Finite Group Databasebeta

← #26109861

Subgroups — #26109861

Order 12032.

378 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 107 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#20246353C211yesmaximal, characteristic
2#20241712C211yesmaximal, characteristic
3#20241289C212yesmaximal
4#20241248C212yesmaximal
5C47×#1801C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689030—21no—
8#15689009—22no—
9#15688991C412yes—
10#15688989C2211yescharacteristic
11C4⋊C4×Dic47—21no—
12C4×C4⋊Dic47C2211yescharacteristic
13#15688969C412yes—
14#15688849—22no—
15C2×C4×Q8×C47C2211yescharacteristic
16C47×#423C2212yes—
17C2×#101×C47—21no—
18C47×#343—21no—
19C2×C4⋊Dic47—21no—
20C2×C4⋊Dic47—21no—
21C2×C4×Dic47—22no—
22C2×C4×Dic47—21no—
23C4⋊(C47⋊C8)—22no—
24C22⋊(C47⋊C8)—22no—
25C4×C47⋊C8—22no—
26C47⋊#95—22no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95—21no—
30C47⋊#95C2×C411yescharacteristic
31C47⋊#95C2×C412yes—
32C47⋊#95C2×C411yescharacteristic
33C22×Q8×C47D411yescharacteristic
34C2×M16×C47—22no—
35C4×Q8×C47—22no—
36C4×Q8×C47—22no—
37C2×C4⋊C4×C47D412yes—
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C42×C47C2×C412yes—
40C2×C42×C47C2311yescharacteristic
41C4⋊C8×C47—22no—
42C4⋊C8×C47—22no—
43#101×C47—42no—
44C2×Q8×C47—22no—
45C2×Q8×C47—22no—
46C2×Q8×C47—21no—
47C2×Q8×C47—21no—
48C22×C4×C47C22⋊C412yes—
49C22×C4×C47C22⋊C412yes—
50C22×C4×C47C22×C411yescharacteristic
51C22×C4×C47C2×D411yescharacteristic
52C22×C4×C47C2×D411yescharacteristic
53C22×Dic47—21no—
54C22×Dic47—22no—
55C22×Dic47—21no—
56M16×C47—42no—
57M16×C47—42no—
58C2×C8×C47—42no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—24no—
61C42×C47—22no—
62C42×C47—22no—
63C42×C47#4112yes—
64C42×C47#4112yes—
65C4⋊Dic47—42no—
66C4⋊Dic47—42no—
67C4×Dic47—44no—
68C4×Dic47—41no—
69C4×Dic47—41no—
70C2×C47⋊C8—42no—
71C23×C47C2×C22⋊C411yescharacteristic
72Q8×C47—42no—
73Q8×C47—42no—
74C2×C4×C47—22no—
75C2×C4×C47—22no—
76C2×C4×C47—22no—
77C2×C4×C47—22no—
78C2×C4×C47—41no—
79C2×C4×C47—41no—
80C2×C4×C47—42no—
81C2×C4×C47#11711yescharacteristic
82C2×C4×C47#11711yescharacteristic
83C2×C4×C47#12312yes—
84C2×C4×C47#12312yes—
85C2×C4×C47#13512yes—
86C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
87C2×C4×C47#13711yescharacteristic
88C2×Dic47—22no—
89C2×Dic47—41no—
90C2×Dic47—41no—
91C2×Dic47—42no—
92C2×Dic47—41no—
93C2×Dic47—41no—
94C2×Dic47—42no—
95C2×Dic47—42no—
96C2×Dic47—41no—
97C8×C47—82no—
98C47⋊C8—82no—
99#8768—471nomaximal, Sylow
100C22×C47—21no—
101C22×C47—21no—
102C22×C47#38111yescharacteristic
103C22×C47#41611yescharacteristic
104C22×C47#41711yescharacteristic
105C4×C47—22no—
106C4×C47—21no—
107C4×C47—21no—
108C4×C47—42no—
109C4×C47—42no—
110C4×C47#48212yes—
111C4×C47#48512yes—
112Dic47—42no—
113Dic47—81no—
114Dic47—82no—
115#2356—471no—
116#1801D4711yescharacteristic
117#1461—471no—
118#1303—472no—
119#1286—472no—
120C2×C47—21no—
121C2×C47—21no—
122C2×C47#182511yescharacteristic
123C2×C47#182711yescharacteristic
124C2×C47#183011yescharacteristic
125C2×C4×Q8C2×D4711yescharacteristic
126#431—472no—
127#423C2×D4712yes—
128C2×#101—21no—
129#391—941no—
130#387—942no—
131#383—472no—
132#382—471no—
133C4×C4⋊C4—471no—
134C4×C4⋊C4—941no—
135#376—472no—
136#343—21no—
137#324—942no—
138C47#876811yescharacteristic, complemented, Sylow
139C22×Q8C22⋊D4711yescharacteristic
140C2×M16—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—941no—
144C2×C4⋊C4—941no—
145C2×C4⋊C4D18812yes—
146C2×C4⋊C4C22⋊D4711yescharacteristic
147C2×C42—942no—
148C2×C42—941no—
149C2×C42C4×D4712yes—
150C2×C42C22×D4711yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—942no—
154#101—42no—
155C22⋊C8—942no—
156C4×C8—942no—
157#95—471no—
158#95—472no—
159#95—471no—
160#95—942no—
161#95—942no—
162#95—942no—
163#95—941no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—941no—
169C22×C4—942no—
170C22×C4—941no—
171C22×C4(C2×D47)⋊C412yes—
172C22×C4(C2×D47)⋊C412yes—
173C22×C4C2×C4×D4711yescharacteristic
174C22×C4C2×D18811yescharacteristic
175C22×C4C2×C22⋊D4711yescharacteristic
176M16—42no—
177M16—42no—
178C2×C8—42no—
179C2×C8—1882no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—1882no—
183C4⋊C4—1882no—
184C42—22no—
185C42—22no—
186C42—1884no—
187C42—1881no—
188C42—1881no—
189C42#1062535212yes—
190C42#1062535212yes—
191C23C2×(C2×D47)⋊C411yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—942no—
202C2×C4—1882no—
203C2×C4—1881no—
204C2×C4—1882no—
205C2×C4—1881no—
206C2×C4—1882no—
207C2×C4—1881no—
208C2×C4—1881no—
209C2×C4—1881no—
210C2×C4#1309398011yescharacteristic
211C2×C4#1309398011yescharacteristic
212C2×C4#1309400512yes—
213C2×C4#1309400812yes—
214C2×C4#1309404412yes—
215C2×C4#1309405011yescharacteristic
216C2×C4#1309405511yescharacteristic
217C8—82no—
218C8—3762no—
219C22—21no—
220C22—21no—
221C22#1568898411yescharacteristic
222C22#1568911711yescharacteristic
223C22#1568912211yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—3762no—
230C4—3761no—
231C4—1882no—
232C4#1568944412yes—
233C4#1568946112yes—
234C2—21no—
235C2—21no—
236C2#2024348011yescharacteristic
237C2#2024349311yescharacteristic
238C2#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#8768
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
138. C47#8768semidirect

Maximal subgroups

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

subgroupindex
#202463532
#202417122
#202412892
#202412482
C47×#18012
#876847