Finite Group Databasebeta

← #20273867

Subgroups — #20273867

Order 6784.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811276C211yesmaximal, characteristic
2#16810975C211yesmaximal, characteristic, complemented
3#16810914C211yesmaximal, characteristic, complemented
4#16810914C211yesmaximal, characteristic, complemented
5#16810350C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#332C211yesmaximal, characteristic, complemented
8C2×#11719949—21no—
9#13133217C2211yescharacteristic
10#13133202—22no—
11#13133193—21no—
12#13133192—21no—
13C4×D212C2211yescharacteristic
14C4×D212C2211yescharacteristic
15C4⋊C4×D53—21no—
16#13133149—21no—
17#13133149—21no—
18#13133140—22no—
19#13133139—21no—
20#13133139—21no—
21C4×#88322C2211yescharacteristic
22C4⋊Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4⋊C8×C53C2211yescharacteristic
25C2×Q8×C53—21no—
26#11719949—42no—
27#11719949—42no—
28C2×D212—22no—
29C2×D212D411yescharacteristic
30C2×D212D411yescharacteristic
31C2×C4×D53—22no—
32C2×C4×D53—21no—
33C2×C4×D53—21no—
34C2×Dic106—21no—
35C2×C8×C53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53D412yes—
39C42×C53C2311yescharacteristic
40#11719921—22no—
41#11719921—21no—
42#11719921—21no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53D411yescharacteristic
46C4⋊Dic53D411yescharacteristic
47#11719919—22no—
48C2×#88322—21no—
49C22×D53—21no—
50C22×D53—21no—
51Q8×C53—41no—
52Q8×C53—41no—
53C2×C4×C53—22no—
54C2×C4×C53C2×D411yescharacteristic
55C2×C4×C53C2×D411yescharacteristic
56C2×C4×C53C2×D411yescharacteristic
57C2×Dic53—21no—
58C2×Dic53—21no—
59C2×Dic53—41no—
60D212—21no—
61D212—21no—
62D212—21no—
63D212—44no—
64D212SD1612yes—
65C4×D53—44no—
66C4×D53—41no—
67C4×D53—41no—
68C4×D53—41no—
69C4×D53—41no—
70Dic106—42no—
71C8×C53—41no—
72C8×C53—41no—
73#88322—41no—
74C22×C53#12011yescharacteristic
75C2×D53—22no—
76C2×D53—41no—
77C2×D53—41no—
78C2×D53—41no—
79C2×D53—41no—
80C2×D53—41no—
81C4×C53—21no—
82C4×C53—42no—
83C4×C53C2×SD1611yescharacteristic
84C4×C53#13511yescharacteristic
85C4×C53AGL(1,ℤ/8)11yescharacteristic
86C4×C53#13711yescharacteristic
87Dic53—41no—
88Dic53—41no—
89Dic53—81no—
90#1635—531nomaximal, Sylow
91C2×C53#45011yescharacteristic
92C2×C53#45211yescharacteristic
93C2×C53#45411yescharacteristic
94D53—42no—
95D53—81no—
96#547—531no—
97#487—531no—
98#476—531no—
99#476—531no—
100#332D5311yescharacteristic, complemented
101D4⋊C8—531no—
102D4⋊C8—531no—
103C53#163511yescharacteristic, complemented, Sylow
104C2×#41—1061no—
105C4⋊Q8—531no—
106C4⋊Q8C2×D5311yescharacteristic
107#123—1062no—
108C22⋊Q8—1061no—
109#121—1061no—
110C4×D4—531no—
111C4×D4—531no—
112C2×C4⋊C4—1061no—
113#106—1061no—
114#106—1061no—
115C4⋊C8C2×D5311yescharacteristic
116C4⋊C8C2×D5311yescharacteristic
117#103—1062no—
118#102—1061no—
119#102—1061no—
120C4×C8—531no—
121#41—2122no—
122#41—2122no—
123C2×Q8—21no—
124C2×Q8—1061no—
125C2×D4—531no—
126C2×D4—531no—
127C2×D4—1062no—
128C22×C4—1062no—
129C22×C4—1061no—
130C22×C4—1061no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—1061no—
134C4⋊C4—21no—
135C4⋊C4—531no—
136C4⋊C4—531no—
137C4⋊C4—1062no—
138C4⋊C4—1061no—
139C4⋊C4—1061no—
140C4⋊C4D21212yes—
141C22⋊C4—1062no—
142C22⋊C4—1061no—
143C22⋊C4—1061no—
144C42C22×D5311yescharacteristic
145C23—1061no—
146C23—1061no—
147Q8—41no—
148Q8—41no—
149Q8—2122no—
150D4—532no—
151D4—1061no—
152D4—1061no—
153D4—1061no—
154D4—2124no—
155C2×C4—22no—
156C2×C4—1061no—
157C2×C4—1061no—
158C2×C4—2124no—
159C2×C4—2121no—
160C2×C4—2121no—
161C2×C4—2121no—
162C2×C4—2121no—
163C2×C4—2121no—
164C2×C4C2×D21211yescharacteristic
165C2×C4D4×D5311yescharacteristic
166C2×C4D4×D5311yescharacteristic
167C8—41no—
168C8—41no—
169C8—2121no—
170C22—1062no—
171C22—2121no—
172C22—2121no—
173C22—2121no—
174C22—2121no—
175C22—2121no—
176C22#1313318711yescharacteristic
177C4—21no—
178C4—42no—
179C4—4241no—
180C4—2121no—
181C4—2121no—
182C4SD16×D5311yescharacteristic
183C4#1313323811yescharacteristic
184C4#1313324111yescharacteristic
185C4#1313324911yescharacteristic
186C2—4241no—
187C2—2122no—
188C2#1681077511yescharacteristic
189C2#1681078811yescharacteristic
190C2#1681079811yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#332C2
socleC22×C53#120
hypercenter#332D53
hypocenterC53#1635
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #16810975C2semidirect
3. #16810914C2semidirect
4. #16810914C2semidirect
5. #16810350C2semidirect
6. #16810350C2semidirect
7. C53×#332C2semidirect
100. #332D53semidirect
103. C53#1635semidirect