Finite Group Databasebeta

← #26129125

Subgroups — #26129125

Order 12032.

336 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 59 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#20251376C211yesmaximal, characteristic
2#20242430C212yesmaximal
3#20241414C212yesmaximal
4#20241322C211yesmaximal, characteristic
5C47×#2028C211yesmaximal, characteristic
6#15689456—22no—
7#15689062C2212yes—
8#15689055—22no—
9#15689030—21no—
10#15689021—21no—
11#15689010—22no—
12#15688860—22no—
13#15688849C2212yes—
14C47×#544C2211yescharacteristic
15C47×#419—21no—
16C47×#391C2211yescharacteristic
17C47×#343C2211yescharacteristic
18C47×#339—21no—
19C2×C4⋊Dic47—21no—
20C2×C4⋊Dic47—21no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47D412yes—
23C2×Dic47⋊C4—22no—
24C2×C4×Dic47—21no—
25#13093957—42no—
26#13093948—42no—
27C22⋊(C47⋊C8)—22no—
28C47⋊#95—22no—
29C47⋊#95—22no—
30C47⋊#95—22no—
31C47⋊#95D412yes—
32C22×Q8×C47D411yescharacteristic
33C2×M16×C47—21no—
34C4⋊Q8×C47—41no—
35#125×C47—41no—
36C22⋊Q8×C47—22no—
37#117×C47—21no—
38C2×C4⋊C4×C47C2311yescharacteristic
39C2×C42×C47D411yescharacteristic
40#103×C47—41no—
41#103×C47—41no—
42#95×C47—22no—
43C2×Q8×C47—21no—
44C2×Q8×C47—22no—
45C2×Q8×C47—41no—
46C2×Q8×C47—41no—
47C2×Q8×C47Q1612yes—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47C2×D411yescharacteristic
51C22×C4×C47C2×D412yes—
52C22×Dic47—22no—
53C22×Dic47—22no—
54M16×C47—41no—
55M16×C47—41no—
56C2×C8×C47—41no—
57C4⋊C4×C47—21no—
58C4⋊C4×C47—21no—
59C4⋊C4×C47—22no—
60C4⋊C4×C47—41no—
61C4⋊C4×C47—42no—
62C22⋊C4×C47—41no—
63C42×C47—41no—
64C42×C47—41no—
65C4⋊Dic47—42no—
66C4⋊Dic47—42no—
67C4⋊Dic47—81no—
68C4⋊Dic47—81no—
69Dic47⋊C4—44no—
70C4×Dic47—81no—
71C2×C47⋊C8—42no—
72C23×C47#12011yescharacteristic
73Q8×C47—42no—
74Q8×C47—81no—
75Q8×C47—82no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—22no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—41no—
82C2×C4×C47—41no—
83C2×C4×C47—41no—
84C2×C4×C47—41no—
85C2×C4×C47—42no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47C2×Q1611yescharacteristic
89C2×C4×C47#13711yescharacteristic
90C2×Dic47—42no—
91C2×Dic47—42no—
92C2×Dic47—42no—
93C2×Dic47—42no—
94C2×Dic47—42no—
95C2×Dic47—42no—
96C8×C47—81no—
97C47⋊C8—82no—
98#12225—471nomaximal, Sylow
99C22×C47—21no—
100C22×C47—21no—
101C22×C47#45111yescharacteristic
102C22×C47#45511yescharacteristic
103C22×C47B(4,𝔽2)11yescharacteristic
104C4×C47—21no—
105C4×C47—21no—
106C4×C47—42no—
107C4×C47—42no—
108C4×C47—42no—
109C4×C47—81no—
110C4×C47—81no—
111Dic47—82no—
112Dic47—82no—
113#2809—471no—
114#2028D4711yescharacteristic
115#1617—472no—
116#1353—472no—
117#1319—471no—
118C2×C47—21no—
119C2×C47—21no—
120C2×C47#160211yescharacteristic
121C2×C47#220111yescharacteristic
122C2×C47#220411yescharacteristic
123#544C2×D4711yescharacteristic
124#484—942no—
125#419—21no—
126#400—472no—
127#398—942no—
128#391—941no—
129#391C2×D4711yescharacteristic
130#389—941no—
131#387—942no—
132#343C2×D4711yescharacteristic
133#339—21no—
134#328—942no—
135#324—472no—
136C47#1222511yescharacteristic, complemented, Sylow
137C22×Q8C22⋊D4711yescharacteristic
138C2×M16—21no—
139C4⋊Q8—41no—
140#125—41no—
141C22⋊Q8—22no—
142#117—21no—
143C2×C4⋊C4—472no—
144C2×C4⋊C4—941no—
145C2×C4⋊C4—941no—
146C2×C4⋊C4—942no—
147C2×C4⋊C4—941no—
148C2×C4⋊C4C22×D4711yescharacteristic
149C2×C42—941no—
150C2×C42C22⋊D4711yescharacteristic
151#107—1882no—
152#103—41no—
153#103—41no—
154#103—1882no—
155C22⋊C8—942no—
156#95—22no—
157#95—472no—
158#95—942no—
159#95—942no—
160#95—942no—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8#1062533712yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—942no—
169C22×C4—942no—
170C22×C4D4×D4712yes—
171C22×C4C2×C22⋊D4711yescharacteristic
172M16—41no—
173M16—41no—
174C2×C8—41no—
175C2×C8—1882no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—3761no—
182C4⋊C4—3761no—
183C4⋊C4—1882no—
184C4⋊C4—1884no—
185C4⋊C4—1882no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42—3761no—
190C23#1309399211yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—1882no—
207C2×C4—1882no—
208C2×C4—1882no—
209C2×C4—1882no—
210C2×C4—1882no—
211C2×C4—1882no—
212C2×C4C2×#1062533711yescharacteristic
213C2×C4#1309405411yescharacteristic
214C8—81no—
215C8—3762no—
216C22—21no—
217C22—21no—
218C22#1568927811yescharacteristic
219C22#1568930011yescharacteristic
220C22#1568930911yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—81no—
227C4—81no—
228C4—3762no—
229C4—3762no—
230C2—21no—
231C2—21no—
232C2#2024234411yescharacteristic
233C2#2024540011yescharacteristic
234C2#2024541311yescharacteristic
subgroupG/H
centerC22#15689309
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2028C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
hypercenter#2028D47
hypocenterC47#12225
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2028

Complements

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

NH ≅ G/Ntype
136. C47#12225semidirect

Maximal subgroups

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

subgroupindex
#202513762
#202424302
#202414142
#202413222
C47×#20282
#1222547