Finite Group Databasebeta

← #14314912

Subgroups — #14314912

Order 1920.

256 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C4×#6650C211yesmaximal, characteristic
2#11746732C211yesmaximal, characteristic, complemented
3C3×#64861C211yesmaximal, characteristic, complemented
4#11746548C212yesmaximal
5#11746544C212yesmaximal, complemented
6#10604155—31nomaximal, Hall
7C42×C5×D3C411yescharacteristic
8C2×#6650C411yescharacteristic
9C2×#6650C2211yescharacteristic
10#91227C2212yes—
11C3×#3998C2212yes—
12C4×C3⋊#168C411yescharacteristic
13C3×C4×#168C2211yescharacteristic
14#90953C411yescharacteristic
15#90953C2211yescharacteristic
16#69298—51nomaximal, Hall
17C4×#681—31no—
18#64912—32no—
19#64861—31no—
20#64861D311yescharacteristic, complemented
21#64860—32no—
22C2×C4×C5×D3—21no—
23C2×C4×C5×D3C2×C411yescharacteristic
24C3×C42×C5C2×C411yescharacteristic
25C2×C3⋊#168C2×C411yescharacteristic
26C2×C3⋊#168C2×C411yescharacteristic
27C4×C5×Dic3—21no—
28C4×C5×Dic3C2×C411yescharacteristic
29C2×C3×#168C2×C411yescharacteristic
30C2×C3×#168C2311yescharacteristic
31#6650—22no—
32#6650—21no—
33#6650—21no—
34C3⋊#674—22no—
35C3×#674—22no—
36C4×ΓL(1,𝔽25)—51no—
37C3×#364—51no—
38#4514—52no—
39#4512—52no—
40#4463—51no—
41C2×C42×C5—31no—
42C2×#681—31no—
43C2×#681—31no—
44#3998—32no—
45#3998C2×D312yes—
46#3935—31no—
47#3935—31no—
48C4×#168—31no—
49C4×#168C2×D311yescharacteristic
50#2165—151noSylow
51C22×C5×D3M1611yescharacteristic
52C2×C3×C4×C5—21no—
53C2×C3×C4×C5C22×C411yescharacteristic
54C2×C5×Dic3—21no—
55C2×C5×Dic3M1611yescharacteristic
56C4×C5×D3—21no—
57C4×C5×D3—21no—
58C4×C5×D3—42no—
59C3⋊#168—21no—
60C3⋊#168—21no—
61C3×#168M1612yes—
62C3×#168#4111yescharacteristic
63C3×#168#4111yescharacteristic
64C2×ΓL(1,𝔽25)—51no—
65C2×ΓL(1,𝔽25)—51no—
66C42×D3AGL(1,𝔽5)11yescharacteristic
67C3×M32—52no—
68C3×C4×C8—51no—
69#815—51no—
70#815—51no—
71#812—52no—
72C4×C3⋊C8—51no—
73C22×C4×C5—31no—
74C22×C4×C5—61no—
75C42×C5—31no—
76C42×C5—61no—
77C42×C5C4×D311yescharacteristic
78#681—62no—
79#681—61no—
80#681—61no—
81C2×#168—31no—
82C2×#168—31no—
83C2×#168C4×D311yescharacteristic
84C2×#168C22×D311yescharacteristic
85#674—22no—
86#674—62no—
87C4×M16—151no—
88#364—51no—
89#364—151no—
90#349—152no—
91#347—152no—
92C22×C3×C5C2×M1611yescharacteristic
93C2×C5×D3—21no—
94C2×C5×D3—41no—
95C3×C4×C5—22no—
96C3×C4×C5#11711yescharacteristic
97C3×C4×C5C2×M1611yescharacteristic
98C5×Dic3—21no—
99C5×Dic3—41no—
100C2×C4×D3—21no—
101C2×C4×D3C2×AGL(1,𝔽5)11yescharacteristic
102C2×C3×C8—51no—
103C2×C3×C8—51no—
104C3×C42C2×AGL(1,𝔽5)11yescharacteristic
105C4×Dic3—21no—
106C4×Dic3C2×AGL(1,𝔽5)11yescharacteristic
107C2×C3⋊C8—51no—
108C2×C3⋊C8—51no—
109ΓL(1,𝔽25)—102no—
110ΓL(1,𝔽25)—101no—
111ΓL(1,𝔽25)—101no—
112C3×C16—102no—
113C3⋊C16—102no—
114C23×C5—31no—
115C2×C4×C5—21no—
116C2×C4×C5—31no—
117C2×C4×C5—61no—
118C2×C4×C5—61no—
119C2×C4×C5—61no—
120C2×C4×C5—122no—
121C2×C4×C5C2×C4×D311yescharacteristic
122#168—61no—
123#168—61no—
124#168ΓL(1,𝔽25)12yes—
125#168#23711yescharacteristic
126#168#23911yescharacteristic
127C2×M16—151no—
128C2×M16—151no—
129C2×C42—31no—
130M32—52no—
131M32—152no—
132#97—151no—
133#97—151no—
134C4×C8—51no—
135C4×C8—151no—
136C2×C3×C5—21no—
137C2×C3×C5#43211yescharacteristic
138C5×D3—41no—
139C22×D3#70411yescharacteristic
140C2×C3×C4—21no—
141C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
142C2×Dic3—21no—
143C2×Dic3#70411yescharacteristic
144C4×D3—21no—
145C4×D3—21no—
146C4×D3—42no—
147C3×C8—52no—
148C3×C8—51no—
149C3×C8—51no—
150C3⋊C8—101no—
151C3⋊C8—101no—
152C22×C5—61no—
153C22×C5—121no—
154C22×C5D3×M1611yescharacteristic
155C4×C5—22no—
156C4×C5—61no—
157C4×C5—121no—
158C4×C5#89211yescharacteristic
159C4×C5C2×ΓL(1,𝔽25)11yescharacteristic
160C22×C4—31no—
161C22×C4—61no—
162M16—301no—
163M16—301no—
164M16—302no—
165C2×C8—51no—
166C2×C8—51no—
167C2×C8—151no—
168C2×C8—151no—
169C42—31no—
170C42—61no—
171C42D3×AGL(1,𝔽5)11yescharacteristic
172C16—102no—
173C16—302no—
174C3×C5#216511yescharacteristic, complemented, Hall
175C22×C3C2×#70411yescharacteristic
176C2×D3—21no—
177C2×D3—41no—
178C3×C4—22no—
179C3×C4D5⋊M1611yescharacteristic
180C3×C4#413011yescharacteristic
181Dic3—21no—
182Dic3—41no—
183C2×C5—21no—
184C2×C5—121no—
185C2×C5#483511yescharacteristic
186C23—31no—
187C2×C4—21no—
188C2×C4—31no—
189C2×C4—61no—
190C2×C4—61no—
191C2×C4—61no—
192C2×C4—122no—
193C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
194C8—51no—
195C8—51no—
196C8—52no—
197C8—301no—
198C8—301no—
199C2×C3—21no—
200C2×C3#6575711yescharacteristic
201D3—41no—
202C5#6929811yescharacteristic, complemented, Sylow
203C22—61no—
204C22—121no—
205C22D3×#70411yescharacteristic
206C4—22no—
207C4—61no—
208C4—121no—
209C4#9190711yescharacteristic
210C4D5⋊ΓL(1,𝔽25)11yescharacteristic
211C3#1060415511yescharacteristic, complemented, Sylow
212C2—21no—
213C2—121no—
214C2#1175180411yescharacteristic
subgroupG/H
centerC4#91907
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C2×D3×AGL(1,𝔽5)
FittingC3×C42×C5C2×C4
socleC2×C3×C5#432
prime radicalC22×C5×D3M16
cosocleC2×C3×#168C23
hypercenterC42D3×AGL(1,𝔽5)
hypocenterC3×C5#2165
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #11746732C2semidirect
3. C3×#64861C2semidirect
5. #11746544C2semidirect
20. #64861D3semidirect
174. C3×C5#2165semidirect
202. C5#69298semidirect
211. C3#10604155semidirect

Maximal subgroups

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

subgroupindex
C4×#66502
#117467322
C3×#648612
#117465482
#117465442
#106041553
#692985