Finite Group Databasebeta

← #20274033

Subgroups — #20274033

Order 6784.

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

normal subgroups only

This group has 208 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#16811367C211yesmaximal, characteristic
2#16811032C211yesmaximal, characteristic, complemented
3#16811018C211yesmaximal, characteristic, complemented
4#16810983C211yesmaximal, characteristic, complemented
5#16810364C211yesmaximal, characteristic, complemented
6#16810362C211yesmaximal, characteristic, complemented
7C53×#329C211yesmaximal, characteristic, complemented
8C2×#11719925—21no—
9C2×#11719923—21no—
10#13133217C2211yescharacteristic
11#13133209C2211yescharacteristic
12#13133205C2211yescharacteristic
13#13133201—21no—
14#13133199—21no—
15#13133194—21no—
16#13133193—21no—
17#13133190—21no—
18#13133189—21no—
19#13133151—21no—
20#13133149—21no—
21#13133142—21no—
22#13133139—21no—
23#13133128C2211yescharacteristic
24#124×C53C2211yescharacteristic
25#97×C53C2211yescharacteristic
26#97×C53C2211yescharacteristic
27C2×Q8×C53D411yescharacteristic
28C2×D4×C53D411yescharacteristic
29C2×C22⋊D53—21no—
30C2×C22⋊D53—21no—
31C22×Dic53—21no—
32C2×D212D411yescharacteristic
33C2×C4×D53—21no—
34C2×Dic106—21no—
35C2×Dic106D411yescharacteristic
36C2×C8×C53—21no—
37C2×C8×C53—21no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53—21no—
40C42×C53C2311yescharacteristic
41C22⋊Dic53—21no—
42#11719925—42no—
43#11719923—42no—
44#11719921—21no—
45#11719921—21no—
46#11719921—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53—21no—
49C4⋊Dic53—21no—
50C4⋊Dic53D411yescharacteristic
51C4⋊Dic53D411yescharacteristic
52#11719919—21no—
53#11719919—21no—
54#11719919—21no—
55C2×#88322—21no—
56C23×C53—21no—
57C22×D53—21no—
58Q8×C53—41no—
59D4×C53—41no—
60C2×C4×C53—21no—
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C22⋊D53—81no—
65C22⋊D53—81no—
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—41no—
70C2×Dic53—81no—
71D212—41no—
72C4×D53—81no—
73Dic106—41no—
74Dic106—41no—
75Dic106—41no—
76C8×C53—22no—
77C8×C53—41no—
78#88322—22no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53—41no—
82C22×C53#12011yescharacteristic
83C2×D53—41no—
84C2×D53—41no—
85C2×D53—41no—
86C4×C53—21no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—41no—
90Dic53—41no—
91Dic53—41no—
92Dic53—41no—
93Dic53—81no—
94#1661—531nomaximal, Sylow
95C2×C53—81no—
96C2×C53#45311yescharacteristic
97C2×C53#45511yescharacteristic
98C2×C53#45611yescharacteristic
99D53—81no—
100#562—531no—
101#501—531no—
102#497—531no—
103#489—531no—
104#330—531no—
105#329—531no—
106#329D5311yescharacteristic, complemented
107C53#166111yescharacteristic, complemented, Sylow
108C2×Q16—1061no—
109C2×SD16—1061no—
110C4⋊Q8—531no—
111#125—531no—
112#124—531no—
113#124C2×D5311yescharacteristic
114#123—1061no—
115#123—1061no—
116C22⋊Q8—1061no—
117C22⋊Q8—1061no—
118#121—1061no—
119#121—1061no—
120#107—1061no—
121#106—1061no—
122#103—1061no—
123#102—1061no—
124#97—531no—
125#97C2×D5311yescharacteristic
126#97C2×D5311yescharacteristic
127C2×Q8—531no—
128C2×Q8—1061no—
129C2×Q8D21211yescharacteristic
130C2×D4—531no—
131C2×D4—1061no—
132C2×D4—1061no—
133C2×D4D21211yescharacteristic
134C22×C4—1061no—
135C22×C4—1061no—
136Q16—2122no—
137SD16—2122no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—1061no—
141C4⋊C4—531no—
142C4⋊C4—531no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4—1061no—
146C4⋊C4—1061no—
147C4⋊C4—1061no—
148C4⋊C4—1061no—
149C22⋊C4—21no—
150C22⋊C4—21no—
151C22⋊C4—1061no—
152C22⋊C4—1061no—
153C22⋊C4—1061no—
154C22⋊C4—1061no—
155C42C22×D5311yescharacteristic
156C23—21no—
157C23—1061no—
158Q8—41no—
159Q8—2121no—
160Q8—2121no—
161Q8—2121no—
162D4—41no—
163D4—4241no—
164D4—4241no—
165D4—2121no—
166C2×C4—21no—
167C2×C4—1061no—
168C2×C4—1061no—
169C2×C4—1061no—
170C2×C4—4241no—
171C2×C4—4241no—
172C2×C4—2121no—
173C2×C4C2×D21211yescharacteristic
174C2×C4D4×D5311yescharacteristic
175C2×C4D4×D5311yescharacteristic
176C8—22no—
177C8—41no—
178C8—1062no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—2121no—
183C22—2121no—
184C22—2121no—
185C22#1313318711yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—41no—
190C4—4241no—
191C4—2121no—
192C4—2121no—
193C4—2121no—
194C2—81no—
195C2—4241no—
196C2#1681079311yescharacteristic
197C2#1681080211yescharacteristic
198C2#1681080711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#329C2
socleC22×C53#120
hypercenter#329D53
hypocenterC53#1661
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. #16811018C2semidirect
4. #16810983C2semidirect
5. #16810364C2semidirect
6. #16810362C2semidirect
7. C53×#329C2semidirect
106. #329D53semidirect
107. C53#1661semidirect