Finite Group Databasebeta

← #20273771

Subgroups — #20273771

Order 6784.

336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D212C211yesmaximal, characteristic, complemented
2#16811000C211yesmaximal, characteristic, complemented
3#16810819C211yesmaximal, characteristic, complemented
4#16810818C211yesmaximal, characteristic, complemented
5#16810366C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C22×D212—21no—
9C2×D4×D53—21no—
10C2×D424—21no—
11C2×D424—22no—
12C2×#11719922—21no—
13#13133214C2211yescharacteristic
14#13133214C2211yescharacteristic, complemented
15#13133192—21no—
16#13133191—21no—
17#13133187—22no—
18C4×D212C2211yescharacteristic
19C4⋊#88322C2211yescharacteristic, complemented
20#13133139—21no—
21#13133137—21no—
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic, complemented
24C4×C8×C53C2211yescharacteristic, complemented
25C23×D53—22no—
26C2×D4×C53D411yescharacteristic
27C22×C4×C53—21no—
28C2×C22⋊D53—22no—
29D4×D53—42no—
30D4×D53—42no—
31C2×D212—22no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—21no—
35C2×D212—21no—
36C2×D212—21no—
37C2×D212—42no—
38C2×D212—42no—
39C2×D212D411yescharacteristic
40C2×D212D412yes—
41C2×C4×D53—21no—
42C2×C8×C53—21no—
43C2×C8×C53—21no—
44C4⋊C4×C53D411yescharacteristic
45C22⋊C4×C53—21no—
46C42×C53C2311yescharacteristic
47#11719922—42no—
48#11719921—22no—
49#11719921—21no—
50C4⋊Dic53D411yescharacteristic
51C2×#88322—21no—
52D424—44no—
53D424—42no—
54C23×C53—21no—
55C22×D53—21no—
56C22×D53—22no—
57C22×D53—44no—
58C22×D53—44no—
59C22×D53—41no—
60C22×D53—82no—
61D4×C53—21no—
62D4×C53D812yescomplemented
63C2×C4×C53—21no—
64C2×C4×C53—41no—
65C2×C4×C53—41no—
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C22⋊D53—44no—
70C2×Dic53—21no—
71D212—21no—
72D212—42no—
73D212—42no—
74D212—44no—
75D212—41no—
76D212—41no—
77D212—41no—
78D212—41no—
79D212—41no—
80D212—41no—
81D212—41no—
82D212—41no—
83D212D812yescomplemented
84C4×D53—41no—
85C4×D53—41no—
86C8×C53—22no—
87C8×C53—41no—
88#88322—41no—
89C22×C53—22no—
90C22×C53—41no—
91C22×C53—41no—
92C22×C53#12011yescharacteristic
93C2×D53—22no—
94C2×D53—41no—
95C2×D53—41no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53—42no—
99C2×D53—88no—
100C2×D53—81no—
101C2×D53—81no—
102C2×D53—81no—
103C4×C53—21no—
104C4×C53—41no—
105C4×C53C2×D811yescharacteristic
106C4×C53C2×D811yescharacteristic
107C4×C53AGL(1,ℤ/8)11yescharacteristic
108C4×C53AGL(1,ℤ/8)11yescharacteristic
109Dic53—41no—
110#1619—531nomaximal, Sylow
111C2×C53—42no—
112C2×C53#44711yescharacteristic
113C2×C53#44711yescharacteristic
114C2×C53#45311yescharacteristic
115D53—42no—
116D53—82no—
117D53—161no—
118D42—531no—
119#493—531no—
120#459—531no—
121#459—531no—
122#331—531no—
123D4⋊C8—531no—
124D4⋊C8D5311yescharacteristic, complemented
125C53#161911yescharacteristic, complemented, Sylow
126C22×D4—1061no—
127C22×D4—1061no—
128C2×D8—1061no—
129C2×D8—1062no—
130C2×D8—1061no—
131#127—531no—
132#127—531no—
133#121—1061no—
134#121—1061no—
135#120—1062no—
136C4×D4—531no—
137C4×D4C2×D5311yescharacteristic
138C4⋊C8—531no—
139C4⋊C8C2×D5311yescharacteristic, complemented
140#102—1061no—
141#102—1061no—
142C4×C8C2×D5311yescharacteristic, complemented
143C24—1062no—
144C2×D4—531no—
145C2×D4—532no—
146C2×D4—1062no—
147C2×D4—1062no—
148C2×D4—1061no—
149C2×D4—1061no—
150C2×D4—1061no—
151C2×D4—1061no—
152C2×D4—1061no—
153C2×D4—2122no—
154C2×D4—2122no—
155C2×D4—2122no—
156C2×D4—2122no—
157C2×D4D21211yescharacteristic
158C22×C4—21no—
159C22×C4—1061no—
160D8—2124no—
161D8—2122no—
162D8—2122no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—1061no—
166C4⋊C4—531no—
167C4⋊C4D21211yescharacteristic
168C22⋊C4—21no—
169C22⋊C4—1062no—
170C22⋊C4—1061no—
171C42C22×D5311yescharacteristic
172C23—21no—
173C23—1061no—
174C23—1062no—
175C23—4242no—
176C23—2124no—
177C23—2124no—
178C23—2121no—
179D4—21no—
180D4—532no—
181D4—1061no—
182D4—2122no—
183D4—2122no—
184D4—2124no—
185D4—2124no—
186D4—2121no—
187D4—2121no—
188D4—2121no—
189D4—2121no—
190D4—2121no—
191D4—2121no—
192D4—2121no—
193D4—2121no—
194D4D42412yescomplemented
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—1061no—
199C2×C4—2121no—
200C2×C4—2121no—
201C2×C4C2×D21211yescharacteristic
202C2×C4D4×D5311yescharacteristic
203C2×C4D4×D5311yescharacteristic
204C8—22no—
205C8—41no—
206C8—2121no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—1062no—
211C22—4248no—
212C22—4241no—
213C22—4241no—
214C22—4241no—
215C22—2122no—
216C22—2122no—
217C22—2122no—
218C22—2121no—
219C22—2121no—
220C22#1313318711yescharacteristic
221C4—21no—
222C4—41no—
223C4—2121no—
224C4D8×D5311yescharacteristic
225C4C2×D42411yescharacteristic
226C4#1313324411yescharacteristic
227C4#1313324511yescharacteristic
228C2—42no—
229C2—8481no—
230C2—4242no—
231C2—2122no—
232C2#1681075811yescharacteristic
233C2#1681075911yescharacteristic
234C2#1681079211yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53#120
hypercenterD4⋊C8D53
hypocenterC53#1619
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
1. D4×D212C2semidirect
2. #16811000C2semidirect
3. #16810819C2semidirect
4. #16810818C2semidirect
5. #16810366C2semidirect
6. #16810352C2semidirect
7. C53×D4⋊C8C2semidirect
14. #13133214C22semidirect
19. C4⋊#88322C22semidirect
23. C4⋊C8×C53C22semidirect
24. C4×C8×C53C22semidirect
62. D4×C53D8semidirect
83. D212D8semidirect
124. D4⋊C8D53semidirect
125. C53#1619semidirect
139. C4⋊C8C2×D53semidirect
142. C4×C8C2×D53semidirect
194. D4D424semidirect