Finite Group Databasebeta

← #20275721

Subgroups — #20275721

Order 6784.

444 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 93 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
1C22×#11719913C211yesmaximal, characteristic, complemented
2C2×#13133197C211yesmaximal, characteristic, complemented
3C2×#13133192C211yesmaximal, characteristic
4C2×#13133140C211yesmaximal, characteristic, complemented
5C2×#13133137C211yesmaximal, characteristic, complemented
6C2×#13133129C211yesmaximal, characteristic, complemented
7C53×#340C211yesmaximal, characteristic, complemented
8C22×Dic106C2211yescharacteristic
9C22×D212—21no—
10C22×D212C2211yescharacteristic
11C22×C4×D53C2211yescharacteristic
12C2×#11719913—22no—
13C2×#11719913—22no—
14C2×#11719913—22no—
15C22×#88322C2211yescharacteristic
16#13133197—22no—
17#13133197—22no—
18#13133192—22no—
19#13133192—22no—
20C2×C4⋊Dic53—21no—
21C2×#11719921—21no—
22C2×#11719921—21no—
23#13133140—22no—
24#13133137—22no—
25#13133129—22no—
26C22×C8×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C23×D53—21no—
30C23×D53D411yescharacteristic
31C22×C4×C53—21no—
32C22×C4×C53—21no—
33C22×C4×C53C2311yescharacteristic
34C22×Dic53—21no—
35C22×Dic53D411yescharacteristic
36C2×D212—21no—
37C2×D212—21no—
38C2×D212—21no—
39C2×D212—21no—
40C2×D212—24no—
41C2×D212—21no—
42C2×D212—42no—
43C2×D212—42no—
44C2×D212D412yes—
45C2×C4×D53—21no—
46C2×C4×D53—21no—
47C2×C4×D53—21no—
48C2×C4×D53—21no—
49C2×C4×D53—21no—
50C2×C4×D53—21no—
51C2×Dic106—21no—
52C2×Dic106—21no—
53C2×Dic106—21no—
54C2×Dic106—21no—
55C2×Dic106—21no—
56C2×Dic106D412yescomplemented
57C2×C8×C53—21no—
58C2×C8×C53—21no—
59C2×C8×C53D412yescomplemented
60C4⋊C4×C53—21no—
61C4⋊C4×C53—21no—
62C4⋊C4×C53D412yes—
63C4⋊C4×C53D412yescomplemented
64#11719921—24no—
65#11719921—24no—
66C4⋊Dic53—24no—
67C2×#88322—21no—
68C2×#88322—21no—
69C2×#88322—21no—
70#11719913—28no—
71C23×C53C2×D411yescharacteristic
72C22×D53—21no—
73C22×D53—21no—
74C22×D53—21no—
75C22×D53—21no—
76C22×D53—21no—
77C22×D53—21no—
78C22×D53—22no—
79C22×D53—41no—
80C22×D53—41no—
81C22×D53—41no—
82C22×D53—41no—
83C22×D53—41no—
84C22×D53—41no—
85C22×D53SD1612yescomplemented
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—41no—
91C2×C4×C53—41no—
92C2×C4×C53C2×D411yescharacteristic
93C2×C4×C53C2×D411yescharacteristic
94C2×C4×C53C2×D411yescharacteristic
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53#4111yescharacteristic
98C2×Dic53—22no—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104D212—22no—
105D212—22no—
106D212—48no—
107D212—41no—
108D212—41no—
109C4×D53—41no—
110C4×D53—41no—
111C4×D53—41no—
112C4×D53—41no—
113Dic106—22no—
114Dic106—22no—
115Dic106—41no—
116Dic106—41no—
117C8×C53—22no—
118#88322—41no—
119C22×C53#12011yescharacteristic
120C22×C53#12011yescharacteristic
121C22×C53#12111yescharacteristic
122C22×C53C2×SD1611yescharacteristic
123C22×C53#13511yescharacteristic
124C22×C53AGL(1,ℤ/8)11yescharacteristic
125C22×C53#13711yescharacteristic
126C2×D53—22no—
127C2×D53—22no—
128C2×D53—22no—
129C2×D53—41no—
130C2×D53—41no—
131C2×D53—41no—
132C2×D53—41no—
133C2×D53—42no—
134C2×D53—42no—
135C2×D53—42no—
136C2×D53—81no—
137C2×D53—81no—
138C2×D53—81no—
139C2×D53—81no—
140C4×C53—42no—
141C4×C53—42no—
142C4×C53#12011yescharacteristic
143C4×C53#12111yescharacteristic
144C4×C53#12111yescharacteristic
145C4×C53#12711yescharacteristic
146Dic53—42no—
147Dic53—41no—
148#2000—531nomaximal, Sylow
149C2×C53#39211yescharacteristic
150C2×C53#44811yescharacteristic
151C2×C53#44911yescharacteristic
152C2×C53#45011yescharacteristic
153C2×C53#45211yescharacteristic
154C2×C53#46511yescharacteristic
155C2×C53#46811yescharacteristic
156D53—42no—
157D53—82no—
158C22×SD16—531no—
159C2×C22⋊Q8—531no—
160C2×#121—531no—
161C2×#103—531no—
162C2×#102—531no—
163C2×C22⋊C8—531no—
164#340D5311yescharacteristic, complemented
165C53#200011yescharacteristic, complemented, Sylow
166C22×Q8—531no—
167C22×D4—531no—
168C22×D4—1061no—
169C23×C4—531no—
170C2×SD16—1062no—
171C2×SD16—1062no—
172C2×SD16—1062no—
173C22×C8—531no—
174C22×C8C2×D5311yescharacteristic
175C22⋊Q8—1062no—
176C22⋊Q8—1062no—
177#121—1062no—
178#121—1062no—
179C2×C4⋊C4—1061no—
180C2×C4⋊C4C2×D5311yescharacteristic
181C2×C4⋊C4C2×D5311yescharacteristic
182C2×C22⋊C4—1061no—
183C2×C22⋊C4—1061no—
184#103—1062no—
185#102—1062no—
186C22⋊C8—1062no—
187C24—531no—
188C24—1061no—
189C2×Q8—532no—
190C2×Q8—1061no—
191C2×Q8—1061no—
192C2×Q8—1061no—
193C2×Q8—1061no—
194C2×Q8—1061no—
195C2×D4—532no—
196C2×D4—1064no—
197C2×D4—1061no—
198C2×D4—1061no—
199C2×D4—1061no—
200C2×D4—1061no—
201C2×D4—1061no—
202C2×D4—2122no—
203C2×D4—2122no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—531no—
207C22×C4—1061no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4—1061no—
211C22×C4—1061no—
212C22×C4—1061no—
213C22×C4—1061no—
214C22×C4C22×D5311yescharacteristic
215SD16—1068no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—1061no—
219C2×C8—1061no—
220C2×C8—1061no—
221C2×C8D21212yescomplemented
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—1064no—
225C4⋊C4D21212yes—
226C4⋊C4D21212yescomplemented
227C22⋊C4—1064no—
228C22⋊C4—1064no—
229C23—532no—
230C23—1061no—
231C23—1061no—
232C23—1061no—
233C23—1061no—
234C23—1061no—
235C23—1061no—
236C23—1062no—
237C23—2121no—
238C23—2121no—
239C23—2121no—
240C23—2121no—
241C23—2121no—
242C23—2121no—
243C23D4×D5311yescharacteristic
244Q8—1062no—
245Q8—1062no—
246Q8—2121no—
247Q8—2121no—
248D4—1062no—
249D4—1062no—
250D4—2128no—
251D4—2121no—
252D4—2121no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—1061no—
260C2×C4—1062no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4C2×D21211yescharacteristic
270C2×C4C2×D21211yescharacteristic
271C2×C4C2×D21211yescharacteristic
272C2×C4D4×D5311yescharacteristic
273C2×C4D4×D5311yescharacteristic
274C2×C4#1171994911yescharacteristic
275C8—22no—
276C8—2121no—
277C22—1062no—
278C22—1062no—
279C22—1062no—
280C22—4241no—
281C22—4241no—
282C22—4241no—
283C22—4241no—
284C22—2122no—
285C22—2122no—
286C22—2122no—
287C22—2121no—
288C22—2121no—
289C22—2121no—
290C22—2121no—
291C22#1313318711yescharacteristic
292C22#1313318711yescharacteristic
293C22#1313319211yescharacteristic
294C22SD16×D5311yescharacteristic
295C22#1313323811yescharacteristic
296C22#1313324111yescharacteristic
297C22#1313324911yescharacteristic
298C4—42no—
299C4—42no—
300C4—2121no—
301C4—2122no—
302C4#1313318711yescharacteristic
303C4#1313319211yescharacteristic
304C4#1313319211yescharacteristic
305C4#1313321411yescharacteristic
306C2—4242no—
307C2—2122no—
308C2#1681053311yescharacteristic
309C2#1681076311yescharacteristic
310C2#1681076911yescharacteristic
311C2#1681077511yescharacteristic
312C2#1681078811yescharacteristic
313C2#1681085311yescharacteristic
314C2#1681087011yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#340C2
socleC23×C53C2×D4
hypercenter#340D53
hypocenterC53#2000
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
1. C22×#11719913C2semidirect
2. C2×#13133197C2semidirect
4. C2×#13133140C2semidirect
5. C2×#13133137C2semidirect
6. C2×#13133129C2semidirect
7. C53×#340C2semidirect
56. C2×Dic106D4semidirect
59. C2×C8×C53D4semidirect
63. C4⋊C4×C53D4semidirect
85. C22×D53SD16semidirect
164. #340D53semidirect
165. C53#2000semidirect
221. C2×C8D212semidirect
226. C4⋊C4D212semidirect