Finite Group Databasebeta

← #26109856

Subgroups — #26109856

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—44no—
20C2×C4⋊Dic47—22no—
21C2×C4⋊Dic47—22no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4×Dic47—22no—
25C2×C4×Dic47—21no—
26C4⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—22no—
28C4×C47⋊C8—22no—
29C47⋊#95—22no—
30C47⋊#95—21no—
31C47⋊#95C2×C411yescharacteristic
32C47⋊#95C2×C412yes—
33C47⋊#95C2×C411yescharacteristic
34C22×Q8×C47D411yescharacteristic
35C2×M16×C47—22no—
36C4×Q8×C47—22no—
37C4×Q8×C47—22no—
38C2×C4⋊C4×C47D412yes—
39C2×C4⋊C4×C47D411yescharacteristic
40C2×C42×C47C2×C412yes—
41C2×C42×C47C2311yescharacteristic
42C4⋊C8×C47—22no—
43C4⋊C8×C47—22no—
44#101×C47—42no—
45C2×Q8×C47—22no—
46C2×Q8×C47—22no—
47C2×Q8×C47SD1612yes—
48C2×Q8×C47Q1612yes—
49C22×C4×C47C22⋊C412yes—
50C22×C4×C47C22⋊C412yes—
51C22×C4×C47C22×C411yescharacteristic
52C22×C4×C47C2×D411yescharacteristic
53C22×C4×C47C2×D411yescharacteristic
54C22×Dic47—21no—
55C22×Dic47—21no—
56C22×Dic47—22no—
57M16×C47—42no—
58M16×C47—42no—
59C2×C8×C47—42no—
60C4⋊C4×C47—24no—
61C4⋊C4×C47—22no—
62C42×C47—22no—
63C42×C47—22no—
64C42×C47#4112yes—
65C42×C47#4112yes—
66C4⋊Dic47—42no—
67C4⋊Dic47—44no—
68C4⋊Dic47—44no—
69C4⋊Dic47—42no—
70C4×Dic47—44no—
71C4×Dic47—42no—
72C2×C47⋊C8—42no—
73C23×C47C2×C22⋊C411yescharacteristic
74Q8×C47—24no—
75Q8×C47—24no—
76C2×C4×C47—22no—
77C2×C4×C47—22no—
78C2×C4×C47—22no—
79C2×C4×C47—41no—
80C2×C4×C47—41no—
81C2×C4×C47—42no—
82C2×C4×C47#10314yes—
83C2×C4×C47#11711yescharacteristic
84C2×C4×C47#11711yescharacteristic
85C2×C4×C47#12312yes—
86C2×C4×C47#12312yes—
87C2×C4×C47C2×SD1611yescharacteristic
88C2×C4×C47C2×Q1611yescharacteristic
89C2×C4×C47AGL(1,ℤ/8)12yes—
90C2×Dic47—22no—
91C2×Dic47—41no—
92C2×Dic47—41no—
93C2×Dic47—41no—
94C2×Dic47—42no—
95C2×Dic47—41no—
96C2×Dic47—42no—
97C2×Dic47—42no—
98C2×Dic47—41no—
99C8×C47—82no—
100C47⋊C8—82no—
101#8767—471nomaximal, Sylow
102C22×C47—21no—
103C22×C47—21no—
104C22×C47#38111yescharacteristic
105C22×C47C2×#10311yescharacteristic
106C22×C47#41811yescharacteristic
107C4×C47—24no—
108C4×C47—22no—
109C4×C47—21no—
110C4×C47—21no—
111C4×C47—42no—
112C4×C47#48112yes—
113C4×C47#48412yes—
114Dic47—42no—
115Dic47—82no—
116Dic47—81no—
117#2351—471no—
118#1801D4711yescharacteristic
119#1461—471no—
120#1303—472no—
121#1286—472no—
122C2×C47—21no—
123C2×C47—21no—
124C2×C47#182411yescharacteristic
125C2×C47#182711yescharacteristic
126C2×C47#183011yescharacteristic
127C2×C4×Q8—941no—
128C2×C4×Q8C2×D4711yescharacteristic
129#431—472no—
130#423C2×D4712yes—
131C2×#101—21no—
132#389—942no—
133#389—941no—
134#382—472no—
135#382—471no—
136C4×C4⋊C4—472no—
137C4×C4⋊C4—471no—
138#343—21no—
139#324—942no—
140C47#876711yescharacteristic, complemented, Sylow
141C22×Q8C22⋊D4711yescharacteristic
142C2×M16—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—1884no—
146C2×C4⋊C4—942no—
147C2×C4⋊C4—942no—
148C2×C4⋊C4—941no—
149C2×C4⋊C4—941no—
150C2×C4⋊C4D18812yes—
151C2×C4⋊C4C22⋊D4711yescharacteristic
152C2×C42—942no—
153C2×C42—941no—
154C2×C42C4×D4712yes—
155C2×C42C22×D4711yescharacteristic
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—942no—
159#101—42no—
160C22⋊C8—942no—
161C4×C8—942no—
162#95—471no—
163#95—472no—
164#95—471no—
165#95—942no—
166#95—941no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8#1062533512yes—
170C2×Q8#1062533712yes—
171C22×C4—941no—
172C22×C4—941no—
173C22×C4—942no—
174C22×C4(C2×D47)⋊C412yes—
175C22×C4(C2×D47)⋊C412yes—
176C22×C4C2×C4×D4711yescharacteristic
177C22×C4C2×D18811yescharacteristic
178C22×C4C2×C22⋊D4711yescharacteristic
179M16—42no—
180M16—42no—
181C2×C8—42no—
182C2×C8—1882no—
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—1884no—
186C4⋊C4—1882no—
187C4⋊C4—1884no—
188C4⋊C4—1882no—
189C42—22no—
190C42—22no—
191C42—1884no—
192C42—1882no—
193C42#1062535212yes—
194C42#1062535212yes—
195C23C2×(C2×D47)⋊C411yescharacteristic
196Q8—24no—
197Q8—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—942no—
205C2×C4—1881no—
206C2×C4—1882no—
207C2×C4—1881no—
208C2×C4—1882no—
209C2×C4—1882no—
210C2×C4—1881no—
211C2×C4—1881no—
212C2×C4—1881no—
213C2×C4#1309394714yes—
214C2×C4#1309398011yescharacteristic
215C2×C4#1309398011yescharacteristic
216C2×C4#1309400512yes—
217C2×C4#1309400812yes—
218C2×C4C2×#1062533511yescharacteristic
219C2×C4C2×#1062533711yescharacteristic
220C2×C4#1309404912yes—
221C8—82no—
222C8—3762no—
223C22—21no—
224C22—21no—
225C22#1568898411yescharacteristic
226C22C2×#1309394711yescharacteristic
227C22#1568912711yescharacteristic
228C4—24no—
229C4—22no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—3762no—
234C4—3761no—
235C4—1882no—
236C4#1568943912yes—
237C4#1568945612yes—
238C2—21no—
239C2—21no—
240C2#2024347611yescharacteristic
241C2#2024349311yescharacteristic
242C2#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#8767
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
140. C47#8767semidirect

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
#876747