Finite Group Databasebeta

← #11752434

Subgroups — #11752434

Order 960.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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×#711C211yesmaximal, characteristic
2C2×#6762C211yesmaximal, characteristic, complemented
3C22×C3⋊#170C211yesmaximal, characteristic, complemented
4#91241C212yesmaximal, complemented
5#91232C212yesmaximal, complemented
6#65775—31nomaximal, Hall
7C2×C3×C5×D4C411yescharacteristic
8C2×C3×C22⋊D5C411yescharacteristic
9C2×C3×C22⋊D5C411yescharacteristic
10C22×C3×Dic5C2211yescharacteristic
11C22×C3×Dic5C2211yescharacteristic
12C3×#711—22no—
13C3×#711—22no—
14C2×C3×C4×D5C2211yescharacteristic
15C2×C3×Dic10C411yescharacteristic
16#6762—22no—
17C2×C3⋊#170—22no—
18C2×C3⋊#170C2212yes—
19C2×C3⋊#170C2212yes—
20#5234—51nomaximal, Hall
21C2×#711D311yescharacteristic
22C2×#704—31no—
23C22×#170—31no—
24C22⋊#170—32no—
25#4003—32no—
26C23×C3×C5C2×C411yescharacteristic
27C23×C3×C5C2×C411yescharacteristic
28C22×C3×D5C2×C411yescharacteristic
29C3×C5×D4—22no—
30C2×C3×C4×C5C2×C411yescharacteristic
31C3×C22⋊D5—22no—
32C3×C22⋊D5—22no—
33C2×C3×Dic5—21no—
34C2×C3×Dic5—21no—
35C2×C3×Dic5C2×C411yescharacteristic
36C2×C3×Dic5C2×C411yescharacteristic
37C2×C3×Dic5D412yes—
38C2×C3×Dic5D412yes—
39C2×C3×Dic5C2311yescharacteristic
40C3×C4×D5—21no—
41C3×C4×D5—21no—
42C3×Dic10—22no—
43C3⋊#170—22no—
44C3⋊#170—22no—
45C2×C3×#41—51no—
46C2×#208—51no—
47C22×C3⋊C8—51no—
48#830—52no—
49#830—52no—
50C2×C5×D4Dic311yescharacteristic
51C2×C22⋊D5Dic311yescharacteristic
52C2×C22⋊D5Dic311yescharacteristic
53C22×Dic5C2×D311yescharacteristic
54C22×Dic5C2×D311yescharacteristic
55#711—22no—
56#711—22no—
57C2×C4×D5C2×D311yescharacteristic
58C2×Dic10Dic311yescharacteristic
59#704—62no—
60C2×#170—32no—
61C2×#170—32no—
62C2×#170—62no—
63#408—151noSylow
64C22×C3×C5—22no—
65C22×C3×C5—21no—
66C22×C3×C5—22no—
67C22×C3×C5C22⋊C412yes—
68C22×C3×C5C22×C411yescharacteristic
69C2×C3×D5—21no—
70C2×C3×D5—22no—
71C3×C4×C5—21no—
72C3×Dic5—21no—
73C3×Dic5C22⋊C412yes—
74C3×Dic5C2×D411yescharacteristic
75C3×Dic5C2×D411yescharacteristic
76C3×#41—102no—
77C3×#41—102no—
78C2×C3×Q8—51no—
79C2×C3×D4—51no—
80C2×C3×D4—51no—
81C2×C3×D4AGL(1,𝔽5)11yescharacteristic
82C22×C3×C4—51no—
83C22×C3×C4—51no—
84C22×C3×C4—51no—
85#208—102no—
86C2×C3⋊C8—52no—
87C2×C3⋊C8—52no—
88C2×C3⋊C8—102no—
89C23×C5C2×Dic311yescharacteristic
90C23×C5C2×Dic311yescharacteristic
91C22×D5C2×Dic311yescharacteristic
92C5×D4—22no—
93C2×C4×C5C2×Dic311yescharacteristic
94C22⋊D5—22no—
95C22⋊D5—22no—
96C2×Dic5—21no—
97C2×Dic5—21no—
98C2×Dic5C2×Dic311yescharacteristic
99C2×Dic5C2×Dic311yescharacteristic
100C2×Dic5#6712yes—
101C2×Dic5#6712yes—
102C2×Dic5C22×D311yescharacteristic
103C4×D5—21no—
104C4×D5—21no—
105Dic10—22no—
106#170—62no—
107#170—62no—
108C2×#41—51no—
109C2×M16—151no—
110C22×C8—151no—
111C22⋊C8—152no—
112C22⋊C8—152no—
113C2×C3×C5—22no—
114C2×C3×C5—41no—
115C2×C3×C5C2×C22⋊C411yescharacteristic
116C2×C3×C5#13112yes—
117C3×D5—41no—
118C23×C3—51no—
119C23×C3C2×AGL(1,𝔽5)11yescharacteristic
120C23×C3C2×AGL(1,𝔽5)11yescharacteristic
121C3×Q8—102no—
122C3×D4—22no—
123C3×D4—102no—
124C3×D4—102no—
125C2×C3×C4—52no—
126C2×C3×C4—52no—
127C2×C3×C4—51no—
128C2×C3×C4—51no—
129C2×C3×C4—51no—
130C2×C3×C4—101no—
131C2×C3×C4—101no—
132C2×C3×C4—101no—
133C2×C3×C4—101no—
134C2×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
135C3⋊C8—102no—
136C3⋊C8—102no—
137C22×C5—22no—
138C22×C5—22no—
139C22×C5—21no—
140C22×C5#21712yes—
141C22×C5C22×Dic311yescharacteristic
142C2×D5—22no—
143C2×D5—21no—
144C4×C5—21no—
145Dic5—21no—
146Dic5#21712yes—
147Dic5C2×#6711yescharacteristic
148Dic5C2×#6711yescharacteristic
149#41—102no—
150#41—102no—
151C2×Q8—51no—
152C2×D4—51no—
153C2×D4—51no—
154C2×D4ΓL(1,𝔽16)11yescharacteristic
155C22×C4—51no—
156C22×C4—51no—
157C22×C4—51no—
158M16—302no—
159C2×C8—152no—
160C2×C8—152no—
161C2×C8—302no—
162C3×C5#40811yescharacteristic, complemented, Hall
163C22×C3—22no—
164C22×C3—22no—
165C22×C3—21no—
166C22×C3—102no—
167C22×C3—101no—
168C22×C3C22⋊AGL(1,𝔽5)12yes—
169C22×C3C22×AGL(1,𝔽5)11yescharacteristic
170C3×C4—21no—
171C3×C4—52no—
172C3×C4—51no—
173C3×C4—51no—
174C3×C4—101no—
175C2×C5—22no—
176C2×C5—41no—
177C2×C5#94812yes—
178C2×C5C2×#21711yescharacteristic
179D5—41no—
180C23—51no—
181C23C2×ΓL(1,𝔽16)11yescharacteristic
182C23C2×ΓL(1,𝔽16)11yescharacteristic
183Q8—102no—
184D4—22no—
185D4—102no—
186D4—102no—
187C2×C4—52no—
188C2×C4—52no—
189C2×C4—51no—
190C2×C4—51no—
191C2×C4—51no—
192C2×C4—101no—
193C2×C4—101no—
194C2×C4—101no—
195C2×C4—101no—
196C2×C4C2×ΓL(1,𝔽16)11yescharacteristic
197C8—302no—
198C8—302no—
199C2×C3—22no—
200C2×C3—41no—
201C2×C3—201no—
202C2×C3#413112yes—
203C2×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
204C5#523411yescharacteristic, complemented, Sylow
205C22—22no—
206C22—22no—
207C22—21no—
208C22—102no—
209C22—101no—
210C22#676312yes—
211C22C22×ΓL(1,𝔽16)11yescharacteristic
212C4—21no—
213C4—51no—
214C4—51no—
215C4—52no—
216C4—101no—
217C3#6577511yescharacteristic, complemented, Sylow
218C2—22no—
219C2—41no—
220C2—201no—
221C2#9199012yes—
222C2C2×#676311yescharacteristic
subgroupG/H
center, FrattiniC22C22×ΓL(1,𝔽16)
commutator, socleC22×C3×C5C22×C4
FittingC2×C3×C5×D4C4
prime radicalC2×C3×#711C2
cosocleC2×C3×Dic5C23
hypercenterC2×D4ΓL(1,𝔽16)
hypocenterC3×C5#408
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×#6762C2semidirect
3. C22×C3⋊#170C2semidirect
4. #91241C2semidirect
5. #91232C2semidirect
162. C3×C5#408semidirect
204. C5#5234semidirect
217. C3#65775semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#7112
C2×#67622
C22×C3⋊#1702
#912412
#912322
#657753
#52345