Finite Group Databasebeta

← #20276366

Subgroups — #20276366

Order 6784.

266 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 65 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#16811187C211yesmaximal, characteristic
2#16810597C211yesmaximal, characteristic, complemented
3#16810411C214yesmaximal, complemented
4C53×#413C211yesmaximal, characteristic, complemented
5C22×D212C411yescharacteristic, complemented
6#13133213—21no—
7#13133206—21no—
8#13133191—21no—
9#13133191—21no—
10#13133185C412yescomplemented
11#13133185C2212yes—
12#13133176—21no—
13#13133176C411yescharacteristic, complemented
14#13133132—21no—
15#13133132C2212yes—
16C2×#41×C53C2211yescharacteristic
17C2×M16×C53—21no—
18#101×C53—21no—
19#100×C53C2212yes—
20C23×D53C2×C412yes—
21#41×C53—41no—
22#41×C53—41no—
23C2×Q8×C53D411yescharacteristic
24C2×D4×C53C2×C411yescharacteristic
25C2×D4×C53D411yescharacteristic
26C2×D4×C53C2311yescharacteristic
27C22×C4×C53C2×C411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53D411yescharacteristic
30C2×C22⋊D53—22no—
31C2×C22⋊D53—22no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—41no—
35C2×D212—41no—
36M16×C53—22no—
37M16×C53—41no—
38C2×C8×C53—41no—
39C22⋊Dic53—22no—
40C22⋊Dic53C2×C412yes—
41#11719921—22no—
42#11719921—22no—
43C4⋊Dic53—21no—
44C4⋊Dic53—41no—
45C4×Dic53—21no—
46C4×Dic53—41no—
47C23×C53C22⋊C411yescharacteristic
48C23×C53C22×C411yescharacteristic
49C23×C53C2×D411yescharacteristic
50C22×D53—22no—
51C22×D53—42no—
52C22×D53—82no—
53Q8×C53—41no—
54D4×C53—41no—
55D4×C53—41no—
56D4×C53—41no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53C22⋊C411yescharacteristic
64C2×C4×C53C22⋊C411yescharacteristic
65C2×C4×C53C22⋊C411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C22⋊D53—82no—
68C22⋊D53—82no—
69C2×Dic53—22no—
70C2×Dic53—42no—
71D212—42no—
72D212—41no—
73D212—41no—
74C8×C53—42no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53—21no—
78C22×C53—41no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53C2×C22⋊C411yescharacteristic
82C2×D53—42no—
83C2×D53—42no—
84C2×D53—42no—
85C2×D53—84no—
86C4×C53—21no—
87C4×C53—21no—
88C4×C53—21no—
89C4×C53#9911yescharacteristic
90C4×C53#9911yescharacteristic
91Dic53—42no—
92Dic53—82no—
93#2122—531nomaximal, Sylow
94C2×C53—21no—
95C2×C53—41no—
96C2×C53—41no—
97C2×C53—41no—
98C2×C53C2×#9911yescharacteristic
99D53—82no—
100#535—531no—
101#413D5311yescharacteristic, complemented
102#410—531no—
103AΣL(1,𝔽16)—534no—
104C53#212211yescharacteristic, complemented, Sylow
105C2×#41C2×D5311yescharacteristic
106C22×D4—531no—
107C2×M16—21no—
108#127—1061no—
109#124—1061no—
110#121—1061no—
111#121—1061no—
112#120—532no—
113#120—532no—
114#117—531no—
115#117—1061no—
116#101—21no—
117#100C2×D5312yes—
118#99—532no—
119#99—1061no—
120C24—532no—
121#41—41no—
122#41—41no—
123C2×Q8C22⋊D5311yescharacteristic
124C2×D4—1062no—
125C2×D4—1062no—
126C2×D4—1061no—
127C2×D4—1061no—
128C2×D4—2121no—
129C2×D4—2121no—
130C2×D4C4×D5311yescharacteristic
131C2×D4D21211yescharacteristic
132C2×D4C22×D5311yescharacteristic
133C22×C4C4×D5311yescharacteristic
134C22×C4D21211yescharacteristic
135C22×C4C22⋊D5311yescharacteristic
136M16—22no—
137M16—41no—
138C2×C8—41no—
139C4⋊C4—1061no—
140C4⋊C4—2121no—
141C22⋊C4—532no—
142C22⋊C4—1062no—
143C22⋊C4—1062no—
144C22⋊C4—1062no—
145C42—1061no—
146C42—2121no—
147C23—1062no—
148C23—4242no—
149C23—2122no—
150C23#1171992111yescharacteristic
151C23C2×C4×D5311yescharacteristic
152C23C2×D21211yescharacteristic
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—4242no—
158D4—4242no—
159D4—2122no—
160D4—2121no—
161D4—2121no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—1062no—
169C2×C4—2122no—
170C2×C4#1171992111yescharacteristic
171C2×C4#1171992111yescharacteristic
172C2×C4#1171992111yescharacteristic
173C2×C4C2×C22⋊D5311yescharacteristic
174C8—42no—
175C22—21no—
176C22—21no—
177C22—21no—
178C22—41no—
179C22—41no—
180C22—41no—
181C22—4244no—
182C22—2122no—
183C22—2122no—
184C22—2122no—
185C22C2×#1171992111yescharacteristic
186C4—21no—
187C4—21no—
188C4—21no—
189C4—4242no—
190C4—2122no—
191C4#1313313111yescharacteristic
192C4#1313313111yescharacteristic
193C2—21no—
194C2—41no—
195C2—41no—
196C2—41no—
197C2—4242no—
198C2C2×#1313313111yescharacteristic
subgroupG/H
centerC2C2×#13133131
commutatorC23×C53C22×C4
FrattiniC2×D4C22×D53
FittingC53×#413C2
socleC2×C53C2×#99
hypercenter#413D53
hypocenterC53#2122
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #16810597C2semidirect
3. #16810411C2semidirect
4. C53×#413C2semidirect
5. C22×D212C4semidirect
10. #13133185C4semidirect
13. #13133176C4semidirect
101. #413D53semidirect
104. C53#2122semidirect