Finite Group Databasebeta

← #20276354

Subgroups — #20276354

Order 6784.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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#16811130C211yesmaximal, characteristic
2#16810607C211yesmaximal, characteristic, complemented
3#16810412C212yesmaximal, complemented
4#16810409C212yesmaximal, complemented
5C53×#410C211yesmaximal, characteristic, complemented
6C22×C4×D53C411yescharacteristic, complemented
7C2×#11719917—21no—
8#13133198C411yescharacteristic, complemented
9#13133198C2211yescharacteristic
10#13133193—21no—
11#13133188—21no—
12#13133185C411yescharacteristic, complemented
13#13133185C2211yescharacteristic
14C4×C22⋊D53—21no—
15C4×C22⋊D53—21no—
16#13133176C411yescharacteristic, complemented
17#13133136—21no—
18#13133134C2212yes—
19C2×#41×C53C2211yescharacteristic
20#117×C53—21no—
21#99×C53—21no—
22#99×C53C2212yes—
23C23×D53C2×C411yescharacteristic
24#41×C53—41no—
25#41×C53—41no—
26C2×Q8×C53D411yescharacteristic
27C2×D4×C53C2×C411yescharacteristic
28C2×D4×C53D411yescharacteristic
29C2×D4×C53C2311yescharacteristic
30C22×C4×C53C2×C411yescharacteristic
31C22×C4×C53D411yescharacteristic
32C22×C4×C53D411yescharacteristic
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35C22×Dic53C2×C411yescharacteristic
36C2×C4×D53—21no—
37C2×C4×D53—21no—
38C2×C4×D53—41no—
39C2×C4×D53—41no—
40C4⋊C4×C53—41no—
41C22⋊C4×C53—22no—
42C42×C53—41no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45C22⋊Dic53C2×C411yescharacteristic
46C22⋊Dic53C2×C411yescharacteristic
47#11719921—21no—
48#11719921—21no—
49C4⋊Dic53—21no—
50#11719919—21no—
51#11719919—21no—
52C4×Dic53—21no—
53#11719917—22no—
54#11719917—41no—
55C2×#88322—41no—
56C23×C53C22⋊C411yescharacteristic
57C23×C53C22×C411yescharacteristic
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60C22×D53—42no—
61C22×D53—41no—
62Q8×C53—41no—
63D4×C53—41no—
64D4×C53—41no—
65D4×C53—41no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—42no—
73C2×C4×C53C22⋊C411yescharacteristic
74C2×C4×C53C22⋊C411yescharacteristic
75C2×C4×C53C22⋊C411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C22⋊D53—42no—
78C22⋊D53—42no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—41no—
83C4×D53—42no—
84C4×D53—41no—
85C4×D53—41no—
86#88322—42no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53C2×C22⋊C411yescharacteristic
94C2×D53—22no—
95C2×D53—41no—
96C2×D53—42no—
97C2×D53—41no—
98C2×D53—42no—
99C4×C53—21no—
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—82no—
103C4×C53#9911yescharacteristic
104C4×C53#9911yescharacteristic
105Dic53—41no—
106Dic53—41no—
107Dic53—41no—
108#2120—531nomaximal, Sylow
109C2×C53—21no—
110C2×C53—41no—
111C2×C53—41no—
112C2×C53—41no—
113C2×C53C2×#9911yescharacteristic
114D53—42no—
115#525—531no—
116#413—531no—
117#410D5311yescharacteristic, complemented
118#352—532no—
119AΣL(1,𝔽16)—532no—
120C53#212011yescharacteristic, complemented, Sylow
121C2×#41C2×D5311yescharacteristic
122C23×C4—531no—
123C2×M16—1061no—
124#123—531no—
125#123—531no—
126C22⋊Q8—1061no—
127#121—1061no—
128#120—531no—
129#120—531no—
130C4×D4—1061no—
131C4×D4—1061no—
132#117—21no—
133#117—531no—
134#101—1061no—
135#100—532no—
136#99—21no—
137#99C2×D5312yes—
138C24—531no—
139#41—41no—
140#41—41no—
141C2×Q8D21211yescharacteristic
142C2×D4—1061no—
143C2×D4—1061no—
144C2×D4C4×D5311yescharacteristic
145C2×D4C22⋊D5311yescharacteristic
146C2×D4C22×D5311yescharacteristic
147C22×C4—531no—
148C22×C4—1061no—
149C22×C4—1061no—
150C22×C4—2121no—
151C22×C4—2121no—
152C22×C4C4×D5311yescharacteristic
153C22×C4D21211yescharacteristic
154C22×C4C22⋊D5311yescharacteristic
155M16—1062no—
156M16—2121no—
157C2×C8—2121no—
158C4⋊C4—41no—
159C4⋊C4—1061no—
160C4⋊C4—1061no—
161C4⋊C4—1061no—
162C22⋊C4—22no—
163C22⋊C4—531no—
164C22⋊C4—531no—
165C22⋊C4—1061no—
166C22⋊C4—1061no—
167C22⋊C4—1061no—
168C22⋊C4—1061no—
169C42—41no—
170C42—1061no—
171C23—1061no—
172C23—2121no—
173C23—2122no—
174C23#1171992111yescharacteristic
175C23C2×C4×D5311yescharacteristic
176C23C2×C22⋊D5311yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—2122no—
182D4—2122no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—2122no—
194C2×C4—2121no—
195C2×C4—2121no—
196C2×C4—2121no—
197C2×C4#1171992111yescharacteristic
198C2×C4#1171992111yescharacteristic
199C2×C4#1171992111yescharacteristic
200C2×C4C2×D21211yescharacteristic
201C8—2122no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—1062no—
209C22—2121no—
210C22—2122no—
211C22—2122no—
212C22—2121no—
213C22C2×#1171992111yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—2121no—
219C4—2121no—
220C4—2121no—
221C4#1313313111yescharacteristic
222C4#1313313111yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—2122no—
228C2C2×#1313313111yescharacteristic
subgroupG/H
centerC4#13133131
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#410C2
socleC2×C53C2×#99
hypercenter#410D53
hypocenterC53#2120
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #16810607C2semidirect
3. #16810412C2semidirect
4. #16810409C2semidirect
5. C53×#410C2semidirect
6. C22×C4×D53C4semidirect
8. #13133198C4semidirect
12. #13133185C4semidirect
16. #13133176C4semidirect
117. #410D53semidirect
120. C53#2120semidirect