Finite Group Databasebeta

← #20279926

Subgroups — #20279926

Order 6784.

468 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 151 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×#13133216C211yesmaximal, characteristic
2C2×#13133182C211yesmaximal, characteristic
3#16810563C211yesmaximal, characteristic
4#16810562C211yesmaximal, characteristic
5#16810561C211yesmaximal, characteristic
6#16810527C212yesmaximal
7#16810527C212yesmaximal
8#16810527C212yesmaximal
9#16810487C212yesmaximal
10#16810474C211yesmaximal, characteristic
11C53×#400C211yesmaximal, characteristic
12C22×Dic106C2212yes—
13C22×Dic106C2211yescharacteristic
14#13133216—24no—
15#13133182—24no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2212yes—
18C2×#11719919C2212yes—
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2211yescharacteristic
21C2×C4×Dic53C2211yescharacteristic
22C2×C4×Dic53C2211yescharacteristic
23C2×C4×Dic53C2212yes—
24C2×C4×Dic53C2211yescharacteristic
25#13133124C2212yes—
26#13133124C2212yes—
27#13133124C2212yes—
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34#95×C53C2212yes—
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C22×C4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×Dic53C2312yes—
43C22×Dic53C2311yescharacteristic
44C22×Dic53C2312yes—
45C22×Dic53C2311yescharacteristic
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C2×Dic106—22no—
49C2×Dic106—22no—
50C2×Dic106—24no—
51C2×Dic106—28no—
52C2×Dic106Q814yescomplemented
53C4⋊C4×C53—22no—
54C4⋊C4×C53—22no—
55C4⋊Dic53—22no—
56#11719919—22no—
57#11719919—22no—
58#11719919—24no—
59#11719919—24no—
60C4×Dic53—22no—
61C4×Dic53—22no—
62C4×Dic53—24no—
63C4×Dic53D414yes—
64C23×C53C2411yescharacteristic
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—22no—
68C2×C4×C53—22no—
69C2×C4×C53—21no—
70C2×C4×C53—24no—
71C2×C4×C53—22no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53C2×D412yes—
76C2×C4×C53C2×Q812yes—
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—22no—
80C2×Dic53—21no—
81C2×Dic53—24no—
82C2×Dic53—22no—
83C2×Dic53—22no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53C2×D412yes—
87C2×Dic53C2×D412yes—
88C2×Dic53C2×Q812yes—
89C2×Dic53C2×Q812yes—
90C2×Dic53#4112yes—
91C2×Dic53#4114yes—
92Dic106—28no—
93Dic106—48no—
94C22×C53C22×D411yescharacteristic
95C22×C53C22×Q811yescharacteristic
96C22×C53C2×#4111yescharacteristic
97C22×C53C2×#4112yes—
98C22×C532−1+412yes—
99C4×C53—22no—
100C4×C53—22no—
101C4×C53—42no—
102C4×C53—41no—
103C4×C53—42no—
104Dic53—22no—
105Dic53—42no—
106Dic53—41no—
107Dic53—42no—
108Dic53C22⋊Q814yes—
109Dic53#12414yes—
110#2568—531nomaximal, Sylow
111C2×C53C2×C22⋊Q811yescharacteristic
112C2×C53C2×#12411yescharacteristic
113C2×C53#53611yescharacteristic
114C2×C53#55612yes—
115C2×C53#55712yes—
116C2×C4⋊Q8—531no—
117C2×C4×Q8—531no—
118#400—531no—
119#400—531no—
120#400—531no—
121#400D5311yescharacteristic
122#391—532no—
123#391—532no—
124#391—532no—
125#382—532no—
126C4×C4⋊C4—531no—
127C53#256811yescharacteristic, complemented, Sylow
128C22×Q8—531no—
129C22×Q8—532no—
130C4⋊Q8—1064no—
131C4×Q8—1064no—
132C2×C4⋊C4—531no—
133C2×C4⋊C4—531no—
134C2×C4⋊C4—532no—
135C2×C4⋊C4—531no—
136C2×C4⋊C4—532no—
137C2×C4⋊C4C2×D5311yescharacteristic
138C2×C4⋊C4C2×D5311yescharacteristic
139C2×C42—531no—
140C2×C42—531no—
141C2×C42—531no—
142C2×C42—532no—
143#95—531no—
144#95—532no—
145#95—532no—
146#95—532no—
147#95—532no—
148#95—532no—
149#95—532no—
150#95C2×D5311yescharacteristic
151#95C2×D5312yes—
152#95C2×D5312yes—
153C2×Q8—534no—
154C2×Q8—1062no—
155C2×Q8—1062no—
156C2×Q8—1064no—
157C2×Q8—1068no—
158C22×C4—531no—
159C22×C4—531no—
160C22×C4—531no—
161C22×C4—532no—
162C22×C4—531no—
163C22×C4—532no—
164C22×C4C22×D5312yes—
165C22×C4C22×D5311yescharacteristic
166C22×C4C22×D5312yes—
167C22×C4C22×D5311yescharacteristic
168C22×C4C22×D5311yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—22no—
171C4⋊C4—1062no—
172C4⋊C4—1064no—
173C4⋊C4—1064no—
174C4⋊C4—1062no—
175C4⋊C4—1062no—
176C42—534no—
177C42—1062no—
178C42—1062no—
179C42—1064no—
180C23C23×D5311yescharacteristic
181Q8—1068no—
182Q8—2128no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—532no—
194C2×C4—532no—
195C2×C4—532no—
196C2×C4—532no—
197C2×C4—534no—
198C2×C4—532no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—1062no—
203C2×C4—1062no—
204C2×C4—1062no—
205C2×C4—1064no—
206C2×C4—1062no—
207C2×C4—1062no—
208C2×C4D4×D5312yes—
209C2×C4Q8×D5312yes—
210C22C2×D4×D5311yescharacteristic
211C22C2×Q8×D5311yescharacteristic
212C22#41×D5311yescharacteristic
213C22#41×D5312yes—
214C22#1313326412yes—
215C4—22no—
216C4—22no—
217C4—42no—
218C4—42no—
219C4—41no—
220C4—534no—
221C4—534no—
222C4—1062no—
223C4—2122no—
224C4—2122no—
225C4—2121no—
226C2C22⋊Q8×D5311yescharacteristic
227C2#124×D5311yescharacteristic
228C2#1681119511yescharacteristic
229C2#1681134012yes—
230C2#1681135012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#400C2
hypercenter#400D53
hypocenterC53#2568
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
52. C2×Dic106Q8semidirect
127. C53#2568semidirect