Finite Group Databasebeta

← #20273207

Subgroups — #20273207

Order 6784.

304 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 77 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#16811134C211yesmaximal, characteristic
2C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
3#16810361C212yesmaximal, complemented
4#16810358C212yesmaximal, complemented
5C2×C22⋊C8×C53C211yesmaximal, characteristic, complemented
6C2×#11719945C411yescharacteristic, complemented
7#13133200—21no—
8#13133196C411yescharacteristic, complemented
9#13133196C2211yescharacteristic
10#13133191C411yescharacteristic, complemented
11#13133191C2211yescharacteristic
12#13133186—21no—
13C4×C22⋊D53—21no—
14C4×C22⋊D53—21no—
15#13133176C411yescharacteristic, complemented
16C2×C22⋊Dic53—21no—
17C2×C4×Dic53—21no—
18#13133124C2212yes—
19C23×C4×C53C2211yescharacteristic
20C22×C8×C53—21no—
21C22⋊C8×C53—21no—
22C22⋊C8×C53C2212yes—
23C24×C53D411yescharacteristic
24C22×C4×C53—42no—
25C22×C4×C53C2×C411yescharacteristic
26C22×C4×C53C2×C411yescharacteristic
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53D411yescharacteristic
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C22×Dic53—22no—
35#11719945—42no—
36C2×D212C2×C411yescharacteristic
37C2×C4×D53—21no—
38C2×Dic106C2×C411yescharacteristic
39C2×C8×C53—22no—
40C2×C8×C53—42no—
41C22⋊Dic53—24no—
42C22⋊Dic53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—21no—
45#11719921—21no—
46#11719921—21no—
47C4⋊Dic53C2×C411yescharacteristic
48C4⋊Dic53C2×C411yescharacteristic
49#11719919—21no—
50#11719919—21no—
51C4×Dic53—21no—
52C4×Dic53—42no—
53C23×C53—22no—
54C23×C53—22no—
55C23×C53—41no—
56C23×C53C22⋊C411yescharacteristic
57C23×C53C22⋊C411yescharacteristic
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60C2×C4×C53—22no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53C22⋊C411yescharacteristic
67C2×C4×C53C22⋊C411yescharacteristic
68C2×C4×C53C22×C411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C22⋊D53—42no—
71C22⋊D53—42no—
72C22⋊D53—42no—
73C2×Dic53—24no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—42no—
78C2×Dic53—42no—
79D212—41no—
80C4×D53—42no—
81Dic106—41no—
82C8×C53—42no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53—44no—
91C22×C53#10412yes—
92C22×C53C2×C22⋊C411yescharacteristic
93C22×C53#13512yes—
94C2×D53—41no—
95C2×D53—41no—
96C2×D53—41no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53#9912yes—
101Dic53—44no—
102Dic53—41no—
103Dic53—41no—
104Dic53—41no—
105#1501—531nomaximal, Sylow
106C2×C53—22no—
107C2×C53—22no—
108C2×C53—41no—
109C2×C53C2×#9911yescharacteristic
110C2×C53#41611yescharacteristic
111C2×C53C2×#10411yescharacteristic
112D53—81no—
113#525—531no—
114C2×C22⋊C8D5311yescharacteristic, complemented
115C4×C22⋊C4—531no—
116#328—532no—
117#327—532no—
118C53#150111yescharacteristic, complemented, Sylow
119C2×#41—531no—
120C23×C4C2×D5311yescharacteristic
121C22×C8—21no—
122#123—1061no—
123C22⋊Q8—531no—
124C22⋊Q8—531no—
125#121—531no—
126#121—531no—
127#120—1061no—
128C4×D4—1061no—
129C4×D4—1061no—
130#117—531no—
131C2×C22⋊C4—1061no—
132C2×C42—1061no—
133C22⋊C8—21no—
134C22⋊C8C2×D5312yes—
135#95—532no—
136C24D21211yescharacteristic
137#41—2122no—
138C2×Q8—531no—
139C2×D4—531no—
140C2×D4—1061no—
141C2×D4—1061no—
142C2×D4—1061no—
143C22×C4—42no—
144C22×C4—1062no—
145C22×C4—1061no—
146C22×C4C4×D5311yescharacteristic
147C22×C4C4×D5311yescharacteristic
148C22×C4D21211yescharacteristic
149C22×C4C22⋊D5311yescharacteristic
150C22×C4C22⋊D5311yescharacteristic
151C22×C4C22×D5311yescharacteristic
152C2×C8—22no—
153C2×C8—42no—
154C4⋊C4—531no—
155C4⋊C4—531no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C22⋊C4—1064no—
159C22⋊C4—1061no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C42—1061no—
165C42—2122no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—1061no—
170C23#1171992111yescharacteristic
171C23#1171992111yescharacteristic
172C23C2×D21211yescharacteristic
173Q8—2121no—
174D4—2121no—
175D4—2122no—
176D4—2122no—
177D4—2122no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—1064no—
185C2×C4—1061no—
186C2×C4—1061no—
187C2×C4—1061no—
188C2×C4—2122no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4#1171992111yescharacteristic
192C2×C4#1171992111yescharacteristic
193C2×C4C2×C4×D5311yescharacteristic
194C2×C4C2×C22⋊D5311yescharacteristic
195C8—42no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—2121no—
205C22—2121no—
206C22—2121no—
207C22#1313314312yes—
208C22C2×#1171992111yescharacteristic
209C22#1313323912yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—2124no—
214C4—2121no—
215C4—2121no—
216C4—2121no—
217C4#1313313112yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—4241no—
222C2C2×#1313313111yescharacteristic
223C2#1681062011yescharacteristic
224C2C2×#1313314311yescharacteristic
subgroupG/H
centerC2×C4#11719921
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×C22⋊C8×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×C22⋊C8D53
hypocenterC53#1501
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Dic53C2semidirect
3. #16810361C2semidirect
4. #16810358C2semidirect
5. C2×C22⋊C8×C53C2semidirect
6. C2×#11719945C4semidirect
8. #13133196C4semidirect
10. #13133191C4semidirect
15. #13133176C4semidirect
114. C2×C22⋊C8D53semidirect
118. C53#1501semidirect