Finite Group Databasebeta

← #20290235

Subgroups — #20290235

Order 6784.

252 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 60 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
2D53⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
3#16811483C212yesmaximal, complemented
4#16811481C212yesmaximal, complemented
5#16811172C211yesmaximal, characteristic
6C2×D53⋊C8—21no—
7#13133284C2212yes—
8#13133284C2212yes—
9C4⋊#88324—21no—
10#13133274—21no—
11#13133272—21no—
12C2×#11719949C411yescharacteristic
13C2×#11719947C411yescharacteristic
14#13133215C2211yescharacteristic
15#13133213C2211yescharacteristic
16#13133205C411yescharacteristic
17#13133190—21no—
18#13133190—21no—
19#13133177—21no—
20#13133177—21no—
21C42×D53C2211yescharacteristic
22#124×C53C411yescharacteristic
23C2×Q8×C53C2×C411yescharacteristic
24C2×D4×C53C2×C411yescharacteristic
25C2×C22⋊D53—21no—
26C2×C22⋊D53—21no—
27C2×C22⋊D53—21no—
28C22×Dic53—21no—
29#11719949—42no—
30#11719947—42no—
31C2×D212—21no—
32C2×D212C2×C411yescharacteristic
33C2×C4×D53—21no—
34C2×C4×D53C2×C411yescharacteristic
35C2×C4×D53D411yescharacteristic
36C2×C4×D53D411yescharacteristic
37C2×Dic106C2×C411yescharacteristic
38C2×#88324—22no—
39C2×#88324—22no—
40D53⋊C8—42no—
41C22⋊C4×C53—21no—
42C22⋊C4×C53—21no—
43C42×C53C2×C411yescharacteristic
44#11719921—21no—
45#11719921—21no—
46#11719919—21no—
47#11719919—21no—
48C4×Dic53D411yescharacteristic
49C4×Dic53D411yescharacteristic
50C4×Dic53C2311yescharacteristic
51C23×C53—21no—
52C22×D53—21no—
53C22×D53C22⋊C411yescharacteristic
54Q8×C53—41no—
55D4×C53—41no—
56C2×C4×C53—21no—
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22⋊C411yescharacteristic
59C2×C4×C53C22×C411yescharacteristic
60C22⋊D53—42no—
61C22⋊D53—42no—
62C22⋊D53—41no—
63C22⋊D53—41no—
64C2×Dic53—21no—
65C2×Dic53—42no—
66C2×Dic53C22⋊C411yescharacteristic
67C2×Dic53C2×D411yescharacteristic
68C2×Dic53C2×D411yescharacteristic
69D212—41no—
70D212—42no—
71C4×D53—22no—
72C4×D53—22no—
73C4×D53—21no—
74C4×D53—21no—
75C4×D53—42no—
76Dic106—41no—
77#88324—42no—
78#88324—42no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53—41no—
82C22×C53C2×C22⋊C411yescharacteristic
83C2×D53—21no—
84C2×D53—21no—
85C2×D53—41no—
86C2×D53—41no—
87C2×D53—41no—
88C2×D53#10012yes—
89C4×C53—21no—
90C4×C53—21no—
91C4×C53—21no—
92C4×C53—41no—
93Dic53—21no—
94Dic53—41no—
95Dic53#13512yes—
96Dic53AGL(1,ℤ/8)11yescharacteristic
97Dic53#13711yescharacteristic
98#1551—531nomaximal, Sylow
99C2×C53—81no—
100C2×C53C2×#10011yescharacteristic
101C2×C53#41611yescharacteristic
102C2×C53#41711yescharacteristic
103D53—22no—
104D53—81no—
105#532—531no—
106#432—531no—
107C2×C4⋊C8—531no—
108#333—532no—
109#331—532no—
110C53#155111yescharacteristic, complemented, Sylow
111C2×#41—531no—
112C2×#41—531no—
113C22×C8—1061no—
114C4⋊Q8—531no—
115#127—531no—
116#124—531no—
117#124#615711yescharacteristic
118#121—1061no—
119#121—1061no—
120C4×D4—1061no—
121C4×D4—1061no—
122C2×C42—531no—
123C4⋊C8—532no—
124C4⋊C8—532no—
125C4⋊C8—1061no—
126C22⋊C8—1061no—
127#97—1061no—
128#41—2122no—
129#41—2122no—
130C2×Q8—531no—
131C2×Q8C2×#615711yescharacteristic
132C2×D4—531no—
133C2×D4—1061no—
134C2×D4—1061no—
135C2×D4—1061no—
136C2×D4—1061no—
137C2×D4C2×#615711yescharacteristic
138C22×C4—531no—
139C22×C4—531no—
140C22×C4—531no—
141C22×C4—1061no—
142C22×C4—1061no—
143C2×C8—1062no—
144C2×C8—1062no—
145C2×C8—2122no—
146C4⋊C4—1061no—
147C4⋊C4—1061no—
148C22⋊C4—21no—
149C22⋊C4—21no—
150C22⋊C4—1061no—
151C22⋊C4—1061no—
152C42—531no—
153C42—531no—
154C42—531no—
155C42C2×#615711yescharacteristic
156C23—21no—
157C23—531no—
158C23—1061no—
159Q8—41no—
160Q8—2121no—
161D4—41no—
162D4—2121no—
163D4—2122no—
164D4—2122no—
165D4—2121no—
166D4—2121no—
167D4—2122no—
168C2×C4—21no—
169C2×C4—531no—
170C2×C4—531no—
171C2×C4—531no—
172C2×C4—1061no—
173C2×C4—1062no—
174C2×C4—1061no—
175C2×C4—1062no—
176C2×C4—1061no—
177C2×C4—2122no—
178C2×C4—2122no—
179C2×C4C22⋊#615711yescharacteristic
180C2×C4C22⋊#615711yescharacteristic
181C2×C4C22×#615711yescharacteristic
182C8—2122no—
183C8—2122no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—532no—
188C22—1061no—
189C22—1061no—
190C22—2121no—
191C22—2121no—
192C22—2121no—
193C22C2×C22⋊#615711yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—41no—
198C4—532no—
199C4—531no—
200C4—531no—
201C4—1061no—
202C4—2121no—
203C2—81no—
204C2—1062no—
205C2—4241no—
206C2D53⋊#10011yescharacteristic
207C2#1681161311yescharacteristic
208C2#1681161411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#124×C53C4
socleC22×C53C2×C22⋊C4
hypercenter#124#6157
hypocenterC53#1551
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #16811646C2semidirect
2. D53⋊(C4⋊C8)C2semidirect
3. #16811483C2semidirect
4. #16811481C2semidirect
110. C53#1551semidirect