Finite Group Databasebeta

← #20273269

Subgroups — #20273269

Order 6784.

284 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 73 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#16811134C211yesmaximal, characteristic
2C2×C22⋊#88322C211yesmaximal, characteristic, complemented
3#16810359C212yesmaximal, complemented
4#16810356C212yesmaximal, complemented
5C53×#381C211yesmaximal, characteristic, complemented
6C2×#11719945C411yescharacteristic, complemented
7C22×#88322—21no—
8#13133200—21no—
9#13133196C411yescharacteristic, complemented
10#13133196C2211yescharacteristic
11#13133191C411yescharacteristic, complemented
12#13133191C2211yescharacteristic
13#13133186—21no—
14C4×C22⋊D53—21no—
15C4×C22⋊D53—21no—
16#13133176C411yescharacteristic, complemented
17C22⋊#88322—21no—
18C22⋊#88322C2212yes—
19C23×C4×C53C2211yescharacteristic
20C2×C22⋊C4×C53—21no—
21#95×C53—21no—
22#95×C53C2212yes—
23C24×C53D411yescharacteristic
24C22×C4×C53—22no—
25C22×C4×C53—42no—
26C22×C4×C53C2×C411yescharacteristic
27C22×C4×C53C2×C411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53D411yescharacteristic
30C22×C4×C53D411yescharacteristic
31C22×C4×C53C2311yescharacteristic
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C2×C22⋊D53—21no—
35#11719945—42no—
36C2×D212C2×C411yescharacteristic
37C2×C4×D53—21no—
38C2×Dic106C2×C411yescharacteristic
39C22⋊C4×C53—42no—
40C22⋊Dic53—21no—
41C22⋊Dic53—21no—
42C22⋊Dic53—21no—
43#11719921—21no—
44#11719921—21no—
45C4⋊Dic53C2×C411yescharacteristic
46C4⋊Dic53C2×C411yescharacteristic
47#11719919—21no—
48#11719919—21no—
49C4×Dic53—21no—
50C2×#88322—22no—
51C2×#88322—42no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—41no—
55C23×C53C22⋊C411yescharacteristic
56C23×C53C22⋊C411yescharacteristic
57C23×C53C2×D411yescharacteristic
58C22×D53—21no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C2×C4×C53C22⋊C411yescharacteristic
69C2×C4×C53C22⋊C411yescharacteristic
70C2×C4×C53C22×C411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C22⋊D53—42no—
73C22⋊D53—42no—
74C22⋊D53—42no—
75C2×Dic53—21no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78D212—41no—
79C4×D53—42no—
80Dic106—41no—
81#88322—42no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—44no—
90C22×C53#9912yes—
91C22×C53C2×C22⋊C411yescharacteristic
92C22×C53#13512yes—
93C2×D53—41no—
94C2×D53—41no—
95C2×D53—41no—
96C4×C53—21no—
97C4×C53—21no—
98C4×C53—21no—
99C4×C53—21no—
100C4×C53—82no—
101Dic53—41no—
102Dic53—41no—
103Dic53—41no—
104#1516—531nomaximal, Sylow
105C2×C53—22no—
106C2×C53—22no—
107C2×C53—41no—
108C2×C53C2×#9911yescharacteristic
109C2×C53#41611yescharacteristic
110C2×C53#42111yescharacteristic
111D53—81no—
112#525—531no—
113C2×C22⋊C8—531no—
114#381D5311yescharacteristic, complemented
115#328—532no—
116#327—532no—
117C53#151611yescharacteristic, complemented, Sylow
118C2×#41—531no—
119C23×C4C2×D5311yescharacteristic
120C22×C8—1061no—
121#123—1061no—
122C22⋊Q8—531no—
123C22⋊Q8—531no—
124#121—531no—
125#121—531no—
126#120—1061no—
127C4×D4—1061no—
128C4×D4—1061no—
129#117—531no—
130C2×C22⋊C4—21no—
131C22⋊C8—532no—
132C22⋊C8—1061no—
133#95—21no—
134#95C2×D5312yes—
135C24C22⋊D5311yescharacteristic
136#41—2122no—
137C2×Q8—531no—
138C2×D4—531no—
139C2×D4—1061no—
140C2×D4—1061no—
141C2×D4—1061no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—1061no—
145C22×C4C4×D5311yescharacteristic
146C22×C4C4×D5311yescharacteristic
147C22×C4D21211yescharacteristic
148C22×C4D21211yescharacteristic
149C22×C4C22⋊D5311yescharacteristic
150C22×C4C22×D5311yescharacteristic
151C2×C8—1062no—
152C2×C8—2122no—
153C4⋊C4—531no—
154C4⋊C4—531no—
155C4⋊C4—1061no—
156C4⋊C4—1061no—
157C22⋊C4—42no—
158C22⋊C4—1061no—
159C22⋊C4—1061no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C22⋊C4—1061no—
163C42—1061no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—1061no—
168C23#1171992111yescharacteristic
169C23#1171992111yescharacteristic
170C23C2×C22⋊D5311yescharacteristic
171Q8—2121no—
172D4—2121no—
173D4—2122no—
174D4—2122no—
175D4—2122no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—1061no—
186C2×C4—1061no—
187C2×C4—1061no—
188C2×C4—2122no—
189C2×C4#1171992111yescharacteristic
190C2×C4#1171992111yescharacteristic
191C2×C4C2×C4×D5311yescharacteristic
192C2×C4C2×D21211yescharacteristic
193C8—2122no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—2121no—
203C22—2121no—
204C22—2121no—
205C22#1313313112yes—
206C22C2×#1171992111yescharacteristic
207C22#1313323712yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—82no—
213C4—2121no—
214C4—2121no—
215C4—2121no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—4241no—
220C2C2×#1313313111yescharacteristic
221C2#1681061711yescharacteristic
222C2#1681064111yescharacteristic
subgroupG/H
centerC22C2×#11719921
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#381C2
socleC22×C53C2×C22⋊C4
hypercenter#381D53
hypocenterC53#1516
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#88322C2semidirect
3. #16810359C2semidirect
4. #16810356C2semidirect
5. C53×#381C2semidirect
6. C2×#11719945C4semidirect
9. #13133196C4semidirect
11. #13133191C4semidirect
16. #13133176C4semidirect
114. #381D53semidirect
117. C53#1516semidirect