Finite Group Databasebeta

← #20290202

Subgroups — #20290202

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#16811644C211yesmaximal, characteristic, complemented
2#16811643C211yesmaximal, characteristic, complemented
3#16811480C212yesmaximal, complemented
4#16811479C212yesmaximal, complemented
5#16811159C211yesmaximal, characteristic
6#13133284—21no—
7C4⋊#88324—21no—
8C22⋊#88324—21no—
9C22⋊#88324—21no—
10#13133272C2212yes—
11#13133272C2212yes—
12#13133212—21no—
13#13133208C2211yescharacteristic
14#13133206C2211yescharacteristic
15#13133201—21no—
16#13133191C411yescharacteristic
17Q8×Dic53C411yescharacteristic
18#13133179C411yescharacteristic
19#13133177—21no—
20#13133175—21no—
21C2×C4×Dic53C2211yescharacteristic
22C22⋊Q8×C53C411yescharacteristic
23C2×Q8×C53C2×C411yescharacteristic
24C22×C4×C53C2×C411yescharacteristic
25C2×C22⋊D53—21no—
26C22×Dic53D411yescharacteristic
27C22×Dic53D411yescharacteristic
28C2×D212C2×C411yescharacteristic
29C2×C4×D53—21no—
30C2×#88324—22no—
31C2×#88324—22no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53C2×C411yescharacteristic
34C22⋊C4×C53—21no—
35#11719921—21no—
36#11719921—21no—
37#11719921—21no—
38#11719921—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53C2×C411yescharacteristic
42#11719919—21no—
43C4×Dic53—21no—
44C4×Dic53C2×C411yescharacteristic
45C4×Dic53D411yescharacteristic
46C4×Dic53D411yescharacteristic
47C4×Dic53C2311yescharacteristic
48C23×C53C22⋊C411yescharacteristic
49C22×D53—21no—
50Q8×C53—41no—
51C2×C4×C53—21no—
52C2×C4×C53—21no—
53C2×C4×C53—41no—
54C2×C4×C53C22⋊C411yescharacteristic
55C2×C4×C53C22×C411yescharacteristic
56C22⋊D53—42no—
57C2×Dic53—21no—
58C2×Dic53—41no—
59C2×Dic53—41no—
60C2×Dic53C22⋊C411yescharacteristic
61C2×Dic53C22⋊C411yescharacteristic
62C2×Dic53C2×D411yescharacteristic
63C2×Dic53C2×D411yescharacteristic
64D212—41no—
65C4×D53—42no—
66#88324—42no—
67#88324—42no—
68C22×C53—21no—
69C22×C53—21no—
70C22×C53—21no—
71C22×C53C2×C22⋊C411yescharacteristic
72C2×D53—41no—
73C2×D53—41no—
74C2×D53—41no—
75C4×C53—21no—
76C4×C53—21no—
77C4×C53—41no—
78C4×C53—41no—
79Dic53—21no—
80Dic53—21no—
81Dic53—21no—
82Dic53—21no—
83Dic53—41no—
84#1535—531nomaximal, Sylow
85C2×C53—41no—
86C2×C53#41311yescharacteristic
87C2×C53#42111yescharacteristic
88C2×C53#42111yescharacteristic
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#615711yescharacteristic
101#121—531no—
102C4×Q8—531no—
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—1061no—
119C22×C4C2×#615711yescharacteristic
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—21no—
129C22⋊C4—1061no—
130C22⋊C4—1061no—
131C22⋊C4—1061no—
132C22⋊C4—1061no—
133C42—531no—
134C42—531no—
135C42—531no—
136C42—531no—
137C42—1061no—
138C23—1061no—
139C23C22⋊#615711yescharacteristic
140Q8—41no—
141D4—2121no—
142D4—2122no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—531no—
147C2×C4—531no—
148C2×C4—531no—
149C2×C4—531no—
150C2×C4—1061no—
151C2×C4—2122no—
152C2×C4—2121no—
153C2×C4—2121no—
154C2×C4C22⋊#615711yescharacteristic
155C2×C4C22×#615711yescharacteristic
156C8—2122no—
157C8—2122no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—2121no—
162C22—2121no—
163C22—2121no—
164C22C2×C22⋊#615711yescharacteristic
165C4—21no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—1061no—
170C4—1061no—
171C4—1061no—
172C4—1061no—
173C4—2121no—
174C2—41no—
175C2—4241no—
176C2#1681161011yescharacteristic
177C2#1681162011yescharacteristic
178C2#1681162211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1535
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #16811644C2semidirect
2. #16811643C2semidirect
3. #16811480C2semidirect
4. #16811479C2semidirect
95. C53#1535semidirect