Finite Group Databasebeta

← #20274940

Subgroups — #20274940

Order 6784.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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#16811104C211yesmaximal, characteristic
2#16810663C211yesmaximal, characteristic, complemented
3C2×#13133145C211yesmaximal, characteristic
4#16810622C211yesmaximal, characteristic
5#16810392C211yesmaximal, characteristic, complemented
6#16810383C211yesmaximal, characteristic, complemented
7C53×#384C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10C2×#11719917C2211yescharacteristic
11#13133182—21no—
12#13133179—21no—
13C4×C22⋊D53—22no—
14#13133176C2211yescharacteristic
15#13133175—21no—
16C4⋊C4×D53—21no—
17#13133150—21no—
18#13133148—21no—
19#13133145C414yescomplemented
20#13133140—21no—
21#13133137—21no—
22C2×C4⋊C4×C53—22no—
23C2×C4⋊C4×C53C2211yescharacteristic
24C2×C4⋊C4×C53C2211yescharacteristic
25C2×C42×C53C2211yescharacteristic
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53D412yes—
29C22×C4×C53C2311yescharacteristic
30C2×C22⋊D53—21no—
31#11719945—21no—
32#11719945—21no—
33#11719945C2×C412yes—
34#11719945Q812yes—
35C2×D212D411yescharacteristic
36C2×C4×D53—21no—
37C2×C4×D53—22no—
38C2×Dic106D411yescharacteristic
39C4⋊C4×C53—21no—
40C4⋊C4×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53—21no—
43C4⋊C4×C53—44no—
44C42×C53—21no—
45C42×C53C2×C412yescomplemented
46C22⋊Dic53—21no—
47#11719921—22no—
48C4⋊Dic53D411yescharacteristic
49C4⋊Dic53D411yescharacteristic
50#11719919—22no—
51C4×Dic53—22no—
52C4×Dic53—21no—
53#11719917—21no—
54#11719917—21no—
55#11719917—21no—
56#11719917C2×C412yescomplemented
57C2×#88322—21no—
58C2×#88322—21no—
59C23×C53C2×D411yescharacteristic
60C22×D53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—22no—
64C2×C4×C53—22no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—41no—
68C2×C4×C53C22×C411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×Q811yescharacteristic
72C2×C4×C53#4111yescharacteristic
73C2×C4×C53#4111yescharacteristic
74C22⋊D53—22no—
75C22⋊D53—22no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—21no—
79D212—21no—
80D212C4⋊C412yescomplemented
81C4×D53—22no—
82C4×D53—22no—
83C4×D53—82no—
84Dic106—21no—
85Dic106C4⋊C412yescomplemented
86#88322—22no—
87#88322—41no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53C4×D411yescharacteristic
91C22×C53#12011yescharacteristic
92C22×C53C22⋊Q811yescharacteristic
93C2×D53—22no—
94C2×D53—41no—
95C2×D53—41no—
96C4×C53—24no—
97C4×C53—42no—
98C4×C53—81no—
99C4×C53C2×C4⋊C411yescharacteristic
100C4×C53C4×D411yescharacteristic
101C4×C53C22⋊Q811yescharacteristic
102C4×C53#12311yescharacteristic
103Dic53—22no—
104Dic53—41no—
105Dic53—41no—
106#1859—531nomaximal, Sylow
107C2×C53—21no—
108C2×C53—21no—
109C2×C53#38511yescharacteristic
110C2×C53#45311yescharacteristic
111C2×C53#45611yescharacteristic
112D53—42no—
113#520—531no—
114#428—531no—
115C2×#104—531no—
116#417—531no—
117#384D5311yescharacteristic, complemented
118#343—531no—
119#339—531no—
120C53#185911yescharacteristic, complemented, Sylow
121C2×#41—531no—
122C2×M16—531no—
123C2×M16—531no—
124C4×Q8—1061no—
125C4×D4—1061no—
126C4×D4—1062no—
127#117—531no—
128#117—1061no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—1061no—
131C2×C4⋊C4C2×D5311yescharacteristic
132C2×C4⋊C4C2×D5311yescharacteristic
133C2×C42C2×D5311yescharacteristic
134#107—1061no—
135#106—1061no—
136#104—534no—
137#103—1061no—
138#102—1061no—
139#41—532no—
140#41—532no—
141#41—1061no—
142#41—1061no—
143C2×Q8—531no—
144C2×D4—531no—
145C2×D4—1061no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—1062no—
149C22×C4—1061no—
150C22×C4C22⋊D5312yes—
151C22×C4C22×D5311yescharacteristic
152M16—532no—
153M16—1061no—
154M16—1061no—
155M16—1061no—
156C2×C8—1061no—
157C2×C8—1061no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—44no—
163C4⋊C4—531no—
164C4⋊C4—531no—
165C4⋊C4—1062no—
166C22⋊C4—1062no—
167C22⋊C4—1061no—
168C42—21no—
169C42—1062no—
170C42—1061no—
171C42C4×D5312yescomplemented
172C23—1061no—
173C23C2×C22⋊D5311yescharacteristic
174Q8—532no—
175Q8—1061no—
176D4—532no—
177D4—1062no—
178D4—1062no—
179D4—1061no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—41no—
187C2×C4—1061no—
188C2×C4—1062no—
189C2×C4—1062no—
190C2×C4—1061no—
191C2×C4—1061no—
192C2×C4—4242no—
193C2×C4C2×C4×D5311yescharacteristic
194C2×C4#1171994511yescharacteristic
195C2×C4D4×D5311yescharacteristic
196C2×C4D4×D5311yescharacteristic
197C2×C4Q8×D5311yescharacteristic
198C2×C4#1171994911yescharacteristic
199C8—1062no—
200C8—2121no—
201C22—21no—
202C22—21no—
203C22—1062no—
204C22—2121no—
205C22—2121no—
206C22C4×C22⋊D5311yescharacteristic
207C22#1313318511yescharacteristic
208C22#1313319311yescharacteristic
209C4—24no—
210C4—42no—
211C4—81no—
212C4—1062no—
213C4—2121no—
214C4—2121no—
215C4C4⋊C4×D5311yescharacteristic
216C4#1313317911yescharacteristic
217C4#1313319511yescharacteristic
218C4#1313320211yescharacteristic
219C2—21no—
220C2—21no—
221C2—2122no—
222C2#1681049411yescharacteristic
223C2#1681079011yescharacteristic
224C2#1681080511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#384C2
socleC22×C53#120
hypercenter#384D53
hypocenterC53#1859
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #16810663C2semidirect
5. #16810392C2semidirect
6. #16810383C2semidirect
7. C53×#384C2semidirect
19. #13133145C4semidirect
45. C42×C53C2×C4semidirect
56. #11719917C2×C4semidirect
80. D212C4⋊C4semidirect
85. Dic106C4⋊C4semidirect
117. #384D53semidirect
120. C53#1859semidirect
171. C42C4×D53semidirect