Finite Group Databasebeta

← #11754752

Subgroups — #11754752

Order 960.

480 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 128 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#91829C211yesmaximal, characteristic, complemented
2C5×#938C211yesmaximal, characteristic, complemented
3C3×D4×Dic5C211yesmaximal, characteristic, complemented
4#91566C212yesmaximal, complemented
5#91562C212yesmaximal, complemented
6#91557C212yesmaximal, complemented
7#91546C212yesmaximal, complemented
8#91460C211yesmaximal, characteristic, complemented
9#91418C211yesmaximal, characteristic, complemented
10#91384C211yesmaximal, characteristic, complemented
11#91371C211yesmaximal, characteristic, complemented
12#65953—31nomaximal, Hall
13C2×C3×C5×D4C2211yescharacteristic
14C2×#1220C2212yes—
15C2×#1220C2212yes—
16C2×D60C2211yescharacteristic
17C2×C5×#67C2212yes—
18C2×C4×C5×D3C2211yescharacteristic
19C22×C3×Dic5C2212yes—
20C2×#1203C2212yes—
21C2×#1200C2212yes—
22C4×Dic15C2211yescharacteristic
23C5×#217C2212yes—
24C5×C4⋊Dic3C2211yescharacteristic
25C3×C22⋊Dic5C2212yes—
26C3×C4⋊Dic5C2211yescharacteristic
27C3×C4×Dic5C2211yescharacteristic
28Dic3⋊Dic5C2211yescharacteristic
29Dic3⋊Dic5C2212yes—
30Dic3⋊Dic5C2211yescharacteristic
31#6668C2212yes—
32#6667C2212yes—
33#6667C2212yes—
34(C2×D3)⋊Dic5C2211yescharacteristic
35(C2×D3)⋊Dic5C2211yescharacteristic
36#5562—51nomaximal, Hall
37C5×#121—31no—
38#4086—31no—
39#4085—32no—
40#4084—32no—
41D4×Dic5D311yescharacteristic, complemented
42#4076—31no—
43#4072—31no—
44#4039—31no—
45#4036—31no—
46#4032—32no—
47#4029—32no—
48C23×C3×C5C2312yes—
49C22×D15C2312yes—
50C22×C5×D3C2311yescharacteristic
51C3×C5×D4—22no—
52C2×C3×C4×C5C2311yescharacteristic
53#1220—42no—
54#1220—42no—
55C2×Dic15C2311yescharacteristic
56C2×Dic15C2311yescharacteristic
57D60—22no—
58C5×#67—42no—
59C2×C5×Dic3C2311yescharacteristic
60C2×C5×Dic3C2312yes—
61C4×C5×D3—22no—
62C2×C3×Dic5—42no—
63C2×C3×Dic5C2311yescharacteristic
64C2×C3×Dic5C2311yescharacteristic
65C2×C3×Dic5C2312yes—
66#1203—42no—
67#1200—42no—
68C3×C4×D4—51no—
69#940—51no—
70#938D511yescharacteristic, complemented
71#930—52no—
72#929—52no—
73#895—51no—
74#890—51no—
75#884—52no—
76#883—52no—
77C4×D12—51no—
78#872—51no—
79C2×C5×D4—32no—
80C2×C5×D4C2×D311yescharacteristic
81C22×C4×C5—31no—
82C2×C22⋊D5—32no—
83C2×C22⋊D5—32no—
84C2×C22⋊D5—32no—
85C22×Dic5C2×D312yes—
86C2×D20—31no—
87C2×C4×D5—32no—
88C5×C4⋊C4—31no—
89C5×C22⋊C4—32no—
90C22⋊Dic5—31no—
91C22⋊Dic5—31no—
92C22⋊Dic5C2×D312yes—
93#685—32no—
94#685—32no—
95C4⋊Dic5—31no—
96C4⋊Dic5—31no—
97C4⋊Dic5—32no—
98C4⋊Dic5C2×D311yescharacteristic
99#683—32no—
100C4×Dic5—31no—
101C4×Dic5C2×D311yescharacteristic
102#540—151noSylow
103C22×C3×C5—22no—
104C22×C3×C5—22no—
105C22×C3×C5—22no—
106C22×C3×C5C2411yescharacteristic
107C2×D15—22no—
108C2×D15—22no—
109C2×D15—22no—
110C2×C5×D3—21no—
111C2×C5×D3—21no—
112C2×C5×D3—21no—
113C3×C4×C5#4112yes—
114Dic15—21no—
115Dic15—21no—
116C3×Dic5—21no—
117C3×Dic5—22no—
118C3×Dic5—21no—
119C5×Dic3—21no—
120C5×Dic3—22no—
121C2×C3×D4C2×D511yescharacteristic
122C22×C3×C4—52no—
123C2×#67—52no—
124C2×#67—52no—
125C2×#67C2×D512yes—
126C2×D12—51no—
127C2×D12—52no—
128C2×C4×D3—52no—
129C2×C4×D3C2×D511yescharacteristic
130C3×C4⋊C4—51no—
131C3×C22⋊C4—52no—
132C3×C42—51no—
133#217C2×D512yes—
134(C2×D3)⋊C4—51no—
135(C2×D3)⋊C4—51no—
136(C2×D3)⋊C4—52no—
137(C2×D3)⋊C4—52no—
138C4⋊Dic3—52no—
139C4⋊Dic3C2×D511yescharacteristic
140Dic3⋊C4—51no—
141Dic3⋊C4—51no—
142Dic3⋊C4—52no—
143C4×Dic3—51no—
144C23×C5—31no—
145C23×C5C22×D312yes—
146C22×D5—32no—
147C5×D4—22no—
148C5×D4—122no—
149C2×C4×C5—32no—
150C2×C4×C5—31no—
151C2×C4×C5—62no—
152C2×C4×C5C22×D311yescharacteristic
153C22⋊D5—122no—
154C22⋊D5—122no—
155C22⋊D5—122no—
156C2×Dic5—31no—
157C2×Dic5—31no—
158C2×Dic5—42no—
159C2×Dic5C22×D311yescharacteristic
160C2×Dic5C22×D311yescharacteristic
161C2×Dic5C22×D312yes—
162D20—62no—
163C4×D5—122no—
164#127—151no—
165#125—151no—
166#123—152no—
167#123—152no—
168#121—31no—
169#121—152no—
170#121—151no—
171#121—152no—
172C4×D4—51no—
173C4×D4—151no—
174#117—151no—
175C2×C3×C5—42no—
176C2×C3×C5C2×#4111yescharacteristic
177C2×C3×C5H(5,𝔽2)11yescharacteristic
178C2×C3×C5H(5,𝔽2)11yescharacteristic
179D15—42no—
180C5×D3—41no—
181C23×C3C22×D512yes—
182C22×D3—52no—
183C22×D3C22×D511yescharacteristic
184C3×D4—22no—
185C2×C3×C4—52no—
186C2×C3×C4—51no—
187C2×C3×C4—51no—
188C2×C3×C4—202no—
189C2×C3×C4C22×D511yescharacteristic
190#67—42no—
191#67—202no—
192#67—202no—
193C2×Dic3—51no—
194C2×Dic3—51no—
195C2×Dic3C22×D511yescharacteristic
196C2×Dic3C22×D512yes—
197D12—102no—
198D12—202no—
199C4×D3—22no—
200C4×D3—202no—
201C22×C5—22no—
202C22×C5—22no—
203C22×C5—22no—
204C22×C5—61no—
205C22×C5—61no—
206C22×C5—61no—
207C22×C5C23×D311yescharacteristic
208C2×D5—62no—
209C2×D5—62no—
210C2×D5—62no—
211C4×C5—61no—
212C4×C5—62no—
213C4×C5#23512yes—
214Dic5—21no—
215Dic5—22no—
216Dic5—21no—
217Dic5—61no—
218Dic5—61no—
219C2×D4—32no—
220C2×D4—152no—
221C2×D4—152no—
222C2×D4—152no—
223C2×D4—151no—
224C2×D4D3×D511yescharacteristic
225C22×C4—31no—
226C22×C4—52no—
227C22×C4—152no—
228C4⋊C4—31no—
229C4⋊C4—51no—
230C4⋊C4—151no—
231C4⋊C4—151no—
232C4⋊C4—152no—
233C4⋊C4—152no—
234C22⋊C4—32no—
235C22⋊C4—52no—
236C22⋊C4—152no—
237C22⋊C4—152no—
238C22⋊C4—151no—
239C22⋊C4—151no—
240C42—51no—
241C42—151no—
242C3×C5#54011yescharacteristic, complemented, Hall
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—22no—
246C22×C3C23×D511yescharacteristic
247C2×D3—21no—
248C2×D3—21no—
249C2×D3—21no—
250C2×D3—102no—
251C2×D3—102no—
252C2×D3—102no—
253C3×C4—101no—
254C3×C4—102no—
255C3×C4—101no—
256C3×C4#71112yes—
257Dic3—21no—
258Dic3—22no—
259Dic3—101no—
260Dic3—101no—
261C2×C5—42no—
262C2×C5—121no—
263C2×C5C2×#23511yescharacteristic
264C2×C5#100411yescharacteristic
265C2×C5#100911yescharacteristic
266D5—122no—
267C23—31no—
268C23—152no—
269C23C2×D3×D512yes—
270D4—22no—
271D4—122no—
272D4—302no—
273D4—602no—
274D4—602no—
275D4—602no—
276C2×C4—31no—
277C2×C4—32no—
278C2×C4—51no—
279C2×C4—51no—
280C2×C4—52no—
281C2×C4—62no—
282C2×C4—151no—
283C2×C4—151no—
284C2×C4—202no—
285C2×C4—602no—
286C2×C4C2×D3×D511yescharacteristic
287C2×C3—42no—
288C2×C3C2×#71111yescharacteristic
289C2×C3#414411yescharacteristic
290C2×C3#414911yescharacteristic
291D3—41no—
292D3—202no—
293C5#556211yescharacteristic, complemented, Sylow
294C22—22no—
295C22—22no—
296C22—22no—
297C22—61no—
298C22—61no—
299C22—61no—
300C22—302no—
301C22—302no—
302C22—302no—
303C22C22×D3×D511yescharacteristic
304C4—62no—
305C4—61no—
306C4—101no—
307C4—102no—
308C4—101no—
309C4—301no—
310C4—301no—
311C4#678212yes—
312C3#6595311yescharacteristic, complemented, Sylow
313C2—42no—
314C2—121no—
315C2—602no—
316C2#9202611yescharacteristic
317C2#9202711yescharacteristic
318C2C2×#678211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC2×C3×C5×D4C22
hypercenterC2×D4D3×D5
hypocenterC3×C5#540
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #91829C2semidirect
2. C5×#938C2semidirect
3. C3×D4×Dic5C2semidirect
4. #91566C2semidirect
5. #91562C2semidirect
6. #91557C2semidirect
7. #91546C2semidirect
8. #91460C2semidirect
9. #91418C2semidirect
10. #91384C2semidirect
11. #91371C2semidirect
41. D4×Dic5D3semidirect
70. #938D5semidirect
242. C3×C5#540semidirect
293. C5#5562semidirect
312. C3#65953semidirect

Maximal subgroups

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

subgroupindex
#918292
C5×#9382
C3×D4×Dic52
#915662
#915622
#915572
#915462
#914602
#914182
#913842
#913712
#659533
#55625