Finite Group Databasebeta

← #20273206

Subgroups — #20273206

Order 6784.

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

normal subgroups only

This group has 304 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
1C2×C22⋊#88322C211yesmaximal, characteristic, complemented
2C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
3#16810360C212yesmaximal, complemented
4#16810357C212yesmaximal, complemented
5C53×#525C211yesmaximal, characteristic
6C22×#88322—21no—
7C2×C22⋊Dic53—21no—
8C2×C4×Dic53—21no—
9C22⋊#88322—21no—
10C22⋊#88322C2212yes—
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—24no—
50C4×Dic53—42no—
51C2×#88322—22no—
52C2×#88322—42no—
53C23×C53—22no—
54C23×C53—22no—
55C23×C53—21no—
56C23×C53—41no—
57C23×C53C22⋊C411yescharacteristic
58C23×C53C22⋊C411yescharacteristic
59C23×C53C2×D411yescharacteristic
60Q8×C53—41no—
61D4×C53—41no—
62D4×C53—42no—
63D4×C53—42no—
64D4×C53—42no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—21no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53—41no—
74C2×C4×C53—42no—
75C2×C4×C53C22⋊C411yescharacteristic
76C2×C4×C53C22⋊C411yescharacteristic
77C2×C4×C53C22×C411yescharacteristic
78C2×C4×C53C2×D411yescharacteristic
79C2×Dic53—24no—
80C2×Dic53—42no—
81C2×Dic53—42no—
82#88322—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—41no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53#10412yes—
95C22×C53C2×C22⋊C411yescharacteristic
96C22×C53#13512yes—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—41no—
101C4×C53—41no—
102C4×C53—41no—
103C4×C53#9912yes—
104Dic53—44no—
105#1501—531nomaximal, Sylow
106C2×C53—22no—
107C2×C53—22no—
108C2×C53—41no—
109C2×C53—81no—
110C2×C53C2×#9911yescharacteristic
111C2×C53#41611yescharacteristic
112C2×C53C2×#10411yescharacteristic
113#525D5311yescharacteristic
114C2×C22⋊C8—531no—
115C4×C22⋊C4—531no—
116#328—532no—
117#327—532no—
118C53#150111yescharacteristic, complemented, Sylow
119C2×#41Dic5311yescharacteristic, complemented
120C23×C4C2×D5311yescharacteristic
121C22×C8—1061no—
122#123—21no—
123C22⋊Q8Dic5311yescharacteristic, complemented
124C22⋊Q8C2×D5311yescharacteristic
125#121Dic5311yescharacteristic, complemented
126#121C2×D5311yescharacteristic
127#120—21no—
128C4×D4—21no—
129C4×D4—21no—
130#117Dic5311yescharacteristic, complemented
131C2×C22⋊C4—1061no—
132C2×C42—1061no—
133C22⋊C8—532no—
134C22⋊C8—1061no—
135#95—532no—
136C24C22⋊D5311yescharacteristic
137#41—42no—
138C2×Q8C2×Dic5311yescharacteristic
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4C2×Dic5311yescharacteristic
143C22×C4—21no—
144C22×C4—42no—
145C22×C4—1062no—
146C22×C4C2×Dic5311yescharacteristic
147C22×C4C2×Dic5311yescharacteristic
148C22×C4C22⋊D5311yescharacteristic
149C22×C4C22⋊D5311yescharacteristic
150C22×C4C22⋊D5311yescharacteristic
151C22×C4C22×D5311yescharacteristic
152C2×C8—1062no—
153C2×C8—2122no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4C2×Dic5311yescharacteristic
157C4⋊C4C2×Dic5311yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—1064no—
164C42—21no—
165C42—2122no—
166C23—22no—
167C23—22no—
168C23—21no—
169C23—41no—
170C23C22⋊Dic5311yescharacteristic
171C23C22⋊Dic5311yescharacteristic
172C23C2×C22⋊D5311yescharacteristic
173Q8—41no—
174D4—41no—
175D4—42no—
176D4—42no—
177D4—42no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—1064no—
189C2×C4—2122no—
190C2×C4—2122no—
191C2×C4C22⋊Dic5311yescharacteristic
192C2×C4C22⋊Dic5311yescharacteristic
193C2×C4C22×Dic5311yescharacteristic
194C2×C4C2×C22⋊D5311yescharacteristic
195C8—2122no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22#1313314412yes—
208C22C2×C22⋊Dic5311yescharacteristic
209C22#1313323712yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—41no—
214C4—41no—
215C4—41no—
216C4—2124no—
217C4#1313313212yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—81no—
222C2C2×#1313313211yescharacteristic
223C2#1681061911yescharacteristic
224C2C2×#1313314411yescharacteristic
subgroupG/H
centerC2×C4C22⋊Dic53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#525C2
socleC22×C53C2×C22⋊C4
hypercenter#525D53
hypocenterC53#1501
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×C22⋊#88322C2semidirect
2. C4×C22⋊Dic53C2semidirect
3. #16810360C2semidirect
4. #16810357C2semidirect
12. C2×#41×C53C4semidirect
15. C22⋊Q8×C53C4semidirect
17. #121×C53C4semidirect
22. #117×C53C4semidirect
118. C53#1501semidirect
119. C2×#41Dic53semidirect
123. C22⋊Q8Dic53semidirect
125. #121Dic53semidirect
130. #117Dic53semidirect