Finite Group Databasebeta

← #26119948

Subgroups — #26119948

Order 12032.

376 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#15689415C211yesmaximal, characteristic
2#20243381C211yesmaximal, characteristic
3#20242163C211yesmaximal, characteristic
4#20242073C211yesmaximal, characteristic
5#20241782C211yesmaximal, characteristic
6#20241283C211yesmaximal, characteristic
7C47×#1548C211yesmaximal, characteristic
8C2×C4×Dic94C2211yescharacteristic
9#15689415—22no—
10#15689415—22no—
11#15689176C2211yescharacteristic
12C2×#13093957—21no—
13#15689147C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093947—21no—
16C2×#13093942—21no—
17#15688891—21no—
18#15688876—22no—
19#15688869—22no—
20#15688853—22no—
21C2×C4⋊Q8×C47C2211yescharacteristic
22C47×#431C2211yescharacteristic
23C47×#423C2211yescharacteristic
24C47×#333—22no—
25C22×Dic94D411yescharacteristic
26C2×#10625329—21no—
27C22×C47⋊C8—21no—
28C4×Dic94—21no—
29C4×Dic94—21no—
30C4×Dic94—21no—
31C4×Dic94—21no—
32C2×C4⋊Dic47D411yescharacteristic
33C2×Dic47⋊C4—21no—
34C2×C4×Dic47—21no—
35#13093957—24no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)D412yes—
41#13093947—24no—
42#13093942—42no—
43C22⋊(C47⋊C8)—21no—
44C4×C47⋊C8—21no—
45C22×Q8×C47—21no—
46C2×M16×C47—21no—
47C4⋊Q8×C47—21no—
48C4⋊Q8×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47D412yes—
51C2×C4⋊C4×C47—21no—
52C2×C4⋊C4×C47—21no—
53C2×C42×C47C2311yescharacteristic
54C4⋊C8×C47—21no—
55C4⋊C8×C47—21no—
56C4⋊C8×C47D412yes—
57C22⋊C8×C47—21no—
58C4×C8×C47—21no—
59C2×Q8×C47—22no—
60C2×Q8×C47—41no—
61C2×Q8×C47—82no—
62C22×C4×C47—41no—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47#4111yescharacteristic
65C22×C4×C47#4111yescharacteristic
66C22×Dic47—21no—
67C22×Dic47—21no—
68C2×Dic94—21no—
69C2×Dic94—21no—
70C2×Dic94—21no—
71C2×Dic94—21no—
72C2×Dic94—21no—
73C2×Dic94Q1612yes—
74M16×C47—41no—
75M16×C47—41no—
76C2×C8×C47—22no—
77C2×C8×C47—41no—
78C4⋊C4×C47—22no—
79C4⋊C4×C47—22no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C42×C47C2×D411yescharacteristic
83C42×C47C2×D411yescharacteristic
84C42×C47C2×D411yescharacteristic
85C42×C47#4111yescharacteristic
86C4⋊Dic47—21no—
87C4⋊Dic47—21no—
88Dic47⋊C4—41no—
89Dic47⋊C4—41no—
90C4×Dic47—41no—
91C4×Dic47—41no—
92#10625329—41no—
93#10625329—41no—
94C2×C47⋊C8—22no—
95C2×C47⋊C8—41no—
96C2×C47⋊C8—41no—
97C2×C47⋊C8—41no—
98C2×C47⋊C8—41no—
99C23×C47#12311yescharacteristic
100Q8×C47—82no—
101Q8×C47—82no—
102C2×C4×C47—22no—
103C2×C4×C47—41no—
104C2×C4×C47—42no—
105C2×C4×C47—82no—
106C2×C4×C47#12011yescharacteristic
107C2×C4×C47#12111yescharacteristic
108C2×C4×C47#12111yescharacteristic
109C2×C4×C47#12111yescharacteristic
110C2×C4×C47#12311yescharacteristic
111C2×C4×C47#12411yescharacteristic
112C2×C4×C47C2×Q1611yescharacteristic
113C2×C4×C47#13511yescharacteristic
114C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C47#13711yescharacteristic
116C2×Dic47—22no—
117C2×Dic47—41no—
118C2×Dic47—41no—
119C2×Dic47—41no—
120C2×Dic47—81no—
121Dic94—22no—
122Dic94—22no—
123Dic94—41no—
124Dic94—41no—
125C8×C47—42no—
126C8×C47—81no—
127C47⋊C8—42no—
128C47⋊C8—81no—
129C47⋊C8—81no—
130#10455—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47#39411yescharacteristic
134C22×C47#48411yescharacteristic
135C22×C47#48511yescharacteristic
136C4×C47—21no—
137C4×C47—21no—
138C4×C47—21no—
139C4×C47—82no—
140C4×C47#44911yescharacteristic
141C4×C47#45111yescharacteristic
142C4×C47#46711yescharacteristic
143C4×C47#46811yescharacteristic
144C4×C47#47312yes—
145Dic47—42no—
146Dic47—81no—
147C2×#477—471no—
148#1801—471no—
149#1571—471no—
150#1549—471no—
151#1548D4711yescharacteristic
152#1479—471no—
153#1301—471no—
154C2×C47—21no—
155C2×C47—21no—
156C2×C47#203611yescharacteristic
157C2×C47#204711yescharacteristic
158C2×C47#204811yescharacteristic
159C2×C4⋊Q8C2×D4711yescharacteristic
160C2×C4×Q8—471no—
161#477—942no—
162#477—942no—
163#431—471no—
164#431C2×D4711yescharacteristic
165C2×#107—941no—
166#423—471no—
167#423C2×D4711yescharacteristic
168C2×C4⋊C8—471no—
169C2×#103—941no—
170C2×#101—941no—
171#343—941no—
172#335—942no—
173#333—22no—
174#332—942no—
175Q8⋊C8—942no—
176C47#1045511yescharacteristic, complemented, Sylow
177C22×Q8—21no—
178C22×Q8—471no—
179C2×M16—21no—
180C2×M16—941no—
181C22×C8—941no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8D18812yes—
186C4×Q8—941no—
187C4×Q8—941no—
188C4×Q8—941no—
189C4×Q8—941no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—471no—
193C2×C4⋊C4—941no—
194C2×C42—941no—
195C2×C42C22×D4711yescharacteristic
196#107—944no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—472no—
200C4⋊C8—941no—
201C4⋊C8—941no—
202C4⋊C8—941no—
203C4⋊C8—941no—
204C4⋊C8C22⋊D4712yes—
205#103—944no—
206#101—1882no—
207C22⋊C8—21no—
208C22⋊C8—941no—
209C4×C8—21no—
210C4×C8—941no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—82no—
214C2×Q8—472no—
215C2×Q8—941no—
216C2×Q8—941no—
217C2×Q8—941no—
218C2×Q8—941no—
219C2×Q8—941no—
220C22×C4—41no—
221C22×C4—941no—
222C22×C4—941no—
223C22×C4#1062535011yescharacteristic
224C22×C4D4×D4711yescharacteristic
225C22×C4#1062535211yescharacteristic
226M16—41no—
227M16—41no—
228M16—1881no—
229M16—1881no—
230C2×C8—22no—
231C2×C8—41no—
232C2×C8—942no—
233C2×C8—1881no—
234C2×C8—1881no—
235C2×C8—1881no—
236C2×C8—1881no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—941no—
242C4⋊C4—941no—
243C4⋊C4—1881no—
244C4⋊C4—1881no—
245C42—1881no—
246C42—1881no—
247C42C2×D18811yescharacteristic
248C42D4×D4711yescharacteristic
249C42#1062535211yescharacteristic
250C42C2×C22⋊D4711yescharacteristic
251C23#1309400611yescharacteristic
252Q8—82no—
253Q8—82no—
254Q8—942no—
255Q8—942no—
256Q8—1881no—
257Q8—1881no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—82no—
262C2×C4—942no—
263C2×C4—3761no—
264C2×C4—1881no—
265C2×C4—1881no—
266C2×C4—1881no—
267C2×C4#1309399411yescharacteristic
268C2×C4#1309399511yescharacteristic
269C2×C4#1309399611yescharacteristic
270C2×C4#1309399811yescharacteristic
271C2×C4#1309400811yescharacteristic
272C2×C4#1309401011yescharacteristic
273C2×C4Q16×D4711yescharacteristic
274C2×C4#1309404711yescharacteristic
275C2×C4#1309405111yescharacteristic
276C2×C4#1309405311yescharacteristic
277C8—42no—
278C8—81no—
279C8—3761no—
280C8—3761no—
281C8—1882no—
282C22—21no—
283C22—21no—
284C22#1568903911yescharacteristic
285C22#1568945711yescharacteristic
286C22#1568946311yescharacteristic
287C4—21no—
288C4—21no—
289C4—21no—
290C4—82no—
291C4—3761no—
292C4—1882no—
293C4#1568927111yescharacteristic
294C4#1568928011yescharacteristic
295C4#1568936111yescharacteristic
296C4#1568936611yescharacteristic
297C4#1568939612yes—
298C2—21no—
299C2—21no—
300C2#2024459011yescharacteristic
301C2#2024465511yescharacteristic
302C2#2024466011yescharacteristic
subgroupG/H
centerC22#15689039
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1548C2
socleC22×C47#394
prime radicalC23×C47#123
hypercenter#1548D47
hypocenterC47#10455
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
176. C47#10455semidirect

Maximal subgroups

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

subgroupindex
C2×#156894152
#202433812
#202421632
#202420732
#202417822
#202412832
C47×#15482
#1045547