Finite Group Databasebeta

← #26109851

Subgroups — #26109851

Order 12032.

422 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 123 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#20246309C211yesmaximal, characteristic
2#20241712C211yesmaximal, characteristic
3#20241289C212yesmaximal
4#20241248C212yesmaximal
5C47×#1801C211yesmaximal, characteristic
6C2×Q8×Dic47—21no—
7#15689176C2212yes—
8#15689022—22no—
9#15689022—21no—
10#15688989C412yes—
11#15688989C2211yescharacteristic
12C4×C4⋊Dic47C412yes—
13C4×C4⋊Dic47C2211yescharacteristic
14#15688849—22no—
15C2×C4×Q8×C47C2211yescharacteristic
16C47×#423C2212yes—
17C2×#101×C47—21no—
18C47×#343—21no—
19Q8×Dic47—42no—
20Q8×Dic47—42no—
21C2×C4⋊Dic47—22no—
22C2×C4⋊Dic47—22no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4×Dic47—22no—
26C2×C4×Dic47—21no—
27C4⋊(C47⋊C8)—22no—
28C22⋊(C47⋊C8)—22no—
29C4×C47⋊C8—22no—
30C47⋊#95—22no—
31C47⋊#95—21no—
32C47⋊#95C2×C411yescharacteristic
33C47⋊#95C2×C412yes—
34C47⋊#95C2×C411yescharacteristic
35C22×Q8×C47D411yescharacteristic
36C2×M16×C47—22no—
37C4×Q8×C47—22no—
38C4×Q8×C47—22no—
39C2×C4⋊C4×C47D412yes—
40C2×C4⋊C4×C47D411yescharacteristic
41C2×C42×C47C2×C412yes—
42C2×C42×C47C2311yescharacteristic
43C4⋊C8×C47—22no—
44C4⋊C8×C47—22no—
45#101×C47—42no—
46C2×Q8×C47—22no—
47C2×Q8×C47—22no—
48C2×Q8×C47D812yes—
49C2×Q8×C47SD1612yes—
50C22×C4×C47C22⋊C412yes—
51C22×C4×C47C22⋊C412yes—
52C22×C4×C47C22×C411yescharacteristic
53C22×C4×C47C2×D411yescharacteristic
54C22×C4×C47C2×D411yescharacteristic
55C22×Dic47—21no—
56C22×Dic47—21no—
57C22×Dic47—22no—
58M16×C47—42no—
59M16×C47—42no—
60C2×C8×C47—42no—
61C4⋊C4×C47—24no—
62C4⋊C4×C47—22no—
63C42×C47—22no—
64C42×C47—22no—
65C42×C47#4112yes—
66C42×C47#4112yes—
67C4⋊Dic47—42no—
68C4⋊Dic47—44no—
69C4⋊Dic47—44no—
70C4⋊Dic47—41no—
71C4⋊Dic47—41no—
72C4×Dic47—44no—
73C4×Dic47—41no—
74C4×Dic47—41no—
75C2×C47⋊C8—42no—
76C23×C47C2×C22⋊C411yescharacteristic
77Q8×C47—24no—
78Q8×C47—24no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—22no—
82C2×C4×C47—41no—
83C2×C4×C47—41no—
84C2×C4×C47—42no—
85C2×C4×C47#10214yes—
86C2×C4×C47#11711yescharacteristic
87C2×C4×C47#11711yescharacteristic
88C2×C4×C47#12312yes—
89C2×C4×C47#12312yes—
90C2×C4×C47C2×D811yescharacteristic
91C2×C4×C47C2×SD1611yescharacteristic
92C2×C4×C47#13712yes—
93C2×Dic47—22no—
94C2×Dic47—41no—
95C2×Dic47—41no—
96C2×Dic47—41no—
97C2×Dic47—42no—
98C2×Dic47—41no—
99C2×Dic47—42no—
100C2×Dic47—42no—
101C2×Dic47—41no—
102C8×C47—82no—
103C47⋊C8—82no—
104#8766—471nomaximal, Sylow
105C22×C47—21no—
106C22×C47—21no—
107C22×C47#38111yescharacteristic
108C22×C47C2×#10211yescharacteristic
109C22×C47#41911yescharacteristic
110C4×C47—24no—
111C4×C47—22no—
112C4×C47—21no—
113C4×C47—21no—
114C4×C47—42no—
115C4×C47#48012yes—
116C4×C47#48312yes—
117Dic47—42no—
118Dic47—82no—
119Dic47—81no—
120#2351—471no—
121#1801D4711yescharacteristic
122#1461—471no—
123#1303—472no—
124#1286—472no—
125C2×C47—21no—
126C2×C47—21no—
127C2×C47#182311yescharacteristic
128C2×C47#182711yescharacteristic
129C2×C47#183011yescharacteristic
130C2×C4×Q8—941no—
131C2×C4×Q8C2×D4711yescharacteristic
132#431—472no—
133#423C2×D4712yes—
134C2×#101—21no—
135#389—942no—
136#389—941no—
137#382—472no—
138#382—471no—
139C4×C4⋊C4—472no—
140C4×C4⋊C4—471no—
141#343—21no—
142#324—942no—
143C47#876611yescharacteristic, complemented, Sylow
144C22×Q8C22⋊D4711yescharacteristic
145C2×M16—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—1882no—
149C4×Q8—1882no—
150C2×C4⋊C4—942no—
151C2×C4⋊C4—942no—
152C2×C4⋊C4—941no—
153C2×C4⋊C4—941no—
154C2×C4⋊C4D18812yes—
155C2×C4⋊C4C22⋊D4711yescharacteristic
156C2×C42—942no—
157C2×C42—941no—
158C2×C42C4×D4712yes—
159C2×C42C22×D4711yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—942no—
163#101—42no—
164C22⋊C8—942no—
165C4×C8—942no—
166#95—471no—
167#95—472no—
168#95—471no—
169#95—942no—
170#95—941no—
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8#1062533412yes—
174C2×Q8#1062533612yes—
175C22×C4—941no—
176C22×C4—941no—
177C22×C4—942no—
178C22×C4(C2×D47)⋊C412yes—
179C22×C4(C2×D47)⋊C412yes—
180C22×C4C2×C4×D4711yescharacteristic
181C22×C4C2×D18811yescharacteristic
182C22×C4C2×C22⋊D4711yescharacteristic
183M16—42no—
184M16—42no—
185C2×C8—42no—
186C2×C8—1882no—
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—1882no—
190C4⋊C4—1884no—
191C4⋊C4—1884no—
192C4⋊C4—1881no—
193C4⋊C4—1881no—
194C42—22no—
195C42—22no—
196C42—1884no—
197C42—1881no—
198C42—1881no—
199C42#1062535212yes—
200C42#1062535212yes—
201C23C2×(C2×D47)⋊C411yescharacteristic
202Q8—24no—
203Q8—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—942no—
211C2×C4—1881no—
212C2×C4—1882no—
213C2×C4—1881no—
214C2×C4—1882no—
215C2×C4—1882no—
216C2×C4—1881no—
217C2×C4—1881no—
218C2×C4—1881no—
219C2×C4#1309394414yes—
220C2×C4#1309398011yescharacteristic
221C2×C4#1309398011yescharacteristic
222C2×C4#1309400512yes—
223C2×C4#1309400812yes—
224C2×C4C2×#1062533411yescharacteristic
225C2×C4C2×#1062533611yescharacteristic
226C2×C4#1309405412yes—
227C8—82no—
228C8—3762no—
229C22—21no—
230C22—21no—
231C22#1568898411yescharacteristic
232C22C2×#1309394411yescharacteristic
233C22#1568913111yescharacteristic
234C4—24no—
235C4—22no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—3762no—
240C4—3761no—
241C4—1882no—
242C4#1568943412yes—
243C4#1568945112yes—
244C2—21no—
245C2—21no—
246C2#2024347211yescharacteristic
247C2#2024349311yescharacteristic
248C2#2024350011yescharacteristic
subgroupG/H
centerC22#15688984
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1801C2
socleC22×C47#381
prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenter#1801D47
hypocenterC47#8766
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
143. C47#8766semidirect

Maximal subgroups

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

subgroupindex
#202463092
#202417122
#202412892
#202412482
C47×#18012
#876647