Finite Group Databasebeta

← #20290585

Subgroups — #20290585

Order 6784.

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

normal subgroups only

This group has 356 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
1D53⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
2C2×#13133279C212yesmaximal
3#16811567C211yesmaximal, characteristic, complemented
4#16811496C212yesmaximal, complemented
5C4×D4×D53C211yesmaximal, characteristic
6C2×D53⋊C8C2212yes—
7C2×C4⋊#6157C2212yes—
8#13133284—21no—
9C4⋊#88324—21no—
10#13133279—24no—
11(C2×#6157)⋊C4—21no—
12(C2×#6157)⋊C4—21no—
13C2×D4×D53C2211yescharacteristic
14C22×C4×D53—21no—
15C4×D212C411yescharacteristic, complemented
16D4×Dic53C411yescharacteristic, complemented
17#13133179C411yescharacteristic, complemented
18C4×C22⋊D53—21no—
19#13133177—21no—
20C4⋊C4×D53C2211yescharacteristic
21C22⋊C4×D53—21no—
22C42×D53C2211yescharacteristic
23C4×D4×C53C411yescharacteristic, complemented
24C23×D53—21no—
25C22×#6157—22no—
26C2×D4×C53C2×C411yescharacteristic
27C22×C4×C53—21no—
28C2×C22⋊D53—21no—
29C22×Dic53—21no—
30D4×D53—21no—
31D4×D53—21no—
32D4×D53D412yes—
33D4×D53Q812yes—
34C2×D212C2×C411yescharacteristic
35C2×C4×D53—21no—
36C2×C4×D53—21no—
37C2×C4×D53—21no—
38C2×C4×D53—44no—
39C2×C4×D53D411yescharacteristic
40C2×C4×D53D411yescharacteristic
41C2×C4×D53C2311yescharacteristic
42C2×#88324—22no—
43C4⋊#6157—22no—
44C4⋊#6157—22no—
45D53⋊C8—22no—
46D53⋊C8—22no—
47C4⋊C4×C53C2×C411yescharacteristic
48C22⋊C4×C53—21no—
49C42×C53C2×C411yescharacteristic
50C22⋊Dic53—21no—
51#11719921—21no—
52#11719921—21no—
53C4⋊Dic53C2×C411yescharacteristic
54#11719919—21no—
55C4×Dic53C2×C411yescharacteristic
56C4×Dic53D411yescharacteristic
57C4×Dic53D411yescharacteristic
58C23×C53—21no—
59C22×D53—21no—
60C22×D53—22no—
61C22×D53—22no—
62C22×D53—42no—
63C22×D53—42no—
64C22×D53C2×D411yescharacteristic
65C2×#6157—42no—
66C2×#6157—42no—
67C2×#6157—42no—
68D4×C53—21no—
69D4×C53C4⋊C412yes—
70C2×C4×C53—21no—
71C2×C4×C53—42no—
72C2×C4×C53C22⋊C411yescharacteristic
73C2×C4×C53C22⋊C411yescharacteristic
74C2×C4×C53C22×C411yescharacteristic
75C22⋊D53—22no—
76C22⋊D53—22no—
77C2×Dic53—21no—
78C2×Dic53—42no—
79C2×Dic53C22⋊C411yescharacteristic
80C2×Dic53C22⋊C411yescharacteristic
81C2×Dic53C2×D411yescharacteristic
82D212—21no—
83D212C4⋊C412yes—
84C4×D53—22no—
85C4×D53—22no—
86C4×D53—42no—
87C4×D53—42no—
88C4×D53—42no—
89C4×D53C2×D411yescharacteristic
90C4×D53C2×Q811yescharacteristic
91C4×D53#4111yescharacteristic
92C4×D53#4111yescharacteristic
93#88324—42no—
94C22×C53—22no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53C2×C22⋊C411yescharacteristic
98C2×D53—22no—
99C2×D53—41no—
100C2×D53—41no—
101C2×D53—44no—
102C2×D53—44no—
103C2×D53#12111yescharacteristic
104C2×D53C22⋊Q811yescharacteristic
105C2×D53#13512yes—
106C2×D53AGL(1,ℤ/8)12yes—
107#6157—82no—
108C4×C53—21no—
109C4×C53—21no—
110C4×C53—41no—
111C4×C53C2×C4⋊C411yescharacteristic
112C4×C53#11711yescharacteristic
113Dic53—21no—
114Dic53—21no—
115Dic53—41no—
116Dic53#12111yescharacteristic
117Dic53C22⋊Q811yescharacteristic
118#1804—531nomaximal, Sylow
119C2×C53—42no—
120C2×C53#38011yescharacteristic
121C2×C53#41611yescharacteristic
122C2×C53#41811yescharacteristic
123D53—42no—
124D53#46312yes—
125D53#47812yes—
126C2×C4×D4—531no—
127C2×C4⋊C8—531no—
128C2×#102—532no—
129#382—531no—
130#340—532no—
131C53#180411yescharacteristic, complemented, Sylow
132C22×D4—531no—
133C23×C4—1061no—
134C22×C8—532no—
135C4×D4—531no—
136C4×D4—531no—
137C4×D4—531no—
138C4×D4—1061no—
139C4×D4—1061no—
140C4×D4#615711yescharacteristic, complemented
141C2×C4⋊C4—531no—
142C2×C4⋊C4—532no—
143C2×C22⋊C4—1061no—
144C2×C42—531no—
145C4⋊C8—1061no—
146C4⋊C8—1061no—
147#102—1064no—
148#95—1061no—
149#95—1061no—
150C24—1061no—
151C2×D4—532no—
152C2×D4—532no—
153C2×D4—531no—
154C2×D4—1061no—
155C2×D4—1061no—
156C2×D4—1061no—
157C2×D4C2×#615711yescharacteristic
158C22×C4—21no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—531no—
162C22×C4—1061no—
163C22×C4—1061no—
164C22×C4—1061no—
165C22×C4—1061no—
166C22×C4—1062no—
167C22×C4—2124no—
168C2×C8—1062no—
169C2×C8—1062no—
170C2×C8—1062no—
171C4⋊C4—531no—
172C4⋊C4—1061no—
173C4⋊C4—1062no—
174C4⋊C4—1062no—
175C4⋊C4C2×#615711yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C22⋊C4—1061no—
180C42—531no—
181C42—531no—
182C42—531no—
183C42C2×#615711yescharacteristic
184C23—21no—
185C23—531no—
186C23—1062no—
187C23—1062no—
188C23—1061no—
189C23—2122no—
190C23—2122no—
191D4—21no—
192D4—532no—
193D4—1062no—
194D4—1062no—
195D4—1061no—
196D4C4⋊#615712yes—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—531no—
200C2×C4—531no—
201C2×C4—531no—
202C2×C4—531no—
203C2×C4—531no—
204C2×C4—531no—
205C2×C4—531no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4—1061no—
209C2×C4—2122no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4—2122no—
215C2×C4—2122no—
216C2×C4C22⋊#615711yescharacteristic
217C2×C4C22⋊#615711yescharacteristic
218C2×C4C22×#615711yescharacteristic
219C8—2122no—
220C22—22no—
221C22—41no—
222C22—41no—
223C22—531no—
224C22—531no—
225C22—532no—
226C22—532no—
227C22—1062no—
228C22—2124no—
229C22—2124no—
230C22—2121no—
231C22—2121no—
232C22C2×C22⋊#615711yescharacteristic
233C4—21no—
234C4—21no—
235C4—41no—
236C4—531no—
237C4—531no—
238C4—1061no—
239C4—1061no—
240C4—4242no—
241C4—2121no—
242C4C2×C4⋊#615711yescharacteristic
243C4#1313329611yescharacteristic
244C2—42no—
245C2—532no—
246C2—532no—
247C2—2122no—
248C2#1681156511yescharacteristic
249C2#1681161311yescharacteristic
250C2#1681161511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×D4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×D4#6157
hypocenterC53#1804
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. D53⋊(C4⋊C8)C2semidirect
3. #16811567C2semidirect
4. #16811496C2semidirect
15. C4×D212C4semidirect
16. D4×Dic53C4semidirect
17. #13133179C4semidirect
23. C4×D4×C53C4semidirect
131. C53#1804semidirect
140. C4×D4#6157semidirect