Finite Group Databasebeta

← #20273271

Subgroups — #20273271

Order 6784.

284 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C2×C22⋊#88322C211yesmaximal, characteristic, complemented
2#16810486C211yesmaximal, characteristic, complemented
3#16810360C212yesmaximal, complemented
4#16810357C212yesmaximal, complemented
5C53×#525C211yesmaximal, characteristic
6C22×#88322—21no—
7C2×C22⋊Dic53—21no—
8C22⋊#88322—21no—
9C22⋊#88322C2212yes—
10#13133124—21no—
11#13133124C2212yes—
12C2×#41×C53C411yescharacteristic, complemented
13C23×C4×C53C2211yescharacteristic
14#123×C53—21no—
15C22⋊Q8×C53C411yescharacteristic, complemented
16C22⋊Q8×C53C2211yescharacteristic
17#121×C53C411yescharacteristic, complemented
18#121×C53C2211yescharacteristic
19#120×C53—21no—
20C4×D4×C53—21no—
21C4×D4×C53—21no—
22#117×C53C411yescharacteristic, complemented
23C24×C53D411yescharacteristic
24#41×C53—42no—
25C2×Q8×C53C2×C411yescharacteristic
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D4×C53C2×C411yescharacteristic
30C22×C4×C53—21no—
31C22×C4×C53—42no—
32C22×C4×C53C2×C411yescharacteristic
33C22×C4×C53C2×C411yescharacteristic
34C22×C4×C53D411yescharacteristic
35C22×C4×C53D411yescharacteristic
36C22×C4×C53D411yescharacteristic
37C22×C4×C53C2311yescharacteristic
38C22×Dic53—22no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53C2×C411yescharacteristic
42C4⋊C4×C53C2×C411yescharacteristic
43C22⋊C4×C53—21no—
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C22⋊C4×C53—21no—
48C42×C53—21no—
49C22⋊Dic53—42no—
50C2×#88322—22no—
51C2×#88322—42no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—21no—
55C23×C53—41no—
56C23×C53C22⋊C411yescharacteristic
57C23×C53C22⋊C411yescharacteristic
58C23×C53C2×D411yescharacteristic
59Q8×C53—41no—
60D4×C53—41no—
61D4×C53—42no—
62D4×C53—42no—
63D4×C53—42no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—41no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53—42no—
74C2×C4×C53C22⋊C411yescharacteristic
75C2×C4×C53C22⋊C411yescharacteristic
76C2×C4×C53C22×C411yescharacteristic
77C2×C4×C53C2×D411yescharacteristic
78C2×Dic53—42no—
79C2×Dic53—42no—
80C2×Dic53—42no—
81#88322—42no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—44no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53#9912yes—
94C22×C53C2×C22⋊C411yescharacteristic
95C22×C53#13512yes—
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—41no—
101C4×C53—41no—
102C4×C53—41no—
103Dic53—82no—
104#1516—531nomaximal, Sylow
105C2×C53—22no—
106C2×C53—22no—
107C2×C53—41no—
108C2×C53—81no—
109C2×C53C2×#9911yescharacteristic
110C2×C53#41611yescharacteristic
111C2×C53#42111yescharacteristic
112#525D5311yescharacteristic
113C2×C22⋊C8—531no—
114#381—531no—
115#328—532no—
116#327—532no—
117C53#151611yescharacteristic, complemented, Sylow
118C2×#41Dic5311yescharacteristic, complemented
119C23×C4C2×D5311yescharacteristic
120C22×C8—1061no—
121#123—21no—
122C22⋊Q8Dic5311yescharacteristic, complemented
123C22⋊Q8C2×D5311yescharacteristic
124#121Dic5311yescharacteristic, complemented
125#121C2×D5311yescharacteristic
126#120—21no—
127C4×D4—21no—
128C4×D4—21no—
129#117Dic5311yescharacteristic, complemented
130C2×C22⋊C4—1061no—
131C22⋊C8—532no—
132C22⋊C8—1061no—
133#95—532no—
134#95—1061no—
135C24C22⋊D5311yescharacteristic
136#41—42no—
137C2×Q8C2×Dic5311yescharacteristic
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4C2×Dic5311yescharacteristic
142C22×C4—21no—
143C22×C4—42no—
144C22×C4—1062no—
145C22×C4C2×Dic5311yescharacteristic
146C22×C4C2×Dic5311yescharacteristic
147C22×C4C22⋊D5311yescharacteristic
148C22×C4C22⋊D5311yescharacteristic
149C22×C4C22⋊D5311yescharacteristic
150C22×C4C22×D5311yescharacteristic
151C2×C8—1062no—
152C2×C8—2122no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4C2×Dic5311yescharacteristic
156C4⋊C4C2×Dic5311yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—2122no—
163C42—21no—
164C23—22no—
165C23—22no—
166C23—21no—
167C23—41no—
168C23C22⋊Dic5311yescharacteristic
169C23C22⋊Dic5311yescharacteristic
170C23C2×C22⋊D5311yescharacteristic
171Q8—41no—
172D4—41no—
173D4—42no—
174D4—42no—
175D4—42no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—2122no—
187C2×C4—2122no—
188C2×C4—2122no—
189C2×C4C22⋊Dic5311yescharacteristic
190C2×C4C22⋊Dic5311yescharacteristic
191C2×C4C22×Dic5311yescharacteristic
192C2×C4C2×C22⋊D5311yescharacteristic
193C8—2122no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#1313313212yes—
206C22C2×C22⋊Dic5311yescharacteristic
207C22#1313323712yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—4242no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—81no—
220C2C2×#1313313211yescharacteristic
221C2#1681061911yescharacteristic
222C2#1681064011yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#525C2
socleC22×C53C2×C22⋊C4
hypercenter#525D53
hypocenterC53#1516
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#88322C2semidirect
2. #16810486C2semidirect
3. #16810360C2semidirect
4. #16810357C2semidirect
12. C2×#41×C53C4semidirect
15. C22⋊Q8×C53C4semidirect
17. #121×C53C4semidirect
22. #117×C53C4semidirect
117. C53#1516semidirect
118. C2×#41Dic53semidirect
122. C22⋊Q8Dic53semidirect
124. #121Dic53semidirect
129. #117Dic53semidirect