Finite Group Databasebeta

← #20273877

Subgroups — #20273877

Order 6784.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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#16811288C211yesmaximal, characteristic
2#16810915C211yesmaximal, characteristic, complemented
3#16810910C211yesmaximal, characteristic
4#16810369C211yesmaximal, characteristic, complemented
5#16810355C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic
7C53×#487C211yesmaximal, characteristic, complemented
8C2×Q8×D53—21no—
9#13133216—21no—
10#13133195—22no—
11#13133193—21no—
12C4×Dic106C2211yescharacteristic
13C4×D212C2211yescharacteristic
14#13133179—21no—
15#13133150—21no—
16#13133148—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133140—21no—
20#13133137—21no—
21C4⋊Q8×C53C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23#103×C53—22no—
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27Q8×D53—42no—
28Q8×D53—42no—
29C2×D212D411yescharacteristic
30C2×C4×D53—22no—
31C2×C4×D53—21no—
32C2×Dic106—22no—
33C2×Dic106D411yescharacteristic
34C2×C8×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53D412yes—
39C42×C53C2311yescharacteristic
40#11719921—22no—
41#11719921—21no—
42C4⋊Dic53D411yescharacteristic
43C4⋊Dic53D411yescharacteristic
44#11719919—22no—
45#11719919—21no—
46C4×Dic53—21no—
47C2×#88322—21no—
48C2×#88322—21no—
49C22×D53—21no—
50Q8×C53—42no—
51Q8×C53—41no—
52Q8×C53—41no—
53C2×C4×C53—22no—
54C2×C4×C53—41no—
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×Dic53—21no—
59C2×Dic53—21no—
60C2×Dic53—21no—
61D212—21no—
62D212Q1612yes—
63C4×D53—44no—
64C4×D53—41no—
65C4×D53—41no—
66Dic106—21no—
67Dic106—44no—
68Dic106SD1612yes—
69C8×C53—41no—
70#88322—41no—
71#88322—41no—
72C22×C53#12011yescharacteristic
73C2×D53—22no—
74C2×D53—41no—
75C2×D53—41no—
76C4×C53—21no—
77C4×C53—42no—
78C4×C53—81no—
79C4×C53C2×SD1611yescharacteristic
80C4×C53C2×Q1611yescharacteristic
81C4×C53AGL(1,ℤ/8)11yescharacteristic
82C4×C53#13711yescharacteristic
83Dic53—22no—
84Dic53—41no—
85Dic53—41no—
86#1637—531nomaximal, Sylow
87C2×C53#45011yescharacteristic
88C2×C53#45111yescharacteristic
89C2×C53#45411yescharacteristic
90D53—42no—
91D4×Q8—531no—
92#487D5311yescharacteristic, complemented
93#477—531no—
94#476—531no—
95#332—531no—
96Q8⋊C8—531no—
97D4⋊C8—531no—
98C53#163711yescharacteristic, complemented, Sylow
99C22×Q8—1061no—
100C4⋊Q8—1061no—
101C4⋊Q8C2×D5311yescharacteristic
102C4⋊Q8C2×D5311yescharacteristic
103C22⋊Q8—1061no—
104C22⋊Q8—1062no—
105C4×Q8—531no—
106C4×D4—531no—
107C4×D4—1061no—
108#107—1061no—
109#106—1061no—
110C4⋊C8—531no—
111C4⋊C8—531no—
112#103—22no—
113#103—1061no—
114#102—1061no—
115C4×C8C2×D5311yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—531no—
119C2×Q8—1062no—
120C2×Q8—2122no—
121C2×Q8—2122no—
122C2×D4—531no—
123C22×C4—1062no—
124C22×C4—1061no—
125C2×C8—21no—
126C2×C8—1061no—
127C2×C8—1061no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—531no—
132C4⋊C4—531no—
133C4⋊C4—1062no—
134C4⋊C4—1061no—
135C4⋊C4C22⋊D5312yes—
136C22⋊C4—1062no—
137C22⋊C4—1061no—
138C42—1061no—
139C42C22×D5311yescharacteristic
140C23—1061no—
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—532no—
145Q8—1061no—
146Q8—2124no—
147D4—532no—
148D4—1061no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—1061no—
152C2×C4—1061no—
153C2×C4—1061no—
154C2×C4—2124no—
155C2×C4—2121no—
156C2×C4—2121no—
157C2×C4D4×D5311yescharacteristic
158C2×C4D4×D5311yescharacteristic
159C2×C4C2×C22⋊D5311yescharacteristic
160C8—41no—
161C8—2121no—
162C8—2121no—
163C22—1062no—
164C22—2121no—
165C22—2121no—
166C22#1313318511yescharacteristic
167C4—21no—
168C4—42no—
169C4—81no—
170C4—1062no—
171C4—2121no—
172C4—2121no—
173C4SD16×D5311yescharacteristic
174C4Q16×D5311yescharacteristic
175C4#1313324411yescharacteristic
176C4#1313324911yescharacteristic
177C2—2122no—
178C2#1681077211yescharacteristic
179C2#1681077711yescharacteristic
180C2#1681079511yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#487C2
socleC22×C53#120
hypercenter#487D53
hypocenterC53#1637
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #16810915C2semidirect
4. #16810369C2semidirect
5. #16810355C2semidirect
7. C53×#487C2semidirect
92. #487D53semidirect
98. C53#1637semidirect