Finite Group Databasebeta

← #26105092

Subgroups — #26105092

Order 12032.

480 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 83 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#20251334C211yesmaximal, characteristic
2#20244526C212yesmaximal, complemented
3#20241213C212yesmaximal, complemented
4#20241209C211yesmaximal, characteristic
5C47×#2013C211yesmaximal, characteristic
6C2×#13093947—22no—
7#15689062—21no—
8#15689035—22no—
9#15689029C2212yes—
10#15689003—21no—
11#15688998—22no—
12#15688885—22no—
13#15688878C2212yes—
14C22×SD16×C47—21no—
15C2×C4⋊Q8×C47C2211yescharacteristic
16C2×#103×C47—21no—
17C47×#390C2211yescharacteristic
18C47×#336C2211yescharacteristic
19C22×Dic94D412yes—
20C23×Dic47—21no—
21C22×C47⋊C8—22no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47D412yes—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—21no—
26#13093947—44no—
27C47⋊#95—22no—
28C47⋊#95—21no—
29C47⋊#95—22no—
30C47⋊#95—22no—
31C22×Q8×C47—21no—
32C22×D4×C47D411yescharacteristic
33C2×SD16×C47—42no—
34C2×SD16×C47—42no—
35C2×SD16×C47—42no—
36C22×C8×C47—21no—
37C4⋊Q8×C47—42no—
38C2×C4⋊C4×C47—22no—
39C2×C4⋊C4×C47D411yescharacteristic
40C2×C22⋊C4×C47—22no—
41C2×C42×C47C2311yescharacteristic
42#103×C47—42no—
43C24×C47—21no—
44C2×Q8×C47—22no—
45C2×Q8×C47—41no—
46C2×Q8×C47—41no—
47C2×Q8×C47—41no—
48C2×Q8×C47—41no—
49C2×Q8×C47—41no—
50C2×D4×C47—21no—
51C2×D4×C47—41no—
52C2×D4×C47—41no—
53C2×D4×C47SD1612yes—
54C22×C4×C47—21no—
55C22×C4×C47—41no—
56C22×C4×C47C2×D411yescharacteristic
57C22×C4×C47C2×D412yes—
58C22×Dic47—22no—
59C22×Dic47—22no—
60C22×Dic47—81no—
61C22×Dic47—82no—
62C2×Dic94—22no—
63C2×Dic94—22no—
64C2×Dic94—42no—
65C2×Dic94—42no—
66SD16×C47—48no—
67C2×C8×C47—22no—
68C2×C8×C47—41no—
69C2×C8×C47—41no—
70C4⋊C4×C47—24no—
71C4⋊C4×C47—42no—
72C4⋊C4×C47—41no—
73C22⋊C4×C47—42no—
74C22⋊C4×C47—42no—
75C42×C47—41no—
76C22⋊Dic47—82no—
77C22⋊Dic47—81no—
78C4⋊Dic47—42no—
79C4⋊Dic47—81no—
80C2×C47⋊C8—42no—
81C2×C47⋊C8—42no—
82C2×C47⋊C8—42no—
83C23×C47—22no—
84C23×C47—42no—
85C23×C47—41no—
86C23×C47—42no—
87C23×C47—41no—
88C23×C47#12011yescharacteristic
89Q8×C47—42no—
90Q8×C47—42no—
91Q8×C47—81no—
92Q8×C47—81no—
93D4×C47—42no—
94D4×C47—81no—
95C2×C4×C47—21no—
96C2×C4×C47—22no—
97C2×C4×C47—22no—
98C2×C4×C47—21no—
99C2×C4×C47—42no—
100C2×C4×C47—41no—
101C2×C4×C47—42no—
102C2×C4×C47—81no—
103C2×C4×C47—81no—
104C2×C4×C47C2×SD1611yescharacteristic
105C2×C4×C47#13512yes—
106C2×C4×C47#13712yes—
107C2×C4×C47#13711yescharacteristic
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—84no—
115Dic94—44no—
116Dic94—82no—
117C8×C47—42no—
118C47⋊C8—82no—
119#7755—471nomaximal, Sylow
120C22×C47—42no—
121C22×C47—44no—
122C22×C47—82no—
123C22×C47—81no—
124C22×C47—81no—
125C22×C47#44811yescharacteristic
126C22×C47#45212yes—
127C22×C47#45311yescharacteristic
128C22×C47#45512yes—
129C22×C47B(4,𝔽2)11yescharacteristic
130C4×C47—21no—
131C4×C47—21no—
132C4×C47—22no—
133C4×C47—22no—
134C4×C47—82no—
135C4×C47—81no—
136Dic47—82no—
137Dic47—82no—
138#2804—471no—
139#2024—472no—
140#2013D4711yescharacteristic
141#1272—472no—
142#1270—471no—
143C2×C47—82no—
144C2×C47#159611yescharacteristic
145C2×C47#160612yes—
146C2×C47#163011yescharacteristic
147C2×C47#164312yes—
148C2×C47#165911yescharacteristic
149C22×SD16—21no—
150C2×C4⋊Q8C2×D4711yescharacteristic
151C2×#103—21no—
152C2×#103—942no—
153#400—941no—
154#393—942no—
155#391—472no—
156#390C2×D4711yescharacteristic
157#386—941no—
158#385—942no—
159#340—942no—
160#336—472no—
161#336C2×D4711yescharacteristic
162C47#775511yescharacteristic, complemented, Sylow
163C22×Q8—21no—
164C22×Q8—472no—
165C22×D4C22⋊D4711yescharacteristic
166C23×C4—941no—
167C2×SD16—42no—
168C2×SD16—42no—
169C2×SD16—42no—
170C22×C8—21no—
171C22×C8—942no—
172C4⋊Q8—42no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—472no—
175C2×C4⋊C4—941no—
176C2×C4⋊C4C22⋊D4711yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—942no—
179C2×C22⋊C4—941no—
180C2×C42C22×D4711yescharacteristic
181#103—42no—
182#103—1884no—
183#95—942no—
184#95—941no—
185#95—942no—
186#95—942no—
187C24—21no—
188C2×Q8—22no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×Q8—942no—
195C2×Q8—942no—
196C2×Q8—1882no—
197C2×Q8—1882no—
198C2×D4—21no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4#1062533512yes—
202C22×C4—21no—
203C22×C4—41no—
204C22×C4—942no—
205C22×C4—942no—
206C22×C4—3761no—
207C22×C4—3762no—
208C22×C4D4×D4712yes—
209C22×C4C2×C22⋊D4711yescharacteristic
210SD16—48no—
211C2×C8—22no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—1882no—
215C2×C8—1882no—
216C2×C8—1882no—
217C4⋊C4—24no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—3761no—
221C4⋊C4—1882no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—3762no—
225C22⋊C4—3761no—
226C42—41no—
227C23—22no—
228C23—42no—
229C23—41no—
230C23—42no—
231C23—41no—
232C23#1309399211yescharacteristic
233Q8—42no—
234Q8—42no—
235Q8—81no—
236Q8—81no—
237Q8—3762no—
238Q8—1884no—
239D4—42no—
240D4—81no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—3764no—
251C2×C4—1882no—
252C2×C4—1882no—
253C2×C4—1882no—
254C2×C4—1882no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4C2×#1062533511yescharacteristic
258C2×C4#1309404512yes—
259C2×C4#1309405312yes—
260C2×C4#1309405411yescharacteristic
261C8—42no—
262C8—3762no—
263C22—42no—
264C22—44no—
265C22—81no—
266C22—81no—
267C22—82no—
268C22#1568926111yescharacteristic
269C22#1568928212yes—
270C22#1568929011yescharacteristic
271C22#1568929912yes—
272C22#1568930911yescharacteristic
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—81no—
278C4—82no—
279C4—3762no—
280C4—3762no—
281C2—82no—
282C2#2024231011yescharacteristic
283C2#2024236512yes—
284C2#2024250711yescharacteristic
285C2#2024259312yes—
286C2#2024269211yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2013C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2013D47
hypocenterC47#7755
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2013

Complements

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

NH ≅ G/Ntype
2. #20244526C2semidirect
3. #20241213C2semidirect
162. C47#7755semidirect

Maximal subgroups

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

subgroupindex
#202513342
#202445262
#202412132
#202412092
C47×#20132
#775547