Finite Group Databasebeta

← #31720179

Subgroups — #31720179

Order 17152.

2604 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22574983C211yesmaximal, characteristic
2#22574920C212yesmaximal, complemented
3#22569127C216yesmaximal, complemented
4#22569120C216yesmaximal, complemented
5#22569113C216yesmaximal, complemented
6C2×#16863519C216yesmaximal, complemented
7#22568820C211yesmaximal, characteristic, complemented
8#22568757C212yesmaximal, complemented
9C67×#2875C211yesmaximal, characteristic, complemented
10#16864232C2212yes—
11#16864213C414yescomplemented
12#16864208C418yescomplemented
13#16864201C414yescomplemented
14#16864196C2216yes—
15C2×#14524065C2216yes—
16#16864163C2212yes—
17#16864162C22112yescomplemented
18#16864161—28no—
19C2×#14524049C22112yescomplemented
20#131×D67—28no—
21#16863519C2214yescomplemented
22#16863519C22112yescomplemented
23#16863519C22112yescomplemented
24#16863519C22112yescomplemented
25#16863514C22124yescomplemented
26#16863445C2216yescomplemented
27C2×#14523979C2216yescomplemented
28#16863414C22112yescomplemented
29C2×#14523971C22112yescomplemented
30C2×#131×C67C2211yescharacteristic
31C2×#131×C67C2211yescharacteristic
32C4×#41×C67C2211yescharacteristic
33#405×C67C2213yescomplemented
34#405×C67C2213yescomplemented
35C4×M16×C67C2213yescomplemented
36C2×#97×C67C2213yescomplemented
37#14524091—22no—
38#14524090C2×C418yes—
39#14524089C2×C418yes—
40#14524088—26no—
41C2×#11761628C23112yes—
42#41×D67—28no—
43#14524076C2×C4124yescomplemented
44#14524071C2×C4124yescomplemented
45#14524065C2×C4112yescomplemented
46#14524065C2×C4112yescomplemented
47#14524065C23112yes—
48#14524065C2314yes—
49C2×Dic268C2×C416yescomplemented
50C2×#11761617C2×C4112yescomplemented
51C2×D536C2×C416yescomplemented
52#14524051—224no—
53#14524050—28no—
54#14524049—212no—
55#14524049—212no—
56#14524049C23124yes—
57#14524049C2318yes—
58C2×#11761612C23112yes—
59C2×#11761611—212no—
60M16×D67—224no—
61C2×C8×D67—212no—
62#14523979C23112yes—
63#14523979C2314yes—
64#14523971C23124yes—
65#14523971C2318yes—
66C2×#41×C67C2311yescharacteristic
67#131×C67C2×C414yes—
68#131×C67C2×C414yes—
69#131×C67D418yes—
70C2×M16×C67C2313yes—
71C2×M16×C67C2313yes—
72C22×C8×C67C2313yes—
73C22×C8×C67C2313yes—
74C4×Q8×C67—21no—
75C4×D4×C67—23no—
76#117×C67C2313yes—
77C2×C42×C67C2313yes—
78#97×C67C2311yescharacteristic
79#97×C67C2313yes—
80#97×C67C2313yes—
81#97×C67C2313yes—
82C4×C8×C67C2316yes—
83#11761629—28no—
84#11761628—212no—
85#11761628—212no—
86#11761628C22×C4124yes—
87#11761628C2418yes—
88#11761627—224no—
89C2×Dic134C22×C4112yes—
90Q8×D67—28no—
91C2×D268C22×C4112yes—
92C2×C22⋊D67—212no—
93D4×D67—224no—
94C2×C4×D67—212no—
95Dic268C22×C414yes—
96Dic268C22×C4112yes—
97#11761617C22×C418yes—
98#11761617C22×C4124yes—
99D536C22×C414yes—
100D536C22×C4112yes—
101#11761612—212no—
102#11761612C2418yes—
103#11761611—224no—
104#11761611—28no—
105C2×C67⋊C8—212no—
106C8×D67—28no—
107C8×D67—224no—
108#41×C67C22×C414yes—
109#41×C67C2×D414yes—
110C2×Q8×C67#4111yescharacteristic
111C2×D4×C67#4113yes—
112C22×C4×C67—23no—
113C22×C4×C67C2413yes—
114M16×C67C22×C4112yes—
115M16×C67C2×D4112yes—
116C2×C8×C67C22×C416yes—
117C2×C8×C67C22×C416yes—
118C2×C8×C67C2×D4112yes—
119C2×C8×C67C2411yescharacteristic
120C2×C8×C67C2413yes—
121C2×C8×C67C2411yescharacteristic
122C2×C8×C67C2413yes—
123C4⋊C4×C67—23no—
124C22⋊C4×C67—23no—
125C42×C67—23no—
126C42×C67C2411yescharacteristic
127C42×C67C2413yes—
128C23×C67C2×#4113yes—
129C22×D67—212no—
130Q8×C67C4×D414yes—
131D4×C67C4×D4112yes—
132C2×C4×C67—21no—
133C2×C4×C67—23no—
134C2×C4×C67—83no—
135C2×C4×C67C23×C416yes—
136C2×C4×C67C22×D416yes—
137C2×C4×C67C2×#4113yes—
138C2×C4×C67C2511yescharacteristic
139C22⋊D67—224no—
140C22⋊D67—28no—
141C2×Dic67—212no—
142D268—212no—
143D268C23×C418yes—
144C4×D67—224no—
145C4×D67—28no—
146Dic134—212no—
147Dic134C23×C418yes—
148C8×C67C23×C411yescharacteristic
149C8×C67C23×C411yescharacteristic
150C8×C67C23×C416yes—
151C8×C67C22×D412yes—
152C8×C67C22×D416yes—
153C67⋊C8—28no—
154C22×C67—43no—
155C22×C67C2×C4×D416yes—
156C22×C67C22×#4111yescharacteristic
157C2×D67—28no—
158C2×D67—812no—
159C4×C67—43no—
160C4×C67—41no—
161C4×C67C2×C4×D416yes—
162C4×C67C24×C411yescharacteristic
163C4×C67C23×D411yescharacteristic
164Dic67—28no—
165#60184—671nomaximal, Sylow
166C2×C67—21no—
167C2×C67—26no—
168C2×C67C22×C4×D411yescharacteristic
169D67—48no—
170#3584—671no—
171#3573—672no—
172#2957—676no—
173#2956—676no—
174#2955—676no—
175C2×#444—676no—
176#2916—671no—
177#2901—672no—
178#2875D6711yescharacteristic, complemented
179C67#6018411yescharacteristic, complemented, Sylow
180#585—672no—
181#578—674no—
182#577—678no—
183#576—674no—
184#575—676no—
185C2×#135—676no—
186#568—672no—
187#568—6712no—
188#568—1348no—
189C2×#131—6712no—
190C2×#131—1348no—
191C2×#131C2×D6711yescharacteristic
192C2×#131C2×D6711yescharacteristic
193C4×#41C2×D6711yescharacteristic
194#444—674no—
195#444—6712no—
196#444—6712no—
197#444—6712no—
198#443—6724no—
199#430—676no—
200C2×#108—676no—
201#421—6712no—
202C2×#104—6712no—
203#405C2×D6713yescomplemented
204#405C2×D6713yescomplemented
205C4×M16C2×D6713yescomplemented
206C2×#97C2×D6713yescomplemented
2072−1+4—678no—
2082−1+4—1342no—
209H(5,𝔽2)—678no—
210H(5,𝔽2)—1346no—
211C2×#41—6712no—
212C2×#41—1348no—
213C2×#41C22×D6711yescharacteristic
214#137—6724no—
215AGL(1,ℤ/8)—6724no—
216#135—6712no—
217#135—6712no—
218#135—674no—
219#135—6712no—
220C2×Q16—676no—
221C2×SD16—6712no—
222C2×D8—676no—
223#131—678no—
224#131—6724no—
225#131—1348no—
226#131—13424no—
227#131—13412no—
228#131—13412no—
229#131C4×D6714yes—
230#131C4×D6714yes—
231#131D26818yes—
232C2×M16—6712no—
233C2×M16—13424no—
234C2×M16—13412no—
235C2×M16C22×D6713yes—
236C2×M16C22×D6713yes—
237C22×C8—13412no—
238C22×C8C22×D6713yes—
239C22×C8C22×D6713yes—
240C4×Q8—21no—
241C4×D4—23no—
242#117C22×D6713yes—
243C2×C42C22×D6713yes—
244#108—674no—
245#108—6712no—
246#104—678no—
247#104—6724no—
248#97C22×D6711yescharacteristic
249#97C22×D6713yes—
250#97C22×D6713yes—
251#97C22×D6713yes—
252C4×C8C22×D6716yes—
253#41—6724no—
254#41—678no—
255#41—1348no—
256#41—13424no—
257#41—13412no—
258#41—13412no—
259#41C2×C4×D6714yes—
260#41C2×D26814yes—
261C2×Q8—6712no—
262C2×Q8—1348no—
263C2×Q8#1176162811yescharacteristic
264C2×D4—6712no—
265C2×D4—13412no—
266C2×D4—13424no—
267C2×D4#1176162813yes—
268C22×C4—23no—
269C22×C4—13412no—
270C22×C4C23×D6713yes—
271Q16—674no—
272Q16—6712no—
273SD16—678no—
274SD16—6724no—
275D8—674no—
276D8—6712no—
277M16—678no—
278M16—13412no—
279M16—1348no—
280M16—13424no—
281M16C2×C4×D67112yes—
282M16C2×D268112yes—
283C2×C8—13412no—
284C2×C8—1348no—
285C2×C8—13424no—
286C2×C8C2×C4×D6716yes—
287C2×C8C2×C4×D6716yes—
288C2×C8C2×D268112yes—
289C2×C8C23×D6713yes—
290C2×C8C23×D6711yescharacteristic
291C2×C8C23×D6713yes—
292C2×C8C23×D6711yescharacteristic
293C4⋊C4—23no—
294C22⋊C4—23no—
295C42—23no—
296C42C23×D6713yes—
297C42C23×D6711yescharacteristic
298C23—13412no—
299C23C2×#1176162813yes—
300Q8—678no—
301Q8—13412no—
302Q8C4×D26814yes—
303D4—678no—
304D4—13424no—
305D4—1348no—
306D4—13412no—
307D4C4×D268112yes—
308C2×C4—21no—
309C2×C4—23no—
310C2×C4—83no—
311C2×C4—13412no—
312C2×C4—13424no—
313C2×C4—1348no—
314C2×C4C22×C4×D6716yes—
315C2×C4C22×D26816yes—
316C2×C4C2×#1176162813yes—
317C2×C4C24×D6711yescharacteristic
318C8—1348no—
319C8C22×C4×D6711yescharacteristic
320C8C22×C4×D6711yescharacteristic
321C8C22×C4×D6716yes—
322C8C22×D26812yes—
323C8C22×D26816yes—
324C22—43no—
325C22—53612no—
326C22—1348no—
327C22C2×C4×D26816yes—
328C22C22×#1176162811yescharacteristic
329C4—41no—
330C4—43no—
331C4—1348no—
332C4C2×C4×D26816yes—
333C4C23×C4×D6711yescharacteristic
334C4C23×D26811yescharacteristic
335C2—21no—
336C2—26no—
337C2—2688no—
338C2C22×C4×D26811yescharacteristic
subgroupG/H
centerC4C23×D268
commutatorC4×C67C24×C4
FrattiniC2×C4C24×D67
FittingC67×#2875C2
socleC2×C67C22×C4×D4
prime radical#22574983C2
cosocleC2×C4×C67C25
hypercenter#2875D67
hypocenterC67#60184
LCSG ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ C2×#131 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #22574920C2semidirect
3. #22569127C2semidirect
4. #22569120C2semidirect
5. #22569113C2semidirect
6. C2×#16863519C2semidirect
7. #22568820C2semidirect
8. #22568757C2semidirect
9. C67×#2875C2semidirect
11. #16864213C4semidirect
12. #16864208C4semidirect
13. #16864201C4semidirect
17. #16864162C22semidirect
19. C2×#14524049C22semidirect
21. #16863519C22semidirect
22. #16863519C22semidirect
23. #16863519C22semidirect
24. #16863519C22semidirect
25. #16863514C22semidirect
26. #16863445C22semidirect
27. C2×#14523979C22semidirect
28. #16863414C22semidirect
29. C2×#14523971C22semidirect
33. #405×C67C22semidirect
34. #405×C67C22semidirect
35. C4×M16×C67C22semidirect
36. C2×#97×C67C22semidirect
43. #14524076C2×C4semidirect
44. #14524071C2×C4semidirect
45. #14524065C2×C4semidirect
46. #14524065C2×C4semidirect
49. C2×Dic268C2×C4semidirect
50. C2×#11761617C2×C4semidirect
51. C2×D536C2×C4semidirect
178. #2875D67semidirect
179. C67#60184semidirect
203. #405C2×D67semidirect
204. #405C2×D67semidirect
205. C4×M16C2×D67semidirect
206. C2×#97C2×D67semidirect

Maximal subgroups

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

subgroupindex
#225749832
#225749202
#225691272
#225691202
#225691132
C2×#168635192
#225688202
#225687572
C67×#28752
#6018467