Finite Group Databasebeta

← #20276122

Subgroups — #20276122

Order 6784.

274 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 81 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×#13133139C211yesmaximal, characteristic
2C2×#13133137C211yesmaximal, characteristic
3#16810524C211yesmaximal, characteristic
4#16810492C211yesmaximal, characteristic, complemented
5#16810385C211yesmaximal, characteristic, complemented
6#16810377C211yesmaximal, characteristic, complemented
7C53×#340C211yesmaximal, characteristic, complemented
8C22×D212C2211yescharacteristic
9C22×#88322C2211yescharacteristic
10C2×C4⋊Dic53C2211yescharacteristic
11C2×#11719919—21no—
12C2×#11719919—21no—
13C2×#11719921—21no—
14C2×#11719921—21no—
15C2×C4×Dic53C2211yescharacteristic
16#13133139—22no—
17#13133137—22no—
18C22×C8×C53C2211yescharacteristic
19C2×C4⋊C4×C53C2211yescharacteristic
20C2×C4⋊C4×C53C2211yescharacteristic
21C23×D53—21no—
22C22×C4×C53—21no—
23C22×C4×C53—21no—
24C22×C4×C53C2311yescharacteristic
25C22×Dic53—21no—
26C22×Dic53D411yescharacteristic
27C22×Dic53D411yescharacteristic
28C2×D212—21no—
29C2×D212—21no—
30C2×D212—21no—
31C2×D212—21no—
32C2×D212—21no—
33C2×D212D412yes—
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41#11719921—42no—
42#11719921—42no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45#11719919—42no—
46#11719919—42no—
47C4×Dic53—21no—
48C4×Dic53—21no—
49C2×#88322—21no—
50C2×#88322—21no—
51C2×#88322—21no—
52C23×C53C2×D411yescharacteristic
53C22×D53—22no—
54C22×D53—41no—
55C22×D53—41no—
56C22×D53—41no—
57C22×D53—41no—
58C22×D53—41no—
59C22×D53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53—41no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53#4111yescharacteristic
68C2×C4×C53#4111yescharacteristic
69C2×C4×C53#4111yescharacteristic
70C2×C4×C53#4111yescharacteristic
71C2×C4×C53#4111yescharacteristic
72C2×Dic53—21no—
73C2×Dic53—21no—
74C2×Dic53—21no—
75C2×Dic53—21no—
76C2×Dic53—41no—
77C2×Dic53—41no—
78C2×Dic53—41no—
79C2×Dic53D812yes—
80C2×Dic53SD1612yes—
81D212—22no—
82D212—22no—
83D212—41no—
84D212—41no—
85C8×C53—41no—
86#88322—41no—
87C22×C53#12111yescharacteristic
88C22×C53#12311yescharacteristic
89C22×C53#12411yescharacteristic
90C22×C53C2×D811yescharacteristic
91C22×C53C2×SD1611yescharacteristic
92C22×C53AGL(1,ℤ/8)11yescharacteristic
93C22×C53#13711yescharacteristic
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—42no—
97C2×D53—81no—
98C2×D53—81no—
99C2×D53—81no—
100C2×D53—81no—
101C4×C53—81no—
102C4×C53—81no—
103C4×C53#12311yescharacteristic
104C4×C53#12311yescharacteristic
105C4×C53#12611yescharacteristic
106C4×C53#12611yescharacteristic
107Dic53—22no—
108Dic53—22no—
109Dic53—81no—
110#2071—531nomaximal, Sylow
111C2×C53#39711yescharacteristic
112C2×C53#45911yescharacteristic
113C2×C53#46111yescharacteristic
114C2×C53#48011yescharacteristic
115C2×C53#48111yescharacteristic
116C2×C53#48611yescharacteristic
117C2×C53#48911yescharacteristic
118D53—82no—
119C2×#102—531no—
120C2×#102—531no—
121#390—531no—
122#384—531no—
123#340—531no—
124#340D5311yescharacteristic, complemented
125#336—531no—
126C53#207111yescharacteristic, complemented, Sylow
127C22×D4—531no—
128C22×C8—531no—
129C22×C8C2×D5311yescharacteristic
130C2×C4⋊C4—531no—
131C2×C4⋊C4—1061no—
132C2×C4⋊C4—1061no—
133C2×C4⋊C4C2×D5311yescharacteristic
134C2×C4⋊C4C2×D5311yescharacteristic
135C2×C22⋊C4—1061no—
136C2×C22⋊C4—1061no—
137C2×C42—531no—
138#102—1062no—
139#102—1062no—
140C24—1061no—
141C2×D4—532no—
142C2×D4—1061no—
143C2×D4—1061no—
144C2×D4—1061no—
145C2×D4—1061no—
146C2×D4—1061no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—531no—
150C22×C4—531no—
151C22×C4—1061no—
152C22×C4C22×D5311yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—1061no—
157C2×C8—1061no—
158C2×C8—1061no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—2122no—
166C4⋊C4—2122no—
167C22⋊C4—2122no—
168C22⋊C4—2122no—
169C42—1061no—
170C42—1061no—
171C23—1062no—
172C23—2121no—
173C23—2121no—
174C23—2121no—
175C23—2121no—
176C23—2121no—
177C23—2121no—
178C23D4×D5311yescharacteristic
179D4—1062no—
180D4—1062no—
181D4—2121no—
182D4—2121no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—532no—
190C2×C4—532no—
191C2×C4—1061no—
192C2×C4—1061no—
193C2×C4—1061no—
194C2×C4—1061no—
195C2×C4—2121no—
196C2×C4—2121no—
197C2×C4—2121no—
198C2×C4#1171994511yescharacteristic
199C2×C4#1171994511yescharacteristic
200C2×C4#1171994511yescharacteristic
201C2×C4D4×D5311yescharacteristic
202C2×C4#1171994711yescharacteristic
203C2×C4#1171994911yescharacteristic
204C8—41no—
205C8—2121no—
206C22—4241no—
207C22—4241no—
208C22—4241no—
209C22—4241no—
210C22—2122no—
211C22—2122no—
212C22—2122no—
213C22#1313319011yescharacteristic
214C22#1313320211yescharacteristic
215C22#1313320311yescharacteristic
216C22D8×D5311yescharacteristic
217C22SD16×D5311yescharacteristic
218C22#1313324411yescharacteristic
219C22#1313324911yescharacteristic
220C4—81no—
221C4—81no—
222C4—1062no—
223C4—1062no—
224C4—4241no—
225C4#1313319911yescharacteristic
226C4#1313320211yescharacteristic
227C4#1313321111yescharacteristic
228C4#1313321211yescharacteristic
229C2—4242no—
230C2#1681055211yescharacteristic
231C2#1681081711yescharacteristic
232C2#1681082711yescharacteristic
233C2#1681093811yescharacteristic
234C2#1681094311yescharacteristic
235C2#1681096811yescharacteristic
236C2#1681098411yescharacteristic
subgroupG/H
centerC23D4×D53
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#340C2
socleC23×C53C2×D4
hypercenter#340D53
hypocenterC53#2071
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
4. #16810492C2semidirect
5. #16810385C2semidirect
6. #16810377C2semidirect
7. C53×#340C2semidirect
124. #340D53semidirect
126. C53#2071semidirect