226118042
#226117982
#226117802
#226117752
#226117172
#226116482
M32×Dic732
C73×#21392
C4×#50973