Finite Group Databasebeta

← #20276019

Subgroups — #20276019

Order 6784.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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
2#16810649C211yesmaximal, characteristic, complemented
3#16810620C211yesmaximal, characteristic, complemented
4#16810597C211yesmaximal, characteristic, complemented
5#16810596C211yesmaximal, characteristic, complemented
6#16810379C211yesmaximal, characteristic, complemented
7C53×#416C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133248—21no—
10#13133247—21no—
11#13133243—21no—
12#13133242—21no—
13#13133237—22no—
14C2×#11719917C2211yescharacteristic
15#13133176C2211yescharacteristic
16#13133176C2211yescharacteristic
17#13133146—22no—
18#13133142—21no—
19#13133139—21no—
20#13133132—22no—
21#13133131—22no—
22C2×#41×C53C2211yescharacteristic
23C22×C8×C53C2211yescharacteristic
24#117×C53C2211yescharacteristic
25#103×C53—21no—
26#102×C53—21no—
27#41×C53—21no—
28#41×C53—21no—
29#41×C53—21no—
30#41×C53—21no—
31C2×Q8×C53D411yescharacteristic
32C2×D4×C53—21no—
33C2×D4×C53D411yescharacteristic
34C22×C4×C53—21no—
35C22×C4×C53C2311yescharacteristic
36C2×C22⋊D53—21no—
37#11719945—21no—
38#11719945—21no—
39#11719945—21no—
40#11719945—21no—
41C2×D212D411yescharacteristic
42C2×C4×D53—21no—
43C2×Dic106D411yescharacteristic
44C2×C8×C53—21no—
45C2×C8×C53—21no—
46C2×C8×C53—21no—
47C4⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C42×C53—21no—
50C22⋊Dic53—21no—
51C22⋊Dic53D412yes—
52#11719925—22no—
53#11719924—22no—
54#11719923—22no—
55#11719922—22no—
56C4⋊Dic53—21no—
57C4⋊Dic53—21no—
58C4×Dic53—21no—
59C4×Dic53—21no—
60#11719917—21no—
61#11719917—21no—
62C2×#88322D412yes—
63C23×C53—21no—
64C23×C53C2×D411yescharacteristic
65C22×D53—21no—
66Q8×C53—21no—
67Q8×C53—21no—
68D4×C53—21no—
69D4×C53—21no—
70D4×C53—41no—
71D4×C53—41no—
72C2×C4×C53—21no—
73C2×C4×C53—41no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53#4111yescharacteristic
80C2×C4×C53#4111yescharacteristic
81C2×C4×C53#4111yescharacteristic
82C22⋊D53—41no—
83C22⋊D53—41no—
84C2×Dic53—21no—
85C2×Dic53—22no—
86C2×Dic53—41no—
87D212—21no—
88D212—21no—
89C4×D53—41no—
90C4×D53—41no—
91Dic106—21no—
92Dic106—21no—
93C8×C53—41no—
94#88322—22no—
95C22×C53—41no—
96C22×C53—41no—
97C22×C53—81no—
98C22×C53#12111yescharacteristic
99C22×C53#12111yescharacteristic
100C22×C53#12111yescharacteristic
101C2×D53—41no—
102C2×D53—81no—
103C4×C53—41no—
104C4×C53—81no—
105C4×C53#12011yescharacteristic
106C4×C53#12311yescharacteristic
107C4×C53#12311yescharacteristic
108C4×C53#12411yescharacteristic
109Dic53—42no—
110Dic53—41no—
111Dic53—81no—
112#2053—531nomaximal, Sylow
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53—81no—
117C2×C53#39411yescharacteristic
118D53—81no—
119#575—531no—
120#424—531no—
121#416—531no—
122#416D5311yescharacteristic, complemented
123#410—531no—
124#410—531no—
125H(3,ℤ/4)—531no—
126C53#205311yescharacteristic, complemented, Sylow
127C2×#41—531no—
128C2×#41C2×D5311yescharacteristic
129#137—1061no—
130#137—1061no—
131AGL(1,ℤ/8)—1061no—
132AGL(1,ℤ/8)—1061no—
133#135—1062no—
134C2×M16—531no—
135C22×C8C2×D5311yescharacteristic
136#117—531no—
137#117—531no—
138#117C2×D5311yescharacteristic
139C4⋊C8—1062no—
140#103—21no—
141#103—1061no—
142#102—21no—
143#102—1061no—
144#99—1062no—
145#99—1062no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—1061no—
151#41—1061no—
152#41—1061no—
153#41—1061no—
154C2×Q8—531no—
155C2×Q8C22⋊D5311yescharacteristic
156C2×D4—21no—
157C2×D4—531no—
158C2×D4—1061no—
159C2×D4C22⋊D5311yescharacteristic
160C22×C4—21no—
161C22×C4—1061no—
162C22×C4C22×D5311yescharacteristic
163Q16—1062no—
164SD16—1062no—
165SD16—1062no—
166D8—1062no—
167M16—1061no—
168M16—1061no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—532no—
173C4⋊C4—21no—
174C4⋊C4—1061no—
175C4⋊C4—1061no—
176C22⋊C4—21no—
177C22⋊C4—532no—
178C22⋊C4—1061no—
179C42—21no—
180C42—1061no—
181C42—1061no—
182C23—21no—
183C23—1061no—
184C23D4×D5311yescharacteristic
185Q8—21no—
186Q8—21no—
187Q8—1061no—
188Q8—1061no—
189D4—21no—
190D4—21no—
191D4—41no—
192D4—41no—
193D4—1061no—
194D4—1061no—
195D4—2121no—
196D4—2121no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—1062no—
202C2×C4—1061no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4—2121no—
206C2×C4#1171994511yescharacteristic
207C2×C4#1171994511yescharacteristic
208C2×C4D4×D5311yescharacteristic
209C2×C4D4×D5311yescharacteristic
210C2×C4#1171994711yescharacteristic
211C2×C4C2×C22⋊D5311yescharacteristic
212C8—41no—
213C8—1062no—
214C22—41no—
215C22—41no—
216C22—81no—
217C22—4241no—
218C22—2121no—
219C22#1313318911yescharacteristic
220C22#1313319011yescharacteristic
221C22#1313319011yescharacteristic
222C4—41no—
223C4—81no—
224C4—4241no—
225C4—2122no—
226C4—2121no—
227C4#1313318511yescharacteristic
228C4#1313319911yescharacteristic
229C4#1313320011yescharacteristic
230C4#1313320311yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—4241no—
236C2#1681053811yescharacteristic
subgroupG/H
centerC4#13133203
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#416C2
socleC2×C53#394
hypercenter#416D53
hypocenterC53#2053
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #16810649C2semidirect
3. #16810620C2semidirect
4. #16810597C2semidirect
5. #16810596C2semidirect
6. #16810379C2semidirect
7. C53×#416C2semidirect
122. #416D53semidirect
126. C53#2053semidirect