Finite Group Databasebeta

← #26134604

Subgroups — #26134604

Order 12032.

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

normal subgroups only

This group has 472 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#20248373C211yesmaximal, characteristic
2C2×#15688921C214yesmaximal
3#20243384C211yesmaximal, characteristic
4C4×#101×C47C211yesmaximal, characteristic
5C2×#13094024C411yescharacteristic
6C2×#13094022C2212yes—
7#15689147—21no—
8C2×#13093943C2212yes—
9#15689053—21no—
10#15689053—21no—
11#15689028C412yes—
12#15689021—21no—
13#15689021—21no—
14C4×C4⋊Dic47C411yescharacteristic
15#15688921—28no—
16#15688892—21no—
17C2×C4×Q8×C47C2211yescharacteristic
18C2×#101×C47C2212yes—
19C4×M16×C47—21no—
20C47×#343—21no—
21C22×Dic94C2×C412yes—
22C2×#10625329—22no—
23#13094024—41no—
24#13094024—41no—
25#13094022—24no—
26#13094022—24no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×Dic47⋊C4—22no—
31C2×Dic47⋊C4—22no—
32C2×C4×Dic47C2×C412yes—
33C4⋊(C47⋊C8)—41no—
34C4⋊(C47⋊C8)—41no—
35#13093943—24no—
36C47⋊#95—21no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C22×Q8×C47C2311yescharacteristic
40C2×M16×C47—22no—
41C4×Q8×C47—41no—
42C4×Q8×C47—41no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47D411yescharacteristic
48C2×C42×C47D411yescharacteristic
49#101×C47—22no—
50#101×C47D414yes—
51#97×C47—41no—
52C4×C8×C47—41no—
53C2×Q8×C47—41no—
54C2×Q8×C47—41no—
55C2×Q8×C47C2×D412yes—
56C2×Q8×C47#4112yes—
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22×C411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×C4×C47C2×D411yescharacteristic
64C22×Dic47—22no—
65C22×Dic47—22no—
66C2×Dic94—24no—
67C2×Dic94—42no—
68C2×Dic94—42no—
69M16×C47—24no—
70M16×C47—42no—
71C2×C8×C47—42no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C42×C47—21no—
76C42×C47—21no—
77C42×C47—21no—
78C42×C47—21no—
79C42×C47—21no—
80C42×C47—21no—
81C4⋊Dic47—24no—
82C4⋊Dic47—42no—
83C4⋊Dic47—41no—
84C4⋊Dic47—41no—
85Dic47⋊C4—44no—
86Dic47⋊C4—42no—
87Dic47⋊C4—42no—
88C4×Dic47—24no—
89#10625329—42no—
90#10625329—42no—
91C2×C47⋊C8—42no—
92C23×C47C2×C22⋊C411yescharacteristic
93Q8×C47—42no—
94Q8×C47—42no—
95C2×C4×C47—21no—
96C2×C4×C47—21no—
97C2×C4×C47—21no—
98C2×C4×C47—21no—
99C2×C4×C47—21no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—41no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47#9911yescharacteristic
109C2×C4×C47#9911yescharacteristic
110C2×C4×C47C4×D412yes—
111C2×C4×C47#12112yes—
112C2×C4×C47#12411yescharacteristic
113C2×C4×C47#12711yescharacteristic
114C2×Dic47—24no—
115C2×Dic47—42no—
116C2×Dic47—42no—
117C2×Dic47—42no—
118C2×Dic47—42no—
119C2×Dic47—42no—
120Dic94—84no—
121C8×C47—44no—
122C47⋊C8—82no—
123#13227—471nomaximal, Sylow
124C22×C47—21no—
125C22×C47—21no—
126C22×C47#39011yescharacteristic
127C22×C47C2×#9911yescharacteristic
128C22×C47#41011yescharacteristic
129C4×C47—21no—
130C4×C47—21no—
131C4×C47—22no—
132C4×C47—22no—
133C4×C47—21no—
134C4×C47—41no—
135C4×C47—41no—
136C4×C47—41no—
137C4×C47#35612yescomplemented
138Dic47—44no—
139Dic47—82no—
140#2546—471no—
141C2×#356—474no—
142#1801—471no—
143C4×#101D4711yescharacteristic
144C2×C47—21no—
145C2×C47—21no—
146C2×C47#196411yescharacteristic
147C2×C47C2×#35611yescharacteristic
148C2×C47#213511yescharacteristic
149C2×C4⋊Q8—472no—
150C2×C4⋊Q8—471no—
151C2×C4×Q8C2×D4711yescharacteristic
152#423—941no—
153C2×#101—472no—
154C2×#101C2×D4712yes—
155C4×M16—21no—
156#398—941no—
157#398—941no—
158#391—472no—
159#389—941no—
160#389—941no—
161C4×C4⋊C4—471no—
162#356—948no—
163#343—21no—
164#343—941no—
165C47#1322711yescharacteristic, complemented, Sylow
166C22×Q8—472no—
167C22×Q8C22×D4711yescharacteristic
168C2×M16—22no—
169C2×M16—942no—
170C4⋊Q8—944no—
171C4⋊Q8—944no—
172C4⋊Q8—1881no—
173C4⋊Q8—1881no—
174C4×Q8—41no—
175C4×Q8—41no—
176C2×C4⋊C4—941no—
177C2×C4⋊C4—941no—
178C2×C4⋊C4—941no—
179C2×C4⋊C4—942no—
180C2×C4⋊C4—942no—
181C2×C4⋊C4C4×D4711yescharacteristic
182C2×C4⋊C4D18811yescharacteristic
183C2×C4⋊C4C22⋊D4711yescharacteristic
184C2×C42—472no—
185C2×C42C4×D4711yescharacteristic
186C2×C42D18811yescharacteristic
187C2×C42C22⋊D4711yescharacteristic
188C4⋊C8—1881no—
189C4⋊C8—1881no—
190#101—22no—
191#101—944no—
192#101D18814yes—
193#97—41no—
194C4×C8—41no—
195#95—941no—
196#95—942no—
197#95—942no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—944no—
201C2×Q8—1882no—
202C2×Q8—1882no—
203C2×Q8C2×D18812yes—
204C2×Q8#1062535012yes—
205C22×C4—942no—
206C22×C4—942no—
207C22×C4(C2×D47)⋊C411yescharacteristic
208C22×C4(C2×D47)⋊C411yescharacteristic
209C22×C4(C2×D47)⋊C411yescharacteristic
210C22×C4(C2×D47)⋊C411yescharacteristic
211C22×C4C2×C4×D4711yescharacteristic
212C22×C4C2×D18811yescharacteristic
213C22×C4C2×C22⋊D4711yescharacteristic
214M16—24no—
215M16—42no—
216M16—1882no—
217M16—1882no—
218C2×C8—42no—
219C2×C8—1882no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—944no—
224C4⋊C4—1882no—
225C4⋊C4—1881no—
226C4⋊C4—1881no—
227C4⋊C4—1884no—
228C4⋊C4—1882no—
229C4⋊C4—1882no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—944no—
237C23C2×(C2×D47)⋊C411yescharacteristic
238Q8—42no—
239Q8—42no—
240Q8—3764no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—944no—
255C2×C4—1882no—
256C2×C4—1882no—
257C2×C4—1882no—
258C2×C4—1882no—
259C2×C4—1882no—
260C2×C4#1309393811yescharacteristic
261C2×C4#1309393811yescharacteristic
262C2×C4C4×D18812yes—
263C2×C4#1309399812yes—
264C2×C4#1309401211yescharacteristic
265C2×C4#1309402111yescharacteristic
266C8—44no—
267C8—3762no—
268C22—21no—
269C22—21no—
270C22#1568902511yescharacteristic
271C22C2×#1309393811yescharacteristic
272C22#1568909611yescharacteristic
273C4—22no—
274C4—21no—
275C4—22no—
276C4—21no—
277C4—21no—
278C4—41no—
279C4—41no—
280C4—41no—
281C4—3762no—
282C4—1884no—
283C4#1568892112yescomplemented
284C2—21no—
285C2—21no—
286C2#2024422411yescharacteristic
287C2C2×#1568892111yescharacteristic
288C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC4×#101×C47C2
socleC22×C47C2×#99
prime radicalC23×C47C2×C22⋊C4
cosocleC22×Q8×C47C23
hypercenterC4×#101D47
hypocenterC47#13227
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ C4×#101

Complements

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

NH ≅ G/Ntype
137. C4×C47#356semidirect
165. C47#13227semidirect
283. C4#15688921semidirect

Maximal subgroups

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

subgroupindex
#202483732
C2×#156889212
#202433842
C4×#101×C472
#1322747