Finite Group Databasebeta

← #20290976

Subgroups — #20290976

Order 6784.

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

normal subgroups only

This group has 266 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#16811609C211yesmaximal, characteristic, complemented
2#16811602C211yesmaximal, characteristic, complemented
3#16811524C214yesmaximal, complemented
4#16811192C211yesmaximal, characteristic
5D53⋊M16—21no—
6#13133296—21no—
7#13133278—21no—
8#13133277C2212yes—
9#13133276—21no—
10#13133275C2212yes—
11C2×#11719945C2211yescharacteristic
12C22×D212C411yescharacteristic, complemented
13#13133214—21no—
14#13133205—21no—
15#13133192—21no—
16#13133192—21no—
17#13133187C412yescomplemented
18#13133187C2212yes—
19#117×C53C411yescharacteristic, complemented
20C23×D53C2×C412yes—
21C22×C4×C53C2×C411yescharacteristic
22C2×C22⋊D53C2×C411yescharacteristic
23C2×C22⋊D53C2311yescharacteristic
24#11719945—41no—
25#11719945—41no—
26C2×D212—21no—
27C2×D212—21no—
28C2×D212—22no—
29C2×D212—22no—
30C2×D212—41no—
31C2×D212—41no—
32C2×D212D411yescharacteristic
33C2×C4×D53D411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×Dic106D411yescharacteristic
36C22⋊#6157—22no—
37#11719940—22no—
38C4⋊#6157—41no—
39C4×#6157—41no—
40#11719936—41no—
41D53⋊C8—41no—
42C4⋊C4×C53—21no—
43C22⋊C4×C53C2×C412yes—
44C42×C53—21no—
45#11719921—22no—
46#11719921—22no—
47C23×C53C22×C411yescharacteristic
48C22×D53—22no—
49C22×D53—42no—
50C22×D53—82no—
51C22×D53C22⋊C411yescharacteristic
52C22×D53C2×D411yescharacteristic
53C2×#6157—42no—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53—22no—
57C2×C4×C53C22⋊C411yescharacteristic
58C2×C4×C53C22⋊C411yescharacteristic
59C22⋊D53—41no—
60C22⋊D53—41no—
61C2×Dic53C22⋊C411yescharacteristic
62C2×Dic53C2×D411yescharacteristic
63D212—41no—
64D212—42no—
65D212—41no—
66D212—41no—
67D212—82no—
68D212—82no—
69C4×D53—21no—
70C4×D53—21no—
71C4×D53—21no—
72C4×D53—21no—
73Dic106—41no—
74#88324—42no—
75C22×C53—21no—
76C22×C53—41no—
77C22×C53C2×C22⋊C411yescharacteristic
78C2×D53—21no—
79C2×D53—21no—
80C2×D53—41no—
81C2×D53—41no—
82C2×D53—42no—
83C2×D53—42no—
84C2×D53—42no—
85C2×D53—84no—
86#6157—82no—
87C4×C53—21no—
88C4×C53—42no—
89C4×C53#9911yescharacteristic
90C4×C53#9911yescharacteristic
91Dic53—21no—
92Dic53—21no—
93#2122—531nomaximal, Sylow
94C2×C53—21no—
95C2×C53—41no—
96C2×C53C2×#9911yescharacteristic
97D53—41no—
98D53—41no—
99D53—82no—
100#535—531no—
101#413—531no—
102#410—531no—
103AΣL(1,𝔽16)—534no—
104C53#212211yescharacteristic, complemented, Sylow
105C2×#41—531no—
106C22×D4—531no—
107C2×M16—1061no—
108#127—1061no—
109#124—1061no—
110#121—1061no—
111#121—1061no—
112#120—532no—
113#120—532no—
114#117—1061no—
115#117#615711yescharacteristic, complemented
116#101—1061no—
117#100—532no—
118#99—532no—
119#99—1061no—
120C24—532no—
121#41—2121no—
122#41—2121no—
123C2×Q8—531no—
124C2×D4—531no—
125C2×D4—531no—
126C2×D4—531no—
127C2×D4—1062no—
128C2×D4—1062no—
129C2×D4—1061no—
130C2×D4—1061no—
131C2×D4—2121no—
132C2×D4—2121no—
133C22×C4—531no—
134C22×C4—531no—
135C22×C4C2×#615711yescharacteristic
136M16—1062no—
137M16—2121no—
138C2×C8—2121no—
139C4⋊C4—21no—
140C4⋊C4—2121no—
141C22⋊C4—1062no—
142C22⋊C4—1062no—
143C22⋊C4—1062no—
144C22⋊C4C2×#615712yes—
145C42—21no—
146C42—2121no—
147C23—531no—
148C23—531no—
149C23—1062no—
150C23—4242no—
151C23—2122no—
152C23C22×#615711yescharacteristic
153Q8—2121no—
154D4—4242no—
155D4—4242no—
156D4—2121no—
157D4—2122no—
158D4—2121no—
159D4—2121no—
160D4—2121no—
161D4—2121no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—531no—
166C2×C4—531no—
167C2×C4—1061no—
168C2×C4—1061no—
169C2×C4—1061no—
170C2×C4—1061no—
171C2×C4—2122no—
172C2×C4C22⋊#615711yescharacteristic
173C2×C4C22⋊#615711yescharacteristic
174C8—2122no—
175C22—21no—
176C22—41no—
177C22—1061no—
178C22—1061no—
179C22—4244no—
180C22—2122no—
181C22—2122no—
182C22—2121no—
183C22—2121no—
184C22—2122no—
185C22C2×C22⋊#615711yescharacteristic
186C4—21no—
187C4—42no—
188C4—1061no—
189C4—1061no—
190C4—4242no—
191C4#1313327611yescharacteristic
192C4#1313327611yescharacteristic
193C2—21no—
194C2—41no—
195C2—4242no—
196C2—2121no—
197C2—2121no—
198C2C2×#1313327611yescharacteristic
subgroupG/H
centerC2C2×#13133276
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
Fitting#117×C53C4
socleC2×C53C2×#99
hypercenter#117#6157
hypocenterC53#2122
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #16811609C2semidirect
2. #16811602C2semidirect
3. #16811524C2semidirect
12. C22×D212C4semidirect
17. #13133187C4semidirect
19. #117×C53C4semidirect
104. C53#2122semidirect
115. #117#6157semidirect