Finite Group Databasebeta

← #20275170

Subgroups — #20275170

Order 6784.

378 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 83 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×#13133198C211yesmaximal, characteristic
2C2×#13133173C211yesmaximal, characteristic, complemented
3C2×#13133132C211yesmaximal, characteristic
4#16810484C211yesmaximal, characteristic, complemented
5#16810389C211yesmaximal, characteristic, complemented
6#16810389C211yesmaximal, characteristic, complemented
7C2×#99×C53C211yesmaximal, characteristic
8C23×Dic53C2211yescharacteristic
9#13133198—22no—
10#13133198—21no—
11#13133198—21no—
12#13133173—22no—
13#13133173—22no—
14C2×#11719919—21no—
15C2×#11719919C412yescomplemented
16C2×C22⋊Dic53C412yescomplemented
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2211yescharacteristic
20C2×C4×Dic53—21no—
21#13133132—22no—
22#13133124—22no—
23C22×D4×C53C2211yescharacteristic
24C2×C22⋊C4×C53C2211yescharacteristic
25C2×C22⋊C4×C53C2211yescharacteristic
26#99×C53—22no—
27C24×C53C2×C411yescharacteristic
28C24×C53C2311yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53—21no—
32C2×D4×C53—21no—
33C2×D4×C53—21no—
34C2×D4×C53—21no—
35C22×C4×C53—21no—
36C22×C4×C53—21no—
37C22×C4×C53C2×C411yescharacteristic
38C22×Dic53—21no—
39C22×Dic53—21no—
40C22×Dic53—21no—
41C22×Dic53—21no—
42C22×Dic53—21no—
43C22×Dic53—21no—
44C22×Dic53C2×C412yes—
45C22×Dic53C2×C412yes—
46C22×Dic53D412yes—
47C22×Dic53D411yescharacteristic
48C22×Dic53D411yescharacteristic
49C22⋊C4×C53—21no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53—21no—
52C22⋊C4×C53D412yes—
53C22⋊Dic53—21no—
54C22⋊Dic53—21no—
55C22⋊Dic53—21no—
56C22⋊Dic53—22no—
57C22⋊Dic53—22no—
58C22⋊Dic53—21no—
59C22⋊Dic53—21no—
60C22⋊Dic53D412yes—
61#11719919—24no—
62#11719919—22no—
63#11719919—22no—
64C4×Dic53—24no—
65C23×C53—21no—
66C23×C53—21no—
67C23×C53—21no—
68C23×C53—42no—
69C23×C53—42no—
70C23×C53C22×C411yescharacteristic
71C23×C53C2×D411yescharacteristic
72C23×C53C2×D411yescharacteristic
73C23×C53C2×D411yescharacteristic
74C23×C53C2×D411yescharacteristic
75C23×C53#4111yescharacteristic
76C23×C53#4111yescharacteristic
77D4×C53—41no—
78D4×C53—41no—
79D4×C53—41no—
80D4×C53—41no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—22no—
85C2×C4×C53—42no—
86C2×C4×C53—42no—
87C2×C4×C53—41no—
88C2×Dic53—22no—
89C2×Dic53—22no—
90C2×Dic53—21no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—42no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—41no—
102C2×Dic53C22⋊C412yes—
103C2×Dic53C22⋊C412yes—
104C22×C53—22no—
105C22×C53—22no—
106C22×C53—22no—
107C22×C53—41no—
108C22×C53—41no—
109C22×C53—41no—
110C22×C53—41no—
111C22×C53—41no—
112C22×C53—42no—
113C22×C53—81no—
114C22×C53—81no—
115C22×C53C2×C22⋊C411yescharacteristic
116C22×C53C4×D411yescharacteristic
117C22×C53C4×D411yescharacteristic
118C22×C53#12011yescharacteristic
119C22×C53#12111yescharacteristic
120C22×C53#12111yescharacteristic
121C22×C53#12311yescharacteristic
122C4×C53—42no—
123C4×C53—41no—
124C4×C53—81no—
125Dic53—22no—
126Dic53—22no—
127Dic53—42no—
128Dic53—42no—
129Dic53—81no—
130#1899—531nomaximal, Sylow
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—21no—
135C2×C53—21no—
136C2×C53—21no—
137C2×C53—81no—
138C2×C53#38611yescharacteristic
139C2×C53#41012yes—
140C2×#123—531no—
141C2×#117—531no—
142C2×#99—531no—
143C2×#99D5311yescharacteristic
144#381—531no—
145#342—531no—
146#342—531no—
147C53#189911yescharacteristic, complemented, Sylow
148C22×D4C2×D5311yescharacteristic
149C23×C4—531no—
150#123—1062no—
151#123—1061no—
152#123—1061no—
153#117—1062no—
154#117—1062no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4—1061no—
157C2×C22⋊C4—531no—
158C2×C22⋊C4—532no—
159C2×C22⋊C4—531no—
160C2×C22⋊C4—531no—
161C2×C22⋊C4C2×D5311yescharacteristic
162C2×C22⋊C4C2×D5311yescharacteristic
163C2×C42—1061no—
164#99—22no—
165#99—1062no—
166#95—1062no—
167C24C4×D5311yescharacteristic
168C24C22×D5311yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—532no—
178C22×C4—532no—
179C22×C4—531no—
180C22×C4—531no—
181C22×C4—532no—
182C22×C4—1061no—
183C22×C4—1061no—
184C22×C4—1061no—
185C22×C4—1061no—
186C22×C4—1061no—
187C22×C4—1061no—
188C22×C4C4×D5311yescharacteristic
189C4⋊C4—1062no—
190C4⋊C4—1062no—
191C4⋊C4—1064no—
192C22⋊C4—21no—
193C22⋊C4—21no—
194C22⋊C4—21no—
195C22⋊C4—532no—
196C22⋊C4—1062no—
197C22⋊C4—1062no—
198C22⋊C4—1061no—
199C22⋊C4—1061no—
200C22⋊C4—1061no—
201C22⋊C4—1061no—
202C22⋊C4—1061no—
203C22⋊C4C22⋊D5312yes—
204C42—1064no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—42no—
209C23—42no—
210C23C2×C4×D5311yescharacteristic
211C23#1171994511yescharacteristic
212C23D4×D5311yescharacteristic
213C23D4×D5311yescharacteristic
214C23D4×D5311yescharacteristic
215C23#1171994711yescharacteristic
216C23C2×C22⋊D5311yescharacteristic
217D4—41no—
218D4—41no—
219D4—41no—
220D4—41no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—532no—
229C2×C4—532no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1061no—
237C2×C4—1061no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—2122no—
241C2×C4—2121no—
242C2×C4—2122no—
243C2×C4—2122no—
244C22—22no—
245C22—22no—
246C22—22no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22—42no—
253C22—81no—
254C22—81no—
255C22C22⋊C4×D5311yescharacteristic
256C22#1313317711yescharacteristic
257C22C4×C22⋊D5311yescharacteristic
258C22#1313318511yescharacteristic
259C22#1313318811yescharacteristic
260C22#1313319011yescharacteristic
261C22#1313319911yescharacteristic
262C4—42no—
263C4—41no—
264C4—81no—
265C4—1062no—
266C4—1062no—
267C4—4241no—
268C4—2122no—
269C4—2122no—
270C2—21no—
271C2—21no—
272C2—21no—
273C2—21no—
274C2—21no—
275C2—21no—
276C2—81no—
277C2#1681049911yescharacteristic
278C2#1681059512yes—
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC2×#99×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#99D53
hypocenterC53#1899
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#13133173C2semidirect
4. #16810484C2semidirect
5. #16810389C2semidirect
6. #16810389C2semidirect
15. C2×#11719919C4semidirect
16. C2×C22⋊Dic53C4semidirect
147. C53#1899semidirect