Finite Group Databasebeta

← #20290201

Subgroups — #20290201

Order 6784.

200 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 54 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#16811646C211yesmaximal, characteristic, complemented
2#16811645C211yesmaximal, characteristic, complemented
3#16811480C212yesmaximal, complemented
4#16811479C212yesmaximal, complemented
5#16811160C211yesmaximal, characteristic
6#13133284—21no—
7C4⋊#88324—21no—
8#13133274—21no—
9#13133274—21no—
10#13133272C2212yes—
11#13133272C2212yes—
12#13133212—21no—
13#13133208C2211yescharacteristic
14#13133206C2211yescharacteristic
15#13133202—21no—
16#13133193C411yescharacteristic
17#13133192C411yescharacteristic
18C4×D212—21no—
19C4×D212C411yescharacteristic
20#13133174—21no—
21C42×D53C2211yescharacteristic
22C4×Q8×C53C411yescharacteristic
23C2×Q8×C53C2×C411yescharacteristic
24C2×D212—21no—
25C2×D212C2×C411yescharacteristic
26C2×C4×D53—21no—
27C2×C4×D53C2×C411yescharacteristic
28C2×C4×D53D411yescharacteristic
29C2×C4×D53D411yescharacteristic
30C2×#88324—22no—
31C2×#88324—22no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53C2×C411yescharacteristic
34C42×C53—21no—
35C42×C53C2×C411yescharacteristic
36#11719921—21no—
37#11719921—21no—
38#11719921—21no—
39#11719921—21no—
40#11719921—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53C2×C411yescharacteristic
43#11719919—21no—
44#11719919—21no—
45C4×Dic53D411yescharacteristic
46C4×Dic53D411yescharacteristic
47C4×Dic53C2311yescharacteristic
48C22×D53—21no—
49C22×D53C22⋊C411yescharacteristic
50Q8×C53—41no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53C22⋊C411yescharacteristic
54C2×C4×C53C22⋊C411yescharacteristic
55C2×C4×C53C22×C411yescharacteristic
56C2×Dic53—21no—
57C2×Dic53C22⋊C411yescharacteristic
58C2×Dic53C2×D411yescharacteristic
59C2×Dic53C2×D411yescharacteristic
60D212—41no—
61D212—42no—
62C4×D53—41no—
63C4×D53—41no—
64C4×D53—41no—
65C4×D53—42no—
66#88324—42no—
67#88324—42no—
68C22×C53C2×C22⋊C411yescharacteristic
69C2×D53—21no—
70C2×D53—21no—
71C2×D53—21no—
72C2×D53—41no—
73C2×D53—41no—
74C2×D53—41no—
75C4×C53—21no—
76C4×C53—21no—
77C4×C53—21no—
78C4×C53—41no—
79C4×C53—41no—
80Dic53—21no—
81Dic53—21no—
82Dic53—21no—
83Dic53—41no—
84#1535—531nomaximal, Sylow
85C2×C53#41311yescharacteristic
86C2×C53#42111yescharacteristic
87C2×C53#42111yescharacteristic
88D53—41no—
89D53—81no—
90#529—531no—
91#432—531no—
92#432—531no—
93#330—532no—
94#329—532no—
95C53#153511yescharacteristic, complemented, Sylow
96#126—1061no—
97#125—531no—
98#124—531no—
99#123—1061no—
100C22⋊Q8—531no—
101#121—531no—
102C4×Q8#615711yescharacteristic
103C4×D4—531no—
104C4×D4—1061no—
105#117—1061no—
106C2×C42—531no—
107C4⋊C8—1061no—
108C4⋊C8—1061no—
109C22⋊C8—1061no—
110C22⋊C8—1061no—
111#97—532no—
112#97—532no—
113C2×Q8C2×#615711yescharacteristic
114C2×D4—531no—
115C2×D4—1061no—
116C22×C4—531no—
117C22×C4—531no—
118C22×C4—531no—
119C22×C4—1061no—
120C2×C8—1062no—
121C2×C8—1062no—
122C4⋊C4—21no—
123C4⋊C4—531no—
124C4⋊C4—1061no—
125C4⋊C4—1061no—
126C4⋊C4—1061no—
127C4⋊C4C2×#615711yescharacteristic
128C22⋊C4—1061no—
129C22⋊C4—1061no—
130C22⋊C4—1061no—
131C22⋊C4—1061no—
132C22⋊C4—1061no—
133C42—21no—
134C42—531no—
135C42—531no—
136C42—531no—
137C42C2×#615711yescharacteristic
138C23—531no—
139C23—1061no—
140Q8—41no—
141D4—2121no—
142D4—2122no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—531no—
146C2×C4—531no—
147C2×C4—531no—
148C2×C4—1061no—
149C2×C4—2121no—
150C2×C4—2121no—
151C2×C4—2121no—
152C2×C4—2122no—
153C2×C4C22⋊#615711yescharacteristic
154C2×C4C22⋊#615711yescharacteristic
155C2×C4C22×#615711yescharacteristic
156C8—2122no—
157C8—2122no—
158C22—1061no—
159C22—1061no—
160C22—1061no—
161C22—2121no—
162C22—2121no—
163C22—2121no—
164C22C2×C22⋊#615711yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—41no—
169C4—41no—
170C4—1061no—
171C4—1061no—
172C4—1061no—
173C4—2121no—
174C2—4241no—
175C2—2121no—
176C2#1681160911yescharacteristic
177C2#1681162011yescharacteristic
178C2#1681162211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4×Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC4×Q8#6157
hypocenterC53#1535
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. #16811646C2semidirect
2. #16811645C2semidirect
3. #16811480C2semidirect
4. #16811479C2semidirect
95. C53#1535semidirect