Finite Group Databasebeta

← #20290588

Subgroups — #20290588

Order 6784.

312 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 80 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#16811627C211yesmaximal, characteristic, complemented
2C2×#13133282C212yesmaximal
3C4×C4⋊#6157C211yesmaximal, characteristic, complemented
4#16811493C212yesmaximal, complemented
5C4×#11719949C211yesmaximal, characteristic
6C2×#11719936C2212yes—
7C2×C4⋊#6157—21no—
8C2×C4×#6157C2212yes—
9C4⋊#88324—21no—
10C4⋊#88324—21no—
11#13133282—24no—
12(C2×#6157)⋊C4—21no—
13C2×#11719949C2211yescharacteristic
14Q8×Dic53C411yescharacteristic, complemented
15C4×D212—21no—
16C4×D212C411yescharacteristic, complemented
17#13133179—21no—
18#13133179C411yescharacteristic, complemented
19#13133175—21no—
20#13133175C2211yescharacteristic
21C42×D53—21no—
22C42×D53C2211yescharacteristic
23C4×Q8×C53C411yescharacteristic, complemented
24C22×#6157—22no—
25C2×Q8×C53C2×C411yescharacteristic
26#11719949—21no—
27#11719949—21no—
28#11719949D412yes—
29#11719949Q812yes—
30C2×D212—21no—
31C2×D212C2×C411yescharacteristic
32C2×C4×D53—21no—
33C2×C4×D53—21no—
34C2×C4×D53—21no—
35C2×C4×D53D411yescharacteristic
36C2×C4×D53D411yescharacteristic
37C2×C4×D53C2311yescharacteristic
38C2×#88324—22no—
39C4⋊#6157—24no—
40C4×#6157—22no—
41C4×#6157—22no—
42#11719936—22no—
43#11719936—22no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53C2×C411yescharacteristic
46C42×C53—21no—
47C42×C53C2×C411yescharacteristic
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51C4⋊Dic53—21no—
52C4⋊Dic53C2×C411yescharacteristic
53C4×Dic53—21no—
54C4×Dic53—21no—
55C4×Dic53C2×C411yescharacteristic
56C4×Dic53D411yescharacteristic
57C4×Dic53D411yescharacteristic
58C22×D53—21no—
59C22×D53C2×D411yescharacteristic
60C2×#6157—24no—
61C2×#6157—42no—
62C2×#6157—42no—
63Q8×C53—21no—
64Q8×C53C4⋊C412yescomplemented
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53C22⋊C411yescharacteristic
68C2×C4×C53C22⋊C411yescharacteristic
69C2×C4×C53C22×C411yescharacteristic
70C2×Dic53—21no—
71C2×Dic53C22⋊C411yescharacteristic
72C2×Dic53C22⋊C411yescharacteristic
73C2×Dic53C2×D411yescharacteristic
74D212—22no—
75D212—22no—
76D212—21no—
77D212C4⋊C412yescomplemented
78C4×D53—22no—
79C4×D53—22no—
80C4×D53—22no—
81C4×D53—41no—
82C4×D53—42no—
83C4×D53—42no—
84C4×D53C2×D411yescharacteristic
85C4×D53C2×Q811yescharacteristic
86C4×D53#4111yescharacteristic
87C4×D53#4111yescharacteristic
88#88324—42no—
89C22×C53C2×C22⋊C411yescharacteristic
90C2×D53—21no—
91C2×D53—21no—
92C2×D53—22no—
93C2×D53—41no—
94C2×D53—41no—
95C2×D53#12111yescharacteristic
96C2×D53C22⋊Q811yescharacteristic
97#6157—44no—
98C4×C53—22no—
99C4×C53—21no—
100C4×C53—41no—
101C4×C53#10412yescomplemented
102C4×C53C2×C4⋊C411yescharacteristic
103C4×C53#11711yescharacteristic
104Dic53—21no—
105Dic53—21no—
106Dic53—41no—
107Dic53#12111yescharacteristic
108Dic53C22⋊Q811yescharacteristic
109#1806—531nomaximal, Sylow
110C2×C53#38011yescharacteristic
111C2×C53C2×#10411yescharacteristic
112C2×C53#42111yescharacteristic
113D53—21no—
114D53—21no—
115D53—42no—
116C4×#41—531no—
117#423—531no—
118C2×#104—532no—
119C4×C4⋊C4—531no—
120#339—532no—
121C53#180611yescharacteristic, complemented, Sylow
122C2×#41—531no—
123C2×M16—532no—
124C4×Q8—531no—
125C4×Q8#615711yescharacteristic, complemented
126C4×D4—531no—
127C4×D4—531no—
128C4×D4—1061no—
129C4×D4—1061no—
130#117—531no—
131#117—1061no—
132C2×C4⋊C4—1061no—
133C2×C42—531no—
134C2×C42—532no—
135C2×C42—1061no—
136C4⋊C8—1061no—
137C4⋊C8—1061no—
138#104—1064no—
139#95—1061no—
140#41—532no—
141#41—532no—
142#41—1061no—
143#41—1061no—
144C2×Q8C2×#615711yescharacteristic
145C2×D4—531no—
146C2×D4—1061no—
147C22×C4—531no—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—1061no—
151C22×C4—1061no—
152C22×C4—1062no—
153C22×C4—1061no—
154M16—1062no—
155M16—1062no—
156C2×C8—1062no—
157C4⋊C4—21no—
158C4⋊C4—531no—
159C4⋊C4—1061no—
160C4⋊C4—1064no—
161C4⋊C4C2×#615711yescharacteristic
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C22⋊C4—1061no—
165C42—21no—
166C42—531no—
167C42—531no—
168C42—531no—
169C42—1061no—
170C42—1061no—
171C42—1062no—
172C42—1062no—
173C42C2×#615711yescharacteristic
174C23—531no—
175C23—1061no—
176Q8—21no—
177Q8C4⋊#615712yescomplemented
178D4—532no—
179D4—1062no—
180D4—1062no—
181D4—1061no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—531no—
185C2×C4—531no—
186C2×C4—531no—
187C2×C4—531no—
188C2×C4—531no—
189C2×C4—531no—
190C2×C4—531no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1062no—
194C2×C4—1064no—
195C2×C4—1061no—
196C2×C4—2121no—
197C2×C4—2122no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4C22⋊#615711yescharacteristic
202C2×C4C22⋊#615711yescharacteristic
203C2×C4C22×#615711yescharacteristic
204C8—2122no—
205C22—531no—
206C22—531no—
207C22—1061no—
208C22—1061no—
209C22—1062no—
210C22—2121no—
211C22—2121no—
212C22C2×C22⋊#615711yescharacteristic
213C4—22no—
214C4—21no—
215C4—41no—
216C4—531no—
217C4—531no—
218C4—1061no—
219C4—1061no—
220C4—2124no—
221C4—2121no—
222C4#1313328212yescomplemented
223C4C2×C4⋊#615711yescharacteristic
224C4#1313329611yescharacteristic
225C2—1061no—
226C2—1061no—
227C2—2122no—
228C2#1681156511yescharacteristic
229C2C2×#1313328211yescharacteristic
230C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×Q8#6157
hypocenterC53#1806
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #16811627C2semidirect
3. C4×C4⋊#6157C2semidirect
4. #16811493C2semidirect
14. Q8×Dic53C4semidirect
16. C4×D212C4semidirect
18. #13133179C4semidirect
23. C4×Q8×C53C4semidirect
64. Q8×C53C4⋊C4semidirect
77. D212C4⋊C4semidirect
101. C4×C53#104semidirect
121. C53#1806semidirect
125. C4×Q8#6157semidirect
177. Q8C4⋊#6157semidirect
222. C4#13133282semidirect