Finite Group Databasebeta

← #14511708

Subgroups — #14511708

Order 1920.

224 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 37 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
1SL(2,𝔽3)×Dic10C211yesmaximal, characteristic
2C5×#5386C211yesmaximal, characteristic
3SL(2,𝔽3)⋊#168C211yesmaximal, characteristic
4#10601922—31nomaximal, Hall
5C4×C5×SL(2,𝔽3)C2211yescharacteristic
6C2×#6661—41nomaximal
7Dic5×SL(2,𝔽3)—21no—
8C5×#859—21no—
9#73099—51nomaximal, Hall
10Q8×Dic10D311yescharacteristic
11C5×#475—31no—
12#65382—31no—
13#65379—31no—
14Q8⋊#168—31no—
15#64704—31no—
16#64701—31no—
17C2×C5×Dic6—41no—
18C2×C3×Dic10—41no—
19C2×C5×SL(2,𝔽3)D411yescharacteristic
20C2×C3⋊#168—41no—
21#6661—44no—
22Q8×SL(2,𝔽3)—51no—
23#5386D511yescharacteristic
24SL(2,𝔽3)⋊C8—51no—
25C5×C4⋊Q8—31no—
26C4×C5×Q8C2×D311yescharacteristic
27Q8×Dic5—21no—
28C2×#689—61no—
29#4034—61no—
30#4015—31no—
31C4×Dic10—31no—
32C5×#106—61no—
33C5×C4⋊C8—31no—
34C5×#103—61no—
35#3967—61no—
36#3942—62no—
37C4⋊#168—31no—
38C4×#168—31no—
39#1645—151noSylow
40C2×C3×C4×C5—41no—
41C2×C5×Dic3—81no—
42C5×Dic6—42no—
43C5×Dic6—81no—
44C2×C3×Dic5—81no—
45C3×Dic10—42no—
46C3×Dic10—81no—
47C5×SL(2,𝔽3)Q1611yescharacteristic, complemented
48C3⋊#168—42no—
49C2×#216—201no—
50C4×SL(2,𝔽3)—101no—
51C4×SL(2,𝔽3)C2×D511yescharacteristic
52#859—21no—
53C2×C5×Q8—61no—
54C2×C5×Q8#6711yescharacteristic
55C2×Dic10—32no—
56C2×Dic10S411yescharacteristic
57C2×C5×C8—61no—
58C5×C4⋊C4—31no—
59C5×C4⋊C4—61no—
60C5×C4⋊C4—61no—
61C42×C5—31no—
62#689—122no—
63C4⋊Dic5—31no—
64C4⋊Dic5—61no—
65#683—61no—
66C4×Dic5—61no—
67C2×#168—61no—
68C2×#168—61no—
69Q82—51no—
70#487—151no—
71#475—31no—
72#462—151no—
73#333—151no—
74Q8⋊C8—151no—
75Q8⋊C8—151no—
76C22×C3×C5—41no—
77C3×C4×C5—41no—
78C3×C4×C5—41no—
79C3×Dic5—82no—
80C5×Dic3—82no—
81C2×C3×Q8—201no—
82C2×Dic6—41no—
83C2×SL(2,𝔽3)C22⋊D511yescharacteristic
84#216—204no—
85C2×C3⋊C8—201no—
86C5×Q8—31no—
87C5×Q8—121no—
88C5×Q8—121no—
89C5×Q8#21611yescharacteristic, complemented
90C2×C4×C5—31no—
91C2×C4×C5—31no—
92C2×C4×C5—121no—
93C2×C4×C5C2×S411yescharacteristic
94C2×Dic5—21no—
95C2×Dic5—61no—
96Dic10—21no—
97Dic10—122no—
98Dic10GL(2,𝔽3)12yes—
99C5×C8—121no—
100#168—121no—
101#168—121no—
102C2×Q16—301no—
103C4⋊Q8—31no—
104C4⋊Q8—151no—
105C4⋊Q8—301no—
106C4×Q8—101no—
107C4×Q8—151no—
108C4×Q8D3×D511yescharacteristic
109#106—61no—
110C4⋊C8—31no—
111C4⋊C8—151no—
112#103—61no—
113#103—302no—
114#103—301no—
115C4×C8—151no—
116C2×C3×C5—41no—
117C2×C3×C5—41no—
118C2×C3×C5—41no—
119C3×Q8—202no—
120C3×Q8—401no—
121C2×C3×C4—41no—
122C2×C3×C4—401no—
123C2×Dic3—81no—
124Dic6—42no—
125Dic6—81no—
126SL(2,𝔽3)#68911yescharacteristic, complemented
127C3⋊C8—202no—
128C22×C5GL(2,ℤ/4)11yescharacteristic
129C4×C5—31no—
130C4×C5—31no—
131C4×C5—61no—
132C4×C5—241no—
133C4×C5C2×GL(2,𝔽3)11yescharacteristic
134C4×C5#98311yescharacteristic
135Dic5—22no—
136Dic5—121no—
137C2×Q8—51no—
138C2×Q8—61no—
139C2×Q8—152no—
140C2×Q8#120111yescharacteristic
141Q16—602no—
142C2×C8—61no—
143C2×C8—301no—
144C2×C8—301no—
145C4⋊C4—31no—
146C4⋊C4—61no—
147C4⋊C4—61no—
148C4⋊C4—151no—
149C4⋊C4—301no—
150C4⋊C4—301no—
151C42—31no—
152C42—301no—
153C3×C5—41noHall
154C22×C3—41no—
155C3×C4—41no—
156C3×C4—41no—
157C3×C4—402no—
158Dic3—82no—
159C2×C5#541611yescharacteristic
160C2×C5#542111yescharacteristic
161C2×C5#542811yescharacteristic
162Q8—31no—
163Q8—52no—
164Q8—101no—
165Q8—121no—
166Q8—121no—
167Q8—602no—
168Q8#666111yescharacteristic, complemented
169C2×C4—31no—
170C2×C4—31no—
171C2×C4—101no—
172C2×C4—121no—
173C2×C4—301no—
174C2×C4D5×S411yescharacteristic
175C8—121no—
176C8—601no—
177C8—601no—
178C2×C3—41no—
179C2×C3—41no—
180C2×C3—41no—
181C5#7309911yescharacteristic, complemented, Sylow
182C22#9190411yescharacteristic
183C4—31no—
184C4—31no—
185C4—61no—
186C4—102no—
187C4—241no—
188C4—601no—
189C4#9189611yescharacteristic
190C4D5×GL(2,𝔽3)11yescharacteristic
191C3—41noSylow
192C2#1175158511yescharacteristic
193C2#1175161311yescharacteristic
194C2#1175163111yescharacteristic
subgroupG/H
centerC22#91904
commutator, cosocleC4×C5×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D5×S4
FittingC4×C5×Q8C2×D3
socleC22×C5GL(2,ℤ/4)
prime radicalC2×C5×SL(2,𝔽3)D4
hypocenterC5×SL(2,𝔽3)Q16
LCSG ▷ C4×C5×SL(2,𝔽3) ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
47. C5×SL(2,𝔽3)Q16semidirect
89. C5×Q8#216semidirect
126. SL(2,𝔽3)#689semidirect
168. Q8#6661semidirect
181. C5#73099semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×Dic102
C5×#53862
SL(2,𝔽3)⋊#1682
#106019223
C2×#66614
#730995