Finite Group Databasebeta

← #20278118

Subgroups — #20278118

Order 6784.

610 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 161 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
3C22×#11719919C211yesmaximal, characteristic, complemented
4#16810554C212yesmaximal, complemented
5#16810546C212yesmaximal, complemented
6#16810519C212yesmaximal, complemented
7#16810511C212yesmaximal, complemented
8#16810503C211yesmaximal, characteristic
9#16810498C211yesmaximal, characteristic, complemented
10#16810486C211yesmaximal, characteristic, complemented
11C2×#123×C53C211yesmaximal, characteristic
12C23×Dic53C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14#13133198—24no—
15#13133173—24no—
16C2×C4⋊Dic53C2212yes—
17C2×#11719919—24no—
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×#11719919C2211yescharacteristic
23C2×C22⋊Dic53C2211yescharacteristic
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2211yescharacteristic
26C2×C22⋊Dic53C2211yescharacteristic
27C2×C22⋊Dic53C2211yescharacteristic
28C2×C4×Dic53C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2212yes—
33C22×D4×C53C2211yescharacteristic
34C23×C4×C53C2211yescharacteristic
35#123×C53—24no—
36C2×C4⋊C4×C53C2212yes—
37C2×C22⋊C4×C53C2212yes—
38C2×C22⋊C4×C53C2211yescharacteristic
39C24×C53C2311yescharacteristic
40C24×C53C2311yescharacteristic
41C2×D4×C53—22no—
42C2×D4×C53—22no—
43C2×D4×C53—22no—
44C22×C4×C53—22no—
45C22×C4×C53—22no—
46C22×C4×C53—22no—
47C22×C4×C53C2312yes—
48C22×C4×C53C2311yescharacteristic
49C22×C4×C53C2312yes—
50C22×Dic53—22no—
51C22×Dic53—22no—
52C22×Dic53—22no—
53C22×Dic53—24no—
54C22×Dic53D414yes—
55C22×Dic53C2311yescharacteristic
56C22×Dic53C2312yes—
57C22×Dic53C2311yescharacteristic
58C22×Dic53C2311yescharacteristic
59C22×Dic53C2311yescharacteristic
60C22×Dic53C2312yes—
61C4⋊C4×C53—24no—
62C22⋊C4×C53—24no—
63C22⋊C4×C53D414yes—
64C22⋊Dic53—22no—
65C22⋊Dic53—22no—
66C22⋊Dic53—22no—
67C22⋊Dic53—22no—
68C22⋊Dic53—24no—
69C4⋊Dic53—24no—
70#11719919—24no—
71#11719919—24no—
72#11719919—24no—
73#11719919—22no—
74#11719919—22no—
75C4×Dic53—24no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53—21no—
79C23×C53—21no—
80C23×C53—21no—
81C23×C53—22no—
82C23×C53—22no—
83C23×C53C2×D412yes—
84C23×C53C2×D412yes—
85C23×C53#4112yes—
86C23×C53C2411yescharacteristic
87D4×C53—44no—
88C2×C4×C53—22no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—21no—
96C2×C4×C53—44no—
97C2×C4×C53C2×D414yes—
98C2×Dic53—28no—
99C2×Dic53—28no—
100C2×Dic53—21no—
101C2×Dic53—22no—
102C2×Dic53—21no—
103C2×Dic53—22no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—22no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53C2×D412yes—
114C2×Dic53C2×D412yes—
115C2×Dic53#4114yes—
116C22×C53—22no—
117C22×C53—22no—
118C22×C53—24no—
119C22×C53—24no—
120C22×C53—41no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—42no—
124C22×C53—42no—
125C22×C53#12114yes—
126C22×C53C22×D411yescharacteristic
127C22×C53C22×D411yescharacteristic
128C22×C53C2×#4112yes—
129C22×C53C2×#4111yescharacteristic
130C22×C532−1+412yes—
131C4×C53—24no—
132C4×C53—42no—
133C4×C53—41no—
134Dic53—22no—
135Dic53—22no—
136Dic53—24no—
137Dic53—41no—
138Dic53—41no—
139Dic53—42no—
140#2413—531nomaximal, Sylow
141C2×C53—24no—
142C2×C53—81no—
143C2×C53C2×#12111yescharacteristic
144C2×C53#52511yescharacteristic
145C2×C53#53611yescharacteristic
146C2×C53#54712yes—
147C2×C53#55212yes—
148C2×#123—531no—
149C2×#123D5311yescharacteristic
150C2×#117—531no—
151C22×C4⋊C4—531no—
152#398—532no—
153#396—532no—
154#389—532no—
155#388—532no—
156#386—531no—
157#385—531no—
158#381—531no—
159C53#241311yescharacteristic, complemented, Sylow
160C22×D4C2×D5311yescharacteristic
161C23×C4—531no—
162C23×C4—531no—
163C23×C4C2×D5311yescharacteristic
164#123—24no—
165#123—1064no—
166#117—1064no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—532no—
169C2×C4⋊C4—532no—
170C2×C4⋊C4—531no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—532no—
173C2×C4⋊C4—1064no—
174C2×C4⋊C4C2×D5312yes—
175C2×C22⋊C4—531no—
176C2×C22⋊C4—531no—
177C2×C22⋊C4—531no—
178C2×C22⋊C4—531no—
179C2×C22⋊C4—532no—
180C2×C22⋊C4C2×D5312yes—
181C2×C22⋊C4C2×D5311yescharacteristic
182C2×C42—532no—
183#95—532no—
184#95—532no—
185#95—532no—
186#95—532no—
187C24C22×D5311yescharacteristic
188C24C22×D5311yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—534no—
196C22×C4—532no—
197C22×C4—531no—
198C22×C4—531no—
199C22×C4—531no—
200C22×C4—532no—
201C22×C4—531no—
202C22×C4—1064no—
203C22×C4—1062no—
204C22×C4—1062no—
205C22×C4—1062no—
206C22×C4C22×D5312yes—
207C22×C4C22×D5311yescharacteristic
208C22×C4C22×D5312yes—
209C4⋊C4—24no—
210C4⋊C4—1064no—
211C4⋊C4—1064no—
212C4⋊C4—1062no—
213C4⋊C4—1062no—
214C4⋊C4—1064no—
215C4⋊C4—1064no—
216C22⋊C4—24no—
217C22⋊C4—1062no—
218C22⋊C4—1062no—
219C22⋊C4—1064no—
220C22⋊C4—1062no—
221C22⋊C4—1062no—
222C22⋊C4C22⋊D5314yes—
223C42—1064no—
224C23—22no—
225C23—22no—
226C23—21no—
227C23—21no—
228C23—22no—
229C23—21no—
230C23—22no—
231C23D4×D5312yes—
232C23#1171994712yes—
233C23C2×C22⋊D5312yes—
234C23C23×D5311yescharacteristic
235D4—44no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—532no—
246C2×C4—532no—
247C2×C4—534no—
248C2×C4—1068no—
249C2×C4—1068no—
250C2×C4—1062no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—1062no—
254C2×C4—1061no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4C2×C22⋊D5314yes—
264C22—24no—
265C22—24no—
266C22—22no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22#1313318814yes—
274C22C2×D4×D5311yescharacteristic
275C22C22×C22⋊D5311yescharacteristic
276C22#41×D5312yes—
277C22C2×#1171994711yescharacteristic
278C22#1313326412yes—
279C4—24no—
280C4—42no—
281C4—41no—
282C4—1062no—
283C4—1062no—
284C4—1064no—
285C4—2122no—
286C4—2121no—
287C4—2121no—
288C2—24no—
289C2—81no—
290C2C2×#1313318811yescharacteristic
291C2#1681113111yescharacteristic
292C2#1681119511yescharacteristic
293C2#1681127212yes—
294C2#1681131012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC2×#123×C53C2
hypercenterC2×#123D53
hypocenterC53#2413
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#13133173C2semidirect
3. C22×#11719919C2semidirect
4. #16810554C2semidirect
5. #16810546C2semidirect
6. #16810519C2semidirect
7. #16810511C2semidirect
9. #16810498C2semidirect
10. #16810486C2semidirect
159. C53#2413semidirect