Finite Group Databasebeta

← #20275843

Subgroups — #20275843

Order 6784.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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×#13133200C211yesmaximal, characteristic, complemented
3C2×#13133132C211yesmaximal, characteristic, complemented
4C2×#13133131C211yesmaximal, characteristic, complemented
5#16810486C211yesmaximal, characteristic, complemented
6#16810389C211yesmaximal, characteristic, complemented
7C2×#123×C53C211yesmaximal, characteristic, complemented
8C22×C22⋊D53C2211yescharacteristic
9#13133200—22no—
10#13133200—21no—
11#13133200—22no—
12#13133198—21no—
13#13133189—22no—
14#13133186—21no—
15#13133185—21no—
16C2×#11719919—21no—
17C2×C22⋊Dic53C2211yescharacteristic
18C2×C22⋊Dic53C2211yescharacteristic
19C2×C22⋊Dic53C2211yescharacteristic
20C2×#11719921—21no—
21#13133132—22no—
22#13133131—22no—
23#13133124—22no—
24C22×D4×C53C2211yescharacteristic
25C23×C4×C53C2211yescharacteristic
26#123×C53—22no—
27#123×C53—22no—
28C2×C4⋊C4×C53—21no—
29C2×C22⋊C4×C53—21no—
30C2×C22⋊C4×C53C2211yescharacteristic
31C24×C53D411yescharacteristic
32C24×C53C2311yescharacteristic
33C23×D53D411yescharacteristic
34C2×D4×C53—21no—
35C2×D4×C53—21no—
36C2×D4×C53—21no—
37C2×D4×C53—21no—
38C2×D4×C53—21no—
39C2×D4×C53—21no—
40C22×C4×C53—21no—
41C22×C4×C53—21no—
42C22×C4×C53—21no—
43C22×C4×C53—21no—
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53D411yescharacteristic
48C22×C4×C53D412yescomplemented
49C2×C22⋊D53—21no—
50C2×C22⋊D53—21no—
51C2×C22⋊D53—21no—
52C2×C22⋊D53—21no—
53C2×C22⋊D53—21no—
54C2×C22⋊D53—21no—
55C2×C22⋊D53—22no—
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53D411yescharacteristic
59C22×Dic53D412yes—
60C4⋊C4×C53—24no—
61C22⋊C4×C53—24no—
62C22⋊C4×C53—21no—
63C22⋊C4×C53D412yescomplemented
64C22⋊Dic53—22no—
65C22⋊Dic53—21no—
66C22⋊Dic53—21no—
67C22⋊Dic53—21no—
68C22⋊Dic53—21no—
69C22⋊Dic53—21no—
70C22⋊Dic53D412yescomplemented
71#11719921—24no—
72#11719921—22no—
73#11719919—24no—
74#11719919—22no—
75C23×C53—21no—
76C23×C53—21no—
77C23×C53—21no—
78C23×C53—42no—
79C23×C53—42no—
80C23×C53C2×D411yescharacteristic
81C23×C53C2×D411yescharacteristic
82C23×C53C2×D411yescharacteristic
83C23×C53C2×D411yescharacteristic
84C23×C53C2×D411yescharacteristic
85C23×C53C2×D411yescharacteristic
86C23×C53#4111yescharacteristic
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—21no—
90C22×D53—42no—
91D4×C53—41no—
92D4×C53—41no—
93D4×C53—41no—
94D4×C53—41no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—22no—
101C2×C4×C53—21no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C22⋊D53—41no—
108C22⋊D53—41no—
109C22⋊D53—41no—
110C22⋊D53—41no—
111C22⋊D53—82no—
112C2×Dic53—22no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—22no—
116C2×Dic53—21no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—41no—
121C22×C53—22no—
122C22×C53—22no—
123C22×C53—22no—
124C22×C53—41no—
125C22×C53—41no—
126C22×C53—41no—
127C22×C53—41no—
128C22×C53—42no—
129C22×C53—41no—
130C22×C53—42no—
131C22×C53—81no—
132C22×C53#12011yescharacteristic
133C22×C53#12011yescharacteristic
134C22×C53#12011yescharacteristic
135C22×C53#12111yescharacteristic
136C22×C53#12111yescharacteristic
137C22×C53#12111yescharacteristic
138C22×C53#12711yescharacteristic
139C2×D53—42no—
140C2×D53—41no—
141C2×D53—42no—
142C2×D53—81no—
143C4×C53—42no—
144C4×C53—42no—
145C4×C53—41no—
146Dic53—42no—
147Dic53—41no—
148Dic53—42no—
149Dic53—81no—
150#2022—531nomaximal, Sylow
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—81no—
158C2×C53#39211yescharacteristic
159C2×C53#45812yes—
160D53—81no—
161#534—531no—
162C2×#123—531no—
163C2×#123D5311yescharacteristic, complemented
164C2×#99—531no—
165C2×#99—531no—
166#381—531no—
167#342—531no—
168C53#202211yescharacteristic, complemented, Sylow
169C22×D4—531no—
170C22×D4C2×D5311yescharacteristic
171C23×C4C2×D5311yescharacteristic
172#123—22no—
173#123—22no—
174#123—1062no—
175#123—1061no—
176#123—1061no—
177#123—1062no—
178#121—1062no—
179#120—1061no—
180#120—1061no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—1061no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—531no—
185C2×C22⋊C4—531no—
186C2×C22⋊C4—531no—
187C2×C22⋊C4—1061no—
188C2×C22⋊C4C2×D5311yescharacteristic
189#99—1062no—
190#99—1062no—
191#95—1062no—
192C24—531no—
193C24C22⋊D5311yescharacteristic
194C24C22×D5311yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—1062no—
202C2×D4—1061no—
203C2×D4—1061no—
204C2×D4—1061no—
205C2×D4—1061no—
206C2×D4—1061no—
207C2×D4—1061no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—531no—
216C22×C4—532no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4C22⋊D5311yescharacteristic
220C22×C4C22⋊D5312yescomplemented
221C4⋊C4—24no—
222C4⋊C4—1064no—
223C4⋊C4—1062no—
224C22⋊C4—24no—
225C22⋊C4—21no—
226C22⋊C4—532no—
227C22⋊C4—1064no—
228C22⋊C4—1062no—
229C22⋊C4—1062no—
230C22⋊C4—1061no—
231C22⋊C4—1061no—
232C22⋊C4—1061no—
233C22⋊C4—1061no—
234C22⋊C4—1061no—
235C22⋊C4C22⋊D5312yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—1061no—
242C23—1061no—
243C23—1061no—
244C23—2122no—
245C23D4×D5311yescharacteristic
246C23D4×D5311yescharacteristic
247C23D4×D5311yescharacteristic
248C23#1171994711yescharacteristic
249C23C2×C22⋊D5311yescharacteristic
250C23C2×C22⋊D5311yescharacteristic
251C23C2×C22⋊D5311yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—4242no—
257D4—2121no—
258D4—2121no—
259D4—2121no—
260D4—2121no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—1062no—
274C2×C4—1061no—
275C2×C4—1062no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—2122no—
279C2×C4—2121no—
280C2×C4—2122no—
281C2×C4—2122no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—81no—
293C22—4241no—
294C22—2122no—
295C22—2121no—
296C22—2122no—
297C22#1313318511yescharacteristic
298C22#1313318511yescharacteristic
299C22#1313318611yescharacteristic
300C22#1313318811yescharacteristic
301C22#1313318911yescharacteristic
302C22#1313318911yescharacteristic
303C22#1313321311yescharacteristic
304C4—42no—
305C4—42no—
306C4—41no—
307C4—4241no—
308C4—2122no—
309C4—2121no—
310C4—2122no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—4241no—
319C2#1681053111yescharacteristic
320C2#1681081212yes—
subgroupG/H
centerC22#13133185
commutatorC24×C53C23
FrattiniC24C22×D53
FittingC2×#123×C53C2
socleC22×C53#120
hypercenterC2×#123D53
hypocenterC53#2022
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#13133200C2semidirect
3. C2×#13133132C2semidirect
4. C2×#13133131C2semidirect
5. #16810486C2semidirect
6. #16810389C2semidirect
7. C2×#123×C53C2semidirect
48. C22×C4×C53D4semidirect
63. C22⋊C4×C53D4semidirect
70. C22⋊Dic53D4semidirect
163. C2×#123D53semidirect
168. C53#2022semidirect
220. C22×C4C22⋊D53semidirect
235. C22⋊C4C22⋊D53semidirect