Finite Group Databasebeta

← #20290973

Subgroups — #20290973

Order 6784.

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

normal subgroups only

This group has 280 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
1D53⋊#100C211yesmaximal, characteristic, complemented
2D53⋊#99C211yesmaximal, characteristic, complemented
3#16811525C212yesmaximal, complemented
4#16811524C212yesmaximal, complemented
5#16811184C211yesmaximal, characteristic
6D53⋊M16—21no—
7C2×C22⋊#6157—21no—
8#13133277—21no—
9#13133277C2212yes—
10#13133276—21no—
11#13133275C2212yes—
12C22×D212C411yescharacteristic, complemented
13C2×D4×D53C2211yescharacteristic
14#13133202—21no—
15#13133200C2211yescharacteristic
16#13133192—21no—
17#13133190—21no—
18#13133187—21no—
19#13133187C2211yescharacteristic
20#13133185C411yescharacteristic, complemented
21C22⋊C4×D53C411yescharacteristic, complemented
22#123×C53C411yescharacteristic, complemented
23C23×D53C2×C411yescharacteristic
24C23×D53C2×C411yescharacteristic, complemented
25C23×D53D411yescharacteristic
26C22×#6157—41no—
27C2×D4×C53C2×C411yescharacteristic
28C22×C4×C53C2×C411yescharacteristic, complemented
29C2×C22⋊D53—21no—
30C2×C22⋊D53D411yescharacteristic
31C2×C22⋊D53C2311yescharacteristic
32D4×D53—41no—
33D4×D53—41no—
34C2×D212—21no—
35C2×D212—21no—
36C2×D212—21no—
37C2×D212—41no—
38C2×D212—41no—
39C2×D212D411yescharacteristic
40C2×C4×D53—21no—
41C2×C4×D53D411yescharacteristic
42C22⋊#6157—22no—
43C22⋊#6157—41no—
44#11719940—22no—
45#11719936—41no—
46D53⋊C8—41no—
47C4⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53C2×C411yescharacteristic, complemented
50C22⋊Dic53C2×C411yescharacteristic, complemented
51#11719921—21no—
52#11719921—21no—
53#11719921—21no—
54#11719921—21no—
55#11719919—21no—
56C23×C53C22⋊C411yescharacteristic
57C23×C53C22×C411yescharacteristic
58C22×D53—21no—
59C22×D53—21no—
60C22×D53—21no—
61C22×D53—21no—
62C22×D53—21no—
63C22×D53—41no—
64C22×D53—81no—
65C22×D53—81no—
66C22×D53—81no—
67C22×D53C22⋊C411yescharacteristic
68C22×D53C22⋊C411yescharacteristic
69C22×D53C2×D411yescharacteristic
70C2×#6157—42no—
71C2×#6157—82no—
72D4×C53—41no—
73C2×C4×C53—21no—
74C2×C4×C53—21no—
75C2×C4×C53—41no—
76C2×C4×C53C22⋊C411yescharacteristic
77C22⋊D53—41no—
78C22⋊D53—41no—
79C22⋊D53—81no—
80C2×Dic53—21no—
81C2×Dic53C2×D411yescharacteristic
82D212—41no—
83D212—42no—
84D212—41no—
85D212—41no—
86D212—81no—
87C4×D53—21no—
88C4×D53—21no—
89C4×D53—81no—
90#88324—42no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—41no—
94C22×C53—41no—
95C22×C53C2×C22⋊C411yescharacteristic
96C2×D53—21no—
97C2×D53—21no—
98C2×D53—41no—
99C2×D53—42no—
100C2×D53—41no—
101C2×D53—41no—
102C2×D53—41no—
103C2×D53—41no—
104C2×D53—41no—
105C2×D53—81no—
106C2×D53—81no—
107C2×D53—81no—
108C2×D53—82no—
109C2×D53#9911yescharacteristic
110C2×D53#9911yescharacteristic
111#6157—82no—
112C4×C53—21no—
113C4×C53—41no—
114C4×C53—41no—
115Dic53—21no—
116Dic53—41no—
117#2121—531nomaximal, Sylow
118C2×C53—21no—
119C2×C53—41no—
120C2×C53—41no—
121C2×C53C2×#9911yescharacteristic
122D53—21no—
123D53—21no—
124D53—41no—
125D53—41no—
126D53—81no—
127#534—531no—
128C2×#100—531no—
129C2×#99—531no—
130#352—532no—
131AΣL(1,𝔽16)—532no—
132C53#212111yescharacteristic, complemented, Sylow
133C22×D4—531no—
134C22×D4—531no—
135C2×M16—1061no—
136#123—531no—
137#123—1061no—
138#123#615711yescharacteristic, complemented
139#121—1061no—
140#121—1061no—
141#120—531no—
142#120—531no—
143#120—1061no—
144C2×C22⋊C4—531no—
145C2×C22⋊C4—1061no—
146#100—532no—
147#100—1061no—
148#99—532no—
149#99—1061no—
150C24—531no—
151C24—531no—
152C24—531no—
153C2×D4—531no—
154C2×D4—531no—
155C2×D4—531no—
156C2×D4—1061no—
157C2×D4—1061no—
158C2×D4—1061no—
159C2×D4—1061no—
160C2×D4—2121no—
161C2×D4—2121no—
162C2×D4—2121no—
163C2×D4—2121no—
164C2×D4C2×#615711yescharacteristic
165C22×C4—531no—
166C22×C4—1061no—
167C22×C4—2121no—
168C22×C4C2×#615711yescharacteristic, complemented
169M16—1062no—
170M16—2121no—
171C2×C8—2121no—
172C4⋊C4—21no—
173C4⋊C4—1061no—
174C22⋊C4—21no—
175C22⋊C4—531no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C22⋊C4—1061no—
180C22⋊C4—1062no—
181C22⋊C4—2121no—
182C22⋊C4C2×#615711yescharacteristic, complemented
183C23—531no—
184C23—531no—
185C23—531no—
186C23—1061no—
187C23—1061no—
188C23—1061no—
189C23—1061no—
190C23—1061no—
191C23—4241no—
192C23—4241no—
193C23—4241no—
194C23—2121no—
195C23C22⋊#615711yescharacteristic
196C23C22×#615711yescharacteristic
197D4—41no—
198D4—4241no—
199D4—4241no—
200D4—2121no—
201D4—2121no—
202D4—2121no—
203D4—2122no—
204D4—2121no—
205D4—2121no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—531no—
210C2×C4—1061no—
211C2×C4—1061no—
212C2×C4—1061no—
213C2×C4—4241no—
214C2×C4—4242no—
215C2×C4—2122no—
216C2×C4C22⋊#615711yescharacteristic
217C8—2122no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—531no—
223C22—531no—
224C22—1061no—
225C22—1061no—
226C22—4241no—
227C22—4241no—
228C22—4241no—
229C22—4242no—
230C22—2121no—
231C22—2121no—
232C22—2121no—
233C22—2121no—
234C22—2121no—
235C22—2122no—
236C22—2121no—
237C22C2×C22⋊#615711yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4—1061no—
242C4—4242no—
243C4—2121no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2—1061no—
248C2—1061no—
249C2—4241no—
250C2—2121no—
251C2—2121no—
252C2D53⋊#9911yescharacteristic
subgroupG/H
centerC2D53⋊#99
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
Fitting#123×C53C4
socleC2×C53C2×#99
hypercenter#123#6157
hypocenterC53#2121
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. D53⋊#100C2semidirect
2. D53⋊#99C2semidirect
3. #16811525C2semidirect
4. #16811524C2semidirect
12. C22×D212C4semidirect
20. #13133185C4semidirect
21. C22⋊C4×D53C4semidirect
22. #123×C53C4semidirect
24. C23×D53C2×C4semidirect
28. C22×C4×C53C2×C4semidirect
49. C22⋊C4×C53C2×C4semidirect
50. C22⋊Dic53C2×C4semidirect
132. C53#2121semidirect
138. #123#6157semidirect
168. C22×C4C2×#6157semidirect
182. C22⋊C4C2×#6157semidirect