Finite Group Databasebeta

← #14458100

Subgroups — #14458100

Order 1920.

920 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756521C214yesmaximal, complemented
2C3×C5×#492C211yesmaximal, characteristic, complemented
3#11751278C214yesmaximal, complemented
4#11751196C212yesmaximal, complemented
5#11751178C212yesmaximal, complemented
6#11750801C212yesmaximal, complemented
7#10613505—31nomaximal, Hall
8#92098C2212yes—
9#92095C2212yescomplemented
10C2×C3×C5×SD16C2214yescomplemented
11C3×C5×C4⋊Q8C2211yescharacteristic, complemented
12C3×C5×#127C2211yescharacteristic
13#91831C2214yescomplemented
14#91821C2214yes—
15#91805—24no—
16#91804—24no—
17#91803—24no—
18SD16×D15—24no—
19#91793C2212yes—
20#91790C2214yescomplemented
21C3×C4×C5×C8C2211yescharacteristic, complemented
22#91120C2212yescomplemented
23#91117C2212yescomplemented
24#91103C2212yescomplemented
25#91089C2214yescomplemented
26#81774—51nomaximal, Hall
27#66096—34no—
28C5×#492D311yescharacteristic, complemented
29#65466—34no—
30#65384—32no—
31#65366—32no—
32#64989—32no—
33C2×C3×C5×Q8C2312yes—
34C2×C3×C5×D4—41no—
35C2×C3×C5×D4C2312yes—
36C2×#1220—42no—
37#6821—24no—
38Q8×D15—24no—
39#6819—24no—
40D4×D15—24no—
41#6817—42no—
42#6817C2314yes—
43C3×C5×SD16—24no—
44C3×C5×SD16—24no—
45C2×C3×C5×C8C2312yes—
46C3×C5×C4⋊C4—41no—
47C3×C42×C5C2311yescharacteristic
48#6718—24no—
49#6717—24no—
50#6716—24no—
51#6715—24no—
52#6710C2314yes—
53Dic60D412yescomplemented
54D120D412yescomplemented
55#6706D414yescomplemented
56#6705—24no—
57C8×D15—24no—
58#5762—54no—
59C3×#492D511yescharacteristic, complemented
60#5177—54no—
61#5095—52no—
62#5077—52no—
63#4700—52no—
64#4147—32no—
65#4144—32no—
66C2×C5×SD16C2×D314yescomplemented
67C5×C4⋊Q8C2×D311yescharacteristic, complemented
68C5×#127C2×D311yescharacteristic
69#4088—34no—
70#4078—34no—
71#4062—64no—
72#4061—64no—
73#4060—64no—
74D5×SD16—64no—
75#4050—32no—
76#4047—34no—
77C4×C5×C8C2×D311yescharacteristic, complemented
78#3968—32no—
79#3965—32no—
80#3951—32no—
81#3937—34no—
82#3291—151noSylow
83C23×C3×C5—22no—
84C22×D15—42no—
85C3×C5×Q8—24no—
86C3×C5×D4—24no—
87C3×C5×D4—82no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—22no—
90C2×C3×C4×C5C2411yescharacteristic
91#1220—24no—
92#1220—44no—
93C2×Dic15—42no—
94D60—42no—
95D60C2×D414yes—
96C4×D15—24no—
97C4×D15—44no—
98Dic30—42no—
99Dic30C2×D414yes—
100C3×C5×C8C2×D412yes—
101C3×C5×C8C2×D412yes—
102C3⋊#168—24no—
103#1007—52no—
104#1004—52no—
105C2×C3×SD16C2×D514yescomplemented
106C3×C4⋊Q8C2×D511yescharacteristic, complemented
107C3×#127C2×D511yescharacteristic
108#942—54no—
109#932—54no—
110#916—104no—
111#915—104no—
112#914—104no—
113D3×SD16—104no—
114#904—52no—
115#901—54no—
116C3×C4×C8C2×D511yescharacteristic, complemented
117#836—52no—
118#833—52no—
119#819—52no—
120#805—54no—
121C2×C5×Q8C22×D312yes—
122C2×C5×D4—41no—
123C2×C5×D4C22×D312yes—
124C2×C22⋊D5—122no—
125#713—64no—
126Q8×D5—64no—
127#711—64no—
128D4×D5—64no—
129#709—34no—
130#709—122no—
131C5×SD16—24no—
132C5×SD16—24no—
133C2×C5×C8C22×D312yes—
134C5×C4⋊C4—41no—
135C42×C5C22×D311yescharacteristic
136#689—64no—
137#688—64no—
138#687—64no—
139#686—64no—
140#681—34no—
141Dic20—32no—
142D40—32no—
143#677—34no—
144#676—64no—
145C8×D5—64no—
146#577—154no—
147#492D1511yescharacteristic, complemented
148#471—154no—
149#456—152no—
150#453—152no—
151#443—152no—
152C22×C3×C5—42no—
153C22×C3×C5—82no—
154C22×C3×C5H(5,𝔽2)11yescharacteristic
155C2×D15—24no—
156C2×D15—162no—
157C3×C4×C5—22no—
158C3×C4×C5—42no—
159C3×C4×C5C22×D412yes—
160Dic15—24no—
161C2×C3×Q8C22×D512yes—
162C2×C3×D4—41no—
163C2×C3×D4C22×D512yes—
164C2×#67—202no—
165#239—104no—
166D3×Q8—104no—
167#237—104no—
168D3×D4—104no—
169#235—54no—
170#235—202no—
171C3×SD16—24no—
172C3×SD16—24no—
173C2×C3×C8C22×D512yes—
174C3×C4⋊C4—41no—
175C3×C42C22×D511yescharacteristic
176#216—104no—
177#215—104no—
178#214—104no—
179#213—104no—
180#208—54no—
181Dic12—52no—
182D24—52no—
183#204—54no—
184ΓL(1,𝔽25)—104no—
185D3×C8—104no—
186C23×C5—22no—
187C22×D5—122no—
188C5×Q8—24no—
189C5×D4—24no—
190C5×D4—82no—
191C2×C4×C5—21no—
192C2×C4×C5—22no—
193C2×C4×C5C23×D311yescharacteristic
194C22⋊D5—64no—
195C22⋊D5—124no—
196C2×Dic5—122no—
197D20—34no—
198D20—122no—
199C4×D5—64no—
200C4×D5—124no—
201Dic10—34no—
202Dic10—122no—
203C5×C8D3×D412yes—
204C5×C8D3×D412yes—
205#168—64no—
2062−1+4—152no—
207H(5,𝔽2)—152no—
208#137—154no—
209#137—304no—
210AGL(1,ℤ/8)—154no—
211AGL(1,ℤ/8)—304no—
212#135—152no—
213#135—304no—
214C2×SD16—304no—
215C2×SD16C2×D1514yescomplemented
216#131—154no—
217C4⋊Q8C2×D1511yescharacteristic, complemented
218#127C2×D1511yescharacteristic
219#108—152no—
220#104—154no—
221#101—152no—
222#100—152no—
223C4×C8C2×D1511yescharacteristic, complemented
224C2×C3×C5—21no—
225C2×C3×C5—82no—
226C2×C3×C5D4211yescharacteristic
227D15—44no—
228C23×C3—22no—
229C22×D3—202no—
230C3×Q8—24no—
231C3×D4—24no—
232C3×D4—82no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4C23×D511yescharacteristic
236#67—104no—
237#67—204no—
238C2×Dic3—202no—
239D12—54no—
240D12—202no—
241C4×D3—104no—
242C4×D3—204no—
243Dic6—54no—
244Dic6—202no—
245C3×C8D4×D512yes—
246C3×C8D4×D512yes—
247C3⋊C8—104no—
248C22×C5—42no—
249C22×C5—82no—
250C22×C5#100411yescharacteristic
251C2×D5—64no—
252C2×D5—482no—
253C4×C5—22no—
254C4×C5—42no—
255C4×C5C2×D3×D412yes—
256Dic5—64no—
257#41—154no—
258#41—304no—
259#41—304no—
260#41—602no—
261C2×Q8—304no—
262C2×Q8C22×D1512yes—
263C2×D4—41no—
264C2×D4—304no—
265C2×D4—602no—
266C2×D4C22×D1512yes—
267Q16—152no—
268Q16—304no—
269SD16—24no—
270SD16—24no—
271SD16—154no—
272SD16—304no—
273SD16—304no—
274D8—152no—
275D8—304no—
276M16—154no—
277M16—304no—
278C2×C8—304no—
279C2×C8C22×D1512yes—
280C4⋊C4—41no—
281C42C22×D1511yescharacteristic
282C3×C5#329111yescharacteristic, complemented, Hall
283C22×C3—42no—
284C22×C3—82no—
285C22×C3#414411yescharacteristic
286C2×D3—104no—
287C2×D3—802no—
288C3×C4—22no—
289C3×C4—42no—
290C3×C4C2×D4×D512yes—
291Dic3—104no—
292C2×C5—21no—
293C2×C5—82no—
294C2×C5#571711yescharacteristic
295D5—124no—
296C23—22no—
297C23—602no—
298Q8—24no—
299Q8—154no—
300Q8—602no—
301D4—24no—
302D4—82no—
303D4—154no—
304D4—304no—
305D4—602no—
306D4—604no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—304no—
310C2×C4—604no—
311C2×C4—602no—
312C2×C4C23×D1511yescharacteristic
313C8—304no—
314C8D4×D1512yes—
315C8D4×D1512yes—
316C2×C3—21no—
317C2×C3—82no—
318C2×C3#6605111yescharacteristic
319D3—204no—
320C5#8177411yescharacteristic, complemented, Sylow
321C22—42no—
322C22—82no—
323C22—304no—
324C22—2402no—
325C22#9209511yescharacteristic
326C4—22no—
327C4—42no—
328C4—304no—
329C4C2×D4×D1512yes—
330C3#1061350511yescharacteristic, complemented, Sylow
331C2—21no—
332C2—82no—
333C2—604no—
334C2#1175647611yescharacteristic
subgroupG/H
centerC2#11756476
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C23×D15
FittingC3×C5×#492C2
socleC2×C3×C5D42
hypercenter#492D15
hypocenterC3×C5#3291
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #11756521C2semidirect
2. C3×C5×#492C2semidirect
3. #11751278C2semidirect
4. #11751196C2semidirect
5. #11751178C2semidirect
6. #11750801C2semidirect
9. #92095C22semidirect
10. C2×C3×C5×SD16C22semidirect
11. C3×C5×C4⋊Q8C22semidirect
13. #91831C22semidirect
20. #91790C22semidirect
21. C3×C4×C5×C8C22semidirect
22. #91120C22semidirect
23. #91117C22semidirect
24. #91103C22semidirect
25. #91089C22semidirect
28. C5×#492D3semidirect
53. Dic60D4semidirect
54. D120D4semidirect
55. #6706D4semidirect
59. C3×#492D5semidirect
66. C2×C5×SD16C2×D3semidirect
67. C5×C4⋊Q8C2×D3semidirect
77. C4×C5×C8C2×D3semidirect
105. C2×C3×SD16C2×D5semidirect
106. C3×C4⋊Q8C2×D5semidirect
116. C3×C4×C8C2×D5semidirect
147. #492D15semidirect
215. C2×SD16C2×D15semidirect
217. C4⋊Q8C2×D15semidirect
223. C4×C8C2×D15semidirect
282. C3×C5#3291semidirect
320. C5#81774semidirect
330. C3#10613505semidirect

Maximal subgroups

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

subgroupindex
#117565212
C3×C5×#4922
#117512782
#117511962
#117511782
#117508012
#106135053
#817745