Finite Group Databasebeta

← #20276361

Subgroups — #20276361

Order 6784.

280 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 73 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×#13133134C211yesmaximal, characteristic, complemented
2C2×#13133132C211yesmaximal, characteristic, complemented
3#16810413C212yesmaximal, complemented
4#16810410C212yesmaximal, complemented
5C53×#534C211yesmaximal, characteristic
6C2×#11719917—21no—
7C2×C22⋊Dic53—21no—
8#13133134—21no—
9#13133134C2212yes—
10#13133132—21no—
11#13133132C2212yes—
12C22×D4×C53C411yescharacteristic, complemented
13C22×D4×C53C2211yescharacteristic
14#123×C53—21no—
15#123×C53C411yescharacteristic, complemented
16#123×C53C2211yescharacteristic
17#121×C53—21no—
18#121×C53—21no—
19#120×C53—21no—
20#120×C53C411yescharacteristic, complemented
21#120×C53C2211yescharacteristic
22C2×C22⋊C4×C53C411yescharacteristic, complemented
23C24×C53C2×C411yescharacteristic
24C24×C53C2×C411yescharacteristic, complemented
25C24×C53D411yescharacteristic
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D4×C53—21no—
30C2×D4×C53—41no—
31C2×D4×C53—41no—
32C2×D4×C53—41no—
33C2×D4×C53—41no—
34C2×D4×C53C2×C411yescharacteristic
35C2×D4×C53D411yescharacteristic
36C2×D4×C53D411yescharacteristic
37C2×D4×C53C2311yescharacteristic
38C22×C4×C53—21no—
39C22×C4×C53C2×C411yescharacteristic, complemented
40C22×C4×C53D411yescharacteristic
41C22×Dic53—41no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53—21no—
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53C2×C411yescharacteristic, complemented
50C22⋊C4×C53C2×C411yescharacteristic, complemented
51C22⋊Dic53—22no—
52C22⋊Dic53—41no—
53#11719917—22no—
54#11719917—41no—
55C2×#88322—41no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—21no—
60C23×C53—21no—
61C23×C53—41no—
62C23×C53—81no—
63C23×C53—81no—
64C23×C53—81no—
65C23×C53C22⋊C411yescharacteristic
66C23×C53C22⋊C411yescharacteristic
67C23×C53C22⋊C411yescharacteristic
68C23×C53C22×C411yescharacteristic
69C23×C53C2×D411yescharacteristic
70D4×C53—41no—
71D4×C53—41no—
72D4×C53—41no—
73D4×C53—41no—
74D4×C53—42no—
75D4×C53—41no—
76D4×C53—41no—
77D4×C53—81no—
78D4×C53—81no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—41no—
85C2×C4×C53—81no—
86C2×C4×C53C22⋊C411yescharacteristic
87C2×C4×C53C2×D411yescharacteristic
88C2×Dic53—42no—
89C2×Dic53—82no—
90#88322—42no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—21no—
94C22×C53—21no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—41no—
98C22×C53—41no—
99C22×C53—41no—
100C22×C53—42no—
101C22×C53—41no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—81no—
105C22×C53—81no—
106C22×C53—81no—
107C22×C53—82no—
108C22×C53#9911yescharacteristic
109C22×C53#9911yescharacteristic
110C22×C53C2×C22⋊C411yescharacteristic
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—41no—
114C4×C53—41no—
115C4×C53—41no—
116Dic53—82no—
117#2121—531nomaximal, Sylow
118C2×C53—21no—
119C2×C53—21no—
120C2×C53—21no—
121C2×C53—41no—
122C2×C53—41no—
123C2×C53—41no—
124C2×C53—41no—
125C2×C53—81no—
126C2×C53C2×#9911yescharacteristic
127#534D5311yescharacteristic
128C2×#100—531no—
129C2×#99—531no—
130#352—532no—
131AΣL(1,𝔽16)—532no—
132C53#212111yescharacteristic, complemented, Sylow
133C22×D4Dic5311yescharacteristic, complemented
134C22×D4C2×D5311yescharacteristic
135C2×M16—1061no—
136#123—21no—
137#123Dic5311yescharacteristic, complemented
138#123C2×D5311yescharacteristic
139#121—21no—
140#121—21no—
141#120—21no—
142#120Dic5311yescharacteristic, complemented
143#120C2×D5311yescharacteristic
144C2×C22⋊C4—1061no—
145C2×C22⋊C4Dic5311yescharacteristic, complemented
146#100—532no—
147#100—1061no—
148#99—532no—
149#99—1061no—
150C24C2×Dic5311yescharacteristic
151C24C2×Dic5311yescharacteristic, complemented
152C24C22⋊D5311yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4C2×Dic5311yescharacteristic
162C2×D4C22⋊D5311yescharacteristic
163C2×D4C22⋊D5311yescharacteristic
164C2×D4C22×D5311yescharacteristic
165C22×C4—21no—
166C22×C4—2121no—
167C22×C4C2×Dic5311yescharacteristic, complemented
168C22×C4C22⋊D5311yescharacteristic
169M16—1062no—
170M16—2121no—
171C2×C8—2121no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—1062no—
180C22⋊C4—2121no—
181C22⋊C4C2×Dic5311yescharacteristic, complemented
182C22⋊C4C2×Dic5311yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—41no—
189C23—81no—
190C23—81no—
191C23—81no—
192C23C22⋊Dic5311yescharacteristic
193C23C22⋊Dic5311yescharacteristic
194C23C22⋊Dic5311yescharacteristic
195C23C22×Dic5311yescharacteristic
196C23C2×C22⋊D5311yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—42no—
202D4—41no—
203D4—41no—
204D4—81no—
205D4—81no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—4242no—
214C2×C4—2122no—
215C2×C4C22⋊Dic5311yescharacteristic
216C2×C4C2×C22⋊D5311yescharacteristic
217C8—2122no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—82no—
235C22#1313313211yescharacteristic
236C22#1313313211yescharacteristic
237C22C2×C22⋊Dic5311yescharacteristic
238C4—21no—
239C4—21no—
240C4—41no—
241C4—41no—
242C4—41no—
243C4—4242no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2C2×#1313313211yescharacteristic
subgroupG/H
centerC2C2×#13133132
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#534C2
socleC2×C53C2×#99
hypercenter#534D53
hypocenterC53#2121
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#13133134C2semidirect
2. C2×#13133132C2semidirect
3. #16810413C2semidirect
4. #16810410C2semidirect
12. C22×D4×C53C4semidirect
15. #123×C53C4semidirect
20. #120×C53C4semidirect
22. C2×C22⋊C4×C53C4semidirect
24. C24×C53C2×C4semidirect
39. C22×C4×C53C2×C4semidirect
49. C22⋊C4×C53C2×C4semidirect
50. C22⋊C4×C53C2×C4semidirect
132. C53#2121semidirect
133. C22×D4Dic53semidirect
137. #123Dic53semidirect
142. #120Dic53semidirect
145. C2×C22⋊C4Dic53semidirect
151. C24C2×Dic53semidirect
167. C22×C4C2×Dic53semidirect
181. C22⋊C4C2×Dic53semidirect
182. C22⋊C4C2×Dic53semidirect