Finite Group Databasebeta

← #27195612

Subgroups — #27195612

Order 13184.

338 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15693000C211yesmaximal, characteristic
2C2×#15692939C212yesmaximal, complemented
3#20267755C211yesmaximal, characteristic
4#20267559C212yesmaximal, complemented
5C22×M16×C103C211yesmaximal, characteristic
6C22×Dic206C2211yescharacteristic
7C2×#13110089C2212yes—
8#15693000—21no—
9#15693000—21no—
10#15693000—21no—
11#15693000—21no—
12C2×C4⋊Dic103C2211yescharacteristic
13C2×Dic103⋊C4C412yes—
14C2×C22⋊Dic103C412yes—
15#15692939—24no—
16C22⋊(C103⋊C8)—21no—
17C22⋊(C103⋊C8)—21no—
18C23×C4×C103C2211yescharacteristic
19C2×M16×C103—21no—
20C2×M16×C103—24no—
21C2×M16×C103C2212yes—
22C22×C8×C103—21no—
23C2×Dic206—22no—
24C2×Dic206—21no—
25C2×Dic206—21no—
26C2×Dic206—22no—
27C22×Dic103C2×C412yes—
28C22×Dic103C2×C412yes—
29#13110089—22no—
30#13110089—22no—
31C2×C103⋊C8—22no—
32C4⋊Dic103—21no—
33C4⋊Dic103—21no—
34Dic103⋊C4—22no—
35Dic103⋊C4—22no—
36C22⋊Dic103—22no—
37C22⋊Dic103—22no—
38C24×C103C2×C411yescharacteristic
39C22×C4×C103—22no—
40C22×C4×C103—22no—
41C22×C4×C103C2×C411yescharacteristic
42C22×C4×C103D411yescharacteristic
43C22×C4×C103D411yescharacteristic
44C22×C4×C103D411yescharacteristic
45C22×C4×C103D411yescharacteristic
46C22×C4×C103C2311yescharacteristic
47M16×C103—24no—
48M16×C103—22no—
49M16×C103D414yes—
50C2×C8×C103—24no—
51C2×C8×C103—22no—
52C23×C103—22no—
53C23×C103—22no—
54C23×C103—41no—
55C23×C103C22⋊C411yescharacteristic
56C23×C103C22⋊C411yescharacteristic
57C23×C103C22×C411yescharacteristic
58C2×C4×C103—22no—
59C2×C4×C103—21no—
60C2×C4×C103—21no—
61C2×C4×C103—22no—
62C2×C4×C103—22no—
63C2×C4×C103—21no—
64C2×C4×C103—21no—
65C2×C4×C103C22⋊C411yescharacteristic
66C2×C4×C103C22⋊C411yescharacteristic
67C2×C4×C103C2×D412yes—
68C2×C4×C103C2×D411yescharacteristic
69C2×C4×C103C2×D411yescharacteristic
70C2×C4×C103#4112yes—
71C2×Dic103—22no—
72C2×Dic103—22no—
73C2×Dic103—24no—
74C2×Dic103—42no—
75Dic206—41no—
76Dic206—42no—
77Dic206—41no—
78C8×C103—24no—
79C103⋊C8—42no—
80C22×C103—21no—
81C22×C103—21no—
82C22×C103—21no—
83C22×C103—21no—
84C22×C103—21no—
85C22×C103—21no—
86C22×C103—21no—
87C22×C103—44no—
88C22×C103#10112yescomplemented
89C22×C103C2×C22⋊C411yescharacteristic
90C22×C103C4×D412yes—
91C4×C103—21no—
92C4×C103—21no—
93C4×C103#12011yescharacteristic
94C4×C103#12112yes—
95C4×C103#12311yescharacteristic
96Dic103—42no—
97Dic103—42no—
98C2×C103—22no—
99C2×C103—22no—
100C2×C103—41no—
101C2×C103#38611yescharacteristic
102C2×C103C2×#10111yescharacteristic
103C2×C103#41311yescharacteristic
104#1901—1031nomaximal, Sylow
105C103#190111yescharacteristic, complemented, Sylow
106C22×M16D10311yescharacteristic
107C2×C22⋊Q8—1031no—
108C2×#101—1032no—
109#407—1031no—
110#343—1032no—
111C22×Q8—1031no—
112C23×C4C2×D10311yescharacteristic
113C2×M16—21no—
114C2×M16—24no—
115C2×M16—1032no—
116C2×M16C2×D10312yes—
117C22×C8—21no—
118C22⋊Q8—2061no—
119C22⋊Q8—2061no—
120C22⋊Q8—2061no—
121C22⋊Q8—2061no—
122C2×C4⋊C4—1032no—
123C2×C4⋊C4—1031no—
124C2×C22⋊C4—1032no—
125#101—2064no—
126C22⋊C8—2061no—
127C22⋊C8—2061no—
128C24C4×D10311yescharacteristic
129C2×Q8—2062no—
130C2×Q8—2061no—
131C2×Q8—2061no—
132C2×Q8—2062no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—1032no—
136C22×C4—1032no—
137C22×C4C4×D10311yescharacteristic
138C22×C4D41211yescharacteristic
139C22×C4D41211yescharacteristic
140C22×C4C22⋊D10311yescharacteristic
141C22×C4C22⋊D10311yescharacteristic
142C22×C4C22×D10311yescharacteristic
143M16—24no—
144M16—22no—
145M16—2062no—
146M16—2062no—
147M16C22⋊D10314yes—
148C2×C8—24no—
149C2×C8—22no—
150C2×C8—2062no—
151C4⋊C4—2062no—
152C4⋊C4—2062no—
153C4⋊C4—2061no—
154C4⋊C4—2061no—
155C22⋊C4—2062no—
156C22⋊C4—2062no—
157C23—22no—
158C23—22no—
159C23—41no—
160C23(C2×D103)⋊C411yescharacteristic
161C23(C2×D103)⋊C411yescharacteristic
162C23C2×C4×D10311yescharacteristic
163Q8—4121no—
164Q8—4122no—
165Q8—4121no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—4122no—
174C2×C4—2062no—
175C2×C4—2062no—
176C2×C4—2064no—
177C2×C4(C2×D103)⋊C411yescharacteristic
178C2×C4(C2×D103)⋊C411yescharacteristic
179C2×C4C2×C22⋊D10312yes—
180C2×C4C2×C22⋊D10311yescharacteristic
181C2×C4C2×D41211yescharacteristic
182C2×C4#1311010512yes—
183C8—24no—
184C8—4122no—
185C22—21no—
186C22—21no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—44no—
193C22#1569293912yescomplemented
194C22C2×(C2×D103)⋊C411yescharacteristic
195C22C4×C22⋊D10312yes—
196C4—21no—
197C4—21no—
198C4—4122no—
199C4—4122no—
200C4#1569299011yescharacteristic
201C4#1569299512yes—
202C4#1569300411yescharacteristic
203C2—22no—
204C2—22no—
205C2—41no—
206C2#2026767011yescharacteristic
207C2C2×#1569293911yescharacteristic
208C2#2026777311yescharacteristic
subgroupG/H
centerC22C2×(C2×D103)⋊C4
commutatorC23×C103C22×C4
FrattiniC22×C4C22×D103
FittingC22×M16×C103C2
socleC22×C103C2×C22⋊C4
prime radicalC24×C103C2×C4
cosocleC22×C4×C103C23
hypercenterC22×M16D103
hypocenterC103#1901
LCSG ▷ C23×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#15692939C2semidirect
4. #20267559C2semidirect
88. C22×C103#101semidirect
105. C103#1901semidirect
193. C22#15692939semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156930002
C2×#156929392
#202677552
#202675592
C22×M16×C1032
#1901103