Finite Group Databasebeta

← #20274049

Subgroups — #20274049

Order 6784.

184 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 49 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#16811033C211yesmaximal, characteristic, complemented
2#16810993C211yesmaximal, characteristic, complemented
3#16810982C211yesmaximal, characteristic
4#16810365C211yesmaximal, characteristic, complemented
5#16810365C211yesmaximal, characteristic, complemented
6#16810363C211yesmaximal, characteristic
7C53×#563C211yesmaximal, characteristic
8#13133217C2211yescharacteristic
9#13133150—21no—
10#13133148—21no—
11#13133141—21no—
12#13133140—21no—
13#13133140—21no—
14#13133138—21no—
15#13133128C2211yescharacteristic
16#13133128C2211yescharacteristic
17#13133128C2211yescharacteristic
18C4⋊Q8×C53—21no—
19#126×C53—21no—
20#126×C53—21no—
21#125×C53C2211yescharacteristic
22#125×C53C2211yescharacteristic
23#124×C53C2211yescharacteristic
24#123×C53—21no—
25C22⋊Q8×C53—21no—
26C22⋊Q8×C53—21no—
27C2×Q8×C53—21no—
28C2×Q8×C53D411yescharacteristic
29C2×D4×C53D411yescharacteristic
30C22×C4×C53—21no—
31C2×Dic106—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53D411yescharacteristic
39C4⋊C4×C53D411yescharacteristic
40C4⋊C4×C53D411yescharacteristic
41C4⋊C4×C53D411yescharacteristic
42C22⋊C4×C53—21no—
43C22⋊C4×C53—21no—
44C22⋊C4×C53—21no—
45C22⋊C4×C53—21no—
46C22⋊C4×C53—21no—
47C42×C53—21no—
48C42×C53C2311yescharacteristic
49C4⋊Dic53—21no—
50C4⋊Dic53—21no—
51C2×#88322—21no—
52C2×#88322—21no—
53C2×#88322—21no—
54C23×C53—21no—
55Q8×C53—41no—
56Q8×C53—81no—
57D4×C53—41no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53—81no—
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×Dic53—41no—
68Dic106—42no—
69#88322—22no—
70#88322—41no—
71#88322—41no—
72C22×C53—41no—
73C22×C53—41no—
74C22×C53—41no—
75C22×C53#12011yescharacteristic
76C4×C53—21no—
77C4×C53—21no—
78C4×C53—21no—
79C4×C53—41no—
80C4×C53—41no—
81C4×C53—41no—
82C4×C53—41no—
83C4×C53—41no—
84Dic53—81no—
85#1664—531nomaximal, Sylow
86C2×C53—81no—
87C2×C53#45411yescharacteristic
88C2×C53#45511yescharacteristic
89C2×C53#45611yescharacteristic
90#563D5311yescharacteristic
91#501—531no—
92#491—531no—
93#489—531no—
94#330—531no—
95#330—531no—
96#329—531no—
97C53#166411yescharacteristic, complemented, Sylow
98C4⋊Q8—21no—
99C4⋊Q8—531no—
100#126—21no—
101#126—21no—
102#125C2×D5311yescharacteristic
103#125C2×D5311yescharacteristic
104#124C2×D5311yescharacteristic
105#123—21no—
106C22⋊Q8—21no—
107C22⋊Q8—21no—
108#107—1061no—
109#106—1061no—
110#103—1061no—
111#103—1061no—
112#103—1061no—
113#102—1061no—
114#97—531no—
115#97—531no—
116#97—531no—
117C2×Q8—21no—
118C2×Q8—1061no—
119C2×Q8C22⋊D5311yescharacteristic
120C2×D4C22⋊D5311yescharacteristic
121C22×C4—21no—
122C2×C8—1061no—
123C2×C8—1061no—
124C2×C8—1061no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—1061no—
132C4⋊C4—1061no—
133C4⋊C4C22⋊D5311yescharacteristic
134C4⋊C4C22⋊D5311yescharacteristic
135C4⋊C4C22⋊D5311yescharacteristic
136C4⋊C4C22⋊D5311yescharacteristic
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C22⋊C4—21no—
141C22⋊C4—21no—
142C42—21no—
143C42C22×D5311yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—81no—
147Q8—2122no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—81no—
155C2×C4—2121no—
156C2×C4C2×C22⋊D5311yescharacteristic
157C2×C4C2×C22⋊D5311yescharacteristic
158C2×C4C2×C22⋊D5311yescharacteristic
159C8—1062no—
160C8—2121no—
161C8—2121no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#1313318611yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—41no—
171C4—41no—
172C4—41no—
173C4—41no—
174C4—4241no—
175C2—81no—
176C2#1681079611yescharacteristic
177C2#1681080111yescharacteristic
178C2#1681080611yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#563C2
socleC22×C53#120
hypercenter#563D53
hypocenterC53#1664
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #16811033C2semidirect
2. #16810993C2semidirect
4. #16810365C2semidirect
5. #16810365C2semidirect
97. C53#1664semidirect