Finite Group Databasebeta

← #26125488

Subgroups — #26125488

Order 12032.

314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 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#20252031C211yesmaximal, characteristic
2#20245184C212yesmaximal, complemented
3#20241330C211yesmaximal, characteristic, complemented
4#20241313C212yesmaximal
5C47×#1323C211yesmaximal, characteristic, complemented
6#15689591C411yescharacteristic
7#15689183—22no—
8#15689175—22no—
9#15689150C411yescharacteristic
10#15689091C411yescharacteristic
11#15689089C2211yescharacteristic
12M16×Dic47C411yescharacteristic
13C2×C8×Dic47C2211yescharacteristic
14Dic47⋊C16—22no—
15C22⋊(C47⋊C16)C2212yes—
16C16×Dic47—22no—
17C47×#424C2211yescharacteristic
18C47×C22⋊C16C2212yes—
19C2×#10625350C2×C411yescharacteristic
20C2×#10625329C2×C411yescharacteristic
21C22×C47⋊C8C2×C411yescharacteristic
22#13093989C812yes—
23#13093986C812yes—
24(C22⋊D47)⋊C4—22no—
25#13093981—21no—
26C2×C4×Dic47C2×C411yescharacteristic
27C42×D47—21no—
28C2×C47⋊C16—22no—
29C4⋊(C47⋊C8)—22no—
30(C2×D47)⋊C8—22no—
31(C2×D47)⋊C8—22no—
32#13093934—21no—
33C8×Dic47—21no—
34C8×Dic47D412yes—
35C8×Dic47D412yes—
36C2×M16×C47C2×C411yescharacteristic
37C22×C8×C47C2311yescharacteristic
38#117×C47C2×C411yescharacteristic
39C2×C16×C47—22no—
40C4⋊C8×C47—22no—
41C22×C4×C47C22×C411yescharacteristic
42C2×C22⋊D47—21no—
43C22×Dic47M1612yes—
44#10625350—41no—
45#10625350—41no—
46C2×D188C2×C811yescharacteristic
47C2×C4×D47—21no—
48C2×C4×D47—22no—
49C2×Dic94C2×C811yescharacteristic
50M16×C47—41no—
51C2×C8×C47—21no—
52C2×C8×C47—41no—
53C2×C8×C47C2×D412yes—
54C4⋊C4×C47C2×C812yes—
55C22⋊C4×C47—21no—
56C42×C47—21no—
57(C2×D47)⋊C4—22no—
58Dic47⋊C4—22no—
59C4×Dic47—22no—
60C4×Dic47C22⋊C411yescharacteristic
61C4×Dic47C22⋊C411yescharacteristic
62C4×Dic47C2×C812yes—
63#10625329—41no—
64C2×C47⋊C8—21no—
65C2×C47⋊C8—41no—
66C2×C47⋊C8C22⋊C412yes—
67C16×C47—42no—
68C47⋊C16—42no—
69C23×C47C2×M1611yescharacteristic
70C22×D47—21no—
71C2×C4×C47—21no—
72C2×C4×C47—21no—
73C2×C4×C47—22no—
74C2×C4×C47C2×C22⋊C411yescharacteristic
75C2×C4×C47C22×C811yescharacteristic
76C22⋊D47—42no—
77C2×Dic47—24no—
78C2×Dic47—21no—
79C2×Dic47C22⋊C812yes—
80C2×Dic47C22⋊C812yes—
81D188—41no—
82C4×D47—41no—
83C4×D47—41no—
84C4×D47—44no—
85Dic94—41no—
86C8×C47—22no—
87C8×C47—41no—
88C47⋊C8—22no—
89C47⋊C8—41no—
90#11462—471nomaximal, Sylow
91C22×C47—22no—
92C22×C47—21no—
93C22×C47C2×C22⋊C811yescharacteristic
94C2×D47—42no—
95C2×D47—41no—
96C4×C47—21no—
97C4×C47—42no—
98C4×C47#41011yescharacteristic
99C4×C47#41311yescharacteristic
100Dic47—22no—
101Dic47—41no—
102Dic47#35014yes—
103#2922—471no—
104#2162—472no—
105#1323—471no—
106#1323D4711yescharacteristic, complemented
107#1315—472no—
108C2×C47—41no—
109C2×C47#145811yescharacteristic
110C2×C47C2×#35012yes—
111D47—81no—
112C4×#41—471no—
113#433—942no—
114#431—942no—
115#424—471no—
116#424C2×D4711yescharacteristic
117#408—471no—
118#408—471no—
119C4×M16—471no—
120C2×C4×C8—471no—
121C4⋊C16—942no—
122C22⋊C16—472no—
123C22⋊C16C2×D4712yes—
124C4×C16—942no—
125C47#1146211yescharacteristic, complemented, Sylow
126C2×#41—471no—
127C2×M16—471no—
128C2×M16C4×D4711yescharacteristic
129C22×C8—471no—
130C22×C8C22×D4711yescharacteristic
131C4×Q8—472no—
132C4×D4—472no—
133C4×D4—942no—
134#117—941no—
135#117C4×D4711yescharacteristic
136C2×C42—471no—
137C2×C42—941no—
138C2×C16—22no—
139C2×C16—942no—
140C4⋊C8—22no—
141C4⋊C8—942no—
142C22⋊C8—942no—
143C22⋊C8—942no—
144#97—941no—
145C4×C8—472no—
146C4×C8—472no—
147C4×C8—941no—
148#41—1881no—
149#41—1881no—
150C2×Q8—471no—
151C2×D4—471no—
152C2×D4—941no—
153C22×C4—472no—
154C22×C4—942no—
155C22×C4—941no—
156C22×C4C2×C4×D4711yescharacteristic
157M16—41no—
158M16—1881no—
159C2×C8—21no—
160C2×C8—41no—
161C2×C8—472no—
162C2×C8—941no—
163C2×C8—1881no—
164C2×C8D4×D4712yes—
165C4⋊C4—942no—
166C4⋊C4C8×D4712yes—
167C22⋊C4—21no—
168C22⋊C4—942no—
169C42—21no—
170C42—471no—
171C42—471no—
172C42—472no—
173C42—942no—
174C16—42no—
175C16—1882no—
176C23—941no—
177C23M16×D4711yescharacteristic
178Q8—1881no—
179D4—1881no—
180D4—1882no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—472no—
185C2×C4—472no—
186C2×C4—944no—
187C2×C4—941no—
188C2×C4—1884no—
189C2×C4—1881no—
190C2×C4—1881no—
191C2×C4C22⋊C4×D4711yescharacteristic
192C2×C4C2×C8×D4711yescharacteristic
193C8—22no—
194C8—41no—
195C8—942no—
196C8—1881no—
197C22—22no—
198C22—21no—
199C22—1882no—
200C22—1881no—
201C22C22⋊C8×D4711yescharacteristic
202C4—21no—
203C4—42no—
204C4—474no—
205C4—942no—
206C4—1881no—
207C4#1568909511yescharacteristic
208C4#1568910611yescharacteristic
209C2—41no—
210C2—3761no—
211C2#2024170111yescharacteristic
212C2#350×D4712yes—
subgroupG/H
centerC2×C4C22⋊C4×D47
commutatorC2×C4×C47C22×C8
FrattiniC22×C8C22×D47
FittingC47×#1323C2
socleC22×C47C2×C22⋊C8
prime radicalC2×#10625350C2×C4
cosocleC22×C8×C47C23
hypercenter#1323D47
hypocenterC47#11462
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #20245184C2semidirect
3. #20241330C2semidirect
5. C47×#1323C2semidirect
106. #1323D47semidirect
125. C47#11462semidirect

Maximal subgroups

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

subgroupindex
#202520312
#202451842
#202413302
#202413132
C47×#13232
#1146247