Finite Group Databasebeta

← #20276365

Subgroups — #20276365

Order 6784.

266 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 73 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#16810607C211yesmaximal, characteristic, complemented
2#16810597C211yesmaximal, characteristic, complemented
3#16810410C214yesmaximal, complemented
4C53×#535C211yesmaximal, characteristic
5C2×#11719917—21no—
6#13133176—21no—
7#13133136—21no—
8#13133134C2212yes—
9#13133132—21no—
10#13133132C2212yes—
11C2×#41×C53C2211yescharacteristic
12C22×D4×C53C411yescharacteristic, complemented
13#127×C53—21no—
14#124×C53—21no—
15#121×C53—21no—
16#121×C53—21no—
17#120×C53C412yescomplemented
18#120×C53C2212yes—
19#117×C53C411yescharacteristic, complemented
20C24×C53C2×C412yes—
21#41×C53—41no—
22#41×C53—41no—
23C2×Q8×C53D411yescharacteristic
24C2×D4×C53—22no—
25C2×D4×C53—22no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—41no—
29C2×D4×C53—41no—
30C2×D4×C53C2×C411yescharacteristic
31C2×D4×C53D411yescharacteristic
32C2×D4×C53C2311yescharacteristic
33C22×C4×C53C2×C411yescharacteristic
34C22×C4×C53D411yescharacteristic
35C22×C4×C53D411yescharacteristic
36C4⋊C4×C53—21no—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39C22⋊C4×C53C2×C412yes—
40C42×C53—21no—
41C22⋊Dic53—22no—
42C4⋊Dic53—41no—
43C4×Dic53—41no—
44#11719917—22no—
45#11719917—41no—
46C2×#88322—41no—
47C23×C53—22no—
48C23×C53—42no—
49C23×C53—82no—
50C23×C53C22⋊C411yescharacteristic
51C23×C53C22×C411yescharacteristic
52C23×C53C2×D411yescharacteristic
53Q8×C53—41no—
54D4×C53—41no—
55D4×C53—41no—
56D4×C53—41no—
57D4×C53—42no—
58D4×C53—41no—
59D4×C53—41no—
60D4×C53—82no—
61D4×C53—82no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—21no—
65C2×C4×C53—21no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—22no—
69C2×C4×C53C22⋊C411yescharacteristic
70C2×C4×C53C22⋊C411yescharacteristic
71C2×C4×C53C22⋊C411yescharacteristic
72C2×C4×C53C2×D411yescharacteristic
73C2×Dic53—42no—
74#88322—42no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53—21no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53—42no—
82C22×C53—42no—
83C22×C53—42no—
84C22×C53—84no—
85C22×C53C2×C22⋊C411yescharacteristic
86C4×C53—21no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53—42no—
90C4×C53#9911yescharacteristic
91C4×C53#9911yescharacteristic
92Dic53—82no—
93#2122—531nomaximal, Sylow
94C2×C53—21no—
95C2×C53—41no—
96C2×C53—41no—
97C2×C53—41no—
98C2×C53—82no—
99C2×C53C2×#9911yescharacteristic
100#535D5311yescharacteristic
101#413—531no—
102#410—531no—
103AΣL(1,𝔽16)—534no—
104C53#212211yescharacteristic, complemented, Sylow
105C2×#41C2×D5311yescharacteristic
106C22×D4Dic5311yescharacteristic, complemented
107C2×M16—1061no—
108#127—21no—
109#124—21no—
110#121—21no—
111#121—21no—
112#120Dic5312yescomplemented
113#120C2×D5312yes—
114#117—1061no—
115#117Dic5311yescharacteristic, complemented
116#101—1061no—
117#100—532no—
118#99—532no—
119#99—1061no—
120C24C2×Dic5312yes—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D5311yescharacteristic
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4C2×Dic5311yescharacteristic
131C2×D4C22⋊D5311yescharacteristic
132C2×D4C22×D5311yescharacteristic
133C22×C4C2×Dic5311yescharacteristic
134C22×C4C22⋊D5311yescharacteristic
135C22×C4C22⋊D5311yescharacteristic
136M16—1062no—
137M16—2121no—
138C2×C8—2121no—
139C4⋊C4—21no—
140C4⋊C4—2121no—
141C22⋊C4—22no—
142C22⋊C4—22no—
143C22⋊C4—1062no—
144C22⋊C4C2×Dic5312yes—
145C42—21no—
146C42—2121no—
147C23—22no—
148C23—42no—
149C23—82no—
150C23C22⋊Dic5311yescharacteristic
151C23C22×Dic5311yescharacteristic
152C23C2×C22⋊D5311yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—42no—
158D4—41no—
159D4—41no—
160D4—82no—
161D4—82no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—2122no—
170C2×C4C22⋊Dic5311yescharacteristic
171C2×C4C22⋊Dic5311yescharacteristic
172C2×C4C22⋊Dic5311yescharacteristic
173C2×C4C2×C22⋊D5311yescharacteristic
174C8—2122no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—84no—
185C22C2×C22⋊Dic5311yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—42no—
190C4—4242no—
191C4#1313313211yescharacteristic
192C4#1313313211yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—82no—
198C2C2×#1313313211yescharacteristic
subgroupG/H
centerC2C2×#13133132
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#535C2
socleC2×C53C2×#99
hypercenter#535D53
hypocenterC53#2122
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #16810607C2semidirect
2. #16810597C2semidirect
3. #16810410C2semidirect
12. C22×D4×C53C4semidirect
17. #120×C53C4semidirect
19. #117×C53C4semidirect
104. C53#2122semidirect
106. C22×D4Dic53semidirect
112. #120Dic53semidirect
115. #117Dic53semidirect