Finite Group Databasebeta

← #20273953

Subgroups — #20273953

Order 6784.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811278C211yesmaximal, characteristic
2#16811028C211yesmaximal, characteristic, complemented
3#16810903C211yesmaximal, characteristic, complemented
4#16810829C211yesmaximal, characteristic, complemented
5#16810370C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719945—21no—
9C2×#11719913—21no—
10#13133217C2211yescharacteristic
11#13133217C2211yescharacteristic
12#13133199—22no—
13#13133196—21no—
14#13133188—21no—
15C4×D212C2211yescharacteristic
16C4⋊C4×D53—21no—
17#13133151—22no—
18#13133150—21no—
19C4⋊#88322C2211yescharacteristic
20#13133142—21no—
21#13133138—21no—
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×D4×C53D411yescharacteristic
26C22×C4×C53—21no—
27C2×C22⋊D53—22no—
28#11719945—42no—
29#11719945—42no—
30C2×D212D411yescharacteristic
31C2×C4×D53—22no—
32C2×C4×D53—21no—
33C2×Dic106—21no—
34C2×Dic106—21no—
35C2×C8×C53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53D411yescharacteristic
38C22⋊C4×C53—21no—
39C42×C53C2311yescharacteristic
40C22⋊Dic53—22no—
41#11719921—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53D411yescharacteristic
46C4⋊Dic53D412yes—
47#11719919—22no—
48C2×#88322—21no—
49#11719913—42no—
50C23×C53—21no—
51C22×D53—21no—
52D4×C53—21no—
53D4×C53—21no—
54C2×C4×C53—21no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C2×C4×C53C2×D411yescharacteristic
60C22⋊D53—44no—
61C2×Dic53—22no—
62C2×Dic53—21no—
63C2×Dic53—41no—
64D212—21no—
65D212D812yes—
66C4×D53—44no—
67C4×D53—41no—
68C4×D53—41no—
69Dic106—41no—
70Dic106—41no—
71Dic106—41no—
72Dic106—41no—
73C8×C53—22no—
74C8×C53—41no—
75#88322—41no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53#12011yescharacteristic
80C2×D53—22no—
81C2×D53—41no—
82C2×D53—41no—
83C4×C53—21no—
84C4×C53—41no—
85C4×C53C2×D811yescharacteristic
86C4×C53#13511yescharacteristic
87C4×C53AGL(1,ℤ/8)11yescharacteristic
88C4×C53#13711yescharacteristic
89Dic53—42no—
90Dic53—41no—
91Dic53—81no—
92#1647—531nomaximal, Sylow
93C2×C53—81no—
94C2×C53#44711yescharacteristic
95C2×C53#45211yescharacteristic
96C2×C53#45611yescharacteristic
97D53—42no—
98#547—531no—
99#500—531no—
100#474—531no—
101#461—531no—
102#332—531no—
103D4⋊C8—531no—
104D4⋊C8D5311yescharacteristic, complemented
105C53#164711yescharacteristic, complemented, Sylow
106C2×#41—1061no—
107C2×SD16—1061no—
108C4⋊Q8—531no—
109C4⋊Q8—531no—
110#123—1062no—
111C22⋊Q8—1061no—
112#121—1061no—
113C4×D4—531no—
114C4×D4C2×D5311yescharacteristic
115C2×C4⋊C4—1061no—
116#107—1062no—
117#107—1061no—
118C4⋊C8—531no—
119C4⋊C8C2×D5311yescharacteristic
120#103—1061no—
121#102—1061no—
122C4×C8C2×D5311yescharacteristic
123#41—2122no—
124#41—2122no—
125C2×Q8—1061no—
126C2×Q8—1061no—
127C2×D4—531no—
128C2×D4—1062no—
129C2×D4D21211yescharacteristic
130C22×C4—21no—
131C22×C4—1062no—
132C22×C4—1061no—
133SD16—2122no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—1061no—
137C4⋊C4—531no—
138C4⋊C4—532no—
139C4⋊C4—1062no—
140C4⋊C4—1061no—
141C4⋊C4—1061no—
142C4⋊C4—1061no—
143C4⋊C4D21211yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—1062no—
146C22⋊C4—1061no—
147C42C22×D5311yescharacteristic
148C23—21no—
149C23—1061no—
150Q8—2121no—
151Q8—2121no—
152Q8—2121no—
153Q8—2121no—
154D4—21no—
155D4—21no—
156D4—532no—
157D4—1061no—
158D4—2124no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—1061no—
163C2×C4—1062no—
164C2×C4—2124no—
165C2×C4—2121no—
166C2×C4—2121no—
167C2×C4—2121no—
168C2×C4C2×D21211yescharacteristic
169C2×C4D4×D5311yescharacteristic
170C2×C4D4×D5311yescharacteristic
171C8—22no—
172C8—41no—
173C8—2121no—
174C22—41no—
175C22—41no—
176C22—41no—
177C22—1062no—
178C22—2121no—
179C22—2121no—
180C22#1313318711yescharacteristic
181C4—21no—
182C4—41no—
183C4—4241no—
184C4—2122no—
185C4—2121no—
186C4D8×D5311yescharacteristic
187C4#1313323911yescharacteristic
188C4#1313324411yescharacteristic
189C4#1313325011yescharacteristic
190C2—81no—
191C2—2122no—
192C2#1681075811yescharacteristic
193C2#1681078711yescharacteristic
194C2#1681080711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53#120
hypercenterD4⋊C8D53
hypocenterC53#1647
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #16811028C2semidirect
3. #16810903C2semidirect
4. #16810829C2semidirect
5. #16810370C2semidirect
6. #16810352C2semidirect
7. C53×D4⋊C8C2semidirect
104. D4⋊C8D53semidirect
105. C53#1647semidirect