Finite Group Databasebeta

← #20275717

Subgroups — #20275717

Order 6784.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 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×#11719924C211yesmaximal, characteristic, complemented
2C2×#13133193C211yesmaximal, characteristic, complemented
3C2×#13133192C211yesmaximal, characteristic
4C2×#13133139C211yesmaximal, characteristic, complemented
5C2×#13133129C211yesmaximal, characteristic, complemented
6#16810385C211yesmaximal, characteristic, complemented
7C2×#103×C53C211yesmaximal, characteristic, complemented
8C22×D212—21no—
9C22×D212C2211yescharacteristic
10C22×C4×D53C2211yescharacteristic
11C2×#11719924—22no—
12C2×#11719924—22no—
13C2×#11719924—22no—
14C22×#88322C2211yescharacteristic
15#13133193—22no—
16#13133193—22no—
17#13133192—22no—
18#13133192—22no—
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719919—21no—
21C2×#11719921—21no—
22C2×#11719921—21no—
23#13133139—22no—
24#13133129—22no—
25C22×Q8×C53C2211yescharacteristic
26C22×C8×C53C2211yescharacteristic
27C2×C4⋊C4×C53C2211yescharacteristic
28#103×C53—22no—
29C23×D53—21no—
30C23×D53D411yescharacteristic
31C2×Q8×C53—21no—
32C2×Q8×C53—21no—
33C2×Q8×C53—21no—
34C2×Q8×C53—21no—
35C2×Q8×C53—21no—
36C2×Q8×C53D412yescomplemented
37C22×C4×C53—21no—
38C22×C4×C53—21no—
39C22×C4×C53C2311yescharacteristic
40C22×Dic53—21no—
41C22×Dic53D411yescharacteristic
42C2×D212—21no—
43C2×D212—21no—
44C2×D212—21no—
45C2×D212—21no—
46C2×D212—24no—
47C2×D212—21no—
48C2×D212—42no—
49C2×D212—42no—
50C2×D212D412yes—
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—21no—
54C2×C4×D53—21no—
55C2×C4×D53—21no—
56C2×C4×D53—21no—
57C2×C8×C53—21no—
58C2×C8×C53—21no—
59C2×C8×C53—21no—
60C4⋊C4×C53—21no—
61C4⋊C4×C53D412yes—
62#11719924—28no—
63#11719921—24no—
64#11719921—24no—
65C4⋊Dic53—21no—
66C4⋊Dic53D412yescomplemented
67#11719919—24no—
68C2×#88322—21no—
69C2×#88322—21no—
70C2×#88322D412yescomplemented
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
86Q8×C53—22no—
87Q8×C53—22no—
88Q8×C53—41no—
89Q8×C53—41no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53C2×D411yescharacteristic
101C2×C4×C53#4111yescharacteristic
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108D212—22no—
109D212—22no—
110D212—48no—
111D212—41no—
112D212—41no—
113C4×D53—41no—
114C4×D53—41no—
115C4×D53—41no—
116C4×D53—41no—
117C8×C53—41no—
118#88322—22no—
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—41no—
147Dic53—42no—
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×#103D5311yescharacteristic, complemented
162C2×#102—531no—
163C2×C22⋊C8—531no—
164#340—531no—
165C53#200011yescharacteristic, complemented, Sylow
166C22×Q8C2×D5311yescharacteristic
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—531no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4C2×D5311yescharacteristic
182C2×C22⋊C4—1061no—
183C2×C22⋊C4—1061no—
184#103—22no—
185#102—1062no—
186C22⋊C8—1062no—
187C24—531no—
188C24—1061no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8D21212yescomplemented
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—21no—
219C2×C8—532no—
220C2×C8—1061no—
221C2×C8—1061no—
222C4⋊C4—21no—
223C4⋊C4—532no—
224C4⋊C4—1064no—
225C4⋊C4—1061no—
226C4⋊C4C22⋊D5312yes—
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—22no—
245Q8—22no—
246Q8—41no—
247Q8—41no—
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×C4#1171994511yescharacteristic
271C2×C4D4×D5311yescharacteristic
272C2×C4D4×D5311yescharacteristic
273C2×C4D4×D5311yescharacteristic
274C2×C4C2×C22⋊D5311yescharacteristic
275C8—41no—
276C8—1062no—
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#1313318511yescharacteristic
292C22#1313318711yescharacteristic
293C22#1313319011yescharacteristic
294C22SD16×D5311yescharacteristic
295C22#1313324011yescharacteristic
296C22#1313324411yescharacteristic
297C22#1313324911yescharacteristic
298C4—42no—
299C4—42no—
300C4—2122no—
301C4—2121no—
302C4#1313318711yescharacteristic
303C4#1313319011yescharacteristic
304C4#1313319111yescharacteristic
305C4#1313321311yescharacteristic
306C2—4242no—
307C2—2122no—
308C2#1681053211yescharacteristic
309C2#1681076311yescharacteristic
310C2#1681077111yescharacteristic
311C2#1681077211yescharacteristic
312C2#1681078411yescharacteristic
313C2#1681085511yescharacteristic
314C2#1681087211yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC2×#103×C53C2
socleC23×C53C2×D4
hypercenterC2×#103D53
hypocenterC53#2000
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C22×#11719924C2semidirect
2. C2×#13133193C2semidirect
4. C2×#13133139C2semidirect
5. C2×#13133129C2semidirect
6. #16810385C2semidirect
7. C2×#103×C53C2semidirect
36. C2×Q8×C53D4semidirect
66. C4⋊Dic53D4semidirect
70. C2×#88322D4semidirect
85. C22×D53SD16semidirect
161. C2×#103D53semidirect
165. C53#2000semidirect
194. C2×Q8D212semidirect