Finite Group Databasebeta

← #20279698

Subgroups — #20279698

Order 6784.

576 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 153 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×#13133215C211yesmaximal, characteristic, complemented
2C2×#13133206C211yesmaximal, characteristic
3C2×#13133205C211yesmaximal, characteristic
4C2×#13133182C211yesmaximal, characteristic, complemented
5#16810547C214yesmaximal, complemented
6#16810524C211yesmaximal, characteristic
7#16810512C214yesmaximal, complemented
8C4×#11719919C211yesmaximal, characteristic, complemented
9C53×#391C211yesmaximal, characteristic, complemented
10C22×Dic106C2211yescharacteristic
11C22×D212C2211yescharacteristic
12#13133215—24no—
13#13133206—24no—
14#13133205—24no—
15#13133182—24no—
16C2×#11719919C2214yes—
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719921C2212yes—
20C2×#11719921C2214yes—
21C2×#11719921C2214yes—
22C2×#11719921C2212yes—
23C2×C4×Dic53C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25C2×C4×Dic53C2211yescharacteristic
26#13133124C2212yes—
27C22×Q8×C53C2211yescharacteristic
28C2×C4⋊C4×C53C2211yescharacteristic
29C2×C42×C53C2211yescharacteristic
30#95×C53C2214yes—
31C23×D53C2312yes—
32C2×Q8×C53—22no—
33C2×Q8×C53—22no—
34C2×Q8×C53—22no—
35C22×C4×C53C2312yes—
36C22×C4×C53C2312yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×Dic53C2311yescharacteristic
40C22×Dic53C2311yescharacteristic
41C22×Dic53C2312yes—
42C22×Dic53C2312yes—
43C2×D212—22no—
44C2×D212—22no—
45C2×D212—22no—
46C2×Dic106—22no—
47C2×Dic106—22no—
48C2×Dic106D414yes—
49C4⋊C4×C53—22no—
50C42×C53—22no—
51#11719921—28no—
52#11719921—24no—
53#11719921—24no—
54#11719921—28no—
55#11719919—24no—
56#11719919—28no—
57#11719919—24no—
58C4×Dic53—22no—
59C4×Dic53—24no—
60C4×Dic53D414yes—
61C23×C53C2411yescharacteristic
62C22×D53—22no—
63C22×D53—22no—
64C22×D53—22no—
65C22×D53—24no—
66C22×D53—24no—
67Q8×C53—44no—
68C2×C4×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—24no—
75C2×C4×C53—22no—
76C2×C4×C53C2×D412yes—
77C2×C4×C53C2×D412yes—
78C2×C4×C53#4112yes—
79C2×Dic53—22no—
80C2×Dic53—22no—
81C2×Dic53—22no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53C2×D414yes—
86C2×Dic53C2×D412yes—
87C2×Dic53C2×D412yes—
88C2×Dic53#4114yes—
89D212—44no—
90Dic106—28no—
91C22×C53C22×D411yescharacteristic
92C22×C53C22×D411yescharacteristic
93C22×C53C2×#4112yes—
94C22×C53C2×#4111yescharacteristic
95C22×C532−1+412yes—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C2×D53—44no—
100C2×D53—44no—
101C4×C53—42no—
102C4×C53—42no—
103C4×C53—42no—
104C4×C53#12114yes—
105Dic53—24no—
106Dic53—22no—
107Dic53—22no—
108Dic53—24no—
109#2545—531nomaximal, Sylow
110C2×C53C2×#12111yescharacteristic
111C2×C53#53211yescharacteristic
112C2×C53#53611yescharacteristic
113C2×C53#54812yes—
114C2×C53#55612yes—
115D53—82no—
116C2×C4⋊Q8—531no—
117C2×#124—531no—
118C2×#124—531no—
119C2×C4×Q8—531no—
120#396—534no—
121#391D5311yescharacteristic, complemented
122#390—531no—
123#388—534no—
124C4×C4⋊C4—531no—
125C53#254511yescharacteristic, complemented, Sylow
126C22×Q8—531no—
127C22×Q8C2×D5311yescharacteristic
128C22×D4—531no—
129C4⋊Q8—1064no—
130#124—1064no—
131#124—1064no—
132C4×Q8—1064no—
133C2×C4⋊C4—532no—
134C2×C4⋊C4—534no—
135C2×C4⋊C4—532no—
136C2×C4⋊C4C2×D5311yescharacteristic
137C2×C22⋊C4—532no—
138C2×C22⋊C4—534no—
139C2×C22⋊C4—532no—
140C2×C22⋊C4—534no—
141C2×C42—531no—
142C2×C42—532no—
143C2×C42—531no—
144C2×C42C2×D5311yescharacteristic
145#95—532no—
146#95C2×D5314yes—
147C24—532no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—534no—
152C2×Q8—1062no—
153C2×Q8—1062no—
154C2×D4—1062no—
155C2×D4—1062no—
156C2×D4—1062no—
157C22×C4—531no—
158C22×C4—532no—
159C22×C4—531no—
160C22×C4—532no—
161C22×C4C22×D5312yes—
162C22×C4C22×D5312yes—
163C22×C4C22×D5312yes—
164C22×C4C22×D5311yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—1064no—
167C4⋊C4—1068no—
168C4⋊C4—1064no—
169C22⋊C4—1068no—
170C22⋊C4—1064no—
171C22⋊C4—1064no—
172C22⋊C4—1068no—
173C42—22no—
174C42—534no—
175C42—1064no—
176C42—1062no—
177C23—1062no—
178C23—1062no—
179C23—1064no—
180C23—1062no—
181C23—1064no—
182C23C23×D5311yescharacteristic
183Q8—44no—
184Q8—1068no—
185D4—2124no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—532no—
195C2×C4—532no—
196C2×C4—534no—
197C2×C4—534no—
198C2×C4—1062no—
199C2×C4—1062no—
200C2×C4—1062no—
201C2×C4—1062no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4#1171994512yes—
205C2×C4D4×D5312yes—
206C2×C4D4×D5312yes—
207C22—2124no—
208C22—2124no—
209C22—2122no—
210C22—2122no—
211C22—2122no—
212C22C2×D4×D5311yescharacteristic
213C22C2×D4×D5311yescharacteristic
214C22#41×D5312yes—
215C22C2×#1171994511yescharacteristic
216C22#1313326512yes—
217C4—42no—
218C4—42no—
219C4—42no—
220C4—1064no—
221C4—1064no—
222C4—1062no—
223C4—1062no—
224C4#1313319014yes—
225C2—4242no—
226C2C2×#1313319011yescharacteristic
227C2#1681117211yescharacteristic
228C2#1681119811yescharacteristic
229C2#1681128312yes—
230C2#1681134512yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2545
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#13133215C2semidirect
4. C2×#13133182C2semidirect
5. #16810547C2semidirect
7. #16810512C2semidirect
8. C4×#11719919C2semidirect
9. C53×#391C2semidirect
121. #391D53semidirect
125. C53#2545semidirect