Finite Group Databasebeta

← #20274032

Subgroups — #20274032

Order 6784.

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

normal subgroups only

This group has 208 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#16811018C211yesmaximal, characteristic, complemented
3#16810982C211yesmaximal, characteristic, complemented
4#16810365C211yesmaximal, characteristic, complemented
5#16810363C211yesmaximal, characteristic, complemented
6#16810363C211yesmaximal, characteristic, complemented
7C53×#562C211yesmaximal, characteristic
8C2×#11719925—21no—
9C2×#11719923—21no—
10#13133217C2211yescharacteristic
11#13133150—21no—
12#13133148—21no—
13#13133141—21no—
14#13133138—21no—
15#13133128C2211yescharacteristic
16#13133128C2211yescharacteristic
17#13133128C2211yescharacteristic
18#125×C53C2211yescharacteristic
19#124×C53C2211yescharacteristic
20#124×C53C2211yescharacteristic
21#123×C53—21no—
22#123×C53—21no—
23C22⋊Q8×C53—21no—
24C22⋊Q8×C53—21no—
25#121×C53—21no—
26#121×C53—21no—
27C2×Q8×C53D411yescharacteristic
28C2×Q8×C53D411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53D411yescharacteristic
32C2×D4×C53D411yescharacteristic
33C22×C4×C53—21no—
34C22×C4×C53—21no—
35C2×Dic106—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—21no—
39C4⋊C4×C53—21no—
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—
47C22⋊C4×C53—21no—
48C42×C53C2311yescharacteristic
49#11719925—42no—
50#11719923—42no—
51C4⋊Dic53—21no—
52C4⋊Dic53—21no—
53C2×#88322—21no—
54C2×#88322—21no—
55C2×#88322—21no—
56C23×C53—21no—
57C23×C53—21no—
58Q8×C53—41no—
59Q8×C53—41no—
60D4×C53—41no—
61D4×C53—41no—
62D4×C53—81no—
63D4×C53—81no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—81no—
69C2×C4×C53—81no—
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×Dic53—41no—
74Dic106—41no—
75Dic106—41no—
76#88322—22no—
77#88322—22no—
78#88322—41no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53—41no—
82C22×C53—41no—
83C22×C53—41no—
84C22×C53—41no—
85C22×C53#12011yescharacteristic
86C4×C53—21no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—41no—
90C4×C53—41no—
91C4×C53—41no—
92C4×C53—41no—
93Dic53—81no—
94#1661—531nomaximal, Sylow
95C2×C53—81no—
96C2×C53—81no—
97C2×C53#45311yescharacteristic
98C2×C53#45511yescharacteristic
99C2×C53#45611yescharacteristic
100#562D5311yescharacteristic
101#501—531no—
102#497—531no—
103#489—531no—
104#330—531no—
105#329—531no—
106#329—531no—
107C53#166111yescharacteristic, complemented, Sylow
108C2×Q16—1061no—
109C2×SD16—1061no—
110C4⋊Q8—531no—
111#125C2×D5311yescharacteristic
112#124C2×D5311yescharacteristic
113#124C2×D5311yescharacteristic
114#123—21no—
115#123—21no—
116C22⋊Q8—21no—
117C22⋊Q8—21no—
118#121—21no—
119#121—21no—
120#107—1061no—
121#106—1061no—
122#103—1061no—
123#102—1061no—
124#97—531no—
125#97—531no—
126#97—531no—
127C2×Q8—1061no—
128C2×Q8C22⋊D5311yescharacteristic
129C2×Q8C22⋊D5311yescharacteristic
130C2×D4—21no—
131C2×D4—21no—
132C2×D4C22⋊D5311yescharacteristic
133C2×D4C22⋊D5311yescharacteristic
134C22×C4—21no—
135C22×C4—21no—
136Q16—2122no—
137SD16—2122no—
138C2×C8—1061no—
139C2×C8—1061no—
140C2×C8—1061no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—1061no—
146C4⋊C4—1061no—
147C4⋊C4C22⋊D5311yescharacteristic
148C4⋊C4C22⋊D5311yescharacteristic
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—21no—
152C22⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C42C22×D5311yescharacteristic
156C23—21no—
157C23—21no—
158Q8—41no—
159Q8—41no—
160Q8—2121no—
161Q8—2121no—
162D4—41no—
163D4—41no—
164D4—81no—
165D4—81no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—2121no—
173C2×C4C2×C22⋊D5311yescharacteristic
174C2×C4C2×C22⋊D5311yescharacteristic
175C2×C4C2×C22⋊D5311yescharacteristic
176C8—1062no—
177C8—1062no—
178C8—2121no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22#1313318611yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—41no—
191C4—41no—
192C4—41no—
193C4—4241no—
194C2—81no—
195C2—81no—
196C2#1681079111yescharacteristic
197C2#1681080111yescharacteristic
198C2#1681080611yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#562C2
socleC22×C53#120
hypercenter#562D53
hypocenterC53#1661
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #16811033C2semidirect
2. #16811018C2semidirect
3. #16810982C2semidirect
4. #16810365C2semidirect
5. #16810363C2semidirect
6. #16810363C2semidirect
107. C53#1661semidirect