Finite Group Databasebeta

← #11752449

Subgroups — #11752449

Order 960.

372 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 113 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×C3×Q8×D5C211yesmaximal, characteristic
2C2×C4⋊ΓL(1,𝔽16)C211yesmaximal, characteristic
3C2×C4×ΓL(1,𝔽16)C211yesmaximal, characteristic
4#91235C214yesmaximal
5(C2×AGL(1,𝔽5))⋊Q8—31nomaximal, Hall
6C22×ΓL(1,𝔽16)C2212yes—
7C22×ΓL(1,𝔽16)C2212yes—
8C2×C3×C5×Q8C411yescharacteristic, complemented
9C3×Q8×D5—22no—
10C3×Q8×D5—22no—
11C2×C3×C4×D5C2212yes—
12C2×C3×C4×D5C2211yescharacteristic
13C2×C3×Dic10C411yescharacteristic, complemented
14C2×C3×Dic10C412yescomplemented
15C4⋊ΓL(1,𝔽16)—22no—
16C4×ΓL(1,𝔽16)—22no—
17#5229—51nomaximal, Hall
18C2×Q8×D5D311yescharacteristic
19C2×C4⋊AGL(1,𝔽5)—31no—
20C2×C4×AGL(1,𝔽5)—31no—
21(C2×AGL(1,𝔽5))⋊C4—34no—
22C22×C3×D5C2311yescharacteristic
23C2×ΓL(1,𝔽16)—22no—
24C2×ΓL(1,𝔽16)—22no—
25C2×ΓL(1,𝔽16)—22no—
26C2×ΓL(1,𝔽16)—24no—
27C2×ΓL(1,𝔽16)Q814yescomplemented
28C3×C5×Q8—22no—
29C2×C3×C4×C5C2×C412yes—
30C2×C3×C4×C5C2×C411yescharacteristic
31C2×C3×Dic5C2×C412yes—
32C2×C3×Dic5C2×C411yescharacteristic
33C3×C4×D5—22no—
34C3×C4×D5—22no—
35C3×C4×D5D412yes—
36C3×C4×D5D412yes—
37C3×Dic10—22no—
38C3×Dic10—24no—
39C22×C3×Q8—51no—
40C2×C4⋊Dic3—51no—
41C2×C4×Dic3—51no—
42C3⋊#95—54no—
43C22×AGL(1,𝔽5)—32no—
44C22×AGL(1,𝔽5)—32no—
45C2×C5×Q8Dic311yescharacteristic, complemented
46Q8×D5—22no—
47Q8×D5—22no—
48C2×C4×D5C2×D312yes—
49C2×C4×D5C2×D311yescharacteristic
50C2×Dic10Dic311yescharacteristic, complemented
51C2×Dic10Dic312yescomplemented
52C4⋊AGL(1,𝔽5)—62no—
53C4×AGL(1,𝔽5)—62no—
54#391—151noSylow
55C22×C3×C5C22×C411yescharacteristic
56C2×C3×D5C2×D411yescharacteristic
57C2×C3×D5C2×D411yescharacteristic
58C2×C3×D5C2×Q812yes—
59C2×C3×D5#4112yes—
60ΓL(1,𝔽16)—24no—
61ΓL(1,𝔽16)—42no—
62C3×C4×C5—22no—
63C3×C4×C5C22⋊C412yes—
64C3×Dic5—22no—
65C3×Dic5C22⋊C412yes—
66C2×C3×Q8—52no—
67C2×C3×Q8—51no—
68C2×C3×Q8—102no—
69C2×C3×Q8—102no—
70C2×C3×Q8AGL(1,𝔽5)11yescharacteristic, complemented
71C22×C3×C4—51no—
72C22×C3×C4—52no—
73C22×Dic3—52no—
74C22×Dic3—52no—
75C4⋊Dic3—102no—
76C4×Dic3—102no—
77C22×D5C22×D311yescharacteristic
78C2×AGL(1,𝔽5)—34no—
79C2×AGL(1,𝔽5)—62no—
80C2×AGL(1,𝔽5)—62no—
81C2×AGL(1,𝔽5)—64no—
82C2×AGL(1,𝔽5)—62no—
83C5×Q8—22no—
84C2×C4×C5C2×Dic312yes—
85C2×C4×C5C2×Dic311yescharacteristic
86C2×Dic5C2×Dic312yes—
87C2×Dic5C2×Dic311yescharacteristic
88C4×D5—22no—
89C4×D5—22no—
90C4×D5#6712yes—
91C4×D5#6712yes—
92Dic10—22no—
93Dic10—24no—
94C22×Q8—51no—
95C2×C4⋊C4—151no—
96C2×C42—151no—
97#95—154no—
98C2×C3×C5C2×C22⋊C411yescharacteristic
99C2×C3×C5C4×Q812yes—
100C3×D5C22⋊Q812yes—
101C3×D5#12411yescharacteristic
102C3×D5C4⋊Q811yescharacteristic
103C23×C3—51no—
104C3×Q8—22no—
105C3×Q8—104no—
106C3×Q8—102no—
107C2×C3×C4—51no—
108C2×C3×C4—52no—
109C2×C3×C4—52no—
110C2×C3×C4—52no—
111C2×C3×C4—102no—
112C2×C3×C4—102no—
113C2×C3×C4C2×AGL(1,𝔽5)12yes—
114C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
115C2×Dic3—54no—
116C2×Dic3—102no—
117C2×Dic3—102no—
118C2×Dic3—102no—
119C2×Dic3—104no—
120C22×C5C22×Dic311yescharacteristic
121C2×D5D3×Q812yes—
122C2×D5#23912yes—
123C2×D5C2×#6711yescharacteristic
124C2×D5C2×#6711yescharacteristic
125AGL(1,𝔽5)—64no—
126AGL(1,𝔽5)—122no—
127C4×C5—22no—
128C4×C5#21712yes—
129Dic5—22no—
130Dic5#21712yes—
131C2×Q8—52no—
132C2×Q8—51no—
133C2×Q8—102no—
134C2×Q8—102no—
135C2×Q8ΓL(1,𝔽16)11yescharacteristic, complemented
136C22×C4—51no—
137C22×C4—52no—
138C22×C4—152no—
139C22×C4—152no—
140C4⋊C4—302no—
141C42—302no—
142C3×C5#39111yescharacteristic, complemented, Hall
143C22×C3—52no—
144C22×C3—52no—
145C22×C3—51no—
146C22×C3—51no—
147C22×C3C22×AGL(1,𝔽5)11yescharacteristic
148C3×C4—22no—
149C3×C4—52no—
150C3×C4—102no—
151C3×C4C22⋊AGL(1,𝔽5)12yes—
152Dic3—104no—
153Dic3—202no—
154C2×C5Q8×Dic312yes—
155C2×C5C2×#21711yescharacteristic
156D5#94411yescharacteristic
157D5#94612yes—
158D5#94711yescharacteristic
159C23—51no—
160Q8—22no—
161Q8—104no—
162Q8—102no—
163C2×C4—51no—
164C2×C4—52no—
165C2×C4—52no—
166C2×C4—52no—
167C2×C4—102no—
168C2×C4—102no—
169C2×C4—154no—
170C2×C4—302no—
171C2×C4—302no—
172C2×C4—302no—
173C2×C4—304no—
174C2×C4C2×ΓL(1,𝔽16)12yes—
175C2×C4C2×ΓL(1,𝔽16)11yescharacteristic
176C2×C3—52no—
177C2×C3—51no—
178C2×C3—51no—
179C2×C3Q8×AGL(1,𝔽5)12yes—
180C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
181C5#522911yescharacteristic, complemented, Sylow
182C22—52no—
183C22—52no—
184C22—51no—
185C22—51no—
186C22C22×ΓL(1,𝔽16)11yescharacteristic
187C4—22no—
188C4—52no—
189C4—102no—
190C4—304no—
191C4—602no—
192C4#676312yes—
193C3(C2×AGL(1,𝔽5))⋊Q811yescharacteristic, complemented, Sylow
194C2—52no—
195C2—51no—
196C2—51no—
197C2Q8×ΓL(1,𝔽16)12yes—
198C2C2×#676311yescharacteristic
subgroupG/H
center, FrattiniC22C22×ΓL(1,𝔽16)
commutator, socleC22×C3×C5C22×C4
FittingC2×C3×C5×Q8C4
prime radical, cosocleC22×C3×D5C23
hypercenterC2×Q8ΓL(1,𝔽16)
hypocenterC3×C5#391
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
8. C2×C3×C5×Q8C4semidirect
13. C2×C3×Dic10C4semidirect
14. C2×C3×Dic10C4semidirect
27. C2×ΓL(1,𝔽16)Q8semidirect
45. C2×C5×Q8Dic3semidirect
50. C2×Dic10Dic3semidirect
51. C2×Dic10Dic3semidirect
70. C2×C3×Q8AGL(1,𝔽5)semidirect
135. C2×Q8ΓL(1,𝔽16)semidirect
142. C3×C5#391semidirect
181. C5#5229semidirect
193. C3(C2×AGL(1,𝔽5))⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D52
C2×C4⋊ΓL(1,𝔽16)2
C2×C4×ΓL(1,𝔽16)2
#912352
(C2×AGL(1,𝔽5))⋊Q83
#52295