Finite Group Databasebeta

← #14512486

Subgroups — #14512486

Order 1920.

260 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 48 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×Dic5×SL(2,𝔽3)C211yesmaximal, characteristic
2SL(2,𝔽3)⋊#170C212yesmaximal, complemented
3#10604968—31nomaximal, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic
5C2×#6762—41nomaximal
6Dic5×SL(2,𝔽3)C411yescharacteristic
7Dic5×SL(2,𝔽3)C2211yescharacteristic
8#73055—51nomaximal, Hall
9C2×Q8×Dic5D311yescharacteristic
10#65758—31no—
11#65754—31no—
12Q8⋊#170—32no—
13Q8⋊#170—32no—
14C22×C3×Dic5—41no—
15C2×C5×SL(2,𝔽3)—21no—
16C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
17#6762—44no—
18C2×C3⋊#170—42no—
19C2×C4×SL(2,𝔽3)—51no—
20SL(2,𝔽3)⋊C8—52no—
21C22×C5×Q8Dic311yescharacteristic
22C2×#704—61no—
23Q8×Dic5—31no—
24Q8×Dic5—31no—
25Q8×Dic5Dic311yescharacteristic
26Q8×Dic5C2×D311yescharacteristic
27C2×C4⋊Dic5—31no—
28C2×C4×Dic5—31no—
29C22⋊#170—61no—
30#4004—61no—
31#4004—61no—
32C4⋊#170—32no—
33C4×#170—32no—
34#1487—151noSylow
35C23×C3×C5—41no—
36C2×C3×Dic5—42no—
37C2×C3×Dic5—42no—
38C2×C3×Dic5—41no—
39C2×C3×Dic5—41no—
40C5×SL(2,𝔽3)M1611yescharacteristic, complemented
41C3⋊#170—44no—
42C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic
43C2×#208—201no—
44C4×SL(2,𝔽3)—51no—
45C4×SL(2,𝔽3)—51no—
46C2×C5×Q8—21no—
47C2×C5×Q8—31no—
48C2×C5×Q8—61no—
49C2×C5×Q8C2×Dic311yescharacteristic
50C22×C4×C5—31no—
51C22×Dic5—31no—
52C22×Dic5S411yescharacteristic
53#704—122no—
54C2×#170—62no—
55C2×#170—62no—
56C4⋊Dic5—31no—
57C4⋊Dic5—31no—
58C4⋊Dic5—61no—
59C4×Dic5—31no—
60C4×Dic5—31no—
61C4×Dic5—31no—
62C4×Dic5—31no—
63C2×C4×Q8—51no—
64#431—151no—
65#423—151no—
66Q8⋊C8—152no—
67Q8⋊C8—152no—
68C22×C3×C5—42no—
69C22×C3×C5—41no—
70C22×C3×C5—81no—
71C22×C3×C5—81no—
72C3×Dic5—42no—
73C3×Dic5—41no—
74C3×Dic5—41no—
75C22×C3×C4—201no—
76C2×SL(2,𝔽3)—21no—
77C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
78#208—204no—
79C2×C3⋊C8—202no—
80C23×C5SL(2,ℤ/4)11yescharacteristic
81C5×Q8—31no—
82C5×Q8—32no—
83C5×Q8—61no—
84C5×Q8#20811yescharacteristic, complemented
85C2×C4×C5—31no—
86C2×C4×C5—31no—
87C2×C4×C5—61no—
88C2×C4×C5—61no—
89C2×Dic5—31no—
90C2×Dic5—31no—
91C2×Dic5—121no—
92C2×Dic5#22612yes—
93C2×Dic5GL(2,𝔽3)12yes—
94C2×Dic5SL(2,ℤ/4)11yescharacteristic
95C2×Dic5C2×S411yescharacteristic
96#170—122no—
97#170—122no—
98C22×Q8ΓL(1,𝔽16)11yescharacteristic
99C2×M16—301no—
100C4×Q8—51no—
101C4×Q8—51no—
102C4×Q8—151no—
103C4×Q8—151no—
104C2×C4⋊C4—151no—
105C2×C42—151no—
106C4⋊C8—152no—
107C4⋊C8—301no—
108C4⋊C8—301no—
109C22⋊C8—301no—
110C4×C8—152no—
111C2×C3×C5—41no—
112C2×C3×C5—41no—
113C2×C3×C5—41no—
114C2×C3×C5—82no—
115C23×C3—41no—
116C2×C3×C4—201no—
117C2×C3×C4—201no—
118C2×C3×C4—202no—
119C2×C3×C4—202no—
120SL(2,𝔽3)#70411yescharacteristic, complemented
121C3⋊C8—204no—
122C22×C5—21no—
123C22×C5—21no—
124C22×C5#85912yes—
125C22×C5C2×SL(2,ℤ/4)11yescharacteristic
126C4×C5—31no—
127C4×C5—31no—
128C4×C5—61no—
129Dic5—61no—
130Dic5—61no—
131Dic5#85912yes—
132Dic5C2×#22611yescharacteristic
133Dic5C2×GL(2,𝔽3)11yescharacteristic
134C2×Q8—21no—
135C2×Q8—31no—
136C2×Q8—61no—
137C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
138C22×C4—31no—
139C22×C4—51no—
140C22×C4—151no—
141M16—602no—
142C2×C8—302no—
143C2×C8—302no—
144C4⋊C4—151no—
145C4⋊C4—151no—
146C4⋊C4—301no—
147C42—151no—
148C42—151no—
149C42—151no—
150C42—151no—
151C3×C5—41noHall
152C22×C3—41no—
153C22×C3—42no—
154C22×C3—81no—
155C22×C3—81no—
156C3×C4—201no—
157C3×C4—202no—
158C3×C4—201no—
159C2×C5—22no—
160C2×C5#540911yescharacteristic
161C2×C5C2×#85911yescharacteristic
162C2×C5#542311yescharacteristic
163C23A4⋊AGL(1,𝔽5)11yescharacteristic
164Q8—31no—
165Q8—32no—
166Q8—61no—
167Q8#676211yescharacteristic, complemented
168C2×C4—31no—
169C2×C4—31no—
170C2×C4—51no—
171C2×C4—51no—
172C2×C4—52no—
173C2×C4—52no—
174C2×C4—61no—
175C2×C4—61no—
176C2×C4—151no—
177C2×C4—151no—
178C2×C4—601no—
179C8—602no—
180C8—602no—
181C2×C3—41no—
182C2×C3—41no—
183C2×C3—41no—
184C2×C3—82no—
185C5#7305511yescharacteristic, complemented, Sylow
186C22—21no—
187C22—21no—
188C22#9188612yes—
189C22C2×A4⋊AGL(1,𝔽5)11yescharacteristic
190C4—31no—
191C4—31no—
192C4—51no—
193C4—52no—
194C4—51no—
195C4—61no—
196C4—301no—
197C4—301no—
198C3—41noSylow
199C2—22no—
200C2#1175672011yescharacteristic
201C2C2×#9188611yescharacteristic
202C2#1175672811yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4⋊AGL(1,𝔽5)
commutatorC2×C5×SL(2,𝔽3)C2×C4
FittingC22×C5×Q8Dic3
socleC22×C5C2×SL(2,ℤ/4)
prime radicalC22×C5×SL(2,𝔽3)C4
cosocleDic5×SL(2,𝔽3)C22
hypercenterC23A4⋊AGL(1,𝔽5)
hypocenterC5×SL(2,𝔽3)M16
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. SL(2,𝔽3)⋊#170C2semidirect
40. C5×SL(2,𝔽3)M16semidirect
84. C5×Q8#208semidirect
120. SL(2,𝔽3)#704semidirect
167. Q8#6762semidirect
185. C5#73055semidirect

Maximal subgroups

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

subgroupindex
C2×Dic5×SL(2,𝔽3)2
SL(2,𝔽3)⋊#1702
#106049683
C2×#67624
#730555