Finite Group Databasebeta

← #26133114

Subgroups — #26133114

Order 12032.

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

normal subgroups only

This group has 492 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#20246500C211yesmaximal, characteristic
2#20243792C211yesmaximal, characteristic
3#20243605C212yesmaximal, complemented
4#20241514C212yesmaximal, complemented
5C2×C47×#355C211yesmaximal, characteristic
6#15689601C2211yescharacteristic
7#15689165—22no—
8#15689128—21no—
9C2×#13093943C2211yescharacteristic
10#15689030C2212yes—
11#15689016—22no—
12#15689003—21no—
13C22⋊C4×Dic47—22no—
14#15688969—21no—
15#15688884—22no—
16#15688883—21no—
17C2×#124×C47C2211yescharacteristic
18C2×#101×C47C2212yes—
19C47×#355C414yescomplemented
20C23×Dic47—21no—
21C2×#10625329—21no—
22Q8×Dic47—42no—
23#13093983—21no—
24#13093983—22no—
25C2×C4⋊Dic47D412yes—
26C2×C22⋊Dic47—22no—
27C2×C22⋊Dic47—21no—
28C2×C4×Dic47—22no—
29C2×C4×Dic47D412yes—
30#13093958—42no—
31#13093956—42no—
32#13093945—42no—
33#13093943—22no—
34C47⋊#95—21no—
35C47⋊#95—21no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C22×Q8×C47C2311yescharacteristic
39C22×D4×C47D411yescharacteristic
40C2×M16×C47—22no—
41#124×C47—22no—
42#124×C47C2×C412yes—
43#124×C47Q812yes—
44C2×C22⋊C4×C47—22no—
45C2×C42×C47D411yescharacteristic
46#101×C47—22no—
47#101×C47C2×C414yescomplemented
48C24×C47—21no—
49C2×Q8×C47—41no—
50C2×Q8×C47—41no—
51C2×Q8×C47C22×C411yescharacteristic
52C2×Q8×C47C2×Q811yescharacteristic
53C2×Q8×C47#4112yes—
54C2×D4×C47—21no—
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47C4⋊C412yes—
58C22×C4×C47—21no—
59C22×C4×C47C2×D411yescharacteristic
60C22×C4×C47C2×D412yes—
61C22×Dic47—22no—
62C22×Dic47—22no—
63C22×Dic47—44no—
64C22×Dic47—81no—
65M16×C47—24no—
66M16×C47—42no—
67C2×C8×C47—42no—
68C22⋊C4×C47—24no—
69C22⋊C4×C47—24no—
70C42×C47—21no—
71C42×C47C4⋊C412yes—
72C22⋊Dic47—44no—
73C22⋊Dic47—41no—
74C22⋊Dic47—81no—
75C4⋊Dic47—22no—
76C4⋊Dic47—22no—
77C4⋊Dic47—22no—
78C4⋊Dic47—42no—
79C4×Dic47—44no—
80C4×Dic47—41no—
81C4×Dic47—42no—
82C4×Dic47—42no—
83#10625329—41no—
84#10625329—41no—
85C2×C47⋊C8—41no—
86C23×C47—22no—
87C23×C47—42no—
88C23×C47—42no—
89C23×C47—41no—
90C23×C47—41no—
91C23×C47#12011yescharacteristic
92Q8×C47—41no—
93Q8×C47—41no—
94Q8×C47—42no—
95D4×C47—42no—
96D4×C47—81no—
97C2×C4×C47—21no—
98C2×C4×C47—22no—
99C2×C4×C47—21no—
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—42no—
103C2×C4×C47C2×C4⋊C411yescharacteristic
104C2×C4×C47C4×D412yes—
105C2×C4×C47C22⋊Q812yes—
106C2×C4×C47#12311yescharacteristic
107C2×Dic47—24no—
108C2×Dic47—42no—
109C2×Dic47—42no—
110C2×Dic47—42no—
111C2×Dic47—42no—
112C2×Dic47—42no—
113C2×Dic47—42no—
114C2×Dic47—84no—
115C8×C47—44no—
116C47⋊C8—81no—
117#12972—471nomaximal, Sylow
118C22×C47—22no—
119C22×C47—44no—
120C22×C47—42no—
121C22×C47—81no—
122C22×C47—81no—
123C22×C47—82no—
124C22×C47#38511yescharacteristic
125C22×C47B(4,𝔽2)11yescharacteristic
126C22×C47#45811yescharacteristic
127C4×C47—21no—
128C4×C47—22no—
129C4×C47—21no—
130C4×C47—22no—
131C4×C47—42no—
132Dic47—44no—
133Dic47—82no—
134Dic47—82no—
135#2372—471no—
136C2×#355D4711yescharacteristic
137#1888—471no—
138#1857—472no—
139#1394—472no—
140C2×C47—21no—
141C2×C47—21no—
142C2×C47—82no—
143C2×C47#186711yescharacteristic
144C2×C47#220412yes—
145C2×#124C2×D4711yescharacteristic
146#519—471no—
147#428—942no—
148#418—941no—
149C2×#101—471no—
150C2×#101C2×D4712yes—
151#391—472no—
152#388—942no—
153#386—941no—
154C4×C22⋊C4—942no—
155#376—941no—
156#355Dic4714yescomplemented
157#339—942no—
158#339—941no—
159C47#1297211yescharacteristic, complemented, Sylow
160C22×Q8C22×D4711yescharacteristic
161C22×D4D18811yescharacteristic
162C23×C4—941no—
163C2×M16—22no—
164C2×M16—941no—
165#124—22no—
166#124Dic9412yes—
167#124C2×Dic4712yes—
168C4×Q8—1882no—
169#117—941no—
170#117—942no—
171C2×C4⋊C4—472no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—941no—
174C2×C22⋊C4—942no—
175C2×C42—472no—
176C2×C42—942no—
177C2×C42D18811yescharacteristic
178#107—1882no—
179#106—1882no—
180#102—1882no—
181#101—22no—
182#101—942no—
183#101C2×Dic4714yescomplemented
184#95—941no—
185#95—941no—
186#95—942no—
187#95—942no—
188C24—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8C2×Dic9411yescharacteristic
192C2×Q8#1062535212yes—
193C2×Q8C22×Dic4711yescharacteristic
194C2×D4—21no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4C4⋊Dic4712yes—
198C22×C4—21no—
199C22×C4—942no—
200C22×C4—942no—
201C22×C4—3761no—
202C22×C4—1884no—
203C22×C4C2×D18811yescharacteristic
204C22×C4D4×D4712yes—
205M16—24no—
206M16—42no—
207M16—1881no—
208M16—1881no—
209C2×C8—42no—
210C2×C8—1881no—
211C4⋊C4—942no—
212C4⋊C4—942no—
213C4⋊C4—942no—
214C4⋊C4—1882no—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—3761no—
218C22⋊C4—1884no—
219C22⋊C4—1881no—
220C42—21no—
221C42—1882no—
222C42—1882no—
223C42—1881no—
224C42—1884no—
225C42C4⋊Dic4712yes—
226C23—22no—
227C23—42no—
228C23—42no—
229C23—41no—
230C23—41no—
231C23#1309399411yescharacteristic
232Q8—41no—
233Q8—42no—
234Q8—41no—
235D4—42no—
236D4—81no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—944no—
244C2×C4—3764no—
245C2×C4—1882no—
246C2×C4—1882no—
247C2×C4—1882no—
248C2×C4—1882no—
249C2×C4—1882no—
250C2×C4—1882no—
251C2×C4C2×C4⋊Dic4711yescharacteristic
252C2×C4D4×Dic4712yes—
253C2×C4#1309400112yes—
254C2×C4#1309400811yescharacteristic
255C8—44no—
256C8—3761no—
257C22—22no—
258C22—44no—
259C22—42no—
260C22—81no—
261C22—81no—
262C22—82no—
263C22#1568899811yescharacteristic
264C22#1568931111yescharacteristic
265C22#1568931411yescharacteristic
266C4—22no—
267C4—21no—
268C4—21no—
269C4—22no—
270C4—42no—
271C4—3762no—
272C4—3762no—
273C4—1884no—
274C2—21no—
275C2—21no—
276C2—82no—
277C2#2024367111yescharacteristic
278C2#2024541512yes—
subgroupG/H
centerC22#15689311
commutatorC2×Q8×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC2×C47×#355C2
socleC22×C47B(4,𝔽2)
prime radicalC22×D4×C47D4
cosocleC22×Q8×C47C23
hypercenterC2×#355D47
hypocenterC47#12972
LCSG ▷ C2×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
3. #20243605C2semidirect
4. #20241514C2semidirect
19. C47×#355C4semidirect
47. #101×C47C2×C4semidirect
156. #355Dic47semidirect
159. C47#12972semidirect
183. #101C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
#202465002
#202437922
#202436052
#202415142
C2×C47×#3552
#1297247