Finite Group Databasebeta

← C5×#85552 #14511290

Subgroups — C5×#85552 #14511290

Order 1920.

186 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 26 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
1C5×Q8⋊SL(2,𝔽3)C211yesmaximal, characteristic
2C5×#1645—31nomaximal, Hall
3C2×C5×#226—41nomaximal
4C5×#859—41nomaximal
5C5×#859—41nomaximal
6#85552C511yesmaximal, characteristic, direct product factor, Hall
7C5×Q82D311yescharacteristic
8C5×#487—31no—
9C5×#475—31no—
10C5×#462—31no—
11C5×#333—31no—
12C5×Q8⋊C8—31no—
13C5×Q8⋊C8—31no—
14C2×C5×SL(2,𝔽3)—41no—
15C2×C5×SL(2,𝔽3)—41no—
16C2×C5×SL(2,𝔽3)—41no—
17C2×C5×SL(2,𝔽3)—81no—
18C5×#226—42no—
19Q8⋊SL(2,𝔽3)C2×C511yescharacteristic
20C2×C5×Q16—61no—
21C5×C4⋊Q8—31no—
22C5×C4⋊Q8—31no—
23C5×C4⋊Q8—61no—
24C4×C5×Q8—31no—
25C4×C5×Q8—31no—
26C5×#106—61no—
27C5×C4⋊C8—31no—
28C5×C4⋊C8—31no—
29C5×#103—61no—
30C5×#103—61no—
31C5×#103—61no—
32C5×#103—61no—
33C4×C5×C8—31no—
34#1645—31noSylow
35C2×C5×Dic3—161no—
36C5×SL(2,𝔽3)—41no—
37C5×SL(2,𝔽3)—41no—
38C5×SL(2,𝔽3)—41no—
39C5×SL(2,𝔽3)—81no—
40C2×#226—41no—
41#859—41no—
42#859—41no—
43C2×C5×Q8—21no—
44C2×C5×Q8—31no—
45C2×C5×Q8—61no—
46C2×C5×Q8S411yescharacteristic
47C2×C5×Q8S411yescharacteristic
48C2×C5×Q8S411yescharacteristic
49C5×Q16—121no—
50C5×Q16—121no—
51C2×C5×C8—61no—
52C2×C5×C8—61no—
53C2×C5×C8—61no—
54C5×C4⋊C4—31no—
55C5×C4⋊C4—31no—
56C5×C4⋊C4—61no—
57C5×C4⋊C4—61no—
58C5×C4⋊C4—61no—
59C5×C4⋊C4—61no—
60C42×C5—31no—
61C42×C5—61no—
62Q82C5×D311yescharacteristic
63#487—31no—
64#475—31no—
65#462—31no—
66#333—31no—
67Q8⋊C8—31no—
68Q8⋊C8—31no—
69C22×C3×C5—161no—
70C5×Dic3—162no—
71C2×SL(2,𝔽3)—41no—
72C2×SL(2,𝔽3)—41no—
73C2×SL(2,𝔽3)—41no—
74C2×SL(2,𝔽3)—81no—
75#226—42no—
76C5×Q8—31no—
77C5×Q8—31no—
78C5×Q8—41no—
79C5×Q8—81no—
80C5×Q8—121no—
81C5×Q8—121no—
82C5×Q8—121no—
83C5×Q8#22611yescharacteristic, complemented
84C5×Q8GL(2,𝔽3)11yescharacteristic
85C2×C4×C5—31no—
86C2×C4×C5—31no—
87C2×C4×C5—31no—
88C2×C4×C5—61no—
89C2×C4×C5—121no—
90C5×C8—121no—
91C5×C8—121no—
92C5×C8—121no—
93C2×Q16—61no—
94C4⋊Q8—31no—
95C4⋊Q8—31no—
96C4⋊Q8—61no—
97C4×Q8—31no—
98C4×Q8—31no—
99#106—61no—
100C4⋊C8—31no—
101C4⋊C8—31no—
102#103—61no—
103#103—61no—
104#103—61no—
105#103—61no—
106C4×C8—31no—
107C2×C3×C5—161no—
108C2×C3×C5—161no—
109C2×C3×C5—161no—
110C2×Dic3—161no—
111SL(2,𝔽3)—41no—
112SL(2,𝔽3)—41no—
113SL(2,𝔽3)—41no—
114SL(2,𝔽3)—81no—
115C22×C5#102011yescharacteristic
116C4×C5—31no—
117C4×C5—31no—
118C4×C5—31no—
119C4×C5—31no—
120C4×C5—61no—
121C4×C5—121no—
122C4×C5—241no—
123C2×Q8—21no—
124C2×Q8—31no—
125C2×Q8—61no—
126C2×Q8C5×S411yescharacteristic
127C2×Q8C5×S411yescharacteristic
128C2×Q8C5×S411yescharacteristic
129Q16—121no—
130Q16—121no—
131C2×C8—61no—
132C2×C8—61no—
133C2×C8—61no—
134C4⋊C4—31no—
135C4⋊C4—31no—
136C4⋊C4—61no—
137C4⋊C4—61no—
138C4⋊C4—61no—
139C4⋊C4—61no—
140C42—31no—
141C42—61no—
142C3×C5—161noHall
143C22×C3—161no—
144Dic3—162no—
145C2×C5#593011yescharacteristic
146C2×C5#593111yescharacteristic
147C2×C5#593311yescharacteristic
148Q8—31no—
149Q8—31no—
150Q8—41no—
151Q8—81no—
152Q8—121no—
153Q8—121no—
154Q8—121no—
155Q8C5×#22611yescharacteristic, complemented
156Q8C5×GL(2,𝔽3)11yescharacteristic
157C2×C4—31no—
158C2×C4—31no—
159C2×C4—31no—
160C2×C4—61no—
161C2×C4—121no—
162C8—121no—
163C8—121no—
164C8—121no—
165C2×C3—161no—
166C2×C3—161no—
167C2×C3—161no—
168C5#8555211yescharacteristic, direct product factor, Sylow
169C22C5×#102011yescharacteristic
170C4—31no—
171C4—31no—
172C4—31no—
173C4—31no—
174C4—61no—
175C4—121no—
176C4—241no—
177C3—161noSylow
178C2C5×#593011yescharacteristic
179C2C5×#593111yescharacteristic
180C2C5×#593311yescharacteristic
subgroupG/H
center, socle, hypercenterC22×C5#1020
commutator, cosocle, hypocenterQ8⋊SL(2,𝔽3)C2×C5
FrattiniC22C5×#1020
FittingC5×Q82D3
prime radicalC5×Q8⋊SL(2,𝔽3)C2
LCSG ▷ Q8⋊SL(2,𝔽3)
UCSC1 ◁ C22×C5

Complements

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

NH ≅ G/Ntype
6. #85552C5direct (N × H)
83. C5×Q8#226semidirect
155. Q8C5×#226semidirect
168. C5#85552direct (N × H)

Maximal subgroups

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

subgroupindex
C5×Q8⋊SL(2,𝔽3)2
C5×#16453
C2×C5×#2264
C5×#8594
#855525