Finite Group Databasebeta

← #20290969

Subgroups — #20290969

Order 6784.

264 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 52 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#16811610C211yesmaximal, characteristic, complemented
2#16811603C211yesmaximal, characteristic, complemented
3#16811525C212yesmaximal, complemented
4#16811524C212yesmaximal, complemented
5#16811131C211yesmaximal, characteristic
6C2×#11719940—21no—
7#13133296—21no—
8#13133278—21no—
9#13133277C2212yes—
10#13133275—21no—
11#13133275C2212yes—
12C2×#11719947C2211yescharacteristic
13C22×C4×D53C411yescharacteristic, complemented
14#13133200C2211yescharacteristic
15#13133195—21no—
16#13133190—21no—
17#13133187C2211yescharacteristic
18#13133185C411yescharacteristic, complemented
19#13133179—21no—
20#13133177—21no—
21#13133173C411yescharacteristic, complemented
22#123×C53C411yescharacteristic, complemented
23C23×D53C2×C411yescharacteristic
24C2×D4×C53C2×C411yescharacteristic
25C22×C4×C53C2×C411yescharacteristic
26C2×C22⋊D53—21no—
27C2×C22⋊D53D411yescharacteristic
28C2×C22⋊D53C2311yescharacteristic
29C22×Dic53C2×C411yescharacteristic
30C22×Dic53D411yescharacteristic
31#11719947—41no—
32#11719947—41no—
33C2×D212—21no—
34C2×C4×D53—21no—
35C2×C4×D53—21no—
36C2×C4×D53—41no—
37C2×C4×D53—41no—
38C2×C4×D53D411yescharacteristic
39C2×Dic106D411yescharacteristic
40C22⋊#6157—22no—
41#11719940—22no—
42#11719940—41no—
43C2×#88324—41no—
44C4⋊#6157—41no—
45C4×#6157—41no—
46C4⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53C2×C411yescharacteristic
49C22⋊Dic53C2×C411yescharacteristic
50#11719921—21no—
51#11719921—21no—
52#11719921—21no—
53#11719919—21no—
54#11719919—21no—
55C4×Dic53—21no—
56C23×C53C22⋊C411yescharacteristic
57C23×C53C22×C411yescharacteristic
58C22×D53—21no—
59C22×D53—42no—
60C22×D53—41no—
61C22×D53C2×D411yescharacteristic
62C2×#6157—42no—
63D4×C53—41no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—41no—
67C2×C4×C53C22⋊C411yescharacteristic
68C22⋊D53—41no—
69C22⋊D53—41no—
70C22⋊D53—42no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53C22⋊C411yescharacteristic
77C2×Dic53C22⋊C411yescharacteristic
78C2×Dic53C2×D411yescharacteristic
79D212—42no—
80C4×D53—21no—
81C4×D53—21no—
82C4×D53—42no—
83C4×D53—41no—
84C4×D53—41no—
85Dic106—41no—
86#88324—42no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—41no—
90C22×C53—41no—
91C22×C53C2×C22⋊C411yescharacteristic
92C2×D53—22no—
93C2×D53—21no—
94C2×D53—41no—
95C2×D53—42no—
96C2×D53—41no—
97C2×D53—41no—
98C2×D53—42no—
99#6157—82no—
100C4×C53—21no—
101C4×C53—41no—
102C4×C53—41no—
103Dic53—21no—
104Dic53—21no—
105Dic53—41no—
106Dic53#9911yescharacteristic
107Dic53#9911yescharacteristic
108#2120—531nomaximal, Sylow
109C2×C53—21no—
110C2×C53—41no—
111C2×C53—41no—
112C2×C53C2×#9911yescharacteristic
113D53—41no—
114D53—42no—
115#525—531no—
116#413—531no—
117#410—531no—
118#352—532no—
119AΣL(1,𝔽16)—532no—
120C53#212011yescharacteristic, complemented, Sylow
121C2×#41—531no—
122C23×C4—531no—
123C2×M16—1061no—
124#123—531no—
125#123#615711yescharacteristic, complemented
126C22⋊Q8—1061no—
127#121—1061no—
128#120—531no—
129#120—531no—
130C4×D4—1061no—
131C4×D4—1061no—
132#117—531no—
133#117—1061no—
134#101—1061no—
135#100—532no—
136#99—532no—
137#99—1061no—
138C24—531no—
139#41—2121no—
140#41—2121no—
141C2×Q8—531no—
142C2×D4—531no—
143C2×D4—531no—
144C2×D4—1061no—
145C2×D4—1061no—
146C2×D4C2×#615711yescharacteristic
147C22×C4—531no—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—1061no—
151C22×C4—1061no—
152C22×C4—2121no—
153C22×C4—2121no—
154C22×C4C2×#615711yescharacteristic
155M16—1062no—
156M16—2121no—
157C2×C8—2121no—
158C4⋊C4—21no—
159C4⋊C4—1061no—
160C4⋊C4—1061no—
161C4⋊C4—2121no—
162C22⋊C4—21no—
163C22⋊C4—531no—
164C22⋊C4—1061no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1062no—
168C22⋊C4C2×#615711yescharacteristic
169C42—1061no—
170C42—2121no—
171C23—531no—
172C23—1061no—
173C23—2121no—
174C23—2122no—
175C23C22⋊#615711yescharacteristic
176C23C22×#615711yescharacteristic
177Q8—2121no—
178D4—41no—
179D4—2122no—
180D4—2121no—
181D4—2121no—
182D4—2122no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—531no—
187C2×C4—531no—
188C2×C4—531no—
189C2×C4—1061no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—2121no—
197C2×C4—2121no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4C22⋊#615711yescharacteristic
201C8—2122no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—1062no—
207C22—1061no—
208C22—2121no—
209C22—2121no—
210C22—2122no—
211C22—2122no—
212C22—2121no—
213C22C2×C22⋊#615711yescharacteristic
214C4—21no—
215C4—41no—
216C4—41no—
217C4—531no—
218C4—531no—
219C4—1061no—
220C4—1061no—
221C4—4242no—
222C4—2121no—
223C2—21no—
224C2—41no—
225C2—41no—
226C2—2121no—
227C2—2122no—
228C2D53⋊#9911yescharacteristic
subgroupG/H
centerC2D53⋊#99
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
Fitting#123×C53C4
socleC2×C53C2×#99
hypercenter#123#6157
hypocenterC53#2120
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #16811610C2semidirect
2. #16811603C2semidirect
3. #16811525C2semidirect
4. #16811524C2semidirect
13. C22×C4×D53C4semidirect
18. #13133185C4semidirect
21. #13133173C4semidirect
22. #123×C53C4semidirect
120. C53#2120semidirect
125. #123#6157semidirect