Finite Group Databasebeta

← #20275775

Subgroups — #20275775

Order 6784.

348 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 71 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#16811421C211yesmaximal, characteristic
2C2×#13133248C211yesmaximal, characteristic, complemented
3#16810640C211yesmaximal, characteristic, complemented
4#16810632C211yesmaximal, characteristic, complemented
5#16810596C211yesmaximal, characteristic, complemented
6#16810393C211yesmaximal, characteristic, complemented
7C53×#536C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133248—22no—
10#13133248—22no—
11#13133248—21no—
12#13133247—21no—
13#13133243—21no—
14#13133242—21no—
15#13133237—22no—
16C2×#11719925—21no—
17C2×#11719924—21no—
18C2×#11719917C2211yescharacteristic
19C2×#11719917C2211yescharacteristic
20#13133176C2211yescharacteristic
21#13133144—22no—
22#13133141—22no—
23#13133131—22no—
24C2×#41×C53C2211yescharacteristic
25C22×Q8×C53C2211yescharacteristic
26C4⋊Q8×C53—21no—
27#124×C53—21no—
28#123×C53—22no—
29C22⋊Q8×C53—22no—
30#117×C53C2211yescharacteristic
31#41×C53—21no—
32#41×C53—21no—
33#41×C53—21no—
34#41×C53—21no—
35C2×Q8×C53—21no—
36C2×Q8×C53—21no—
37C2×Q8×C53—21no—
38C2×Q8×C53—21no—
39C2×Q8×C53—21no—
40C2×Q8×C53D412yes—
41C2×Q8×C53D411yescharacteristic
42C2×D4×C53—21no—
43C2×D4×C53D411yescharacteristic
44C22×C4×C53—21no—
45C22×C4×C53—21no—
46C22×C4×C53C2311yescharacteristic
47C2×C22⋊D53—21no—
48#11719945—21no—
49#11719945—21no—
50#11719945—21no—
51#11719945—21no—
52C2×D212D411yescharacteristic
53C2×C4×D53—21no—
54C2×Dic106D411yescharacteristic
55C4⋊C4×C53—24no—
56C4⋊C4×C53—21no—
57C22⋊C4×C53—24no—
58C22⋊C4×C53D412yes—
59C42×C53—21no—
60C22⋊Dic53—21no—
61#11719925—24no—
62#11719925—22no—
63#11719924—24no—
64#11719924—22no—
65#11719923—22no—
66#11719922—22no—
67C4⋊Dic53—21no—
68C4×Dic53—21no—
69#11719917—21no—
70#11719917—21no—
71#11719917—21no—
72#11719917D412yes—
73C2×#88322—21no—
74C2×#88322D412yes—
75C23×C53—21no—
76C23×C53C2×D411yescharacteristic
77C22×D53—21no—
78Q8×C53—22no—
79Q8×C53—21no—
80Q8×C53—22no—
81Q8×C53—21no—
82Q8×C53—41no—
83Q8×C53—41no—
84D4×C53—21no—
85D4×C53—21no—
86D4×C53—41no—
87D4×C53—41no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53C2×D411yescharacteristic
96C2×C4×C53C2×D411yescharacteristic
97C2×C4×C53C2×D411yescharacteristic
98C2×C4×C53C2×D411yescharacteristic
99C2×C4×C53C2×D411yescharacteristic
100C2×C4×C53#4111yescharacteristic
101C22⋊D53—41no—
102C22⋊D53—41no—
103C2×Dic53—21no—
104C2×Dic53—41no—
105D212—21no—
106D212—21no—
107C4×D53—41no—
108C4×D53—41no—
109Dic106—21no—
110Dic106—21no—
111#88322—22no—
112#88322—22no—
113C22×C53—41no—
114C22×C53—42no—
115C22×C53—41no—
116C22×C53#12011yescharacteristic
117C22×C53#12111yescharacteristic
118C22×C53#12711yescharacteristic
119C2×D53—41no—
120C2×D53—81no—
121C4×C53—42no—
122C4×C53—41no—
123C4×C53—42no—
124C4×C53#12011yescharacteristic
125C4×C53#12011yescharacteristic
126C4×C53#12111yescharacteristic
127C4×C53#12111yescharacteristic
128Dic53—41no—
129Dic53—81no—
130#2009—531nomaximal, Sylow
131C2×C53—21no—
132C2×C53—21no—
133C2×C53—21no—
134C2×C53—81no—
135C2×C53#39211yescharacteristic
136D53—81no—
137#575—531no—
138C2×#137—531no—
139#536D5311yescharacteristic, complemented
140#421—531no—
141#419—531no—
142#410—531no—
143#344—531no—
144C53#200911yescharacteristic, complemented, Sylow
145C2×#41—531no—
146C2×#41C2×D5311yescharacteristic
147C22×Q8C2×D5311yescharacteristic
148#137—1062no—
149#137—1061no—
150#137—1062no—
151#137—1061no—
152AGL(1,ℤ/8)—1061no—
153AGL(1,ℤ/8)—1061no—
154#135—1062no—
155C2×Q16—1061no—
156C2×SD16—1061no—
157C2×M16—531no—
158C2×M16—531no—
159C4⋊Q8—21no—
160#124—21no—
161#123—22no—
162C22⋊Q8—22no—
163#117—531no—
164#117C2×D5311yescharacteristic
165#104—1062no—
166#103—1062no—
167#99—1062no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—1061no—
173#41—1061no—
174#41—1061no—
175#41—1061no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—531no—
182C2×Q8C22⋊D5312yes—
183C2×Q8C22⋊D5311yescharacteristic
184C2×D4—21no—
185C2×D4—531no—
186C2×D4—1061no—
187C2×D4C22⋊D5311yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—1061no—
191C22×C4C22×D5311yescharacteristic
192Q16—1064no—
193Q16—1062no—
194SD16—1062no—
195SD16—1064no—
196SD16—1062no—
197D8—1062no—
198M16—532no—
199M16—1061no—
200M16—1061no—
201M16—1061no—
202C2×C8—532no—
203C2×C8—1061no—
204C4⋊C4—24no—
205C4⋊C4—21no—
206C4⋊C4—1061no—
207C22⋊C4—24no—
208C22⋊C4—1061no—
209C22⋊C4C22⋊D5312yes—
210C42—21no—
211C42—1061no—
212C23—21no—
213C23—1061no—
214C23C2×C22⋊D5311yescharacteristic
215Q8—22no—
216Q8—21no—
217Q8—22no—
218Q8—21no—
219Q8—41no—
220Q8—41no—
221Q8—1061no—
222Q8—1061no—
223D4—21no—
224D4—21no—
225D4—41no—
226D4—41no—
227D4—1061no—
228D4—1061no—
229D4—2121no—
230D4—2121no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—1061no—
239C2×C4—2121no—
240C2×C4—2121no—
241C2×C4—2121no—
242C2×C4D4×D5311yescharacteristic
243C2×C4D4×D5311yescharacteristic
244C2×C4D4×D5311yescharacteristic
245C2×C4#1171994711yescharacteristic
246C2×C4C2×C22⋊D5311yescharacteristic
247C2×C4C2×C22⋊D5311yescharacteristic
248C8—1062no—
249C8—1062no—
250C22—41no—
251C22—42no—
252C22—41no—
253C22—4241no—
254C22—2121no—
255C22#1313318611yescharacteristic
256C22#1313318811yescharacteristic
257C22#1313321311yescharacteristic
258C4—42no—
259C4—41no—
260C4—42no—
261C4—4241no—
262C4—2121no—
263C4#1313318511yescharacteristic
264C4#1313318511yescharacteristic
265C4#1313318911yescharacteristic
266C4#1313318911yescharacteristic
267C2—21no—
268C2—21no—
269C2—21no—
270C2—81no—
271C2—4241no—
272C2#1681053111yescharacteristic
subgroupG/H
centerC2#16810531
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#536C2
socleC2×C53#392
hypercenter#536D53
hypocenterC53#2009
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#13133248C2semidirect
3. #16810640C2semidirect
4. #16810632C2semidirect
5. #16810596C2semidirect
6. #16810393C2semidirect
7. C53×#536C2semidirect
139. #536D53semidirect
144. C53#2009semidirect