Finite Group Databasebeta

← #20273345

Subgroups — #20273345

Order 6784.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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#16811158C211yesmaximal, characteristic
2#16810678C211yesmaximal, characteristic, complemented
3#16810678C211yesmaximal, characteristic, complemented
4#16810365C211yesmaximal, characteristic, complemented
5#16810362C211yesmaximal, characteristic, complemented
6#16810362C211yesmaximal, characteristic, complemented
7C53×#330C211yesmaximal, characteristic, complemented
8#13133212—21no—
9#13133205C2211yescharacteristic
10#13133202—21no—
11#13133197C411yescharacteristic
12#13133192C411yescharacteristic
13#13133182C411yescharacteristic
14#13133179—21no—
15#13133179C411yescharacteristic
16#13133175—21no—
17C42×D53C2211yescharacteristic
18#13133146—21no—
19#13133146—21no—
20#13133129—21no—
21#13133129—21no—
22#13133128C2211yescharacteristic
23#13133128C2211yescharacteristic
24#125×C53C2211yescharacteristic
25#97×C53C2211yescharacteristic
26#97×C53C2211yescharacteristic
27C2×D212—21no—
28C2×D212C2×C411yescharacteristic
29C2×C4×D53—21no—
30C2×C4×D53C2×C411yescharacteristic
31C2×C4×D53D411yescharacteristic
32C2×C4×D53D411yescharacteristic
33C2×Dic106C2×C411yescharacteristic
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53C2×C411yescharacteristic
39C4⋊C4×C53C2×C411yescharacteristic
40C42×C53C2311yescharacteristic
41#11719921—21no—
42#11719921—21no—
43#11719921—21no—
44#11719921—21no—
45#11719921—21no—
46C4⋊Dic53—21no—
47#11719919—21no—
48C4×Dic53—21no—
49C4×Dic53C2×C411yescharacteristic
50C4×Dic53D411yescharacteristic
51C4×Dic53D411yescharacteristic
52C2×#88322—21no—
53C2×#88322—21no—
54C22×D53—21no—
55C22×D53C22⋊C411yescharacteristic
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53C22×C411yescharacteristic
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C2×Dic53—21no—
62C2×Dic53C22⋊C411yescharacteristic
63C2×Dic53C22⋊C411yescharacteristic
64C2×Dic53C22⋊C411yescharacteristic
65D212—41no—
66D212—42no—
67C4×D53—41no—
68C4×D53—41no—
69C4×D53—41no—
70C4×D53—42no—
71Dic106—41no—
72C8×C53—41no—
73C8×C53—41no—
74#88322—41no—
75#88322—41no—
76C22×C53C2×C22⋊C411yescharacteristic
77C2×D53—21no—
78C2×D53—21no—
79C2×D53—21no—
80C2×D53—41no—
81C2×D53—41no—
82C2×D53—41no—
83C4×C53—21no—
84C4×C53—21no—
85C4×C53—21no—
86C4×C53—41no—
87C4×C53—41no—
88Dic53—21no—
89Dic53—21no—
90Dic53—21no—
91Dic53—41no—
92#1535—531nomaximal, Sylow
93C2×C53#41311yescharacteristic
94C2×C53#42111yescharacteristic
95C2×C53#42111yescharacteristic
96D53—41no—
97D53—81no—
98#529—531no—
99#432—531no—
100#432—531no—
101#330—531no—
102#330D5311yescharacteristic, complemented
103#329—531no—
104#329—531no—
105C53#153511yescharacteristic, complemented, Sylow
106#126—1061no—
107#125C2×D5311yescharacteristic
108#124—531no—
109#123—1061no—
110C22⋊Q8—531no—
111#121—531no—
112C4×Q8—531no—
113C4×D4—531no—
114C4×D4—1061no—
115#117—1061no—
116C2×C42—531no—
117C4⋊C8—1061no—
118C4⋊C8—1061no—
119C22⋊C8—1061no—
120C22⋊C8—1061no—
121#97—531no—
122#97—531no—
123#97C2×D5311yescharacteristic
124#97C2×D5311yescharacteristic
125C2×Q8—531no—
126C2×D4—531no—
127C2×D4—1061no—
128C22×C4—531no—
129C22×C4—531no—
130C22×C4—531no—
131C22×C4—1061no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—1061no—
135C2×C8—1061no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—1061no—
139C4⋊C4—1061no—
140C4⋊C4C4×D5311yescharacteristic
141C4⋊C4C4×D5311yescharacteristic
142C22⋊C4—1061no—
143C22⋊C4—1061no—
144C22⋊C4—1061no—
145C22⋊C4—1061no—
146C22⋊C4—1061no—
147C42—531no—
148C42—531no—
149C42—531no—
150C42—1061no—
151C42C22×D5311yescharacteristic
152C23—531no—
153C23—1061no—
154Q8—2121no—
155D4—2121no—
156D4—2122no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—531no—
160C2×C4—531no—
161C2×C4—531no—
162C2×C4—1061no—
163C2×C4—2121no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4—2122no—
167C2×C4C2×C4×D5311yescharacteristic
168C2×C4D4×D5311yescharacteristic
169C2×C4D4×D5311yescharacteristic
170C8—41no—
171C8—41no—
172C8—2121no—
173C8—2121no—
174C22—1061no—
175C22—1061no—
176C22—1061no—
177C22—2121no—
178C22—2121no—
179C22—2121no—
180C22C22⋊C4×D5311yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—41no—
185C4—41no—
186C4—1061no—
187C4—1061no—
188C4—1061no—
189C4—2121no—
190C2—4241no—
191C2—2121no—
192C2#1681060611yescharacteristic
193C2#1681063811yescharacteristic
194C2#1681063811yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC2×C4×C53C22×C4
FrattiniC42C22×D53
FittingC53×#330C2
socleC22×C53C2×C22⋊C4
hypercenter#330D53
hypocenterC53#1535
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #16810678C2semidirect
3. #16810678C2semidirect
4. #16810365C2semidirect
5. #16810362C2semidirect
6. #16810362C2semidirect
7. C53×#330C2semidirect
102. #330D53semidirect
105. C53#1535semidirect