Finite Group Databasebeta

← #14511709

Subgroups — #14511709

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
2#11752105C211yesmaximal, characteristic
3C5×SL(2,𝔽3)⋊C8C211yesmaximal, characteristic
4#10599048—31nomaximal, Hall
5C4×C5×SL(2,𝔽3)C2211yescharacteristic
6C2×#6662—41nomaximal
7Dic5×SL(2,𝔽3)—21no—
8#91184—21no—
9#73099—51nomaximal, Hall
10Q8×Dic10D311yescharacteristic
11#65309—31no—
12#65138—31no—
13#64970—31no—
14C5×Q8⋊C8—31no—
15#64701—31no—
16#64676—31no—
17C2×Dic30—41no—
18C2×C3×Dic10—41no—
19C2×C5×SL(2,𝔽3)D411yescharacteristic
20C2×C5×C3⋊C8—41no—
21#6662—44no—
22Q8×SL(2,𝔽3)—51no—
23#5386—51no—
24SL(2,𝔽3)⋊C8D511yescharacteristic
25C4×C5×Q8C2×D311yescharacteristic
26Q8×Dic5—21no—
27C2×Dic20—61no—
28#4034—61no—
29#4015—31no—
30#4015—31no—
31C4×Dic10—31no—
32C5×C4⋊C8—31no—
33C4×C5×C8—31no—
34#3967—61no—
35#3948—61no—
36#3948—62no—
37#3940—61no—
38C4⋊#168—31no—
39#1645—151noSylow
40C2×C3×C4×C5—41no—
41C2×Dic15—81no—
42Dic30—42no—
43Dic30—81no—
44C2×C3×Dic5—81no—
45C3×Dic10—42no—
46C3×Dic10—81no—
47C5×SL(2,𝔽3)Q1611yescharacteristic, complemented
48C5×C3⋊C8—42no—
49C2×#216—201no—
50C4×SL(2,𝔽3)—101no—
51C4×SL(2,𝔽3)C2×D511yescharacteristic
52#859—101no—
53C2×C5×Q8#6711yescharacteristic
54C2×Dic10—32no—
55C2×Dic10—61no—
56C2×Dic10S411yescharacteristic
57C2×C5×C8—61no—
58C2×C5×C8—61no—
59C5×C4⋊C4—31no—
60C42×C5—31no—
61C4⋊Dic5—31no—
62C4⋊Dic5—61no—
63C4⋊Dic5—61no—
64C4⋊Dic5—61no—
65#683—61no—
66C4×Dic5—61no—
67C2×#168—61no—
68Dic20—122no—
69Q82—51no—
70#487—151no—
71#475—151no—
72#462—151no—
73#333—151no—
74Q8⋊C8—31no—
75Q8⋊C8—151no—
76C22×C3×C5—41no—
77C3×C4×C5—41no—
78C3×C4×C5—41no—
79Dic15—82no—
80C3×Dic5—82no—
81C2×C3×Q8—201no—
82C2×Dic6—201no—
83C2×SL(2,𝔽3)D2011yescharacteristic
84#216—204no—
85C2×C3⋊C8—41no—
86C5×Q8—31no—
87C5×Q8#21611yescharacteristic, complemented
88C2×C4×C5—31no—
89C2×C4×C5—31no—
90C2×C4×C5C2×S411yescharacteristic
91C2×Dic5—21no—
92C2×Dic5—61no—
93C2×Dic5—121no—
94Dic10—21no—
95Dic10—122no—
96Dic10—121no—
97Dic10—121no—
98Dic10GL(2,𝔽3)12yes—
99C5×C8—121no—
100C5×C8—121no—
101#168—121no—
102C2×Q16—301no—
103C4⋊Q8—151no—
104C4⋊Q8—151no—
105C4⋊Q8—301no—
106C4×Q8—101no—
107C4×Q8—151no—
108C4×Q8D3×D511yescharacteristic
109#106—301no—
110C4⋊C8—31no—
111C4⋊C8—151no—
112#103—301no—
113#103—302no—
114#103—301no—
115C4×C8—31no—
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—401no—
124Dic6—202no—
125Dic6—401no—
126SL(2,𝔽3)Dic2011yescharacteristic, complemented
127C3⋊C8—42no—
128C22×C5GL(2,ℤ/4)11yescharacteristic
129C4×C5—31no—
130C4×C5—31no—
131C4×C5—61no—
132C4×C5C2×GL(2,𝔽3)11yescharacteristic
133C4×C5#98311yescharacteristic
134Dic5—22no—
135Dic5—121no—
136Dic5—241no—
137C2×Q8—51no—
138C2×Q8—152no—
139C2×Q8—301no—
140C2×Q8#120211yescharacteristic
141Q16—602no—
142C2×C8—61no—
143C2×C8—61no—
144C2×C8—301no—
145C4⋊C4—31no—
146C4⋊C4—151no—
147C4⋊C4—301no—
148C4⋊C4—301no—
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—402no—
159C2×C5#541611yescharacteristic
160C2×C5#542111yescharacteristic
161C2×C5#542811yescharacteristic
162Q8—31no—
163Q8—52no—
164Q8—101no—
165Q8—602no—
166Q8—601no—
167Q8—601no—
168Q8#666211yescharacteristic, complemented
169C2×C4—31no—
170C2×C4—31no—
171C2×C4—101no—
172C2×C4—301no—
173C2×C4—601no—
174C2×C4D5×S411yescharacteristic
175C8—121no—
176C8—121no—
177C8—601no—
178C2×C3—41no—
179C2×C3—41no—
180C2×C3—41no—
181C5#7309911yescharacteristic, complemented, Sylow
182C22#9190511yescharacteristic
183C4—31no—
184C4—31no—
185C4—61no—
186C4—102no—
187C4—601no—
188C4—1201no—
189C4#9189711yescharacteristic
190C4D5×GL(2,𝔽3)11yescharacteristic
191C3—41noSylow
192C2#1175158611yescharacteristic
193C2#1175161411yescharacteristic
194C2#1175163211yescharacteristic
subgroupG/H
centerC22#91905
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
87. C5×Q8#216semidirect
126. SL(2,𝔽3)Dic20semidirect
168. Q8#6662semidirect
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
#117521052
C5×SL(2,𝔽3)⋊C82
#105990483
C2×#66624
#730995