Finite Group Databasebeta

← #20276362

Subgroups — #20276362

Order 6784.

280 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#16811182C211yesmaximal, characteristic
2C2×#13133132C211yesmaximal, characteristic, complemented
3#16810414C212yesmaximal, complemented
4#16810411C212yesmaximal, complemented
5C2×#100×C53C211yesmaximal, characteristic, complemented
6C22×C22⋊D53C411yescharacteristic, complemented
7#13133200—21no—
8#13133198C411yescharacteristic, complemented
9#13133198C2211yescharacteristic
10#13133189—21no—
11#13133189—21no—
12#13133186—21no—
13#13133185C411yescharacteristic, complemented
14#13133185C2211yescharacteristic
15C2×C22⋊Dic53—21no—
16C2×C22⋊Dic53C411yescharacteristic, complemented
17#13133132—21no—
18#13133132C2212yes—
19C22×D4×C53C2211yescharacteristic
20C2×M16×C53—21no—
21#100×C53—21no—
22#100×C53C2212yes—
23C24×C53C2×C411yescharacteristic
24C24×C53D411yescharacteristic
25C23×D53C2×C411yescharacteristic, complemented
26C2×D4×C53—41no—
27C2×D4×C53—41no—
28C2×D4×C53C2×C411yescharacteristic
29C2×D4×C53D411yescharacteristic
30C2×D4×C53D411yescharacteristic
31C2×D4×C53C2311yescharacteristic
32C22×C4×C53D411yescharacteristic
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35C2×C22⋊D53—21no—
36C2×C22⋊D53—21no—
37C2×C22⋊D53—41no—
38C2×C22⋊D53—41no—
39C22×Dic53—21no—
40C22×Dic53—41no—
41C22×Dic53C2×C411yescharacteristic, complemented
42M16×C53—22no—
43M16×C53—41no—
44C2×C8×C53—41no—
45C22⋊Dic53—21no—
46C22⋊Dic53—21no—
47C22⋊Dic53—21no—
48C22⋊Dic53—22no—
49C22⋊Dic53—41no—
50C22⋊Dic53C2×C411yescharacteristic, complemented
51C22⋊Dic53C2×C411yescharacteristic, complemented
52#11719921—21no—
53#11719921—21no—
54#11719919—21no—
55#11719919—21no—
56C23×C53—21no—
57C23×C53—21no—
58C23×C53—21no—
59C23×C53—21no—
60C23×C53—81no—
61C23×C53—81no—
62C23×C53C22⋊C411yescharacteristic
63C23×C53C22⋊C411yescharacteristic
64C23×C53C22⋊C411yescharacteristic
65C23×C53C22×C411yescharacteristic
66C23×C53C2×D411yescharacteristic
67C22×D53—21no—
68C22×D53—41no—
69C22×D53—81no—
70D4×C53—41no—
71D4×C53—41no—
72D4×C53—41no—
73D4×C53—41no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53C22⋊C411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C22⋊D53—42no—
79C22⋊D53—41no—
80C22⋊D53—41no—
81C22⋊D53—81no—
82C22⋊D53—81no—
83C2×Dic53—21no—
84C2×Dic53—21no—
85C2×Dic53—21no—
86C2×Dic53—41no—
87C2×Dic53—42no—
88C2×Dic53—81no—
89C2×Dic53—82no—
90C8×C53—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—81no—
102C22×C53—81no—
103C22×C53—81no—
104C22×C53#9911yescharacteristic
105C22×C53#9911yescharacteristic
106C22×C53C2×C22⋊C411yescharacteristic
107C2×D53—41no—
108C2×D53—41no—
109C2×D53—41no—
110C2×D53—82no—
111C4×C53—21no—
112C4×C53—21no—
113Dic53—41no—
114Dic53—41no—
115Dic53—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×C53C2×#9911yescharacteristic
126D53—81no—
127#534—531no—
128C2×#100D5311yescharacteristic, complemented
129C2×#99—531no—
130#352—532no—
131AΣL(1,𝔽16)—532no—
132C53#212111yescharacteristic, complemented, Sylow
133C22×D4—531no—
134C22×D4C2×D5311yescharacteristic
135C2×M16—21no—
136#123—531no—
137#123—531no—
138#123—1061no—
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—21no—
147#100C2×D5312yes—
148#99—532no—
149#99—1061no—
150C24—531no—
151C24C4×D5311yescharacteristic
152C24D21211yescharacteristic
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—1061no—
156C2×D4—1061no—
157C2×D4—1061no—
158C2×D4—1061no—
159C2×D4—2121no—
160C2×D4—2121no—
161C2×D4C4×D5311yescharacteristic
162C2×D4D21211yescharacteristic
163C2×D4C22⋊D5311yescharacteristic
164C2×D4C22×D5311yescharacteristic
165C22×C4—531no—
166C22×C4—1061no—
167C22×C4—2121no—
168C22×C4C22⋊D5311yescharacteristic
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C4⋊C4—1061no—
173C4⋊C4—1061no—
174C22⋊C4—531no—
175C22⋊C4—531no—
176C22⋊C4—1061no—
177C22⋊C4—1061no—
178C22⋊C4—1061no—
179C22⋊C4—1061no—
180C22⋊C4—1061no—
181C22⋊C4—1062no—
182C22⋊C4—2121no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—81no—
188C23—81no—
189C23—1061no—
190C23—4241no—
191C23—2121no—
192C23#1171992111yescharacteristic
193C23#1171992111yescharacteristic
194C23#1171992111yescharacteristic
195C23C2×C4×D5311yescharacteristic
196C23C2×D21211yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—4241no—
202D4—4241no—
203D4—2122no—
204D4—2121no—
205D4—2121no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—1061no—
209C2×C4—1061no—
210C2×C4—1061no—
211C2×C4—4242no—
212C2×C4—4241no—
213C2×C4—2122no—
214C2×C4—2121no—
215C2×C4#1171992111yescharacteristic
216C2×C4C2×C22⋊D5311yescharacteristic
217C8—42no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—4242no—
232C22—2121no—
233C22—2121no—
234C22—2121no—
235C22#1313313111yescharacteristic
236C22#1313313111yescharacteristic
237C22C2×#1171992111yescharacteristic
238C4—21no—
239C4—21no—
240C4—4242no—
241C4—2121no—
242C4—2121no—
243C4—2121no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—4241no—
252C2C2×#1313313111yescharacteristic
subgroupG/H
centerC2C2×#13133131
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC2×#100×C53C2
socleC2×C53C2×#99
hypercenterC2×#100D53
hypocenterC53#2121
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×#13133132C2semidirect
3. #16810414C2semidirect
4. #16810411C2semidirect
5. C2×#100×C53C2semidirect
6. C22×C22⋊D53C4semidirect
8. #13133198C4semidirect
13. #13133185C4semidirect
16. C2×C22⋊Dic53C4semidirect
25. C23×D53C2×C4semidirect
41. C22×Dic53C2×C4semidirect
50. C22⋊Dic53C2×C4semidirect
51. C22⋊Dic53C2×C4semidirect
128. C2×#100D53semidirect
132. C53#2121semidirect