Finite Group Databasebeta

← #30544361

Subgroups — #30544361

Order 15616.

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#21460593C211yesmaximal, characteristic
2#21460530C212yesmaximal, complemented
3#21454737C216yesmaximal, complemented
4#21454730C216yesmaximal, complemented
5#21454723C216yesmaximal, complemented
6C2×#16839885C216yesmaximal, complemented
7#21454430C211yesmaximal, characteristic, complemented
8#21454367C212yesmaximal, complemented
9C61×#2875C211yesmaximal, characteristic, complemented
10#16840598C2212yes—
11#16840579C414yescomplemented
12#16840574C418yescomplemented
13#16840567C414yescomplemented
14#16840562C2216yes—
15C2×#14519799C2216yes—
16#16840529C2212yes—
17#16840528C22112yescomplemented
18#16840527—28no—
19C2×#14519783C22112yescomplemented
20#131×D61—28no—
21#16839885C2214yescomplemented
22#16839885C22112yescomplemented
23#16839885C22112yescomplemented
24#16839885C22112yescomplemented
25#16839880C22124yescomplemented
26#16839811C2216yescomplemented
27C2×#14519713C2216yescomplemented
28#16839780C22112yescomplemented
29C2×#14519705C22112yescomplemented
30C2×#131×C61C2211yescharacteristic
31C2×#131×C61C2211yescharacteristic
32C4×#41×C61C2211yescharacteristic
33C61×#405C2213yescomplemented
34C61×#405C2213yescomplemented
35C4×M16×C61C2213yescomplemented
36C2×#97×C61C2213yescomplemented
37#14519825—22no—
38#14519824C2×C418yes—
39#14519823C2×C418yes—
40#14519822—26no—
41C2×#11758222C23112yes—
42#41×D61—28no—
43#14519810C2×C4124yescomplemented
44#14519805C2×C4124yescomplemented
45#14519799C2×C4112yescomplemented
46#14519799C2×C4112yescomplemented
47#14519799C2314yes—
48#14519799C23112yes—
49C2×Dic244C2×C416yescomplemented
50C2×#11758190C2×C4112yescomplemented
51C2×D488C2×C416yescomplemented
52#14519785—224no—
53#14519784—28no—
54#14519783—212no—
55#14519783—212no—
56#14519783C2318yes—
57#14519783C23124yes—
58C2×#11758194C23112yes—
59C2×#11758189—212no—
60M16×D61—224no—
61C2×C8×D61—212no—
62#14519713C2314yes—
63#14519713C23112yes—
64#14519705C2318yes—
65#14519705C23124yes—
66C2×#41×C61C2311yescharacteristic
67#131×C61C2×C414yes—
68#131×C61C2×C414yes—
69#131×C61D418yes—
70C2×M16×C61C2313yes—
71C2×M16×C61C2313yes—
72C22×C8×C61C2313yes—
73C22×C8×C61C2313yes—
74C4×Q8×C61—21no—
75C4×D4×C61—23no—
76#117×C61C2313yes—
77C2×C42×C61C2313yes—
78#97×C61C2311yescharacteristic
79#97×C61C2313yes—
80#97×C61C2313yes—
81#97×C61C2313yes—
82C4×C8×C61C2316yes—
83#41×C61C22×C414yes—
84#41×C61C2×D414yes—
85C2×Q8×C61#4111yescharacteristic
86C2×D4×C61#4113yes—
87C22×C4×C61—23no—
88C22×C4×C61C2413yes—
89C2×C22⋊D61—212no—
90#11758226—28no—
91Q8×D61—28no—
92#11758224—224no—
93D4×D61—224no—
94#11758222—212no—
95#11758222—212no—
96#11758222C22×C4124yes—
97#11758222C2418yes—
98C2×D244C22×C4112yes—
99C2×C4×D61—212no—
100C2×Dic122C22×C4112yes—
101M16×C61C22×C4112yes—
102M16×C61C2×D4112yes—
103C2×C8×C61C22×C416yes—
104C2×C8×C61C22×C416yes—
105C2×C8×C61C2×D4112yes—
106C2×C8×C61C2413yes—
107C2×C8×C61C2411yescharacteristic
108C2×C8×C61C2413yes—
109C2×C8×C61C2411yescharacteristic
110C4⋊C4×C61—23no—
111C22⋊C4×C61—23no—
112C42×C61—23no—
113C42×C61C2413yes—
114C42×C61C2411yescharacteristic
115#11758194—212no—
116#11758194C2418yes—
117C2×#92418—212no—
118Dic244C22×C4112yes—
119Dic244C22×C414yes—
120D488C22×C414yes—
121D488C22×C4112yes—
122#11758190C22×C4124yes—
123#11758190C22×C418yes—
124#11758189—224no—
125#11758189—28no—
126C8×D61—28no—
127C8×D61—224no—
128C23×C61C2×#4113yes—
129C22×D61—212no—
130Q8×C61C4×D414yes—
131D4×C61C4×D4112yes—
132C2×C4×C61—21no—
133C2×C4×C61—23no—
134C2×C4×C61—83no—
135C2×C4×C61C23×C416yes—
136C2×C4×C61C22×D416yes—
137C2×C4×C61C2×#4113yes—
138C2×C4×C61C2511yescharacteristic
139C22⋊D61—224no—
140C22⋊D61—28no—
141C2×Dic61—212no—
142D244—212no—
143D244C23×C418yes—
144C4×D61—224no—
145C4×D61—28no—
146Dic122—212no—
147Dic122C23×C418yes—
148C8×C61C23×C411yescharacteristic
149C8×C61C23×C416yes—
150C8×C61C23×C411yescharacteristic
151C8×C61C22×D412yes—
152C8×C61C22×D416yes—
153#92418—28no—
154#60184—611nomaximal, Sylow
155C22×C61—43no—
156C22×C61C2×C4×D416yes—
157C22×C61C22×#4111yescharacteristic
158C2×D61—28no—
159C2×D61—812no—
160C4×C61—43no—
161C4×C61—41no—
162C4×C61C2×C4×D416yes—
163C4×C61C24×C411yescharacteristic
164C4×C61C23×D411yescharacteristic
165Dic61—28no—
166#3584—611no—
167#3573—612no—
168#2957—616no—
169#2956—616no—
170#2955—616no—
171C2×#444—616no—
172#2916—611no—
173#2901—612no—
174#2875D6111yescharacteristic, complemented
175C2×C61—21no—
176C2×C61—26no—
177C2×C61C22×C4×D411yescharacteristic
178D61—48no—
179#585—612no—
180#578—614no—
181#577—618no—
182#576—614no—
183#575—616no—
184C2×#135—616no—
185#568—612no—
186#568—6112no—
187#568—1228no—
188C2×#131—6112no—
189C2×#131—1228no—
190C2×#131C2×D6111yescharacteristic
191C2×#131C2×D6111yescharacteristic
192C4×#41C2×D6111yescharacteristic
193#444—614no—
194#444—6112no—
195#444—6112no—
196#444—6112no—
197#443—6124no—
198#430—616no—
199C2×#108—616no—
200#421—6112no—
201C2×#104—6112no—
202#405C2×D6113yescomplemented
203#405C2×D6113yescomplemented
204C4×M16C2×D6113yescomplemented
205C2×#97C2×D6113yescomplemented
206C61#6018411yescharacteristic, complemented, Sylow
2072−1+4—618no—
2082−1+4—1222no—
209H(5,𝔽2)—618no—
210H(5,𝔽2)—1226no—
211C2×#41—6112no—
212C2×#41—1228no—
213C2×#41C22×D6111yescharacteristic
214#137—6124no—
215AGL(1,ℤ/8)—6124no—
216#135—6112no—
217#135—6112no—
218#135—614no—
219#135—6112no—
220C2×Q16—616no—
221C2×SD16—6112no—
222C2×D8—616no—
223#131—618no—
224#131—6124no—
225#131—1228no—
226#131—12224no—
227#131—12212no—
228#131—12212no—
229#131C4×D6114yes—
230#131C4×D6114yes—
231#131D24418yes—
232C2×M16—6112no—
233C2×M16—12224no—
234C2×M16—12212no—
235C2×M16C22×D6113yes—
236C2×M16C22×D6113yes—
237C22×C8—12212no—
238C22×C8C22×D6113yes—
239C22×C8C22×D6113yes—
240C4×Q8—21no—
241C4×D4—23no—
242#117C22×D6113yes—
243C2×C42C22×D6113yes—
244#108—614no—
245#108—6112no—
246#104—618no—
247#104—6124no—
248#97C22×D6111yescharacteristic
249#97C22×D6113yes—
250#97C22×D6113yes—
251#97C22×D6113yes—
252C4×C8C22×D6116yes—
253#41—6124no—
254#41—618no—
255#41—1228no—
256#41—12224no—
257#41—12212no—
258#41—12212no—
259#41C2×C4×D6114yes—
260#41C2×D24414yes—
261C2×Q8—6112no—
262C2×Q8—1228no—
263C2×Q8#1175822211yescharacteristic
264C2×D4—6112no—
265C2×D4—12212no—
266C2×D4—12224no—
267C2×D4#1175822213yes—
268C22×C4—23no—
269C22×C4—12212no—
270C22×C4C23×D6113yes—
271Q16—614no—
272Q16—6112no—
273SD16—618no—
274SD16—6124no—
275D8—614no—
276D8—6112no—
277M16—618no—
278M16—12212no—
279M16—1228no—
280M16—12224no—
281M16C2×C4×D61112yes—
282M16C2×D244112yes—
283C2×C8—12212no—
284C2×C8—1228no—
285C2×C8—12224no—
286C2×C8C2×C4×D6116yes—
287C2×C8C2×C4×D6116yes—
288C2×C8C2×D244112yes—
289C2×C8C23×D6113yes—
290C2×C8C23×D6111yescharacteristic
291C2×C8C23×D6113yes—
292C2×C8C23×D6111yescharacteristic
293C4⋊C4—23no—
294C22⋊C4—23no—
295C42—23no—
296C42C23×D6113yes—
297C42C23×D6111yescharacteristic
298C23—12212no—
299C23C2×#1175822213yes—
300Q8—618no—
301Q8—12212no—
302Q8C4×D24414yes—
303D4—618no—
304D4—12224no—
305D4—1228no—
306D4—12212no—
307D4C4×D244112yes—
308C2×C4—21no—
309C2×C4—23no—
310C2×C4—83no—
311C2×C4—12212no—
312C2×C4—12224no—
313C2×C4—1228no—
314C2×C4C22×C4×D6116yes—
315C2×C4C22×D24416yes—
316C2×C4C2×#1175822213yes—
317C2×C4C24×D6111yescharacteristic
318C8—1228no—
319C8C22×C4×D6111yescharacteristic
320C8C22×C4×D6111yescharacteristic
321C8C22×C4×D6116yes—
322C8C22×D24412yes—
323C8C22×D24416yes—
324C22—43no—
325C22—1228no—
326C22—48812no—
327C22C2×C4×D24416yes—
328C22C22×#1175822211yescharacteristic
329C4—43no—
330C4—41no—
331C4—1228no—
332C4C2×C4×D24416yes—
333C4C23×C4×D6111yescharacteristic
334C4C23×D24411yescharacteristic
335C2—21no—
336C2—26no—
337C2—2448no—
338C2C22×C4×D24411yescharacteristic
subgroupG/H
centerC4C23×D244
commutatorC4×C61C24×C4
FrattiniC2×C4C24×D61
FittingC61×#2875C2
socleC2×C61C22×C4×D4
prime radical#21460593C2
cosocleC2×C4×C61C25
hypercenter#2875D61
hypocenterC61#60184
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×#131 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #21460530C2semidirect
3. #21454737C2semidirect
4. #21454730C2semidirect
5. #21454723C2semidirect
6. C2×#16839885C2semidirect
7. #21454430C2semidirect
8. #21454367C2semidirect
9. C61×#2875C2semidirect
11. #16840579C4semidirect
12. #16840574C4semidirect
13. #16840567C4semidirect
17. #16840528C22semidirect
19. C2×#14519783C22semidirect
21. #16839885C22semidirect
22. #16839885C22semidirect
23. #16839885C22semidirect
24. #16839885C22semidirect
25. #16839880C22semidirect
26. #16839811C22semidirect
27. C2×#14519713C22semidirect
28. #16839780C22semidirect
29. C2×#14519705C22semidirect
33. C61×#405C22semidirect
34. C61×#405C22semidirect
35. C4×M16×C61C22semidirect
36. C2×#97×C61C22semidirect
43. #14519810C2×C4semidirect
44. #14519805C2×C4semidirect
45. #14519799C2×C4semidirect
46. #14519799C2×C4semidirect
49. C2×Dic244C2×C4semidirect
50. C2×#11758190C2×C4semidirect
51. C2×D488C2×C4semidirect
174. #2875D61semidirect
202. #405C2×D61semidirect
203. #405C2×D61semidirect
204. C4×M16C2×D61semidirect
205. C2×#97C2×D61semidirect
206. C61#60184semidirect

Maximal subgroups

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

subgroupindex
#214605932
#214605302
#214547372
#214547302
#214547232
C2×#168398852
#214544302
#214543672
C61×#28752
#6018461