Finite Group Databasebeta

← #26105104

Subgroups — #26105104

Order 12032.

468 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 87 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#20250420C211yesmaximal, characteristic
2#20244713C212yesmaximal
3#20241214C211yesmaximal, characteristic
4#20241212C212yesmaximal
5C47×#2016C211yesmaximal, characteristic
6C2×#13094022—21no—
7C2×#13093957—22no—
8#15689053—21no—
9#15689044—22no—
10#15689021C2212yes—
11#15689008—22no—
12#15688885—22no—
13#15688878C2212yes—
14C22×Q16×C47—21no—
15C2×C4⋊Q8×C47C2211yescharacteristic
16C2×#103×C47—21no—
17C47×#391C2211yescharacteristic
18C47×#336C2211yescharacteristic
19C22×Dic94—21no—
20C22×C47⋊C8—22no—
21#13094022—42no—
22#13094022—42no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47D412yes—
25C2×C4⋊Dic47D412yes—
26C2×Dic47⋊C4—22no—
27C2×Dic47⋊C4—22no—
28C2×C4×Dic47—21no—
29#13093957—44no—
30C47⋊#95—21no—
31C47⋊#95—22no—
32C47⋊#95—22no—
33C22×Q8×C47—21no—
34C22×Q8×C47D411yescharacteristic
35C2×Q16×C47—42no—
36C2×Q16×C47—42no—
37C2×Q16×C47—42no—
38C22×C8×C47—21no—
39C4⋊Q8×C47—42no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C42×C47C2311yescharacteristic
43#103×C47—42no—
44#95×C47—22no—
45C2×Q8×C47—21no—
46C2×Q8×C47—22no—
47C2×Q8×C47—41no—
48C2×Q8×C47—41no—
49C2×Q8×C47—41no—
50C2×Q8×C47—41no—
51C2×Q8×C47—41no—
52C2×Q8×C47—41no—
53C2×Q8×C47—41no—
54C2×Q8×C47D812yes—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—41no—
58C22×C4×C47C2×D411yescharacteristic
59C22×C4×C47C2×D412yes—
60C22×Dic47—22no—
61C22×Dic47—22no—
62C2×Dic94—42no—
63C2×Dic94—81no—
64C2×Dic94—81no—
65Q16×C47—48no—
66C2×C8×C47—22no—
67C2×C8×C47—41no—
68C2×C8×C47—41no—
69C4⋊C4×C47—24no—
70C4⋊C4×C47—42no—
71C4⋊C4×C47—41no—
72C42×C47—41no—
73C4⋊Dic47—22no—
74C4⋊Dic47—42no—
75C4⋊Dic47—81no—
76C4⋊Dic47Q1614yescomplemented
77Dic47⋊C4—44no—
78Dic47⋊C4—42no—
79Dic47⋊C4—42no—
80C4×Dic47—42no—
81C2×C47⋊C8—24no—
82C2×C47⋊C8—42no—
83C2×C47⋊C8—42no—
84C23×C47#12011yescharacteristic
85Q8×C47—42no—
86Q8×C47—42no—
87Q8×C47—42no—
88Q8×C47—81no—
89Q8×C47—81no—
90Q8×C47—81no—
91C2×C4×C47—21no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—21no—
95C2×C4×C47—22no—
96C2×C4×C47—41no—
97C2×C4×C47—41no—
98C2×C4×C47—41no—
99C2×C4×C47—42no—
100C2×C4×C47—42no—
101C2×C4×C47—81no—
102C2×C4×C47—81no—
103C2×C4×C47C2×D811yescharacteristic
104C2×C4×C47C2×Q1612yes—
105C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
106C2×C4×C47#13712yes—
107C2×Dic47—24no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113Dic94—84no—
114C8×C47—42no—
115C47⋊C8—44no—
116#7757—471nomaximal, Sylow
117C22×C47#44711yescharacteristic
118C22×C47#45112yes—
119C22×C47#45311yescharacteristic
120C22×C47#45612yes—
121C22×C47B(4,𝔽2)11yescharacteristic
122C4×C47—21no—
123C4×C47—21no—
124C4×C47—22no—
125C4×C47—22no—
126C4×C47—42no—
127C4×C47—82no—
128C4×C47—81no—
129Dic47—44no—
130Dic47—82no—
131#2734—471no—
132#2057—472no—
133#2016D4711yescharacteristic
134#1272—471no—
135#1271—472no—
136C2×C47#159511yescharacteristic
137C2×C47#160512yes—
138C2×C47#163311yescharacteristic
139C2×C47#164912yes—
140C2×C47#166011yescharacteristic
141C22×Q16—21no—
142C2×C4⋊Q8—941no—
143C2×C4⋊Q8C2×D4711yescharacteristic
144C2×#107—942no—
145C2×#103—21no—
146#398—941no—
147#395—942no—
148#391C2×D4711yescharacteristic
149#389—472no—
150#387—942no—
151#340—942no—
152#336—472no—
153#336C2×D4711yescharacteristic
154C47#775711yescharacteristic, complemented, Sylow
155C22×Q8—21no—
156C22×Q8—941no—
157C22×Q8C22⋊D4711yescharacteristic
158C2×Q16—42no—
159C2×Q16—42no—
160C2×Q16—42no—
161C22×C8—21no—
162C22×C8—942no—
163C4⋊Q8—42no—
164C4⋊Q8—1882no—
165C4⋊Q8—1882no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—472no—
168C2×C4⋊C4—472no—
169C2×C4⋊C4—941no—
170C2×C4⋊C4—942no—
171C2×C4⋊C4—942no—
172C2×C4⋊C4C22⋊D4711yescharacteristic
173C2×C42—941no—
174C2×C42C22×D4711yescharacteristic
175#107—1884no—
176#103—42no—
177#95—22no—
178#95—941no—
179#95—942no—
180#95—942no—
181C2×Q8—21no—
182C2×Q8—22no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—41no—
190C2×Q8—3761no—
191C2×Q8—3761no—
192C2×Q8—1882no—
193C2×Q8#1062533412yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—41no—
197C22×C4—942no—
198C22×C4—942no—
199C22×C4D4×D4712yes—
200C22×C4C2×C22⋊D4711yescharacteristic
201Q16—48no—
202C2×C8—22no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—944no—
206C2×C8—1882no—
207C2×C8—1882no—
208C4⋊C4—24no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—474no—
212C4⋊C4—942no—
213C4⋊C4—3761no—
214C4⋊C4—1882no—
215C4⋊C4—1884no—
216C4⋊C4—1882no—
217C4⋊C4—1882no—
218C42—41no—
219C42—1882no—
220C23#1309399211yescharacteristic
221Q8—42no—
222Q8—42no—
223Q8—42no—
224Q8—81no—
225Q8—81no—
226Q8—81no—
227Q8—3764no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—944no—
241C2×C4—1882no—
242C2×C4—1882no—
243C2×C4—1882no—
244C2×C4—1882no—
245C2×C4—1882no—
246C2×C4C2×#1062533411yescharacteristic
247C2×C4Q16×D4712yes—
248C2×C4#1309404911yescharacteristic
249C2×C4#1309405612yes—
250C8—42no—
251C8—1884no—
252C22#1568925611yescharacteristic
253C22#1568927712yes—
254C22#1568929011yescharacteristic
255C22#1568930412yes—
256C22#1568930911yescharacteristic
257C4—21no—
258C4—21no—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—81no—
263C4—82no—
264C4—3762no—
265C4—1884no—
266C2#2024230511yescharacteristic
267C2#2024236012yes—
268C2#2024252611yescharacteristic
269C2#2024263512yes—
270C2#2024269711yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2016C2
socle, prime radicalC23×C47#120
hypercenter#2016D47
hypocenterC47#7757
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2016

Complements

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

NH ≅ G/Ntype
76. C4⋊Dic47Q16semidirect
154. C47#7757semidirect

Maximal subgroups

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

subgroupindex
#202504202
#202447132
#202412142
#202412122
C47×#20162
#775747