Finite Group Databasebeta

← #26133109

Subgroups — #26133109

Order 12032.

424 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 103 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#20246569C211yesmaximal, characteristic
2#20243782C211yesmaximal, characteristic
3#20243605C212yesmaximal
4#20241514C212yesmaximal
5C2×C47×#356C211yesmaximal, characteristic
6#15689601C2211yescharacteristic
7#15689165—22no—
8#15689132—21no—
9C2×#13093943C2211yescharacteristic
10#15689030—21no—
11#15689030C2212yes—
12#15689009—22no—
13#15689009—22no—
14#15688991—21no—
15#15688884—22no—
16#15688883—21no—
17C2×C4⋊Q8×C47C2211yescharacteristic
18C2×#101×C47C2212yes—
19C47×#356C414yescomplemented
20C2×#10625329—21no—
21Q8×Dic47—42no—
22#13093983—21no—
23#13093983—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47D412yes—
26C2×C4×Dic47—21no—
27C2×C4×Dic47D412yes—
28#13093958—42no—
29#13093956—42no—
30#13093948—42no—
31#13093943—22no—
32C47⋊#95—21no—
33C47⋊#95—22no—
34C47⋊#95—21no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C22×Q8×C47D411yescharacteristic
39C22×Q8×C47C2311yescharacteristic
40C2×M16×C47—22no—
41C4⋊Q8×C47—22no—
42C4⋊Q8×C47C2×C412yes—
43C4⋊Q8×C47Q812yes—
44C2×C4⋊C4×C47—22no—
45C2×C42×C47D411yescharacteristic
46#101×C47—22no—
47#101×C47C2×C414yes—
48C2×Q8×C47—21no—
49C2×Q8×C47—41no—
50C2×Q8×C47—41no—
51C2×Q8×C47—41no—
52C2×Q8×C47—41no—
53C2×Q8×C47C4⋊C412yes—
54C2×Q8×C47C22×C411yescharacteristic
55C2×Q8×C47C2×Q811yescharacteristic
56C2×Q8×C47#4112yes—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D412yes—
61C22×Dic47—22no—
62C22×Dic47—22no—
63M16×C47—24no—
64M16×C47—42no—
65C2×C8×C47—42no—
66C4⋊C4×C47—24no—
67C4⋊C4×C47—24no—
68C42×C47—21no—
69C42×C47C4⋊C412yes—
70C22⋊Dic47—41no—
71C4⋊Dic47—22no—
72C4⋊Dic47—22no—
73C4⋊Dic47—22no—
74C4⋊Dic47—42no—
75C4⋊Dic47—81no—
76C4×Dic47—41no—
77C4×Dic47—42no—
78C4×Dic47—42no—
79C4×Dic47—81no—
80#10625329—41no—
81#10625329—41no—
82C2×C47⋊C8—41no—
83C23×C47#12011yescharacteristic
84Q8×C47—41no—
85Q8×C47—41no—
86Q8×C47—42no—
87Q8×C47—42no—
88Q8×C47—81no—
89C2×C4×C47—21no—
90C2×C4×C47—22no—
91C2×C4×C47—21no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—22no—
95C2×C4×C47—42no—
96C2×C4×C47—41no—
97C2×C4×C47—41no—
98C2×C4×C47C2×C4⋊C411yescharacteristic
99C2×C4×C47C4×D412yes—
100C2×C4×C47C22⋊Q812yes—
101C2×C4×C47#12311yescharacteristic
102C2×Dic47—44no—
103C2×Dic47—44no—
104C2×Dic47—42no—
105C2×Dic47—42no—
106C2×Dic47—42no—
107C8×C47—44no—
108C47⋊C8—81no—
109#12971—471nomaximal, Sylow
110C22×C47—22no—
111C22×C47#38511yescharacteristic
112C22×C47B(4,𝔽2)11yescharacteristic
113C22×C47#45811yescharacteristic
114C4×C47—21no—
115C4×C47—22no—
116C4×C47—21no—
117C4×C47—22no—
118C4×C47—42no—
119C4×C47—42no—
120Dic47—82no—
121Dic47—82no—
122Dic47—82no—
123#2381—471no—
124C2×#356D4711yescharacteristic
125#1886—471no—
126#1857—472no—
127#1394—472no—
128C2×C47—21no—
129C2×C47—21no—
130C2×C47#186711yescharacteristic
131C2×C47#220312yes—
132C2×C4⋊Q8C2×D4711yescharacteristic
133#519—471no—
134#428—942no—
135#419—941no—
136C2×#101—471no—
137C2×#101C2×D4712yes—
138#391—472no—
139#391—941no—
140#387—942no—
141#387—942no—
142#383—941no—
143#356Dic4714yescomplemented
144#339—942no—
145#339—941no—
146C47#1297111yescharacteristic, complemented, Sylow
147C22×Q8D18811yescharacteristic
148C22×Q8C22×D4711yescharacteristic
149C2×M16—22no—
150C2×M16—941no—
151C4⋊Q8—22no—
152C4⋊Q8Dic9412yes—
153C4⋊Q8C2×Dic4712yes—
154C4×Q8—1882no—
155#117—941no—
156#117—942no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—472no—
159C2×C4⋊C4—941no—
160C2×C42—472no—
161C2×C42—941no—
162C2×C42D18811yescharacteristic
163#107—1882no—
164#106—1882no—
165#103—1882no—
166#101—22no—
167#101—942no—
168#101C2×Dic4714yes—
169#95—941no—
170#95—942no—
171#95—941no—
172#95—942no—
173#95—942no—
174#95—942no—
175C2×Q8—21no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8C4⋊Dic4712yes—
181C2×Q8C2×Dic9411yescharacteristic
182C2×Q8#1062535212yes—
183C2×Q8C22×Dic4711yescharacteristic
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—942no—
187C22×C4—942no—
188C22×C4C2×D18811yescharacteristic
189C22×C4D4×D4712yes—
190M16—24no—
191M16—42no—
192M16—1881no—
193M16—1881no—
194C2×C8—42no—
195C2×C8—1881no—
196C4⋊C4—24no—
197C4⋊C4—24no—
198C4⋊C4—942no—
199C4⋊C4—942no—
200C4⋊C4—942no—
201C4⋊C4—3761no—
202C4⋊C4—1882no—
203C22⋊C4—1881no—
204C42—21no—
205C42—3761no—
206C42—1882no—
207C42—1882no—
208C42—1881no—
209C42C4⋊Dic4712yes—
210C23#1309399411yescharacteristic
211Q8—41no—
212Q8—42no—
213Q8—41no—
214Q8—42no—
215Q8—81no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—1884no—
226C2×C4—1884no—
227C2×C4—1882no—
228C2×C4—1882no—
229C2×C4—1882no—
230C2×C4C2×C4⋊Dic4711yescharacteristic
231C2×C4D4×Dic4712yes—
232C2×C4#1309400112yes—
233C2×C4#1309400811yescharacteristic
234C8—44no—
235C8—3761no—
236C22—22no—
237C22#1568899811yescharacteristic
238C22#1568931111yescharacteristic
239C22#1568931411yescharacteristic
240C4—22no—
241C4—21no—
242C4—21no—
243C4—22no—
244C4—42no—
245C4—42no—
246C4—3762no—
247C4—3762no—
248C4—3762no—
249C2—21no—
250C2—21no—
251C2#2024367111yescharacteristic
252C2#2024541012yes—
subgroupG/H
centerC22#15689311
commutatorC2×Q8×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC2×C47×#356C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
cosocleC22×Q8×C47C23
hypercenterC2×#356D47
hypocenterC47#12971
LCSG ▷ C2×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
19. C47×#356C4semidirect
143. #356Dic47semidirect
146. C47#12971semidirect

Maximal subgroups

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

subgroupindex
#202465692
#202437822
#202436052
#202415142
C2×C47×#3562
#1297147