Finite Group Databasebeta

← #20273719

Subgroups — #20273719

Order 6784.

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

normal subgroups only

This group has 254 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#16811200C211yesmaximal, characteristic
2#16810937C211yesmaximal, characteristic, complemented
3#16810934C211yesmaximal, characteristic, complemented
4#16810555C211yesmaximal, characteristic, complemented
5#16810356C211yesmaximal, characteristic, complemented
6#16810348C211yesmaximal, characteristic, complemented
7C53×#327C211yesmaximal, characteristic, complemented
8C2×#11719947—21no—
9#13133195—21no—
10#13133194—21no—
11#13133191C2211yescharacteristic
12#13133190—21no—
13#13133189—21no—
14#13133189—21no—
15C2×C4⋊Dic53—21no—
16C2×C4⋊Dic53C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18#13133151—21no—
19#13133150—21no—
20#13133139—21no—
21#13133138—21no—
22C22⋊#88322C2211yescharacteristic
23#13133124—21no—
24#121×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C2×D4×C53—21no—
28C2×D4×C53D411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C22×Dic53—21no—
35C22×Dic53—21no—
36C22×Dic53D411yescharacteristic
37C22×Dic53D411yescharacteristic
38#11719947—42no—
39#11719947—42no—
40C2×D212D411yescharacteristic
41C2×C4×D53—21no—
42C2×Dic106—21no—
43C2×C8×C53—21no—
44C4⋊C4×C53D411yescharacteristic
45C22⋊C4×C53—21no—
46C22⋊Dic53—21no—
47#11719921—21no—
48#11719921—21no—
49C4⋊Dic53—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53—42no—
52C4⋊Dic53D411yescharacteristic
53#11719919—21no—
54#11719919—21no—
55#11719919—21no—
56C2×#88322—21no—
57C23×C53—21no—
58C23×C53C2×D411yescharacteristic
59C22×D53—21no—
60D4×C53—41no—
61D4×C53—41no—
62D4×C53—41no—
63C2×C4×C53—22no—
64C2×C4×C53—21no—
65C2×C4×C53—41no—
66C2×C4×C53—41no—
67C2×C4×C53—41no—
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C22⋊D53—42no—
71C22⋊D53—41no—
72C22⋊D53—41no—
73C22⋊D53—81no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—42no—
79C2×Dic53—41no—
80C2×Dic53—41no—
81C2×Dic53—41no—
82C2×Dic53—41no—
83C2×Dic53D812yes—
84D212—41no—
85C4×D53—42no—
86Dic106—42no—
87C8×C53—41no—
88#88322—41no—
89C22×C53—21no—
90C22×C53—21no—
91C22×C53—41no—
92C22×C53—41no—
93C22×C53—41no—
94C22×C53#12011yescharacteristic
95C22×C53C2×D811yescharacteristic
96C22×C53AGL(1,ℤ/8)11yescharacteristic
97C2×D53—41no—
98C2×D53—41no—
99C2×D53—41no—
100C4×C53—21no—
101C4×C53—21no—
102C4×C53—42no—
103C4×C53—41no—
104Dic53—22no—
105Dic53—41no—
106Dic53—41no—
107Dic53—81no—
108#1610—531nomaximal, Sylow
109C2×C53—21no—
110C2×C53—21no—
111C2×C53—81no—
112C2×C53#44711yescharacteristic
113C2×C53#45611yescharacteristic
114C2×C53#45811yescharacteristic
115D53—81no—
116#537—531no—
117#480—531no—
118#480—531no—
119#398—531no—
120#327—531no—
121#327D5311yescharacteristic, complemented
122#324—531no—
123C53#161011yescharacteristic, complemented, Sylow
124C2×#41—1061no—
125C22⋊Q8—1061no—
126C22⋊Q8—1061no—
127#121—531no—
128#121—1061no—
129#121—1061no—
130#121—1061no—
131#121C2×D5311yescharacteristic
132C2×C4⋊C4—531no—
133C2×C4⋊C4—531no—
134C2×C4⋊C4—1061no—
135#107—1061no—
136#107—1061no—
137#102—1061no—
138#102—1061no—
139C22⋊C8—531no—
140C22⋊C8C2×D5311yescharacteristic
141#95—1061no—
142#95C2×D5311yescharacteristic
143#41—2122no—
144#41—2122no—
145C2×Q8—1061no—
146C2×D4—21no—
147C2×D4—531no—
148C2×D4—1061no—
149C2×D4—1061no—
150C2×D4—1061no—
151C2×D4D21211yescharacteristic
152C22×C4—21no—
153C22×C4—531no—
154C22×C4—531no—
155C22×C4—1061no—
156C22×C4—1061no—
157C22×C4—1061no—
158C22×C4C22×D5311yescharacteristic
159C2×C8—21no—
160C2×C8—1061no—
161C4⋊C4—531no—
162C4⋊C4—1061no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—1061no—
166C4⋊C4—1061no—
167C4⋊C4—2122no—
168C4⋊C4D21211yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—1061no—
171C22⋊C4—1061no—
172C22⋊C4—1061no—
173C23—21no—
174C23—1061no—
175C23D4×D5311yescharacteristic
176Q8—2122no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—4241no—
181D4—2121no—
182D4—2122no—
183D4—2121no—
184D4—2121no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—532no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—2122no—
196C2×C4—2122no—
197C2×C4—2121no—
198C2×C4—2121no—
199C2×C4—2121no—
200C2×C4—2121no—
201C2×C4C2×D21211yescharacteristic
202C2×C4D4×D5311yescharacteristic
203C8—41no—
204C8—2121no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—2121no—
211C22—2121no—
212C22—2121no—
213C22#1313318711yescharacteristic
214C22D8×D5311yescharacteristic
215C22#1313324411yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—1062no—
221C4—4241no—
222C4—2121no—
223C4—2121no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—4241no—
228C2#1681075811yescharacteristic
229C2#1681080711yescharacteristic
230C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#327C2
socleC22×C53#120
hypercenter#327D53
hypocenterC53#1610
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #16810937C2semidirect
3. #16810934C2semidirect
4. #16810555C2semidirect
5. #16810356C2semidirect
6. #16810348C2semidirect
7. C53×#327C2semidirect
121. #327D53semidirect
123. C53#1610semidirect