Finite Group Databasebeta

← #20275874

Subgroups — #20275874

Order 6784.

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

normal subgroups only

This group has 246 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
1#16811251C211yesmaximal, characteristic
2#16810632C211yesmaximal, characteristic
3#16810530C211yesmaximal, characteristic
4#16810391C211yesmaximal, characteristic
5#16810384C211yesmaximal, characteristic
6#16810382C211yesmaximal, characteristic
7C53×#419C211yesmaximal, characteristic
8C2×#11719917C2211yescharacteristic
9#13133217—21no—
10#13133216—21no—
11#13133209—21no—
12#13133208—21no—
13#13133194—22no—
14#13133176C2211yescharacteristic
15C2×C4⋊Dic53C2211yescharacteristic
16C2×C4×Dic53C2211yescharacteristic
17#13133141—21no—
18#13133141—21no—
19#13133124—22no—
20C22×Q8×C53C2211yescharacteristic
21C2×M16×C53C2211yescharacteristic
22#117×C53C2211yescharacteristic
23#103×C53—21no—
24#103×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27C2×Q8×C53—21no—
28C2×Q8×C53—21no—
29C2×Q8×C53—21no—
30C2×Q8×C53Q812yescomplemented
31C22×C4×C53—21no—
32C22×C4×C53C2311yescharacteristic
33C22×Dic53—21no—
34C22×Dic53D412yes—
35C2×Dic106—22no—
36M16×C53—21no—
37M16×C53—21no—
38C2×C8×C53—21no—
39C4⋊C4×C53D411yescharacteristic
40C4⋊C4×C53D411yescharacteristic
41C22⋊C4×C53—21no—
42C42×C53—21no—
43C22⋊Dic53—21no—
44C4⋊Dic53—22no—
45C4⋊Dic53—21no—
46C4⋊Dic53—21no—
47C4⋊Dic53D411yescharacteristic
48C4⋊Dic53D411yescharacteristic
49#11719919—22no—
50#11719919—22no—
51C4×Dic53—21no—
52C4×Dic53—21no—
53C4×Dic53—21no—
54#11719917—21no—
55#11719917—21no—
56C2×#88322—21no—
57C23×C53C2×D411yescharacteristic
58Q8×C53—22no—
59Q8×C53—22no—
60Q8×C53—41no—
61Q8×C53—41no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—22no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×Q811yescharacteristic
70C2×C4×C53#4111yescharacteristic
71C2×C4×C53#4111yescharacteristic
72C2×C4×C53#4111yescharacteristic
73C2×Dic53—22no—
74C2×Dic53—21no—
75C2×Dic53—22no—
76C2×Dic53—21no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—42no—
80Dic106—82no—
81C8×C53—41no—
82#88322—41no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53#12011yescharacteristic
86C22×C53C22⋊Q811yescharacteristic
87C22×C53#12411yescharacteristic
88C4×C53—42no—
89C4×C53—41no—
90C4×C53—41no—
91C4×C53C22⋊Q811yescharacteristic
92C4×C53C22⋊Q811yescharacteristic
93C4×C53#12411yescharacteristic
94C4×C53#12411yescharacteristic
95Dic53—42no—
96Dic53—41no—
97Dic53—42no—
98Dic53—41no—
99#2028—531nomaximal, Sylow
100C2×C53—21no—
101C2×C53—21no—
102C2×C53#39311yescharacteristic
103C2×C53#45511yescharacteristic
104C2×C53#45611yescharacteristic
105#544—531no—
106#419—531no—
107#419D5311yescharacteristic
108#391—531no—
109#343—531no—
110#339—531no—
111#339—531no—
112C53#202811yescharacteristic, complemented, Sylow
113C22×Q8C2×D5311yescharacteristic
114C2×M16—531no—
115C2×M16C2×D5311yescharacteristic
116C4⋊Q8—1061no—
117C4⋊Q8—1061no—
118#125—1061no—
119#125—1061no—
120C22⋊Q8—1062no—
121#117—531no—
122#117C2×D5311yescharacteristic
123C2×C4⋊C4—531no—
124C2×C42—531no—
125#103—21no—
126#103—21no—
127#103—1061no—
128#103—1061no—
129#95—1062no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—1062no—
136C2×Q8Dic10612yescomplemented
137C22×C4—21no—
138C22×C4—532no—
139C22×C4—1061no—
140C22×C4C22×D5311yescharacteristic
141M16—21no—
142M16—21no—
143M16—1061no—
144M16—1061no—
145C2×C8—21no—
146C2×C8—1061no—
147C4⋊C4—531no—
148C4⋊C4—531no—
149C4⋊C4—1062no—
150C4⋊C4—1062no—
151C4⋊C4—1062no—
152C4⋊C4—1061no—
153C4⋊C4—1061no—
154C4⋊C4C22⋊D5311yescharacteristic
155C4⋊C4C22⋊D5311yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—1061no—
158C42—21no—
159C42—1061no—
160C42—1061no—
161C42—1061no—
162C23D4×D5311yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—4242no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—1062no—
174C2×C4—1061no—
175C2×C4—1061no—
176C2×C4—1062no—
177C2×C4—2121no—
178C2×C4—2121no—
179C2×C4—2122no—
180C2×C4C2×Dic10611yescharacteristic
181C2×C4#1171994511yescharacteristic
182C2×C4D4×D5311yescharacteristic
183C2×C4#1171994711yescharacteristic
184C2×C4#1171994711yescharacteristic
185C2×C4C2×C22⋊D5311yescharacteristic
186C8—41no—
187C8—2121no—
188C22—21no—
189C22—21no—
190C22#1313318511yescharacteristic
191C22#1313319411yescharacteristic
192C22#1313320311yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—2122no—
197C4—2122no—
198C4—2121no—
199C4—2121no—
200C4#1313319411yescharacteristic
201C4#1313319611yescharacteristic
202C4#1313320311yescharacteristic
203C4#1313320411yescharacteristic
204C2—21no—
205C2—21no—
206C2#1681053511yescharacteristic
207C2#1681079911yescharacteristic
208C2#1681080411yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#419C2
socleC22×C53#120
hypercenter#419D53
hypocenterC53#2028
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
30. C2×Q8×C53Q8semidirect
112. C53#2028semidirect
136. C2×Q8Dic106semidirect