Finite Group Databasebeta

← #20273763

Subgroups — #20273763

Order 6784.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#16810956C211yesmaximal, characteristic, complemented
2#16810939C211yesmaximal, characteristic, complemented
3#16810559C211yesmaximal, characteristic, complemented
4#16810360C211yesmaximal, characteristic, complemented
5#16810357C211yesmaximal, characteristic, complemented
6#16810347C211yesmaximal, characteristic, complemented
7C53×#538C211yesmaximal, characteristic
8C2×C4⋊Dic53C2211yescharacteristic
9C2×#11719919—21no—
10C2×C22⋊Dic53—21no—
11#13133150—21no—
12#13133148—21no—
13#13133141—21no—
14#13133138—21no—
15C22⋊#88322C2211yescharacteristic
16C22⋊#88322C2211yescharacteristic
17#13133124C2211yescharacteristic
18#126×C53—21no—
19#124×C53—21no—
20#123×C53—21no—
21C22⋊Q8×C53C2211yescharacteristic
22#121×C53—21no—
23#121×C53C2211yescharacteristic
24#120×C53—21no—
25C4×D4×C53—21no—
26C2×C22⋊C4×C53C2211yescharacteristic
27C24×C53D411yescharacteristic
28C2×Q8×C53D411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53—21no—
32C2×D4×C53D411yescharacteristic
33C22×C4×C53—21no—
34C22×C4×C53D411yescharacteristic
35C22×C4×C53C2311yescharacteristic
36C22×Dic53—21no—
37C22×Dic53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53D411yescharacteristic
41C4⋊C4×C53D411yescharacteristic
42C22⋊C4×C53—21no—
43C22⋊C4×C53—21no—
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C42×C53—21no—
50C22⋊Dic53—42no—
51C4⋊Dic53—21no—
52C4⋊Dic53—21no—
53#11719919—42no—
54C2×#88322—21no—
55C2×#88322—21no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—21no—
60C23×C53—41no—
61C23×C53—41no—
62C23×C53C2×D411yescharacteristic
63Q8×C53—41no—
64D4×C53—41no—
65D4×C53—41no—
66D4×C53—41no—
67D4×C53—81no—
68D4×C53—81no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—81no—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×Dic53—22no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53—41no—
82C2×Dic53—41no—
83C2×Dic53—41no—
84#88322—41no—
85#88322—41no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—42no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53—41no—
95C22×C53—81no—
96C22×C53#12011yescharacteristic
97C22×C53#13511yescharacteristic
98C22×C53#13711yescharacteristic
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—41no—
102C4×C53—41no—
103C4×C53—41no—
104C4×C53—41no—
105Dic53—42no—
106Dic53—81no—
107#1618—531nomaximal, Sylow
108C2×C53—21no—
109C2×C53—21no—
110C2×C53—81no—
111C2×C53—81no—
112C2×C53#45211yescharacteristic
113C2×C53#45311yescharacteristic
114C2×C53#45811yescharacteristic
115#538D5311yescharacteristic
116#484—531no—
117#481—531no—
118#399—531no—
119#328—531no—
120#327—531no—
121#323—531no—
122C53#161811yescharacteristic, complemented, Sylow
123#126—21no—
124#124—21no—
125#123—21no—
126C22⋊Q8C2×D5311yescharacteristic
127#121—21no—
128#121C2×D5311yescharacteristic
129#120—21no—
130C4×D4—21no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—1061no—
133C2×C22⋊C4—1061no—
134C2×C22⋊C4C2×D5311yescharacteristic
135#107—1061no—
136#106—1061no—
137#103—1061no—
138#102—1061no—
139C22⋊C8—531no—
140C22⋊C8—531no—
141#95—531no—
142C24C22⋊D5311yescharacteristic
143C2×Q8C22⋊D5311yescharacteristic
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4C22⋊D5311yescharacteristic
148C22×C4—21no—
149C22×C4—1061no—
150C22×C4—1061no—
151C22×C4C22⋊D5311yescharacteristic
152C22×C4C22×D5311yescharacteristic
153C2×C8—1061no—
154C2×C8—1061no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—1061no—
158C4⋊C4—1061no—
159C4⋊C4—2122no—
160C4⋊C4C22⋊D5311yescharacteristic
161C4⋊C4C22⋊D5311yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C22⋊C4—2122no—
170C42—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—21no—
175C23—41no—
176C23—41no—
177C23C2×C22⋊D5311yescharacteristic
178Q8—41no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—81no—
183D4—81no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4—1062no—
192C2×C4—2121no—
193C2×C4—2121no—
194C2×C4—2121no—
195C2×C4—2121no—
196C2×C4—2121no—
197C2×C4C2×C22⋊D5311yescharacteristic
198C2×C4C2×C22⋊D5311yescharacteristic
199C8—2121no—
200C8—2121no—
201C22—21no—
202C22—21no—
203C22—42no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—81no—
211C22#1313318611yescharacteristic
212C22#1313323711yescharacteristic
213C22#1313324711yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—4241no—
221C4—2122no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—81no—
226C2#1681078511yescharacteristic
227C2#1681079111yescharacteristic
228C2#1681081311yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#538C2
socleC22×C53#120
hypercenter#538D53
hypocenterC53#1618
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #538

Complements

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

NH ≅ G/Ntype
1. #16810956C2semidirect
2. #16810939C2semidirect
3. #16810559C2semidirect
4. #16810360C2semidirect
5. #16810357C2semidirect
6. #16810347C2semidirect
122. C53#1618semidirect