Finite Group Databasebeta

← #20276080

Subgroups — #20276080

Order 6784.

208 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 69 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×#13133153C211yesmaximal, characteristic
2#16810663C211yesmaximal, characteristic
3#16810649C211yesmaximal, characteristic
4#16810632C211yesmaximal, characteristic
5C2×#13133136C211yesmaximal, characteristic
6#16810392C211yesmaximal, characteristic
7C2×#103×C53C211yesmaximal, characteristic
8C2×#11719917C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10C2×#11719917C2211yescharacteristic
11#13133176C2211yescharacteristic
12#13133153—22no—
13#13133150—21no—
14#13133148—21no—
15#13133146—22no—
16#13133141—21no—
17#13133141—21no—
18#13133136—22no—
19C22×Q8×C53C2211yescharacteristic
20C22×C8×C53C2211yescharacteristic
21C2×C4⋊C4×C53C2211yescharacteristic
22#103×C53—22no—
23C2×Q8×C53—21no—
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27C2×Q8×C53—21no—
28C2×Q8×C53Q812yes—
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C22×C4×C53C2311yescharacteristic
32C2×C8×C53—21no—
33C2×C8×C53—21no—
34C2×C8×C53D412yes—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C22⋊Dic53—21no—
38C4⋊Dic53D411yescharacteristic
39C4⋊Dic53D411yescharacteristic
40C4×Dic53—21no—
41#11719917—21no—
42#11719917—21no—
43#11719917—21no—
44#11719917—21no—
45#11719917—21no—
46#11719917D412yes—
47C2×#88322—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C23×C53#4111yescharacteristic
51Q8×C53—22no—
52Q8×C53—22no—
53Q8×C53—41no—
54Q8×C53—41no—
55C2×C4×C53—22no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53—81no—
60C2×C4×C53C2×D411yescharacteristic
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×Q811yescharacteristic
64C2×C4×C53#4111yescharacteristic
65C2×C4×C53#4111yescharacteristic
66C2×Dic53—21no—
67C2×Dic53—21no—
68C8×C53—22no—
69#88322—22no—
70#88322—41no—
71#88322—41no—
72C22×C53—21no—
73C22×C53—21no—
74C22×C53#12111yescharacteristic
75C22×C53C22⋊Q811yescharacteristic
76C22×C53#12611yescharacteristic
77C4×C53—42no—
78C4×C53—81no—
79C4×C53#12111yescharacteristic
80C4×C53#12111yescharacteristic
81C4×C53C22⋊Q811yescharacteristic
82C4×C53C22⋊Q811yescharacteristic
83Dic53—41no—
84Dic53—41no—
85#2064—531nomaximal, Sylow
86C2×C53—21no—
87C2×C53—21no—
88C2×C53#39611yescharacteristic
89C2×C53#47111yescharacteristic
90C2×C53#47311yescharacteristic
91C2×#108—531no—
92#428—531no—
93#424—531no—
94#419—531no—
95C2×#103D5311yescharacteristic
96C2×#101—531no—
97#343—531no—
98C53#206411yescharacteristic, complemented, Sylow
99C22×Q8C2×D5311yescharacteristic
100C2×M16—531no—
101C2×M16—531no—
102C2×M16—531no—
103C22×C8C2×D5311yescharacteristic
104#117—531no—
105C2×C4⋊C4C2×D5311yescharacteristic
106#108—1062no—
107#107—1061no—
108#106—1061no—
109C4⋊C8—1062no—
110#103—22no—
111#103—1061no—
112#103—1061no—
113#101—1062no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—21no—
119C2×Q8Dic10612yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4C22×D5311yescharacteristic
123M16—532no—
124M16—1061no—
125M16—1061no—
126M16—1061no—
127M16—1061no—
128M16—1061no—
129C2×C8—21no—
130C2×C8—21no—
131C2×C8—1061no—
132C2×C8—1061no—
133C2×C8—1061no—
134C2×C8C22⋊D5312yes—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—531no—
138C4⋊C4—531no—
139C22⋊C4—1061no—
140C42—1061no—
141C23#1171994711yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—41no—
145Q8—41no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—81no—
151C2×C4—1061no—
152C2×C4—1061no—
153C2×C4C2×Dic10611yescharacteristic
154C2×C4#1171994511yescharacteristic
155C2×C4D4×D5311yescharacteristic
156C2×C4D4×D5311yescharacteristic
157C2×C4#1171994711yescharacteristic
158C2×C4C2×C22⋊D5311yescharacteristic
159C8—22no—
160C8—1062no—
161C8—2121no—
162C8—2121no—
163C22—21no—
164C22—21no—
165C22#1313318811yescharacteristic
166C22#1313319411yescharacteristic
167C22#1313321011yescharacteristic
168C4—42no—
169C4—81no—
170C4—2121no—
171C4—2121no—
172C4#1313318911yescharacteristic
173C4#1313319011yescharacteristic
174C4#1313319411yescharacteristic
175C4#1313319611yescharacteristic
176C2—21no—
177C2—21no—
178C2#1681054611yescharacteristic
179C2#1681088311yescharacteristic
180C2#1681089511yescharacteristic
subgroupG/H
centerC22#13133188
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC2×#103×C53C2
socleC22×C53#121
hypercenterC2×#103D53
hypocenterC53#2064
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
98. C53#2064semidirect