Finite Group Databasebeta

← #26125692

Subgroups — #26125692

Order 12032.

418 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 99 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#20241722C212yesmaximal
2#20241334C214yesmaximal, complemented
3C47×#2317C211yesmaximal, characteristic
4#15689180—22no—
5#15688849C2214yes—
6#15688847—22no—
7C2×#117×C47C411yescharacteristic
8C2×#117×C47C2211yescharacteristic
9C22×C4⋊C4×C47C411yescharacteristic
10C47×#389—22no—
11C47×#385—22no—
12C47×#384C2212yes—
13C47×#382C412yes—
14C4⋊(C47⋊C8)—42no—
15C22⋊(C47⋊C8)—24no—
16#13093935—42no—
17C23×C4×C47—21no—
18C23×C4×C47C2×C411yescharacteristic
19#117×C47—24no—
20#117×C47—42no—
21C2×C4⋊C4×C47—21no—
22C2×C4⋊C4×C47—21no—
23C2×C4⋊C4×C47—24no—
24C2×C4⋊C4×C47—42no—
25C2×C4⋊C4×C47C2×C412yes—
26C2×C4⋊C4×C47C2×C411yescharacteristic
27C2×C4⋊C4×C47C2311yescharacteristic
28C2×C22⋊C4×C47—21no—
29C2×C22⋊C4×C47D412yes—
30C2×C42×C47C2×C412yes—
31C2×C42×C47D412yes—
32#95×C47—24no—
33C24×C47C22⋊C411yescharacteristic
34C22×C4×C47—22no—
35C22×C4×C47—22no—
36C22×C4×C47—22no—
37C22×C4×C47—22no—
38C22×C4×C47—22no—
39C22×C4×C47—42no—
40C22×C4×C47—82no—
41C22×C4×C47C22⋊C411yescharacteristic
42C22×C4×C47C22⋊C412yes—
43C22×C4×C47C22×C411yescharacteristic
44C22×C4×C47C2×D412yes—
45C4⋊C4×C47—24no—
46C4⋊C4×C47—41no—
47C4⋊C4×C47—41no—
48C4⋊C4×C47—42no—
49C4⋊C4×C47—42no—
50C4⋊C4×C47—84no—
51C22⋊C4×C47—24no—
52C22⋊C4×C47—42no—
53C42×C47—22no—
54C42×C47—22no—
55C42×C47—22no—
56C42×C47M1614yes—
57C2×C47⋊C8—44no—
58C23×C47—21no—
59C23×C47—21no—
60C23×C47—41no—
61C23×C47—41no—
62C23×C47#10112yes—
63C23×C47C2×C22⋊C411yescharacteristic
64C2×C4×C47—28no—
65C2×C4×C47—21no—
66C2×C4×C47—21no—
67C2×C4×C47—21no—
68C2×C4×C47—21no—
69C2×C4×C47—24no—
70C2×C4×C47—22no—
71C2×C4×C47—22no—
72C2×C4×C47—42no—
73C2×C4×C47—42no—
74C2×C4×C47—42no—
75C2×C4×C47—42no—
76C2×C4×C47—44no—
77C2×C4×C47—42no—
78C2×C4×C47—84no—
79C2×C4×C47#9912yes—
80C2×C4×C47C2×M1612yes—
81C47⋊C8—84no—
82#11516—471nomaximal, Sylow
83C22×C47—21no—
84C22×C47—21no—
85C22×C47—22no—
86C22×C47—42no—
87C22×C47—42no—
88C22×C47—41no—
89C22×C47—41no—
90C22×C47#40711yescharacteristic
91C22×C47C2×#9911yescharacteristic
92C22×C47C2×#10111yescharacteristic
93C4×C47—22no—
94C4×C47—22no—
95C4×C47—44no—
96C4×C47—42no—
97C4×C47—42no—
98C4×C47—82no—
99#2317D4711yescharacteristic
100#1464—472no—
101#1325—474no—
102C2×C47—21no—
103C2×C47—21no—
104C2×C47—42no—
105C2×C47#146311yescharacteristic
106C2×C47#212811yescharacteristic
107C2×C47#213511yescharacteristic
108C2×#117Dic4711yescharacteristic
109C2×#117C2×D4711yescharacteristic
110C22×C4⋊C4Dic4711yescharacteristic
111#432—942no—
112#389—22no—
113#385—22no—
114#384C2×D4712yes—
115#382Dic4712yes—
116#324—474no—
117#323—942no—
118C47#1151611yescharacteristic, complemented, Sylow
119C23×C4—21no—
120C23×C4C2×Dic4711yescharacteristic
121#117—24no—
122#117—42no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4C2×Dic4712yes—
128C2×C4⋊C4C2×Dic4711yescharacteristic
129C2×C4⋊C4C22×D4711yescharacteristic
130C2×C22⋊C4—21no—
131C2×C22⋊C4C22⋊D4712yes—
132C2×C42C2×Dic4712yes—
133C2×C42C22⋊D4712yes—
134C4⋊C8—1882no—
135C22⋊C8—944no—
136#97—1882no—
137#95—24no—
138C24C22⋊Dic4711yescharacteristic
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—82no—
146C22×C4C22⋊Dic4712yes—
147C22×C4C22⋊Dic4711yescharacteristic
148C22×C4C22×Dic4711yescharacteristic
149C22×C4C2×C22⋊D4712yes—
150C2×C8—1884no—
151C4⋊C4—24no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—84no—
157C22⋊C4—24no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C42—22no—
162C42#1062532914yes—
163C23—21no—
164C23—21no—
165C23—41no—
166C23—41no—
167C23#1309394312yes—
168C23C2×C22⋊Dic4711yescharacteristic
169C2×C4—28no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—42no—
183C2×C4—84no—
184C2×C4#1309393912yes—
185C2×C4C2×#1062532912yes—
186C8—3764no—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22#1568908711yescharacteristic
195C22C2×#1309393911yescharacteristic
196C22C2×#1309394311yescharacteristic
197C4—22no—
198C4—22no—
199C4—44no—
200C4—42no—
201C4—42no—
202C4—82no—
203C2—21no—
204C2—21no—
205C2—42no—
206C2#2024171811yescharacteristic
207C2#2024506011yescharacteristic
208C2#2024508311yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2317C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#2317D47
hypocenterC47#11516
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. #20241334C2semidirect
118. C47#11516semidirect

Maximal subgroups

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

subgroupindex
#202417222
#202413342
C47×#23172
#1151647