Finite Group Databasebeta

← #20290593

Subgroups — #20290593

Order 6784.

316 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 84 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×#13133286C211yesmaximal, characteristic, complemented
2C2×#13133285C211yesmaximal, characteristic, complemented
3#16811565C212yesmaximal
4#16811497C212yesmaximal, complemented
5C2×M16×D53C211yesmaximal, characteristic
6C2×C4⋊#6157C2212yes—
7C2×C4⋊#6157C2212yes—
8C2×C22⋊#6157—22no—
9#13133286—22no—
10#13133285—22no—
11(C2×#6157)⋊C4—22no—
12C22×C4×D53C2211yescharacteristic
13C2×#11719917C411yescharacteristic, complemented
14C2×#11719912C411yescharacteristic, complemented
15C2×#11719912C411yescharacteristic, complemented
16M16×D53—22no—
17M16×D53—22no—
18C2×C8×D53C2211yescharacteristic
19C2×C8×D53C2211yescharacteristic
20C2×M16×C53C411yescharacteristic, complemented
21C23×D53D411yescharacteristic
22C22×#6157—22no—
23C22×#6157—22no—
24C22×C4×C53C2×C411yescharacteristic
25C22×Dic53Q811yescharacteristic
26C2×C4×D53—21no—
27C2×C4×D53—21no—
28C2×C4×D53—21no—
29C2×C4×D53—21no—
30C2×C4×D53C2×C411yescharacteristic
31C2×C4×D53D411yescharacteristic
32C2×C4×D53Q811yescharacteristic
33C2×C4×D53C2311yescharacteristic
34C22⋊#6157—44no—
35C4⋊#6157—22no—
36C4⋊#6157—22no—
37C4⋊#6157—22no—
38C4⋊#6157—22no—
39M16×C53—22no—
40C2×C8×C53C2×C411yescharacteristic
41C2×C8×C53C2×C411yescharacteristic
42#11719917—22no—
43C2×#88322C2×C411yescharacteristic
44C2×#88322C2×C411yescharacteristic
45#11719912—22no—
46#11719912—22no—
47C8×D53—21no—
48C8×D53—21no—
49C8×D53D412yes—
50C8×D53D412yes—
51C23×C53C4⋊C411yescharacteristic
52C22×D53—21no—
53C22×D53—21no—
54C22×D53—22no—
55C22×D53—22no—
56C22×D53C4⋊C411yescharacteristic
57C22×D53C2×D411yescharacteristic
58C2×#6157—42no—
59C2×#6157—42no—
60C2×#6157—42no—
61C2×#6157—42no—
62C2×#6157—42no—
63C2×#6157—42no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53C4⋊C411yescharacteristic
67C2×C4×C53C22×C411yescharacteristic
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53C4⋊C411yescharacteristic
71C2×Dic53C2×Q811yescharacteristic
72C4×D53—21no—
73C4×D53—21no—
74C4×D53—21no—
75C4×D53—21no—
76C4×D53—21no—
77C4×D53—21no—
78C4×D53C2×D411yescharacteristic
79C4×D53C2×D411yescharacteristic
80C4×D53#4111yescharacteristic
81C4×D53#4111yescharacteristic
82C8×C53—21no—
83C8×C53C22⋊C412yescomplemented
84#88322—21no—
85#88322C22⋊C412yescomplemented
86C22×C53—21no—
87C22×C53—22no—
88C22×C53C2×C4⋊C411yescharacteristic
89C2×D53—21no—
90C2×D53—22no—
91C2×D53—42no—
92C2×D53—41no—
93C2×D53—41no—
94C2×D53#12111yescharacteristic
95C2×D53#12111yescharacteristic
96C2×D53AGL(1,ℤ/8)12yes—
97C2×D53#13712yes—
98#6157—82no—
99#6157—82no—
100C4×C53—21no—
101C4×C53C2×C22⋊C411yescharacteristic
102C4×C53#11711yescharacteristic
103Dic53—21no—
104Dic53C22⋊Q811yescharacteristic
105Dic53C22⋊Q811yescharacteristic
106#1809—531nomaximal, Sylow
107C2×C53—41no—
108C2×C53#38011yescharacteristic
109C2×C53#42812yes—
110D53—41no—
111D53#46812yes—
112D53#46911yescharacteristic
113D53#47011yescharacteristic
114C22×M16—531no—
115C2×#107—531no—
116C2×#106—531no—
117#380—532no—
118#340—532no—
119C53#180911yescharacteristic, complemented, Sylow
120C23×C4—531no—
121C2×M16—531no—
122C2×M16—531no—
123C2×M16—531no—
124C2×M16—1062no—
125C2×M16—1062no—
126C2×M16#615711yescharacteristic, complemented
127C22×C8—531no—
128C22×C8—531no—
129C2×C4⋊C4—532no—
130C2×C4⋊C4—532no—
131C2×C22⋊C4—1062no—
132#107—1062no—
133#106—1062no—
134#95—1062no—
135C24—531no—
136C22×C4—531no—
137C22×C4—531no—
138C22×C4—531no—
139C22×C4—531no—
140C22×C4—531no—
141C22×C4—1061no—
142C22×C4—1061no—
143C22×C4—1061no—
144C22×C4—1061no—
145C22×C4—1062no—
146C22×C4—1062no—
147C22×C4C2×#615711yescharacteristic
148M16—22no—
149M16—1062no—
150M16—1062no—
151M16—1062no—
152C2×C8—531no—
153C2×C8—532no—
154C2×C8—532no—
155C2×C8—531no—
156C2×C8—1061no—
157C2×C8—1061no—
158C2×C8C2×#615711yescharacteristic
159C2×C8C2×#615711yescharacteristic
160C4⋊C4—1062no—
161C4⋊C4—1062no—
162C4⋊C4—1062no—
163C4⋊C4—1062no—
164C22⋊C4—2124no—
165C23—531no—
166C23—531no—
167C23—1062no—
168C23—1061no—
169C23—1061no—
170C23—1062no—
171C23C4⋊#615711yescharacteristic
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—531no—
175C2×C4—531no—
176C2×C4—531no—
177C2×C4—531no—
178C2×C4—531no—
179C2×C4—531no—
180C2×C4—1061no—
181C2×C4—1061no—
182C2×C4—1061no—
183C2×C4—1061no—
184C2×C4—1061no—
185C2×C4—1061no—
186C2×C4—1061no—
187C2×C4—1061no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4—2122no—
192C2×C4—2122no—
193C2×C4—2122no—
194C2×C4C4⋊#615711yescharacteristic
195C2×C4C22×#615711yescharacteristic
196C8—21no—
197C8—532no—
198C8—1061no—
199C8C22⋊#615712yescomplemented
200C22—21no—
201C22—22no—
202C22—531no—
203C22—531no—
204C22—532no—
205C22—532no—
206C22—1061no—
207C22—1062no—
208C22—2121no—
209C22—2122no—
210C22—2121no—
211C22C2×C4⋊#615711yescharacteristic
212C4—21no—
213C4—531no—
214C4—531no—
215C4—1061no—
216C4—4242no—
217C4—4242no—
218C4C2×C22⋊#615711yescharacteristic
219C4#1313329611yescharacteristic
220C2—41no—
221C2—531no—
222C2—532no—
223C2—531no—
224C2—2121no—
225C2#1681156511yescharacteristic
226C2#1681163412yes—
subgroupG/H
centerC22C2×C4⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×M16×C53C4
socleC22×C53C2×C4⋊C4
hypercenterC2×M16#6157
hypocenterC53#1809
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#13133286C2semidirect
2. C2×#13133285C2semidirect
4. #16811497C2semidirect
13. C2×#11719917C4semidirect
14. C2×#11719912C4semidirect
15. C2×#11719912C4semidirect
20. C2×M16×C53C4semidirect
83. C8×C53C22⋊C4semidirect
85. #88322C22⋊C4semidirect
119. C53#1809semidirect
126. C2×M16#6157semidirect
199. C8C22⋊#6157semidirect