Finite Group Databasebeta

← C5×#2984⋊C3 #14510978

Subgroups — C5×#2984⋊C3 #14510978

Order 1920.

238 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 68 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×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic
2C5×C8×SL(2,𝔽3)C212yesmaximal, complemented
3C5×#2984C311yesmaximal, characteristic, complemented, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic
5C4×C5×SL(2,𝔽3)C411yescharacteristic
6C4×C5×SL(2,𝔽3)C2211yescharacteristic
7C3×C5×C22⋊C8—41nomaximal
8#2984⋊C3C511yesmaximal, characteristic, direct product factor, Hall
9C2×C4×C5×Q8C2×C311yescharacteristic
10C5×#446—32no—
11C5×C8×Q8C2×C312yescomplemented
12C5×#432—31no—
13C5×#431—31no—
14C22×C3×C4×C5—41no—
15C2×C5×SL(2,𝔽3)—21no—
16C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
17C2×C3×C5×C8—42no—
18C2×C4×SL(2,𝔽3)C2×C511yescharacteristic
19C8×SL(2,𝔽3)C2×C512yescomplemented
20C22×C5×Q8C3×C411yescharacteristic
21C4×C5×Q8—31no—
22C4×C5×Q8—31no—
23C4×C5×Q8C3×C411yescharacteristic
24C4×C5×Q8C22×C311yescharacteristic
25C2×C5×C4⋊C4—31no—
26C2×C42×C5—31no—
27C5×C4⋊C8—32no—
28C5×C4⋊C8—32no—
29C5×C22⋊C8—31no—
30C5×C22⋊C8A411yescharacteristic
31C5×#97—32no—
32C4×C5×C8—32no—
33#2984C3×C511yescharacteristic, complemented, Sylow
34C23×C3×C5—41no—
35C2×C3×C4×C5—41no—
36C2×C3×C4×C5—41no—
37C2×C3×C4×C5—82no—
38C5×SL(2,𝔽3)M1611yescharacteristic
39C3×C5×C8—82no—
40C22×SL(2,𝔽3)C4×C511yescharacteristic
41C4×SL(2,𝔽3)C4×C511yescharacteristic
42C4×SL(2,𝔽3)C22×C511yescharacteristic
43C3×C22⋊C8—41no—
44C2×C5×Q8—21no—
45C2×C5×Q8—31no—
46C2×C5×Q8—61no—
47C2×C5×Q8C2×C3×C411yescharacteristic
48C22×C4×C5—31no—
49C22×C4×C5—31no—
50C22×C4×C5C2×A411yescharacteristic
51C2×C5×C8—32no—
52C2×C5×C8C2×A412yes—
53C5×C4⋊C4—31no—
54C5×C4⋊C4—31no—
55C5×C4⋊C4—32no—
56C42×C5—31no—
57C42×C5—31no—
58C42×C5—31no—
59C42×C5—31no—
60C2×C4×Q8C2×C3×C511yescharacteristic
61#446—32no—
62C8×Q8C2×C3×C512yescomplemented
63#432—31no—
64#431—31no—
65C22×C3×C5—42no—
66C22×C3×C5—41no—
67C22×C3×C5—81no—
68C22×C3×C5—81no—
69C3×C4×C5—42no—
70C3×C4×C5—81no—
71C22×C3×C4—41no—
72C2×SL(2,𝔽3)—21no—
73C2×SL(2,𝔽3)C2×C4×C511yescharacteristic
74C2×C3×C8—42no—
75C23×C5C4×A411yescharacteristic
76C5×Q8—31no—
77C5×Q8—61no—
78C5×Q8—61no—
79C5×Q8C3×M1611yescharacteristic
80C2×C4×C5—22no—
81C2×C4×C5—31no—
82C2×C4×C5—31no—
83C2×C4×C5—31no—
84C2×C4×C5—31no—
85C2×C4×C5—61no—
86C2×C4×C5—61no—
87C2×C4×C5—62no—
88C2×C4×C5C4×A411yescharacteristic
89C2×C4×C5C22×A411yescharacteristic
90C5×C8—22no—
91C5×C8—62no—
92C22×Q8C3×C4×C511yescharacteristic
93C4×Q8—31no—
94C4×Q8—31no—
95C4×Q8C3×C4×C511yescharacteristic
96C4×Q8C22×C3×C511yescharacteristic
97C2×C4⋊C4—31no—
98C2×C42—31no—
99C4⋊C8—32no—
100C4⋊C8—32no—
101C22⋊C8—31no—
102C22⋊C8C5×A411yescharacteristic
103#97—32no—
104C4×C8—32no—
105C2×C3×C5—41no—
106C2×C3×C5—41no—
107C2×C3×C5—41no—
108C2×C3×C5—82no—
109C23×C3—41no—
110C2×C3×C4—41no—
111C2×C3×C4—41no—
112C2×C3×C4—82no—
113SL(2,𝔽3)C5×M1611yescharacteristic
114C3×C8—82no—
115C22×C5—21no—
116C22×C5—21no—
117C22×C5#131⋊C312yes—
118C22×C5C2×C4×A411yescharacteristic
119C4×C5—21no—
120C4×C5—31no—
121C4×C5—31no—
122C4×C5—61no—
123C4×C5—61no—
124C4×C5—61no—
125C4×C52−1+4⋊C312yes—
126C2×Q8—21no—
127C2×Q8—31no—
128C2×Q8—61no—
129C2×Q8C2×C3×C4×C511yescharacteristic
130C22×C4—31no—
131C22×C4—31no—
132C22×C4C2×C5×A411yescharacteristic
133C2×C8—32no—
134C2×C8C2×C5×A412yes—
135C4⋊C4—31no—
136C4⋊C4—31no—
137C4⋊C4—32no—
138C42—31no—
139C42—31no—
140C42—31no—
141C42—31no—
142C3×C5—41noHall
143C22×C3—41no—
144C22×C3—42no—
145C22×C3—81no—
146C22×C3—81no—
147C3×C4—42no—
148C3×C4—81no—
149C2×C5—22no—
150C2×C5#519⋊C311yescharacteristic
151C2×C5A4×M1611yescharacteristic
152C2×C5C2×#131⋊C311yescharacteristic
153C23C4×C5×A411yescharacteristic
154Q8—31no—
155Q8—61no—
156Q8—61no—
157Q8C3×C5×M1611yescharacteristic
158C2×C4—22no—
159C2×C4—31no—
160C2×C4—31no—
161C2×C4—31no—
162C2×C4—31no—
163C2×C4—61no—
164C2×C4—61no—
165C2×C4—62no—
166C2×C4C4×C5×A411yescharacteristic
167C2×C4C22×C5×A411yescharacteristic
168C8—22no—
169C8—62no—
170C2×C3—41no—
171C2×C3—41no—
172C2×C3—41no—
173C2×C3—82no—
174C5#2984⋊C311yescharacteristic, direct product factor, Sylow
175C22—21no—
176C22—21no—
177C22C5×#131⋊C312yes—
178C22C2×C4×C5×A411yescharacteristic
179C4—21no—
180C4—31no—
181C4—31no—
182C4—61no—
183C4—61no—
184C4—61no—
185C4C5×2−1+4⋊C312yes—
186C3—41noSylow
187C2—22no—
188C2C5×#519⋊C311yescharacteristic
189C2C5×A4×M1611yescharacteristic
190C2C2×C5×#131⋊C311yescharacteristic
subgroupG/H
centerC2×C4×C5C22×A4
commutatorC2×Q8C2×C3×C4×C5
FrattiniC2×C4C22×C5×A4
FittingC5×#2984C3
socleC22×C5C2×C4×A4
prime radicalC22×C5×SL(2,𝔽3)C4
cosocleC4×Q8C22×C3×C5
hypercenterC5×C22⋊C8A4
hypocenterQ8C3×C5×M16
LCSG ▷ C2×Q8 ▷ Q8
UCSC1 ◁ C2×C4×C5 ◁ C5×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C5×C8×SL(2,𝔽3)C2semidirect
3. C5×#2984C3semidirect
8. #2984⋊C3C5direct (N × H)
11. C5×C8×Q8C2×C3semidirect
19. C8×SL(2,𝔽3)C2×C5semidirect
33. #2984C3×C5semidirect
62. C8×Q8C2×C3×C5semidirect
174. C5#2984⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C5×SL(2,𝔽3)2
C5×C8×SL(2,𝔽3)2
C5×#29843
C3×C5×C22⋊C84
#2984⋊C35