Finite Group Databasebeta

← #20290721

Subgroups — #20290721

Order 6784.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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#16811691C211yesmaximal, characteristic
2#16811616C212yesmaximal
3D53⋊#101C211yesmaximal, characteristic
4#16811494C212yesmaximal
5C4⋊Q8×D53C211yesmaximal, characteristic
6D53⋊M16C2212yes—
7Q8×#6157—24no—
8#13133296—22no—
9#13133296C2212yes—
10#13133280—24no—
11#13133278—22no—
12C2×Q8×D53C2211yescharacteristic
13C2×Q8×D53C2211yescharacteristic
14#13133217C411yescharacteristic, complemented
15#13133216—22no—
16#13133215C411yescharacteristic, complemented
17#13133215C411yescharacteristic, complemented
18C4⋊C4×D53—22no—
19C42×D53C2211yescharacteristic
20C4⋊Q8×C53C411yescharacteristic, complemented
21C2×Q8×C53C2×C411yescharacteristic
22C2×Q8×C53C2×C411yescharacteristic
23Q8×D53—22no—
24Q8×D53—22no—
25Q8×D53—22no—
26Q8×D53—22no—
27C2×C4×D53—21no—
28C2×C4×D53—21no—
29C2×C4×D53D412yes—
30C2×C4×D53C2311yescharacteristic
31C2×Dic106—21no—
32C2×Dic106C2×C411yescharacteristic
33C2×Dic106C2×C411yescharacteristic
34C2×Dic106D412yes—
35C22⋊#6157—22no—
36#11719940—22no—
37C4⋊#6157—24no—
38C4⋊#6157D414yes—
39C4×#6157—24no—
40C4×#6157—22no—
41#11719936—22no—
42D53⋊C8—22no—
43C4⋊C4×C53—22no—
44C42×C53C2×C411yescharacteristic
45C4⋊Dic53—22no—
46#11719919—22no—
47#11719919—22no—
48C4×Dic53—21no—
49C4×Dic53C2×C411yescharacteristic
50C22×D53C2×D411yescharacteristic
51C2×#6157—24no—
52Q8×C53—22no—
53Q8×C53—22no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53C22⋊C412yes—
57C2×C4×C53C22×C411yescharacteristic
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53C22⋊C412yes—
61C2×Dic53C2×D411yescharacteristic
62C4×D53—24no—
63C4×D53—22no—
64C4×D53—22no—
65C4×D53—41no—
66C4×D53—41no—
67C4×D53C2×D412yes—
68C4×D53#4112yes—
69Dic106—22no—
70Dic106—24no—
71Dic106—22no—
72Dic106—42no—
73#88324—42no—
74C22×C53C2×C22⋊C411yescharacteristic
75C2×D53—22no—
76C2×D53#12011yescharacteristic
77C2×D53#12311yescharacteristic
78#6157—44no—
79C4×C53—22no—
80C4×C53—22no—
81C4×C53—41no—
82C4×C53C4×D412yes—
83Dic53—22no—
84Dic53—22no—
85Dic53—41no—
86Dic53#12112yes—
87#1886—531nomaximal, Sylow
88C2×C53—21no—
89C2×C53#38611yescharacteristic
90D53—21no—
91D53#45612yes—
92C2×C4⋊Q8—531no—
93#519—531no—
94#419—532no—
95C2×#101—531no—
96#339—532no—
97C53#188611yescharacteristic, complemented, Sylow
98C22×Q8—531no—
99C22×Q8—531no—
100C2×M16—532no—
101C4⋊Q8—531no—
102C4⋊Q8—531no—
103C4⋊Q8—531no—
104C4⋊Q8—1062no—
105C4⋊Q8#615711yescharacteristic, complemented
106C4×Q8—1064no—
107#117—532no—
108#117—1062no—
109C2×C4⋊C4—1062no—
110C2×C42—531no—
111#103—1064no—
112#101—1062no—
113C2×Q8—532no—
114C2×Q8—531no—
115C2×Q8—531no—
116C2×Q8—1061no—
117C2×Q8—1062no—
118C2×Q8—1062no—
119C2×Q8—1062no—
120C2×Q8—1062no—
121C2×Q8C2×#615711yescharacteristic
122C2×Q8C2×#615711yescharacteristic
123C22×C4—531no—
124C22×C4—532no—
125C22×C4—1061no—
126C22×C4—1061no—
127M16—1062no—
128M16—1062no—
129C2×C8—1062no—
130C4⋊C4—22no—
131C4⋊C4—534no—
132C4⋊C4—1062no—
133C4⋊C4—1062no—
134C4⋊C4—1064no—
135C4⋊C4—1062no—
136C22⋊C4—1062no—
137C42—531no—
138C42—1061no—
139C42—1062no—
140C42—1064no—
141C42C2×#615711yescharacteristic
142C23—531no—
143Q8—22no—
144Q8—22no—
145Q8—1064no—
146Q8—1062no—
147Q8—1062no—
148Q8—2122no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—531no—
152C2×C4—532no—
153C2×C4—532no—
154C2×C4—532no—
155C2×C4—1062no—
156C2×C4—1062no—
157C2×C4—1064no—
158C2×C4—1061no—
159C2×C4—1064no—
160C2×C4—1061no—
161C2×C4—2121no—
162C2×C4—2121no—
163C2×C4C22⋊#615712yes—
164C2×C4C22×#615711yescharacteristic
165C8—2122no—
166C22—531no—
167C22—531no—
168C22—1062no—
169C22C2×C22⋊#615711yescharacteristic
170C4—22no—
171C4—22no—
172C4—41no—
173C4—532no—
174C4—1062no—
175C4—1062no—
176C4—2124no—
177C4—2121no—
178C4D4×#615712yes—
179C2—21no—
180C2—532no—
181C2—1061no—
182C2#1681157111yescharacteristic
subgroupG/H
centerC2#16811571
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4⋊Q8×C53C4
socleC2×C53#386
hypercenterC4⋊Q8#6157
hypocenterC53#1886
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
14. #13133217C4semidirect
16. #13133215C4semidirect
17. #13133215C4semidirect
20. C4⋊Q8×C53C4semidirect
97. C53#1886semidirect
105. C4⋊Q8#6157semidirect