Finite Group Databasebeta

← #20275557

Subgroups — #20275557

Order 6784.

292 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#13133215C211yesmaximal, characteristic
2#16810646C211yesmaximal, characteristic
3C2×#13133136C211yesmaximal, characteristic
4C2×#13133135C211yesmaximal, characteristic
5C2×#13133135C211yesmaximal, characteristic
6M16×Dic53C211yesmaximal, characteristic
7C2×#101×C53C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10C2×#11719917C2211yescharacteristic
11#13133215—21no—
12#13133215—21no—
13#13133215—22no—
14C2×#11719919C414yes—
15C2×C4×Dic53C2211yescharacteristic
16#13133146—22no—
17#13133136—22no—
18#13133135—22no—
19#13133135—22no—
20#13133127—21no—
21C8×Dic53—21no—
22C22×Q8×C53C2211yescharacteristic
23C2×M16×C53C2211yescharacteristic
24C2×M16×C53C2211yescharacteristic
25#101×C53—22no—
26C2×Q8×C53—21no—
27C2×Q8×C53—21no—
28C2×Q8×C53—21no—
29C2×Q8×C53—21no—
30C2×Q8×C53—21no—
31C2×Q8×C53—21no—
32C22×C4×C53C2×C412yes—
33C22×C4×C53C2311yescharacteristic
34C22×Dic53C2×C412yes—
35C22×Dic53C2×C412yes—
36C2×Dic106—21no—
37C2×Dic106—21no—
38C2×Dic106—21no—
39C2×Dic106—21no—
40C2×Dic106—21no—
41C2×Dic106—21no—
42M16×C53—21no—
43M16×C53—21no—
44M16×C53—21no—
45M16×C53D412yes—
46C2×C8×C53—21no—
47C2×C8×C53—21no—
48#11719919—24no—
49#11719919—24no—
50C4×Dic53D411yescharacteristic
51C4×Dic53D411yescharacteristic
52C4×Dic53D412yes—
53#11719917—21no—
54#11719917—21no—
55#11719917—21no—
56#11719917D412yes—
57C2×#88322—21no—
58C2×#88322—21no—
59C23×C53C22×C411yescharacteristic
60Q8×C53—41no—
61Q8×C53—41no—
62Q8×C53—41no—
63Q8×C53—41no—
64C2×C4×C53—22no—
65C2×C4×C53—22no—
66C2×C4×C53—22no—
67C2×C4×C53C2×D411yescharacteristic
68C2×C4×C53C2×D411yescharacteristic
69C2×C4×C53C2×D411yescharacteristic
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53#4111yescharacteristic
72C2×C4×C53#4111yescharacteristic
73C2×Dic53—24no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76C2×Dic53—22no—
77C2×Dic53C22⋊C412yes—
78C2×Dic53C22⋊C412yes—
79Dic106—41no—
80Dic106—41no—
81Dic106—41no—
82Dic106—41no—
83C8×C53—22no—
84C8×C53—41no—
85#88322—22no—
86#88322—41no—
87C22×C53—22no—
88C22×C53C2×C22⋊C411yescharacteristic
89C22×C53C4×D411yescharacteristic
90C22×C53C4×D411yescharacteristic
91C4×C53—42no—
92C4×C53#12111yescharacteristic
93C4×C53#12111yescharacteristic
94C4×C53#12411yescharacteristic
95C4×C53#12711yescharacteristic
96Dic53—22no—
97Dic53—42no—
98Dic53#10114yescomplemented
99#1966—531nomaximal, Sylow
100C2×C53—21no—
101C2×C53—21no—
102C2×C53#39011yescharacteristic
103C2×C53C2×#10112yes—
104C2×C4⋊Q8—531no—
105#423—531no—
106C2×#101—531no—
107C2×#101—531no—
108C2×#101—531no—
109C2×#101D5311yescharacteristic
110C4×M16—531no—
111C53#196611yescharacteristic, complemented, Sylow
112C22×Q8—531no—
113C22×Q8C2×D5311yescharacteristic
114C2×M16—531no—
115C2×M16—531no—
116C2×M16C2×D5311yescharacteristic
117C2×M16C2×D5311yescharacteristic
118C4⋊Q8—1061no—
119C4⋊Q8—1061no—
120C4⋊Q8—1062no—
121C2×C4⋊C4—534no—
122C2×C42—531no—
123C4⋊C8—1062no—
124#101—22no—
125#101—1062no—
126#101—1062no—
127#101—1062no—
128#97—1061no—
129C4×C8—1061no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—1061no—
137C2×Q8—1061no—
138C2×Q8—1061no—
139C2×Q8—1061no—
140C2×Q8—1061no—
141C2×Q8—1061no—
142C22×C4—532no—
143C22×C4—532no—
144C22×C4C4×D5312yes—
145C22×C4C22×D5311yescharacteristic
146M16—21no—
147M16—21no—
148M16—21no—
149M16—532no—
150M16—1061no—
151M16—1061no—
152M16—1061no—
153M16C22⋊D5312yes—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—1061no—
157C2×C8—1061no—
158C4⋊C4—1064no—
159C4⋊C4—1064no—
160C42—531no—
161C42—531no—
162C42—532no—
163C23C2×C4×D5311yescharacteristic
164Q8—41no—
165Q8—41no—
166Q8—41no—
167Q8—41no—
168Q8—2121no—
169Q8—2121no—
170Q8—2121no—
171Q8—2121no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—532no—
176C2×C4—532no—
177C2×C4—1064no—
178C2×C4—1062no—
179C2×C4—1062no—
180C2×C4—1062no—
181C2×C4#1171994511yescharacteristic
182C2×C4D4×D5311yescharacteristic
183C2×C4D4×D5311yescharacteristic
184C2×C4D4×D5311yescharacteristic
185C2×C4#1171994711yescharacteristic
186C2×C4C2×C22⋊D5311yescharacteristic
187C8—22no—
188C8—41no—
189C8—1062no—
190C8—2121no—
191C22—22no—
192C22C22⋊C4×D5311yescharacteristic
193C22#1313317711yescharacteristic
194C22C4×C22⋊D5311yescharacteristic
195C4—42no—
196C4—534no—
197C4—1062no—
198C4—2122no—
199C4#1313318911yescharacteristic
200C4#1313319011yescharacteristic
201C4#1313320311yescharacteristic
202C4#1313321311yescharacteristic
203C2—21no—
204C2—21no—
205C2#1681052311yescharacteristic
206C2#101×D5312yes—
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC2×#101×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#101D53
hypocenterC53#1966
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
98. Dic53#101semidirect
111. C53#1966semidirect