Finite Group Databasebeta

← #20273679

Subgroups — #20273679

Order 6784.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#16811218C211yesmaximal, characteristic
2#16810781C211yesmaximal, characteristic, complemented
3#16810761C211yesmaximal, characteristic
4#16810544C211yesmaximal, characteristic, complemented
5#16810359C211yesmaximal, characteristic, complemented
6#16810348C211yesmaximal, characteristic, complemented
7C53×#327C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C2×Dic212—21no—
10C2×#11719923—21no—
11#13133210—21no—
12#13133207—21no—
13#13133198—21no—
14#13133198—21no—
15#13133196C2211yescharacteristic
16#13133194—21no—
17D4×Dic53—21no—
18C2×C4⋊Dic53—21no—
19C2×#11719919C2211yescharacteristic
20#13133142—21no—
21#13133140—21no—
22C22⋊#88322C2211yescharacteristic
23#13133124—21no—
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C2×D4×C53—21no—
28C2×D4×C53D411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C22×Dic53—21no—
32C22×Dic53—21no—
33C22×Dic53D411yescharacteristic
34C22×Dic53D411yescharacteristic
35C2×Dic106—21no—
36C2×Dic106—21no—
37C2×Dic106—42no—
38C2×Dic106D411yescharacteristic
39C2×C8×C53—21no—
40C4⋊C4×C53D411yescharacteristic
41C22⋊C4×C53—21no—
42C22⋊Dic53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45C22⋊Dic53—21no—
46#11719923—42no—
47C4⋊Dic53—21no—
48C4⋊Dic53—42no—
49C4⋊Dic53D411yescharacteristic
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53#11719919—21no—
54#11719919—21no—
55C4×Dic53—21no—
56C2×#88322—21no—
57Dic212—42no—
58C23×C53—21no—
59C23×C53C2×D411yescharacteristic
60D4×C53—41no—
61D4×C53—41no—
62D4×C53—41no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×Dic53—22no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C2×Dic53—81no—
81Dic106—22no—
82Dic106—22no—
83Dic106—41no—
84Dic106—41no—
85Dic106—41no—
86C8×C53—41no—
87#88322—41no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53#12011yescharacteristic
94C22×C53#13511yescharacteristic
95C22×C53#13711yescharacteristic
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—42no—
99C4×C53—41no—
100Dic53—41no—
101Dic53—41no—
102Dic53—42no—
103Dic53—41no—
104Dic53—41no—
105#1603—531nomaximal, Sylow
106C2×C53—21no—
107C2×C53—21no—
108C2×C53—81no—
109C2×C53#45211yescharacteristic
110C2×C53#45611yescharacteristic
111C2×C53B(4,𝔽2)11yescharacteristic
112#539—531no—
113#451—531no—
114#448—531no—
115#395—531no—
116#328—531no—
117#327D5311yescharacteristic
118#324—531no—
119C53#160311yescharacteristic, complemented, Sylow
120C22×Q8—531no—
121C2×Q16—1061no—
122C2×SD16—1061no—
123#126—1061no—
124#125—1061no—
125#123—1061no—
126#123—1061no—
127C22⋊Q8—531no—
128C22⋊Q8—1061no—
129#121C2×D5311yescharacteristic
130C4×D4—1061no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—1061no—
133#103—1061no—
134#103—1061no—
135C22⋊C8—531no—
136C22⋊C8C2×D5311yescharacteristic
137#95—1061no—
138#95C2×D5311yescharacteristic
139C2×Q8—531no—
140C2×Q8—1061no—
141C2×Q8—1061no—
142C2×Q8—2122no—
143C2×D4—21no—
144C2×D4D21211yescharacteristic
145C22×C4—21no—
146C22×C4—531no—
147C22×C4—531no—
148C22×C4—1061no—
149C22×C4—1061no—
150C22×C4C22×D5311yescharacteristic
151Q16—2122no—
152SD16—2122no—
153C2×C8—21no—
154C2×C8—1061no—
155C4⋊C4—531no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C4⋊C4—1061no—
159C4⋊C4—1061no—
160C4⋊C4—1061no—
161C4⋊C4—1061no—
162C4⋊C4—2122no—
163C4⋊C4D21211yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C22⋊C4—1061no—
169C42—1061no—
170C23—21no—
171C23D4×D5311yescharacteristic
172Q8—1062no—
173Q8—1062no—
174Q8—2121no—
175Q8—2121no—
176Q8—2121no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—1062no—
186C2×C4—1061no—
187C2×C4—1061no—
188C2×C4—1061no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—4241no—
193C2×C4—2121no—
194C2×C4—2121no—
195C2×C4—2121no—
196C2×C4C2×D21211yescharacteristic
197C2×C4D4×D5311yescharacteristic
198C8—41no—
199C8—2121no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#1313318711yescharacteristic
206C22#1313323611yescharacteristic
207C22#1313324611yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—2121no—
213C4—2121no—
214C4—2121no—
215C4—2121no—
216C4—2122no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#1681078611yescharacteristic
221C2#1681080711yescharacteristic
222C2#1681081111yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53#120
hypercenter#327D53
hypocenterC53#1603
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #16810781C2semidirect
4. #16810544C2semidirect
5. #16810359C2semidirect
6. #16810348C2semidirect
119. C53#1603semidirect