Finite Group Databasebeta

← #11749310

Subgroups — #11749310

Order 960.

296 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 70 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
1C2×Q8×D15C211yesmaximal, characteristic
2C2×C3×#689C211yesmaximal, characteristic, complemented
3#91390C211yesmaximal, characteristic
4C2×#6662C211yesmaximal, characteristic, complemented
5C5×#835C211yesmaximal, characteristic, complemented
6#90975C211yesmaximal, characteristic, complemented
7#90957C211yesmaximal, characteristic
8#65097—31nomaximal, Hall
9C2×C3×C5×Q8C2211yescharacteristic
10Q8×D15—22no—
11Q8×D15—22no—
12C2×C4×D15—21no—
13C2×C4×D15C2211yescharacteristic
14C2×Dic30—21no—
15C2×Dic30C2211yescharacteristic
16C2×C3×Dic10C2211yescharacteristic
17C5×C4⋊Dic3C2211yescharacteristic
18C2×C5×C3⋊C8C2211yescharacteristic
19C3×#689—22no—
20C2×C3×#168C2211yescharacteristic
21Dic3⋊Dic5—21no—
22#6667—21no—
23#6662—22no—
24#5186—51nomaximal, Hall
25C2×Q8×D5—31no—
26C2×#689D311yescharacteristic, complemented
27C2×Dic20—31no—
28#4043—31no—
29C5×#103—31no—
30#3952—31no—
31#3942—31no—
32C22×D15D411yescharacteristic
33C3×C5×Q8—21no—
34C3×C5×Q8D412yes—
35C2×C3×C4×C5—21no—
36C2×C3×C4×C5C2311yescharacteristic
37C2×Dic15—21no—
38C2×Dic15D411yescharacteristic
39C4×D15—24no—
40C4×D15—21no—
41C4×D15—21no—
42Dic30—24no—
43Dic30—21no—
44Dic30D412yes—
45C2×C5×Dic3—21no—
46C2×C3×Dic5—21no—
47C3×Dic10—21no—
48C3×Dic10—21no—
49C3×#168—21no—
50C5×C3⋊C8—21no—
51C2×D3×Q8—51no—
52C2×C3×Q16—51no—
53#946—51no—
54C2×#216—51no—
55#835D511yescharacteristic, complemented
56(C2×D3)⋊C8—51no—
57#816—51no—
58C2×C5×Q8C2×D311yescharacteristic
59Q8×D5—62no—
60Q8×D5—62no—
61C2×C4×D5—31no—
62C2×C4×D5—61no—
63C2×Dic10—31no—
64C2×Dic10—61no—
65C2×Dic10C2×D311yescharacteristic
66C2×C5×C8—31no—
67C5×C4⋊C4—31no—
68#689—22no—
69#685—61no—
70C4⋊Dic5—61no—
71C2×#168C2×D311yescharacteristic
72Dic20—62no—
73#451—151noSylow
74C22×C3×C5C2×D411yescharacteristic
75C2×D15—21no—
76C2×D15—21no—
77C2×D15Q1612yescomplemented
78C3×C4×C5—22no—
79C3×C4×C5C2×D411yescharacteristic
80C3×C4×C5C2×D411yescharacteristic
81Dic15—21no—
82Dic15—22no—
83C3×Dic5—41no—
84C5×Dic3—41no—
85C2×C3×Q8—51no—
86C2×C3×Q8C2×D511yescharacteristic
87D3×Q8—102no—
88D3×Q8—102no—
89C2×C4×D3—51no—
90C2×C4×D3—101no—
91C2×Dic6—51no—
92C2×Dic6—101no—
93C3×Q16—102no—
94C2×C3×C8—51no—
95#216—102no—
96(C2×D3)⋊C4—101no—
97C4⋊Dic3C2×D511yescharacteristic
98Dic3⋊C4—101no—
99C2×C3⋊C8C2×D511yescharacteristic
100C22×D5—31no—
101C5×Q8—21no—
102C5×Q8#6712yes—
103C2×C4×C5—21no—
104C2×C4×C5—61no—
105C2×C4×C5C22×D311yescharacteristic
106C2×Dic5—21no—
107C2×Dic5—31no—
108C2×Dic5—61no—
109C4×D5—64no—
110C4×D5—61no—
111C4×D5—61no—
112Dic10—21no—
113Dic10—21no—
114Dic10—32no—
115Dic10—64no—
116Dic10—61no—
117C5×C8—61no—
118#168—21no—
119C22×Q8—151no—
120C2×Q16—51no—
121C2×Q16—151no—
122C22⋊Q8—151no—
123#103—31no—
124#103—151no—
125C22⋊C8—151no—
126C2×C3×C5#12011yescharacteristic
127C2×C3×C5C2×Q1611yescharacteristic
128C2×C3×C5#13711yescharacteristic
129D15—22no—
130C22×D3—51no—
131C3×Q8—21no—
132C3×Q8—101no—
133C3×Q8—101no—
134C3×Q8D2012yes—
135C2×C3×C4—21no—
136C2×C3×C4—101no—
137C2×C3×C4C22×D511yescharacteristic
138C2×Dic3—21no—
139C2×Dic3—51no—
140C2×Dic3—101no—
141C4×D3—104no—
142C4×D3—101no—
143C4×D3—101no—
144Dic6—52no—
145Dic6—104no—
146Dic6—101no—
147C3×C8—101no—
148C3⋊C8—21no—
149C22×C5D3×D411yescharacteristic
150C2×D5—32no—
151C2×D5—61no—
152C2×D5—61no—
153C4×C5—22no—
154C4×C5—121no—
155C4×C5D3×D411yescharacteristic
156C4×C5C2×#6711yescharacteristic
157Dic5—41no—
158Dic5—62no—
159Dic5—61no—
160C2×Q8—51no—
161C2×Q8—151no—
162C2×Q8—301no—
163C2×Q8—302no—
164C2×Q8—302no—
165C2×Q8D3×D511yescharacteristic
166C22×C4—151no—
167C22×C4—301no—
168Q16—102no—
169Q16—302no—
170C2×C8—31no—
171C2×C8—51no—
172C4⋊C4—31no—
173C4⋊C4—301no—
174C22⋊C4—301no—
175C3×C5#45111yescharacteristic, complemented, Hall
176C22×C3D4×D511yescharacteristic
177C2×D3—52no—
178C2×D3—101no—
179C2×D3—101no—
180C3×C4—22no—
181C3×C4—201no—
182C3×C4C2×D2011yescharacteristic
183C3×C4D4×D511yescharacteristic
184Dic3—41no—
185Dic3—102no—
186Dic3—101no—
187C2×C5D3×Q1611yescharacteristic
188C2×C5#91811yescharacteristic
189C2×C5#93711yescharacteristic
190D5—62no—
191C23—151no—
192Q8—21no—
193Q8—101no—
194Q8—101no—
195Q8—152no—
196Q8—304no—
197Q8—301no—
198Q8#120212yes—
199C2×C4—21no—
200C2×C4—61no—
201C2×C4—101no—
202C2×C4—151no—
203C2×C4—304no—
204C2×C4—301no—
205C2×C4—301no—
206C2×C4—301no—
207C2×C4C2×D3×D511yescharacteristic
208C8—61no—
209C8—101no—
210C2×C3#402811yescharacteristic
211C2×C3#406111yescharacteristic
212C2×C3D5×Q1611yescharacteristic
213D3—102no—
214C5#518611yescharacteristic, complemented, Sylow
215C22—152no—
216C22—301no—
217C22—301no—
218C22#679011yescharacteristic
219C4—22no—
220C4—121no—
221C4—201no—
222C4—301no—
223C4—302no—
224C4C2×#120211yescharacteristic
225C4#679011yescharacteristic
226C3#6509711yescharacteristic, complemented, Sylow
227C2—302no—
228C2#9151111yescharacteristic
229C2#9152411yescharacteristic
230C2#9157211yescharacteristic
subgroupG/H
centerC22#6790
commutator, cosocleC2×C3×C4×C5C23
FrattiniC2×C4C2×D3×D5
FittingC2×C3×C5×Q8C22
socleC22×C3×C5C2×D4
prime radicalC22×D15D4
hypercenterC2×Q8D3×D5
hypocenterC3×C5#451
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#689C2semidirect
4. C2×#6662C2semidirect
5. C5×#835C2semidirect
6. #90975C2semidirect
26. C2×#689D3semidirect
55. #835D5semidirect
77. C2×D15Q16semidirect
175. C3×C5#451semidirect
214. C5#5186semidirect
226. C3#65097semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×D152
C2×C3×#6892
#913902
C2×#66622
C5×#8352
#909752
#909572
#650973
#51865