Finite Group Databasebeta

← #20273653

Subgroups — #20273653

Order 6784.

266 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 51 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#16811252C211yesmaximal, characteristic, complemented
2#16810774C211yesmaximal, characteristic
3#16810537C211yesmaximal, characteristic
4#16810361C211yesmaximal, characteristic, complemented
5#16810360C211yesmaximal, characteristic, complemented
6#16810348C211yesmaximal, characteristic, complemented
7C53×#450C211yesmaximal, characteristic
8C2×#11719923—21no—
9#13133216—21no—
10#13133215—21no—
11#13133207—21no—
12#13133207—21no—
13#13133196C2211yescharacteristic
14#13133194—21no—
15#13133173—21no—
16C2×#11719919C2211yescharacteristic
17C2×C22⋊Dic53—21no—
18C2×C22⋊Dic53—21no—
19#13133138—21no—
20C22⋊#88322C2211yescharacteristic
21#13133124C2211yescharacteristic
22C22×D4×C53C2211yescharacteristic
23C2×SD16×C53—21no—
24C22⋊Q8×C53C2211yescharacteristic
25#102×C53—21no—
26C22⋊C8×C53C2211yescharacteristic
27C24×C53—21no—
28C2×Q8×C53D411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53—42no—
32C22×C4×C53C2311yescharacteristic
33C22×Dic53—21no—
34C22×Dic53D411yescharacteristic
35C22×Dic53D411yescharacteristic
36C2×Dic106—21no—
37C2×Dic106D411yescharacteristic
38SD16×C53—42no—
39C2×C8×C53—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53D411yescharacteristic
42C22⋊C4×C53—21no—
43C22⋊Dic53—21no—
44C22⋊Dic53—42no—
45C22⋊Dic53—42no—
46#11719923—42no—
47C4⋊Dic53—21no—
48C4⋊Dic53D411yescharacteristic
49#11719919—21no—
50#11719919—21no—
51#11719919—21no—
52#11719919—21no—
53#11719919—21no—
54#11719919—21no—
55#11719919—21no—
56C4×Dic53—21no—
57C4×Dic53—21no—
58C2×#88322—21no—
59C23×C53—22no—
60C23×C53—41no—
61C23×C53—41no—
62C23×C53—41no—
63C23×C53—41no—
64C23×C53—41no—
65C23×C53—41no—
66C23×C53C2×D411yescharacteristic
67Q8×C53—41no—
68D4×C53—22no—
69D4×C53—22no—
70D4×C53—41no—
71D4×C53—41no—
72C2×C4×C53—21no—
73C2×C4×C53—21no—
74C2×C4×C53—41no—
75C2×C4×C53C2×D411yescharacteristic
76C2×C4×C53C2×D411yescharacteristic
77C2×Dic53—21no—
78C2×Dic53—21no—
79C2×Dic53—21no—
80C2×Dic53—21no—
81C2×Dic53—21no—
82C2×Dic53—41no—
83C2×Dic53—41no—
84C2×Dic53—41no—
85C2×Dic53—41no—
86C2×Dic53SD1612yescomplemented
87Dic106—41no—
88Dic106—81no—
89C8×C53—41no—
90#88322—41no—
91C22×C53—21no—
92C22×C53—21no—
93C22×C53—42no—
94C22×C53—42no—
95C22×C53—42no—
96C22×C53—81no—
97C22×C53—81no—
98C22×C53—81no—
99C22×C53—81no—
100C22×C53#12011yescharacteristic
101C22×C53C2×SD1611yescharacteristic
102C22×C53#13711yescharacteristic
103C4×C53—21no—
104C4×C53—21no—
105C4×C53—41no—
106C4×C53—41no—
107Dic53—22no—
108Dic53—41no—
109Dic53—41no—
110Dic53—41no—
111Dic53—81no—
112#1599—531nomaximal, Sylow
113C2×C53—21no—
114C2×C53—21no—
115C2×C53—82no—
116C2×C53#44811yescharacteristic
117C2×C53#45411yescharacteristic
118C2×C53B(4,𝔽2)11yescharacteristic
119#544—531no—
120#450—531no—
121#450D5311yescharacteristic
122#394—531no—
123#328—531no—
124#328—531no—
125#324—531no—
126C53#159911yescharacteristic, complemented, Sylow
127C22×D4C2×D5311yescharacteristic
128C2×SD16—21no—
129C2×SD16—1061no—
130C4⋊Q8—1061no—
131C4⋊Q8—1061no—
132#125—1061no—
133#125—1061no—
134C22⋊Q8—531no—
135C22⋊Q8—1061no—
136C22⋊Q8C2×D5311yescharacteristic
137#117—1061no—
138C2×C4⋊C4—531no—
139C2×C22⋊C4—1061no—
140C2×C22⋊C4—1061no—
141#102—21no—
142#102—1061no—
143C22⋊C8—531no—
144C22⋊C8C2×D5311yescharacteristic
145#95—531no—
146C24—21no—
147C2×Q8—531no—
148C2×Q8—1061no—
149C2×Q8C22⋊D5311yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—1061no—
156C22×C4C22×D5311yescharacteristic
157SD16—42no—
158SD16—2122no—
159C2×C8—21no—
160C2×C8—1061no—
161C4⋊C4—21no—
162C4⋊C4—531no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—1061no—
166C4⋊C4—1061no—
167C4⋊C4—1061no—
168C4⋊C4—1061no—
169C4⋊C4—1061no—
170C4⋊C4—1061no—
171C4⋊C4C22⋊D5311yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—1061no—
174C22⋊C4—2122no—
175C22⋊C4—2122no—
176C42—1061no—
177C42—1061no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23D4×D5311yescharacteristic
186Q8—41no—
187Q8—4241no—
188Q8—2121no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—532no—
197C2×C4—1061no—
198C2×C4—1061no—
199C2×C4—1061no—
200C2×C4—1061no—
201C2×C4—1061no—
202C2×C4—2121no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4—2121no—
206C2×C4D4×D5311yescharacteristic
207C2×C4C2×C22⋊D5311yescharacteristic
208C8—41no—
209C8—2121no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22#1313318511yescharacteristic
220C22SD16×D5311yescharacteristic
221C22#1313324611yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—1062no—
227C4—4241no—
228C4—2121no—
229C4—2121no—
230C4—2121no—
231C2—21no—
232C2—21no—
233C2—82no—
234C2#1681076011yescharacteristic
235C2#1681079411yescharacteristic
236C2#1681080911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#450C2
socleC22×C53#120
hypercenter#450D53
hypocenterC53#1599
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #16811252C2semidirect
4. #16810361C2semidirect
5. #16810360C2semidirect
6. #16810348C2semidirect
86. C2×Dic53SD16semidirect
126. C53#1599semidirect