Finite Group Databasebeta

← #31009173

Subgroups — #31009173

Order 17152.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863832C211yesmaximal, characteristic
2#22569086C211yesmaximal, characteristic, complemented
3#22562083C212yesmaximal, complemented
4#22562073C212yesmaximal, complemented
5C4×#14523987C212yesmaximal, complemented
6C4×#14523984C212yesmaximal, complemented
7#22561998C211yesmaximal, characteristic, complemented
8#22561914C211yesmaximal, characteristic, complemented
9C2×#16863197C211yesmaximal, characteristic, complemented
10#22561806C211yesmaximal, characteristic, complemented
11C67×#2948C211yesmaximal, characteristic, complemented
12C2×#14524065C2211yescharacteristic
13#16863876—21no—
14#16863832C418yescomplemented
15C2×#14523987C2212yes—
16C2×#14523984C2212yes—
17C2×#14523983C2211yescharacteristic
18C22×C67⋊C16C2211yescharacteristic
19#16863498—22no—
20C4×Dic268C2212yescomplemented
21C4×D536C2212yescomplemented
22C2×#14523977C2211yescharacteristic
23#16863398—21no—
24#16863223C2212yescomplemented
25#16863221C2212yescomplemented
26#16863199C2211yescharacteristic, complemented
27#16863199C2211yescharacteristic, complemented
28#16863197C2212yes—
29#16863197C2212yes—
30#16863196C2211yescharacteristic, complemented
31#16863196C2211yescharacteristic, complemented
32#16863172C2212yescomplemented
33C4×C67⋊C16C2212yescomplemented
34C22×D8×C67C2211yescharacteristic
35#518×C67—21no—
36#442×C67—22no—
37C4×D8×C67C2212yes—
38C4×D8×C67C2212yes—
39#427×C67C2211yescharacteristic
40#418×C67—21no—
41#405×C67C2211yescharacteristic
42C2×#11761628—21no—
43#14524065—22no—
44#14524065C2×C414yescomplemented
45C2×Dic268C2311yescharacteristic
46C2×#11761617—21no—
47C2×D536C2311yescharacteristic
48C4×Dic134—22no—
49C4×D268—22no—
50(C22⋊D67)⋊C4—42no—
51#14524001—41no—
52C2×C4⋊Dic67—21no—
53C4⋊C4×D67—41no—
54#14523987C2×C418yescomplemented
55#14523984C2×C418yescomplemented
56#14523983C2×C414yescomplemented
57C2×C67⋊C16—22no—
58C2×C67⋊C16C2311yescharacteristic
59C2×C67⋊C16C2311yescharacteristic
60C2×C67⋊C16C2312yes—
61#14523977—21no—
62#14523977C2312yes—
63#14523968—22no—
64#14523965—22no—
65C22×D4×C67—21no—
66C2×D8×C67—21no—
67C2×D8×C67—22no—
68C2×D8×C67C2×C414yescomplemented
69C2×D8×C67C2312yes—
70C2×M16×C67D411yescharacteristic
71C22×C8×C67C2311yescharacteristic
72C4×D4×C67—22no—
73C4×D4×C67—22no—
74C4×D4×C67—41no—
75C4×D4×C67—41no—
76#117×C67—21no—
77#117×C67D411yescharacteristic
78C2×C22⋊C4×C67—42no—
79#107×C67C2311yescharacteristic
80#107×C67C2311yescharacteristic
81#106×C67—21no—
82#102×C67—22no—
83#102×C67—22no—
84#97×C67D412yes—
85C4×C8×C67C2312yes—
86#11761628—24no—
87#11761628—42no—
88C2×Dic134—21no—
89C2×D268—21no—
90C2×C22⋊D67—41no—
91C22×Dic67—41no—
92C2×C4×D67—41no—
93C2×C4×D67—42no—
94Dic268—21no—
95Dic268C22×C412yes—
96#11761617—24no—
97D536—21no—
98D536C22×C412yes—
99C4⋊Dic67—22no—
100C4⋊Dic67—41no—
101Dic67⋊C4—42no—
102(C2×D67)⋊C4—42no—
103C4×Dic67—42no—
104C67⋊C16—21no—
105C67⋊C16—21no—
106C67⋊C16C22×C414yes—
107C24×C67—41no—
108C2×D4×C67—24no—
109C2×D4×C67—22no—
110C2×D4×C67—41no—
111C2×D4×C67—42no—
112C22×C4×C67—42no—
113C22×C4×C67—42no—
114C22×C4×C67C2×D411yescharacteristic
115D8×C67—24no—
116D8×C67—22no—
117D8×C67C22×C414yes—
118M16×C67—22no—
119C2×C8×C67C22×C412yes—
120C2×C8×C67C2×D412yes—
121C2×C8×C67C2×D411yescharacteristic
122C2×C8×C67#4112yes—
123C2×C8×C67C2411yescharacteristic
124C4⋊C4×C67—21no—
125C4⋊C4×C67—21no—
126C4⋊C4×C67D811yescharacteristic
127C4⋊C4×C67D811yescharacteristic
128C22⋊C4×C67—41no—
129C22⋊C4×C67—42no—
130C22⋊C4×C67—42no—
131C22⋊C4×C67D812yes—
132C42×C67—41no—
133C42×C67C2×D412yes—
134C23×C67—42no—
135C23×C67—44no—
136C23×C67—161no—
137C23×C67—161no—
138C23×C67C2×D811yescharacteristic
139C22×D67—41no—
140D4×C67—24no—
141D4×C67—42no—
142D4×C67—44no—
143C2×C4×C67—41no—
144C2×C4×C67—41no—
145C2×C4×C67—162no—
146C2×C4×C67—162no—
147C2×C4×C67C4×D412yes—
148C2×C4×C67C2×D812yes—
149C2×C4×C67C2×D812yes—
150C2×C4×C67C2×D811yescharacteristic
151C2×C4×C67#13512yes—
152C2×C4×C67C22×D411yescharacteristic
153C22⋊D67—44no—
154C2×Dic67—41no—
155C2×Dic67—42no—
156C2×Dic67—161no—
157D268—22no—
158D268—41no—
159C4×D67—44no—
160C4×D67—162no—
161Dic134—22no—
162Dic134—41no—
163C8×C67—22no—
164C8×C67C4×D412yes—
165C8×C67C23×C411yescharacteristic
166C8×C67C2×#4111yescharacteristic
167C22×C67—21no—
168C22×C67—21no—
169C22×C67—44no—
170C22×C67—82no—
171C22×C67—82no—
172C22×C67—164no—
173C22×C67C4×D812yes—
174C22×C67C22×D811yescharacteristic
175C2×D67—42no—
176C2×D67—81no—
177C2×D67—81no—
178C4×C67—22no—
179C4×C67—22no—
180C4×C67—81no—
181C4×C67—81no—
182C4×C67C4×D812yes—
183C4×C67C2×C4×D411yescharacteristic
184C4×C67C2×#13511yescharacteristic
185Dic67—42no—
186Dic67—82no—
187#33454—671nomaximal, Sylow
188C2×C67—22no—
189C2×C67—84no—
190C2×C67C2×C4×D811yescharacteristic
191C2×C67#341511yescharacteristic
192C2×C67#341611yescharacteristic
193D67—82no—
194C2×#509—671no—
195#2951—671no—
196#2948D6711yescharacteristic, complemented
197#2177—672no—
198#2175—672no—
199C4×SD32—672no—
200C4×D16—672no—
201#2159—671no—
202#2138—671no—
203C2×#357—671no—
204#2108—671no—
205C67#3345411yescharacteristic, complemented, Sylow
206C2×#135—671no—
207C22×D8C2×D6711yescharacteristic
208#520—1341no—
209#518—21no—
210#509—678no—
211C2×SD32—672no—
212C2×D16—672no—
213C2×M32—671no—
214C22×C16—671no—
215#442—22no—
216#440—1342no—
217C4×Q16—672no—
218C4×D8—672no—
219C4×D8C2×D6712yes—
220C4×D8C2×D6712yes—
221#427C2×D6711yescharacteristic
222C2×#107—671no—
223#418—21no—
224#417—1341no—
225#405C2×D6711yescharacteristic
226#367—672no—
227#366—672no—
228#358—671no—
229#358—671no—
230#357—672no—
231#357—672no—
232#357—671no—
233#357—671no—
234#346—672no—
235C4×C16—672no—
236C2×#41—1341no—
237C22×D4—21no—
238#135—674no—
239#135—1342no—
240C2×Q16—671no—
241C2×SD16—1341no—
242C2×D8—21no—
243C2×D8—22no—
244C2×D8—671no—
245C2×D8C4×D6714yescomplemented
246C2×D8C22×D6712yes—
247C2×M16C22⋊D6711yescharacteristic
248C22×C8C22×D6711yescharacteristic
249C4×Q8—1342no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—2682no—
255C4×D4—1342no—
256#117—21no—
257#117—2681no—
258#117C22⋊D6711yescharacteristic
259C2×C4⋊C4—2681no—
260C2×C4⋊C4—1341no—
261C2×C22⋊C4—42no—
262SD32—678no—
263D16—678no—
264M32—674no—
265C2×C16—672no—
266C2×C16—671no—
267C2×C16—671no—
268C2×C16—1342no—
269#107—672no—
270#107—1341no—
271#107C22×D6711yescharacteristic
272#107C22×D6711yescharacteristic
273#106—21no—
274#103—1342no—
275#102—22no—
276#102—22no—
277#102—1342no—
278#97C22⋊D6712yes—
279C4×C8C22×D6712yes—
280C24—41no—
281#41—2682no—
282#41—1344no—
283C2×Q8—1341no—
284C2×D4—24no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—2681no—
289C2×D4—1341no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—2681no—
293C22×C4—2681no—
294C22×C4—2682no—
295C22×C4C2×C22⋊D6711yescharacteristic
296Q16—672no—
297Q16—1341no—
298SD16—1344no—
299D8—24no—
300D8—22no—
301D8—672no—
302D8—1341no—
303D8C2×C4×D6714yes—
304M16—22no—
305C2×C8C2×C4×D6712yes—
306C2×C8C2×C22⋊D6712yes—
307C2×C8C2×C22⋊D6711yescharacteristic
308C2×C8#1176162812yes—
309C2×C8C23×D6711yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—2681no—
313C4⋊C4—2682no—
314C4⋊C4—1342no—
315C4⋊C4#1176161311yescharacteristic
316C4⋊C4#1176161311yescharacteristic
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—2682no—
321C22⋊C4#1176161312yes—
322C42—41no—
323C42—2682no—
324C42C2×C22⋊D6712yes—
325C16—674no—
326C16—1341no—
327C16—1341no—
328C23—42no—
329C23—44no—
330C23—2681no—
331C23—161no—
332C23—161no—
333C23C2×#1176161311yescharacteristic
334Q8—2681no—
335Q8—1342no—
336D4—24no—
337D4—42no—
338D4—44no—
339D4—2684no—
340D4—2681no—
341D4—1342no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—2681no—
345C2×C4—2684no—
346C2×C4—2682no—
347C2×C4—162no—
348C2×C4—162no—
349C2×C4—10722no—
350C2×C4—10721no—
351C2×C4C4×C22⋊D6712yes—
352C2×C4C2×#1176161312yes—
353C2×C4C2×#1176161312yes—
354C2×C4C2×#1176161311yescharacteristic
355C2×C4#1452406312yes—
356C2×C4C22×C22⋊D6711yescharacteristic
357C8—22no—
358C8C4×C22⋊D6712yes—
359C8C22×C4×D6711yescharacteristic
360C8C2×#1176162811yescharacteristic
361C22—21no—
362C22—21no—
363C22—44no—
364C22—82no—
365C22—82no—
366C22—2682no—
367C22—164no—
368C22—5361no—
369C22—5361no—
370C22C4×#1176161312yes—
371C22C22×#1176161311yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—81no—
376C4—2682no—
377C4—5362no—
378C4C4×#1176161312yes—
379C4C2×C4×C22⋊D6711yescharacteristic
380C4C2×#1452406311yescharacteristic
381C2—22no—
382C2—84no—
383C2—5362no—
384C2C2×C4×#1176161311yescharacteristic
385C2#2257334811yescharacteristic
386C2#2257335611yescharacteristic
subgroupG/H
centerC22C22×#11761613
commutatorC8×C67C23×C4
FrattiniC2×C8C23×D67
FittingC67×#2948C2
socleC22×C67C22×D8
prime radicalC2×#16863832C2
cosocleC2×C8×C67C24
hypercenter#2948D67
hypocenterC67#33454
LCSG ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2948

Complements

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

NH ≅ G/Ntype
2. #22569086C2semidirect
3. #22562083C2semidirect
4. #22562073C2semidirect
5. C4×#14523987C2semidirect
6. C4×#14523984C2semidirect
7. #22561998C2semidirect
8. #22561914C2semidirect
9. C2×#16863197C2semidirect
10. #22561806C2semidirect
11. C67×#2948C2semidirect
14. #16863832C4semidirect
20. C4×Dic268C22semidirect
21. C4×D536C22semidirect
24. #16863223C22semidirect
25. #16863221C22semidirect
26. #16863199C22semidirect
27. #16863199C22semidirect
30. #16863196C22semidirect
31. #16863196C22semidirect
32. #16863172C22semidirect
33. C4×C67⋊C16C22semidirect
44. #14524065C2×C4semidirect
54. #14523987C2×C4semidirect
55. #14523984C2×C4semidirect
56. #14523983C2×C4semidirect
68. C2×D8×C67C2×C4semidirect
196. #2948D67semidirect
205. C67#33454semidirect
245. C2×D8C4×D67semidirect

Maximal subgroups

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

subgroupindex
C2×#168638322
#225690862
#225620832
#225620732
C4×#145239872
C4×#145239842
#225619982
#225619142
C2×#168631972
#225618062
C67×#29482
#3345467