Finite Group Databasebeta

← #20290404

Subgroups — #20290404

Order 6784.

448 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×D53⋊M16C211yesmaximal, characteristic
2C22×C4⋊#6157C211yesmaximal, characteristic
3#16811496C214yesmaximal, complemented
4#117×D53C211yesmaximal, characteristic
5C23×#6157—21no—
6C2×#11719936—21no—
7D53⋊M16—22no—
8D53⋊M16C414yescomplemented
9C2×#11719940—21no—
10C2×D53⋊C8C2212yes—
11C2×C4⋊#6157—21no—
12C2×C4⋊#6157—22no—
13C2×C4⋊#6157C414yescomplemented
14C2×C4⋊#6157C2212yes—
15C22×C4×D53C2211yescharacteristic
16#13133176C411yescharacteristic, complemented
17#13133175C412yescomplemented
18#13133174—21no—
19#13133173—21no—
20C4⋊C4×D53C2212yes—
21C22⋊C4×D53—21no—
22C42×D53—21no—
23#117×C53C411yescharacteristic, complemented
24C23×D53D411yescharacteristic
25C22×#6157—24no—
26C22×#6157—22no—
27C22×#6157—44no—
28C22×C4×C53C2×C411yescharacteristic
29C22×Dic53D411yescharacteristic
30C2×C4×D53—22no—
31C2×C4×D53C2×C411yescharacteristic
32C2×C4×D53C2×C412yes—
33C2×C4×D53C2×C412yes—
34C2×C4×D53D411yescharacteristic
35C2×C4×D53D411yescharacteristic
36C2×C4×D53D412yes—
37C2×C4×D53Q812yes—
38C2×C4×D53C2311yescharacteristic
39#11719940—24no—
40C2×#88324—22no—
41C4⋊#6157—24no—
42C4⋊#6157—22no—
43C4⋊#6157C2×C414yes—
44#11719936—24no—
45D53⋊C8—22no—
46D53⋊C8C2×C414yescomplemented
47C4⋊C4×C53C2×C412yescomplemented
48C22⋊C4×C53—21no—
49C42×C53—21no—
50C22⋊Dic53—21no—
51#11719921—22no—
52C4⋊Dic53C2×C412yescomplemented
53#11719919—22no—
54C4×Dic53—21no—
55C4×Dic53—22no—
56C23×C53C22⋊C411yescharacteristic
57C22×D53—22no—
58C22×D53—22no—
59C22×D53C22⋊C411yescharacteristic
60C22×D53C22⋊C412yes—
61C22×D53C4⋊C412yes—
62C22×D53C2×D411yescharacteristic
63C2×#6157—24no—
64C2×#6157—48no—
65C2×#6157—44no—
66C2×C4×C53—22no—
67C2×C4×C53C4212yes—
68C2×C4×C53C22⋊C411yescharacteristic
69C2×C4×C53C4⋊C412yes—
70C2×C4×C53C22×C411yescharacteristic
71C2×Dic53—22no—
72C2×Dic53C22⋊C411yescharacteristic
73C2×Dic53C22⋊C412yes—
74C2×Dic53C4⋊C412yes—
75C2×Dic53C2×D411yescharacteristic
76C4×D53—42no—
77C4×D53—42no—
78C4×D53C4212yes—
79C4×D53C22⋊C412yes—
80C4×D53C22⋊C412yes—
81C4×D53C4⋊C412yes—
82C4×D53C4⋊C412yes—
83C4×D53C4⋊C412yes—
84C4×D53C22×C411yescharacteristic
85C4×D53C22×C411yescharacteristic
86C4×D53C2×D411yescharacteristic
87C4×D53C2×Q811yescharacteristic
88#88324—24no—
89C22×C53—22no—
90C22×C53#9512yes—
91C22×C53C2×C22⋊C411yescharacteristic
92C2×D53—22no—
93C2×D53—22no—
94C2×D53—22no—
95C2×D53—24no—
96C2×D53#9512yes—
97C2×D53C2×C22⋊C411yescharacteristic
98C2×D53C2×C4⋊C411yescharacteristic
99C2×D53AGL(1,ℤ/8)12yes—
100C2×D53#13712yes—
101#6157—44no—
102C4×C53—42no—
103C4×C53#9512yes—
104C4×C53C2×C4211yescharacteristic
105C4×C53C2×C4⋊C411yescharacteristic
106Dic53—42no—
107Dic53#9512yes—
108Dic53C2×C22⋊C411yescharacteristic
109Dic53C2×C4⋊C411yescharacteristic
110#1736—531nomaximal, Sylow
111C2×C53—22no—
112C2×C53C2×#9511yescharacteristic
113C2×C53#41712yes—
114D53—22no—
115D53#41811yescharacteristic
116D53#41911yescharacteristic
117D53#42812yes—
118C22×M16—531no—
119C2×#117—531no—
120C22×C4⋊C4—531no—
121#340—534no—
122C53#173611yescharacteristic, complemented, Sylow
123C23×C4—531no—
124C23×C4—1061no—
125C2×M16—534no—
126C2×M16—1061no—
127C2×M16—1062no—
128C2×M16—1061no—
129C22×C8—532no—
130#117—531no—
131#117—532no—
132#117—1061no—
133#117—1061no—
134#117#615711yescharacteristic, complemented
135C2×C4⋊C4—532no—
136C2×C4⋊C4—534no—
137C2×C4⋊C4—532no—
138C2×C4⋊C4—1062no—
139C2×C4⋊C4—1061no—
140C2×C22⋊C4—1061no—
141C2×C42—1061no—
142C24—531no—
143C22×C4—531no—
144C22×C4—532no—
145C22×C4—532no—
146C22×C4—532no—
147C22×C4—532no—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—531no—
151C22×C4—531no—
152C22×C4—1062no—
153C22×C4—1062no—
154C22×C4—1064no—
155C22×C4—2124no—
156C22×C4C2×#615711yescharacteristic
157M16—1064no—
158M16—1064no—
159C2×C8—534no—
160C2×C8—1062no—
161C2×C8—1062no—
162C4⋊C4—534no—
163C4⋊C4—532no—
164C4⋊C4—1062no—
165C4⋊C4—1064no—
166C4⋊C4—1062no—
167C4⋊C4C2×#615712yescomplemented
168C22⋊C4—21no—
169C22⋊C4—1062no—
170C22⋊C4—1061no—
171C42—21no—
172C42—1062no—
173C42—1061no—
174C23—531no—
175C23—532no—
176C23—532no—
177C23—531no—
178C23—1062no—
179C23—1062no—
180C23C22⋊#615711yescharacteristic
181C2×C4—22no—
182C2×C4—532no—
183C2×C4—532no—
184C2×C4—531no—
185C2×C4—531no—
186C2×C4—532no—
187C2×C4—532no—
188C2×C4—532no—
189C2×C4—532no—
190C2×C4—532no—
191C2×C4—531no—
192C2×C4—531no—
193C2×C4—532no—
194C2×C4—531no—
195C2×C4—531no—
196C2×C4—1064no—
197C2×C4—1062no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2124no—
201C2×C4—2128no—
202C2×C4C4×#615712yes—
203C2×C4C4⋊#615712yes—
204C2×C4C22⋊#615711yescharacteristic
205C2×C4C22×#615711yescharacteristic
206C8—1064no—
207C22—22no—
208C22—531no—
209C22—531no—
210C22—532no—
211C22—532no—
212C22—532no—
213C22—1062no—
214C22—1062no—
215C22—1064no—
216C22—1062no—
217C22(C2×#6157)⋊C412yes—
218C22C2×C22⋊#615711yescharacteristic
219C4—42no—
220C4—531no—
221C4—532no—
222C4—531no—
223C4—2124no—
224C4—2122no—
225C4(C2×#6157)⋊C412yes—
226C4C2×C4×#615711yescharacteristic
227C4C2×C4⋊#615711yescharacteristic
228C2—22no—
229C2—531no—
230C2—531no—
231C2—532no—
232C2—1062no—
233C2C2×(C2×#6157)⋊C411yescharacteristic
234C2#1681161412yes—
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC4×C53C2×C42
FrattiniC2×C4C22×#6157
Fitting#117×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#117#6157
hypocenterC53#1736
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #16811496C2semidirect
8. D53⋊M16C4semidirect
13. C2×C4⋊#6157C4semidirect
16. #13133176C4semidirect
17. #13133175C4semidirect
23. #117×C53C4semidirect
46. D53⋊C8C2×C4semidirect
47. C4⋊C4×C53C2×C4semidirect
52. C4⋊Dic53C2×C4semidirect
122. C53#1736semidirect
134. #117#6157semidirect
167. C4⋊C4C2×#6157semidirect