Finite Group Databasebeta

← #26109871

Subgroups — #26109871

Order 12032.

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

normal subgroups only

This group has 426 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#20246361C211yesmaximal, characteristic
2#20241709C211yesmaximal, characteristic
3#20241289C212yesmaximal, complemented
4#20241248C212yesmaximal, complemented
5C47×#1800C211yesmaximal, characteristic
6#15689176C2212yes—
7#15689026—21no—
8#15689016—22no—
9#15688989C2211yescharacteristic
10#15688986C412yes—
11C4⋊C4×Dic47—21no—
12C4×C4⋊Dic47C2211yescharacteristic
13C4×C22⋊Dic47C412yes—
14#15688847—22no—
15C2×C4×D4×C47C2211yescharacteristic
16C47×#423C2212yes—
17C2×#100×C47—21no—
18C47×#343—21no—
19C2×C4⋊Dic47—21no—
20C2×C4⋊Dic47—21no—
21C2×C22⋊Dic47—22no—
22C2×C22⋊Dic47—22no—
23C2×C4×Dic47—21no—
24C2×C4×Dic47—22no—
25C4⋊(C47⋊C8)—22no—
26C22⋊(C47⋊C8)—22no—
27C4×C47⋊C8—22no—
28C47⋊#95—22no—
29C47⋊#95—21no—
30C47⋊#95C2×C411yescharacteristic
31C47⋊#95C2×C412yes—
32C47⋊#95C2×C411yescharacteristic
33C22×D4×C47D411yescharacteristic
34C23×C4×C47C2×C412yes—
35C2×M16×C47—22no—
36C4×D4×C47—22no—
37C4×D4×C47—22no—
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C22⋊C4×C47D412yes—
40C2×C42×C47C2311yescharacteristic
41C4⋊C8×C47—22no—
42C4⋊C8×C47—22no—
43#100×C47—42no—
44C24×C47C22⋊C412yes—
45C2×D4×C47—22no—
46C2×D4×C47—22no—
47C2×D4×C47—21no—
48C2×D4×C47—21no—
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—82no—
52C22×C4×C47C22⋊C412yes—
53C22×C4×C47C22×C411yescharacteristic
54C22×C4×C47C2×D411yescharacteristic
55C22×C4×C47C2×D411yescharacteristic
56C22×Dic47—21no—
57C22×Dic47—22no—
58C22×Dic47—21no—
59M16×C47—42no—
60M16×C47—42no—
61C2×C8×C47—42no—
62C4⋊C4×C47—22no—
63C22⋊C4×C47—24no—
64C42×C47#4112yes—
65C42×C47#4112yes—
66C22⋊Dic47—44no—
67C22⋊Dic47—82no—
68C4⋊Dic47—42no—
69C4⋊Dic47—42no—
70C4×Dic47—41no—
71C4×Dic47—41no—
72C4×Dic47—82no—
73C2×C47⋊C8—42no—
74C23×C47—22no—
75C23×C47—22no—
76C23×C47—22no—
77C23×C47—42no—
78C23×C47—42no—
79C23×C47C2×C22⋊C411yescharacteristic
80D4×C47—42no—
81D4×C47—42no—
82C2×C4×C47—22no—
83C2×C4×C47—41no—
84C2×C4×C47—41no—
85C2×C4×C47—42no—
86C2×C4×C47—82no—
87C2×C4×C47—82no—
88C2×C4×C47#11711yescharacteristic
89C2×C4×C47#11711yescharacteristic
90C2×C4×C47#12312yes—
91C2×C4×C47#12312yes—
92C2×C4×C47#13512yes—
93C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
94C2×C4×C47#13711yescharacteristic
95C2×Dic47—22no—
96C2×Dic47—41no—
97C2×Dic47—41no—
98C2×Dic47—42no—
99C2×Dic47—41no—
100C2×Dic47—42no—
101C2×Dic47—41no—
102C2×Dic47—42no—
103C2×Dic47—41no—
104C8×C47—82no—
105C47⋊C8—82no—
106#8770—471nomaximal, Sylow
107C22×C47—21no—
108C22×C47—21no—
109C22×C47—42no—
110C22×C47—42no—
111C22×C47—42no—
112C22×C47—82no—
113C22×C47—82no—
114C22×C47#38111yescharacteristic
115C22×C47#41611yescharacteristic
116C22×C47#41711yescharacteristic
117C4×C47—22no—
118C4×C47—21no—
119C4×C47—21no—
120C4×C47—42no—
121C4×C47#48212yes—
122C4×C47#48512yes—
123Dic47—42no—
124Dic47—81no—
125Dic47—82no—
126#2357—471no—
127#1800D4711yescharacteristic
128#1460—471no—
129#1303—472no—
130#1286—472no—
131C2×C47—21no—
132C2×C47—21no—
133C2×C47—82no—
134C2×C47#182511yescharacteristic
135C2×C47#182711yescharacteristic
136C2×C47#182911yescharacteristic
137C2×C4×D4C2×D4711yescharacteristic
138#431—472no—
139#423C2×D4712yes—
140C2×#100—21no—
141#390—941no—
142#388—942no—
143#382—471no—
144#381—472no—
145C4×C4⋊C4—471no—
146C4×C4⋊C4—941no—
147C4×C22⋊C4—472no—
148#343—21no—
149#323—942no—
150C47#877011yescharacteristic, complemented, Sylow
151C22×D4C22⋊D4711yescharacteristic
152C23×C4C4×D4712yes—
153C2×M16—22no—
154C4×D4—22no—
155C4×D4—22no—
156C2×C4⋊C4—941no—
157C2×C4⋊C4—941no—
158C2×C4⋊C4C22⋊D4711yescharacteristic
159C2×C22⋊C4—942no—
160C2×C22⋊C4—942no—
161C2×C22⋊C4D18812yes—
162C2×C42—942no—
163C2×C42—941no—
164C2×C42C22×D4711yescharacteristic
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—942no—
168#100—42no—
169C22⋊C8—942no—
170C4×C8—942no—
171#95—471no—
172#95—472no—
173#95—471no—
174#95—942no—
175#95—941no—
176C24(C2×D47)⋊C412yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—21no—
180C2×D4—21no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—82no—
184C22×C4—941no—
185C22×C4—942no—
186C22×C4—941no—
187C22×C4(C2×D47)⋊C412yes—
188C22×C4C2×C4×D4711yescharacteristic
189C22×C4C2×D18811yescharacteristic
190C22×C4C2×C22⋊D4711yescharacteristic
191M16—42no—
192M16—42no—
193C2×C8—42no—
194C2×C8—1882no—
195C4⋊C4—22no—
196C4⋊C4—1882no—
197C4⋊C4—1882no—
198C22⋊C4—24no—
199C22⋊C4—3762no—
200C22⋊C4—1884no—
201C42—3762no—
202C42—1881no—
203C42—1881no—
204C42#1062535212yes—
205C42#1062535212yes—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—42no—
210C23—42no—
211C23C2×(C2×D47)⋊C411yescharacteristic
212D4—42no—
213D4—42no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—82no—
219C2×C4—82no—
220C2×C4—942no—
221C2×C4—1882no—
222C2×C4—1881no—
223C2×C4—1882no—
224C2×C4—1881no—
225C2×C4—1882no—
226C2×C4—1881no—
227C2×C4—1881no—
228C2×C4—1881no—
229C2×C4#1309398011yescharacteristic
230C2×C4#1309398011yescharacteristic
231C2×C4#1309400512yes—
232C2×C4#1309400812yes—
233C2×C4#1309404412yes—
234C2×C4#1309405011yescharacteristic
235C2×C4#1309405511yescharacteristic
236C8—82no—
237C8—3762no—
238C22—21no—
239C22—21no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—82no—
244C22—82no—
245C22#1568898411yescharacteristic
246C22#1568911711yescharacteristic
247C22#1568912211yescharacteristic
248C4—22no—
249C4—21no—
250C4—21no—
251C4—42no—
252C4—3762no—
253C4—3761no—
254C4—1882no—
255C4#1568944412yes—
256C4#1568946112yes—
257C2—21no—
258C2—21no—
259C2—82no—
260C2#2024348011yescharacteristic
261C2#2024349311yescharacteristic
262C2#2024349711yescharacteristic
subgroupG/H
centerC22#15688984
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1800C2
socleC22×C47#381
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenter#1800D47
hypocenterC47#8770
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #20241289C2semidirect
4. #20241248C2semidirect
150. C47#8770semidirect

Maximal subgroups

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

subgroupindex
#202463612
#202417092
#202412892
#202412482
C47×#18002
#877047