Finite Group Databasebeta

← #20275244

Subgroups — #20275244

Order 6784.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#16811187C211yesmaximal, characteristic
2#16811097C211yesmaximal, characteristic, complemented
3#16810640C211yesmaximal, characteristic, complemented
4C2×#13133133C211yesmaximal, characteristic, complemented
5#16810380C211yesmaximal, characteristic, complemented
6#16810379C211yesmaximal, characteristic, complemented
7C53×#421C211yesmaximal, characteristic, complemented
8C22×D212C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10#13133213C412yescomplemented
11#13133206C412yescomplemented
12#13133191—22no—
13#13133185—22no—
14C4×D212—22no—
15#13133179—22no—
16#13133176C2211yescharacteristic
17#13133176C2211yescharacteristic
18C22⋊C4×D53—22no—
19#13133144—22no—
20#13133133—22no—
21C2×#41×C53C2211yescharacteristic
22C2×M16×C53C2211yescharacteristic
23#117×C53C2211yescharacteristic
24#104×C53—22no—
25C23×D53D412yes—
26#41×C53—21no—
27#41×C53—21no—
28#41×C53—21no—
29#41×C53—21no—
30C2×Q8×C53C2×C411yescharacteristic, complemented
31C2×D4×C53—21no—
32C2×D4×C53C2×C411yescharacteristic, complemented
33C22×C4×C53—21no—
34C22×C4×C53C2311yescharacteristic
35C2×C22⋊D53—24no—
36C2×D212—21no—
37C2×D212—21no—
38C2×D212—21no—
39C2×D212—21no—
40C2×D212C2×C412yes—
41C2×D212D412yes—
42C2×C4×D53—24no—
43M16×C53—21no—
44M16×C53—21no—
45C2×C8×C53—21no—
46C4⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C42×C53D412yescomplemented
49C22⋊Dic53—21no—
50C22⋊Dic53—21no—
51#11719921—24no—
52#11719921—24no—
53C4⋊Dic53—21no—
54C4⋊Dic53—21no—
55C4×Dic53C2×C412yescomplemented
56C4×Dic53D412yescomplemented
57#11719917—21no—
58#11719917—21no—
59C2×#88322—21no—
60C23×C53—21no—
61C23×C53C2×D411yescharacteristic
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—82no—
65C22×D53C22⋊C414yes—
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—22no—
74C2×C4×C53—41no—
75C2×C4×C53—41no—
76C2×C4×C53C22×C411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×C4×C53C2×D411yescharacteristic
80C2×C4×C53#4111yescharacteristic
81C2×C4×C53#4111yescharacteristic
82C22⋊D53—84no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85D212—22no—
86D212—22no—
87D212—22no—
88D212—22no—
89C4×D53—84no—
90C8×C53—41no—
91#88322—41no—
92C22×C53—41no—
93C22×C53—42no—
94C22×C53—41no—
95C22×C53C2×C22⋊C411yescharacteristic
96C22×C53#12011yescharacteristic
97C22×C53#12311yescharacteristic
98C2×D53—24no—
99C2×D53—44no—
100C2×D53—44no—
101C2×D53—82no—
102C4×C53—42no—
103C4×C53—41no—
104C4×C53C4×D411yescharacteristic
105C4×C53C4×D411yescharacteristic
106C4×C53#12111yescharacteristic
107C4×C53#12111yescharacteristic
108Dic53—42no—
109Dic53—42no—
110#1911—531nomaximal, Sylow
111C2×C53—21no—
112C2×C53—21no—
113C2×C53—21no—
114C2×C53—81no—
115C2×C53#38611yescharacteristic
116D53—44no—
117#535—531no—
118#518—531no—
119#421—531no—
120#421D5311yescharacteristic, complemented
121C2×#100—531no—
122H(3,ℤ/4)—531no—
123H(3,ℤ/4)—531no—
124C53#191111yescharacteristic, complemented, Sylow
125C2×#41C2×D5311yescharacteristic
126C22×D4—531no—
127C2×M16—531no—
128C2×M16C2×D5311yescharacteristic
129#127—532no—
130#124—532no—
131#121—1062no—
132#120—1062no—
133C4×D4—1062no—
134C4×D4—1062no—
135#117—531no—
136#117—531no—
137#117C2×D5311yescharacteristic
138C2×C22⋊C4—1062no—
139#104—22no—
140#104—1062no—
141#100—1062no—
142C24—532no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8C4×D5311yescharacteristic, complemented
148C2×D4—21no—
149C2×D4—532no—
150C2×D4—532no—
151C2×D4—1064no—
152C2×D4—1061no—
153C2×D4—1061no—
154C2×D4—1061no—
155C2×D4—1061no—
156C2×D4C4×D5311yescharacteristic, complemented
157C22×C4—21no—
158C22×C4—1064no—
159C22×C4C22×D5311yescharacteristic
160M16—21no—
161M16—21no—
162M16—1061no—
163M16—1061no—
164C2×C8—21no—
165C2×C8—1061no—
166C4⋊C4—21no—
167C4⋊C4—1061no—
168C4⋊C4—1061no—
169C22⋊C4—21no—
170C22⋊C4—1064no—
171C22⋊C4—1064no—
172C22⋊C4—1061no—
173C22⋊C4—1061no—
174C42—532no—
175C42—532no—
176C42C22⋊D5312yescomplemented
177C23—21no—
178C23—534no—
179C23—1062no—
180C23—1062no—
181C23—4242no—
182C23D4×D5311yescharacteristic
183Q8—21no—
184Q8—21no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—1062no—
190D4—1062no—
191D4—1062no—
192D4—1062no—
193D4—4244no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—4244no—
201C2×C4C2×C4×D5311yescharacteristic
202C2×C4#1171994511yescharacteristic
203C2×C4D4×D5311yescharacteristic
204C2×C4D4×D5311yescharacteristic
205C2×C4#1171994711yescharacteristic
206C2×C4C2×C22⋊D5311yescharacteristic
207C8—41no—
208C8—2121no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—1064no—
213C22—4242no—
214C22—2124no—
215C22—2124no—
216C22C22⋊C4×D5311yescharacteristic
217C22#1313318511yescharacteristic
218C22#1313319911yescharacteristic
219C4—42no—
220C4—41no—
221C4—2122no—
222C4—2122no—
223C4#1313317711yescharacteristic
224C4C4×C22⋊D5311yescharacteristic
225C4#1313318811yescharacteristic
226C4#1313319011yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—81no—
231C2—2124no—
232C2#1681049911yescharacteristic
subgroupG/H
centerC2#16810499
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#421C2
socleC2×C53#386
hypercenter#421D53
hypocenterC53#1911
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #16811097C2semidirect
3. #16810640C2semidirect
4. C2×#13133133C2semidirect
5. #16810380C2semidirect
6. #16810379C2semidirect
7. C53×#421C2semidirect
10. #13133213C4semidirect
11. #13133206C4semidirect
30. C2×Q8×C53C2×C4semidirect
32. C2×D4×C53C2×C4semidirect
48. C42×C53D4semidirect
55. C4×Dic53C2×C4semidirect
56. C4×Dic53D4semidirect
120. #421D53semidirect
124. C53#1911semidirect
147. C2×Q8C4×D53semidirect
156. C2×D4C4×D53semidirect
176. C42C22⋊D53semidirect