Finite Group Databasebeta

← #20276399

Subgroups — #20276399

Order 6784.

250 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811174C211yesmaximal, characteristic
2#16810606C212yesmaximal, complemented
3#16810422C211yesmaximal, characteristic, complemented
4#16810419C212yesmaximal, complemented
5C53×#356C211yesmaximal, characteristic, complemented
6C2×#11719949C411yescharacteristic
7C2×#11719949C2211yescharacteristic
8M16×D53—22no—
9#13133214C411yescharacteristic
10#13133206C411yescharacteristic
11#13133206C2211yescharacteristic
12#13133192—22no—
13#13133179—22no—
14C42×D53C411yescharacteristic
15#13133136C2212yes—
16#13133133—22no—
17C4⋊Q8×C53C2211yescharacteristic
18#101×C53C2212yes—
19C2×Q8×C53C2×C411yescharacteristic
20C2×Q8×C53C2311yescharacteristic
21#11719949—41no—
22#11719949—41no—
23#11719949—41no—
24#11719949—41no—
25C2×D212—22no—
26C2×D212—21no—
27C2×D212C2×C411yescharacteristic
28C2×D212C2×C411yescharacteristic
29C2×D212D412yes—
30C2×C4×D53—21no—
31C2×C4×D53—21no—
32C2×C4×D53C2×C411yescharacteristic
33C2×C4×D53D412yes—
34M16×C53—22no—
35C4⋊C4×C53—22no—
36C42×C53C2×C411yescharacteristic
37#11719921—22no—
38#11719921—22no—
39C4×Dic53—21no—
40C4×Dic53C2×C411yescharacteristic
41#11719917—22no—
42#11719912—42no—
43C8×D53—42no—
44C22×D53—21no—
45C22×D53C22⋊C411yescharacteristic
46C22×D53C22⋊C412yes—
47Q8×C53—41no—
48Q8×C53—41no—
49C2×C4×C53—21no—
50C2×C4×C53—21no—
51C2×C4×C53C22×C411yescharacteristic
52C2×C4×C53C2×D412yes—
53C2×Dic53—21no—
54C2×Dic53C22⋊C411yescharacteristic
55D212—41no—
56D212—42no—
57D212—41no—
58D212—44no—
59D212—42no—
60C4×D53—21no—
61C4×D53—21no—
62C4×D53—22no—
63C4×D53—22no—
64C4×D53—42no—
65C4×D53—41no—
66C4×D53—41no—
67C8×C53—42no—
68#88322—42no—
69C22×C53C2×C22⋊C411yescharacteristic
70C2×D53—21no—
71C2×D53—22no—
72C2×D53—41no—
73C2×D53—42no—
74C2×D53—42no—
75C2×D53—41no—
76C4×C53—22no—
77C4×C53—22no—
78C4×C53—21no—
79C4×C53—41no—
80Dic53—41no—
81Dic53#9911yescharacteristic
82Dic53#9911yescharacteristic
83#2132—531nomaximal, Sylow
84C2×C53—21no—
85C2×C53C2×#9911yescharacteristic
86D53—41no—
87D53—42no—
88D53—81no—
89#532—531no—
90#413—532no—
91#356—531no—
92#356D5311yescharacteristic, complemented
93#355—532no—
94C53#213211yescharacteristic, complemented, Sylow
95C2×#41—531no—
96C2×#41—531no—
97C2×M16—1062no—
98C4⋊Q8C2×D5311yescharacteristic
99#127—531no—
100#124—531no—
101#124—531no—
102#121—1062no—
103C4×D4—1062no—
104C2×C42—531no—
105#101—532no—
106#101C2×D5312yes—
107#100—1062no—
108#41—2121no—
109#41—2121no—
110#41—2121no—
111#41—2121no—
112C2×Q8C4×D5311yescharacteristic
113C2×Q8C22×D5311yescharacteristic
114C2×D4—531no—
115C2×D4—531no—
116C2×D4—532no—
117C2×D4—1062no—
118C2×D4—1061no—
119C22×C4—531no—
120C22×C4—532no—
121C22×C4—1061no—
122C22×C4—1061no—
123M16—22no—
124M16—1062no—
125M16—2122no—
126C2×C8—2122no—
127C4⋊C4—22no—
128C22⋊C4—1062no—
129C22⋊C4—1062no—
130C42—531no—
131C42—1061no—
132C42C4×D5311yescharacteristic
133C23—532no—
134C23—531no—
135C23—1061no—
136Q8—41no—
137Q8—41no—
138D4—2122no—
139D4—2121no—
140D4—2121no—
141D4—2124no—
142D4—2122no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—531no—
146C2×C4—1062no—
147C2×C4—1062no—
148C2×C4—1061no—
149C2×C4—1061no—
150C2×C4—1061no—
151C2×C4—2122no—
152C2×C4—2121no—
153C2×C4—2121no—
154C2×C4C2×C4×D5311yescharacteristic
155C2×C4D4×D5312yes—
156C8—42no—
157C8—2122no—
158C22—1061no—
159C22—1062no—
160C22—2122no—
161C22—2121no—
162C22—2122no—
163C22—2121no—
164C22C22⋊C4×D5311yescharacteristic
165C4—22no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—531no—
170C4—531no—
171C4—2121no—
172C2—21no—
173C2—4241no—
174C2—2121no—
175C2—2122no—
176C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×Q8C22×D53
FittingC53×#356C2
socleC2×C53C2×#99
hypercenter#356D53
hypocenterC53#2132
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #16810606C2semidirect
3. #16810422C2semidirect
4. #16810419C2semidirect
5. C53×#356C2semidirect
92. #356D53semidirect
94. C53#2132semidirect