Finite Group Databasebeta

← #20276405

Subgroups — #20276405

Order 6784.

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

normal subgroups only

This group has 262 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#16811191C211yesmaximal, characteristic
2#16810606C212yesmaximal, complemented
3#16810420C211yesmaximal, characteristic, complemented
4#16810419C212yesmaximal, complemented
5C53×#355C211yesmaximal, characteristic, complemented
6C2×#11719949C2211yescharacteristic
7C2×D4×D53C411yescharacteristic
8M16×D53—22no—
9#13133214C411yescharacteristic
10#13133213C411yescharacteristic
11#13133206C2211yescharacteristic
12#13133190—22no—
13#13133187—22no—
14#13133174C411yescharacteristic
15#13133136C2212yescomplemented
16#13133133—22no—
17#124×C53C2211yescharacteristic
18#101×C53C2212yescomplemented
19C23×D53—21no—
20C2×Q8×C53C2311yescharacteristic
21C2×D4×C53C2×C411yescharacteristic
22C2×C22⋊D53—22no—
23C2×C22⋊D53—21no—
24#11719949—42no—
25D4×D53—42no—
26C2×D212—22no—
27C2×D212C2×C411yescharacteristic
28C2×D212C2×C411yescharacteristic
29C2×D212D412yes—
30C2×C4×D53C2×C411yescharacteristic
31C2×C4×D53D412yes—
32M16×C53—22no—
33C22⋊C4×C53—22no—
34C42×C53C2×C411yescharacteristic
35#11719921—21no—
36#11719921—22no—
37#11719919—21no—
38C4×Dic53C2×C411yescharacteristic
39#11719917—22no—
40#11719912—42no—
41C8×D53—42no—
42C23×C53—21no—
43C22×D53—21no—
44C22×D53—42no—
45C22×D53—82no—
46C22×D53C22⋊C411yescharacteristic
47C22×D53C22⋊C412yes—
48Q8×C53—41no—
49D4×C53—41no—
50C2×C4×C53—21no—
51C2×C4×C53C22×C411yescharacteristic
52C2×C4×C53C2×D412yes—
53C22⋊D53—42no—
54C22⋊D53—82no—
55C2×Dic53—21no—
56C2×Dic53C22⋊C411yescharacteristic
57D212—41no—
58D212—42no—
59D212—41no—
60D212—82no—
61C4×D53—22no—
62C4×D53—22no—
63C4×D53—22no—
64C8×C53—42no—
65#88322—42no—
66C22×C53—42no—
67C22×C53—41no—
68C22×C53C2×C22⋊C411yescharacteristic
69C2×D53—21no—
70C2×D53—22no—
71C2×D53—42no—
72C2×D53—41no—
73C2×D53—42no—
74C2×D53—41no—
75C2×D53—84no—
76C4×C53—22no—
77C4×C53—21no—
78C4×C53—41no—
79Dic53—41no—
80Dic53#9912yes—
81#2134—531nomaximal, Sylow
82C2×C53—21no—
83C2×C53—81no—
84C2×C53C2×#9911yescharacteristic
85D53—42no—
86D53—41no—
87D53—81no—
88#535—531no—
89#413—532no—
90#355—531no—
91#355—532no—
92#355D5311yescharacteristic, complemented
93C53#213411yescharacteristic, complemented, Sylow
94C2×#41—531no—
95C22×D4—531no—
96C2×M16—1062no—
97#127—531no—
98#127—531no—
99#124—531no—
100#124C2×D5311yescharacteristic
101#121—1062no—
102#120—1062no—
103#117—531no—
104#101—532no—
105#101C2×D5312yescomplemented
106#100—1062no—
107C24—1061no—
108#41—2122no—
109C2×Q8C22×D5311yescharacteristic
110C2×D4—531no—
111C2×D4—531no—
112C2×D4—532no—
113C2×D4—1062no—
114C2×D4—1061no—
115C2×D4—1062no—
116C2×D4—2122no—
117C2×D4C4×D5311yescharacteristic
118C22×C4—531no—
119C22×C4—532no—
120M16—22no—
121M16—1062no—
122M16—2122no—
123C2×C8—2122no—
124C4⋊C4—1061no—
125C22⋊C4—22no—
126C22⋊C4—1061no—
127C22⋊C4—1062no—
128C42—531no—
129C42C4×D5311yescharacteristic
130C23—21no—
131C23—531no—
132C23—532no—
133C23—1061no—
134C23—4242no—
135C23—2122no—
136Q8—41no—
137D4—41no—
138D4—4242no—
139D4—4242no—
140D4—2122no—
141D4—2121no—
142D4—2121no—
143D4—2122no—
144C2×C4—21no—
145C2×C4—531no—
146C2×C4—1062no—
147C2×C4—1062no—
148C2×C4—1062no—
149C2×C4—1061no—
150C2×C4C2×C4×D5311yescharacteristic
151C2×C4D4×D5312yes—
152C8—42no—
153C8—2122no—
154C22—42no—
155C22—41no—
156C22—1061no—
157C22—1062no—
158C22—4244no—
159C22—2122no—
160C22—2122no—
161C22—2121no—
162C22—2121no—
163C22C22⋊C4×D5311yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—532no—
168C4—2121no—
169C2—21no—
170C2—81no—
171C2—4241no—
172C2—2121no—
173C2—2122no—
174C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×Q8C22×D53
FittingC53×#355C2
socleC2×C53C2×#99
hypercenter#355D53
hypocenterC53#2134
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #16810606C2semidirect
3. #16810420C2semidirect
4. #16810419C2semidirect
5. C53×#355C2semidirect
15. #13133136C22semidirect
18. #101×C53C22semidirect
92. #355D53semidirect
93. C53#2134semidirect
105. #101C2×D53semidirect