Finite Group Databasebeta

← #20274052

Subgroups — #20274052

Order 6784.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#16811373C211yesmaximal, characteristic
2#16811032C211yesmaximal, characteristic, complemented
3#16810992C211yesmaximal, characteristic, complemented
4#16810982C211yesmaximal, characteristic
5#16810364C211yesmaximal, characteristic, complemented
6#16810364C211yesmaximal, characteristic, complemented
7C53×#329C211yesmaximal, characteristic
8#13133217C2211yescharacteristic
9#13133215—21no—
10#13133210—21no—
11#13133210—21no—
12#13133209C2211yescharacteristic
13#13133209C2211yescharacteristic
14#13133198—21no—
15#13133194—21no—
16#13133194—21no—
17#13133151—21no—
18#13133149—21no—
19#13133142—21no—
20#13133142—21no—
21#13133141—21no—
22#13133138—21no—
23#13133128C2211yescharacteristic
24#124×C53C2211yescharacteristic
25#97×C53C2211yescharacteristic
26#97×C53C2211yescharacteristic
27C2×Q8×C53D411yescharacteristic
28C2×D4×C53D411yescharacteristic
29C22×Dic53—21no—
30C2×Dic106—21no—
31C2×Dic106—21no—
32C2×C8×C53—21no—
33C2×C8×C53—21no—
34C22⋊C4×C53—21no—
35C22⋊C4×C53—21no—
36C42×C53C2311yescharacteristic
37C22⋊Dic53—21no—
38C22⋊Dic53—21no—
39C22⋊Dic53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53D411yescharacteristic
43C4⋊Dic53D411yescharacteristic
44C4⋊Dic53D411yescharacteristic
45C4⋊Dic53D411yescharacteristic
46#11719919—21no—
47#11719919—21no—
48#11719919—21no—
49#11719919—21no—
50#11719919—21no—
51#11719919—21no—
52C4×Dic53—21no—
53C2×#88322—21no—
54C23×C53—21no—
55Q8×C53—41no—
56D4×C53—41no—
57C2×C4×C53—21no—
58C2×C4×C53C2×D411yescharacteristic
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C2×Dic53—21no—
62C2×Dic53—21no—
63C2×Dic53—21no—
64C2×Dic53—21no—
65C2×Dic53—41no—
66C2×Dic53—81no—
67Dic106—42no—
68Dic106—81no—
69C8×C53—22no—
70C8×C53—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—
80Dic53—41no—
81Dic53—41no—
82Dic53—41no—
83Dic53—41no—
84Dic53—81no—
85#1664—531nomaximal, Sylow
86C2×C53—81no—
87C2×C53#45411yescharacteristic
88C2×C53#45511yescharacteristic
89C2×C53#45611yescharacteristic
90#563—531no—
91#501—531no—
92#491—531no—
93#489—531no—
94#330—531no—
95#330—531no—
96#329D5311yescharacteristic
97C53#166411yescharacteristic, complemented, Sylow
98C4⋊Q8—531no—
99C4⋊Q8—1061no—
100#126—1061no—
101#126—1061no—
102#125—531no—
103#125—531no—
104#124C2×D5311yescharacteristic
105#123—1061no—
106C22⋊Q8—1061no—
107C22⋊Q8—1061no—
108#107—1061no—
109#106—1061no—
110#103—1061no—
111#103—1061no—
112#103—1061no—
113#102—1061no—
114#97—531no—
115#97C2×D5311yescharacteristic
116#97C2×D5311yescharacteristic
117C2×Q8—1061no—
118C2×Q8—1061no—
119C2×Q8D21211yescharacteristic
120C2×D4D21211yescharacteristic
121C22×C4—1061no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—1061no—
125C4⋊C4—531no—
126C4⋊C4—531no—
127C4⋊C4—531no—
128C4⋊C4—531no—
129C4⋊C4—1061no—
130C4⋊C4—1061no—
131C4⋊C4—1061no—
132C4⋊C4—1061no—
133C4⋊C4—1061no—
134C4⋊C4—1061no—
135C4⋊C4—1061no—
136C4⋊C4—1061no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—1061no—
140C22⋊C4—1061no—
141C22⋊C4—1061no—
142C42—1061no—
143C42C22×D5311yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—4241no—
147Q8—2122no—
148D4—41no—
149C2×C4—21no—
150C2×C4—1061no—
151C2×C4—1061no—
152C2×C4—1061no—
153C2×C4—1061no—
154C2×C4—4241no—
155C2×C4—2121no—
156C2×C4C2×D21211yescharacteristic
157C2×C4D4×D5311yescharacteristic
158C2×C4D4×D5311yescharacteristic
159C8—22no—
160C8—41no—
161C8—2121no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#1313318711yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—4241no—
171C4—2121no—
172C4—2121no—
173C4—2121no—
174C4—2121no—
175C2—81no—
176C2#1681079711yescharacteristic
177C2#1681080311yescharacteristic
178C2#1681080711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#329C2
socleC22×C53#120
hypercenter#329D53
hypocenterC53#1664
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #16811032C2semidirect
3. #16810992C2semidirect
5. #16810364C2semidirect
6. #16810364C2semidirect
97. C53#1664semidirect