Finite Group Databasebeta

← #20275075

Subgroups — #20275075

Order 6784.

310 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 85 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#16811390C211yesmaximal, characteristic, complemented
3#16810641C211yesmaximal, characteristic, complemented
4#16810589C211yesmaximal, characteristic, complemented
5#16810393C211yesmaximal, characteristic, complemented
6#16810388C211yesmaximal, characteristic, complemented
7C53×#416C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9#13133248—21no—
10#13133247C412yescomplemented
11#13133243—21no—
12#13133242C412yescomplemented
13#13133237—22no—
14#13133224—22no—
15#13133224—22no—
16C2×#11719917—22no—
17C2×#11719917C2211yescharacteristic
18C2×#11719917C2211yescharacteristic
19C2×#11719917C2211yescharacteristic
20#13133145—22no—
21#13133129—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×C53C2×C412yes—
30#41×C53D412yes—
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×D212C2×C411yescharacteristic
42C2×C4×D53—21no—
43C2×Dic106C2×C411yescharacteristic
44C2×C8×C53—21no—
45C2×C8×C53—21no—
46C2×C8×C53—21no—
47C4⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C42×C53—21no—
50#11719925—22no—
51#11719924—22no—
52#11719923—22no—
53#11719922—22no—
54#11719917—24no—
55#11719917—21no—
56#11719917—21no—
57#11719917—21no—
58#11719917—21no—
59#11719917C2×C412yes—
60#11719917D412yes—
61C2×#88322—24no—
62C2×#88322—21no—
63C2×#88322—21no—
64C2×#88322D412yes—
65C23×C53—21no—
66C23×C53#4111yescharacteristic
67C22×D53—21no—
68Q8×C53—21no—
69Q8×C53C22⋊C412yes—
70D4×C53—22no—
71D4×C53—22no—
72D4×C53—21no—
73D4×C53C22⋊C412yes—
74C2×C4×C53—21no—
75C2×C4×C53—22no—
76C2×C4×C53—22no—
77C2×C4×C53—41no—
78C2×C4×C53C22×C411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2×D411yescharacteristic
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53#4111yescharacteristic
84C22⋊D53—41no—
85C22⋊D53—41no—
86C2×Dic53—21no—
87D212—21no—
88D212—21no—
89C4×D53—41no—
90C4×D53—41no—
91Dic106—21no—
92Dic106—21no—
93C8×C53—41no—
94#88322—22no—
95#88322—22no—
96#88322—22no—
97C22×C53—22no—
98C22×C53—41no—
99C22×C53—81no—
100C22×C53C4×D411yescharacteristic
101C22×C53#12111yescharacteristic
102C22×C53#12311yescharacteristic
103C2×D53—42no—
104C2×D53—41no—
105C4×C53—22no—
106C4×C53—81no—
107C4×C53C2×C22⋊C411yescharacteristic
108C4×C53C4×D411yescharacteristic
109C4×C53#12011yescharacteristic
110C4×C53#12111yescharacteristic
111Dic53—41no—
112#1881—531nomaximal, Sylow
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—21no—
116C2×C53—42no—
117C2×C53#38611yescharacteristic
118D53—81no—
119#575—531no—
120#568—531no—
121#421—531no—
122#416D5311yescharacteristic, complemented
123#408—531no—
124#344—531no—
125#341—531no—
126C53#188111yescharacteristic, complemented, Sylow
127C2×#41—531no—
128C2×#41C2×D5311yescharacteristic
129#137—532no—
130#137—1061no—
131AGL(1,ℤ/8)—532no—
132AGL(1,ℤ/8)—1061no—
133#135—1062no—
134#131—1062no—
135#131—1062no—
136C2×M16—531no—
137C2×M16—531no—
138C2×M16—531no—
139C2×M16—1062no—
140C22×C8C2×D5311yescharacteristic
141#117C2×D5311yescharacteristic
142#104—1062no—
143#103—21no—
144#102—21no—
145C22⋊C8—1062no—
146#41—21no—
147#41—21no—
148#41—1061no—
149#41—1061no—
150#41—1061no—
151#41—1061no—
152#41C4×D5312yes—
153#41D21212yes—
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—532no—
168M16—532no—
169M16—1064no—
170M16—1061no—
171M16—1061no—
172M16—1061no—
173M16—1061no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—532no—
178C2×C8—1064no—
179C2×C8—1061no—
180C2×C8—1061no—
181C4⋊C4—21no—
182C22⋊C4—21no—
183C42—21no—
184C23—21no—
185C23—1061no—
186C23#1171994711yescharacteristic
187Q8—21no—
188Q8—1061no—
189Q8—1061no—
190Q8#1171992112yes—
191D4—22no—
192D4—22no—
193D4—21no—
194D4—1061no—
195D4—1061no—
196D4—2121no—
197D4—2121no—
198D4#1171992112yes—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—1061no—
204C2×C4—2121no—
205C2×C4—2121no—
206C2×C4C2×C4×D5311yescharacteristic
207C2×C4C2×D21211yescharacteristic
208C2×C4D4×D5311yescharacteristic
209C2×C4D4×D5311yescharacteristic
210C2×C4#1171994711yescharacteristic
211C2×C4C2×C22⋊D5311yescharacteristic
212C8—41no—
213C8—1062no—
214C8—1062no—
215C8—1062no—
216C22—22no—
217C22—41no—
218C22—81no—
219C22—2122no—
220C22—2121no—
221C22#1313317711yescharacteristic
222C22#1313318911yescharacteristic
223C22#1313320111yescharacteristic
224C4—22no—
225C4—81no—
226C4—2121no—
227C4C2×#1171992111yescharacteristic
228C4#1313317711yescharacteristic
229C4#1313318711yescharacteristic
230C4#1313318911yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—42no—
235C2—4241no—
236C2#1681050011yescharacteristic
subgroupG/H
centerC4#13133187
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#416C2
socleC2×C53#386
hypercenter#416D53
hypocenterC53#1881
LCSG ▷ C2×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. #16811390C2semidirect
3. #16810641C2semidirect
4. #16810589C2semidirect
5. #16810393C2semidirect
6. #16810388C2semidirect
7. C53×#416C2semidirect
10. #13133247C4semidirect
12. #13133242C4semidirect
122. #416D53semidirect
126. C53#1881semidirect