Finite Group Databasebeta

← #26133129

Subgroups — #26133129

Order 12032.

416 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 99 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#20252455C211yesmaximal, characteristic
2#20243772C211yesmaximal, characteristic
3#20243610C212yesmaximal, complemented
4#20241518C212yesmaximal, complemented
5C47×#2132C211yesmaximal, characteristic
6#15689890C2211yescharacteristic
7#15689201—22no—
8D4×C47⋊C8—22no—
9#15689176—21no—
10#15689172—22no—
11#15689140—21no—
12#15689097C2211yescharacteristic
13#15689090—21no—
14#15689090C2212yes—
15#15688887—22no—
16#15688883—21no—
17C47×#532C2211yescharacteristic
18C47×#413C2212yes—
19C47×#356C412yescomplemented
20C47×#355C412yescomplemented
21#13094031—42no—
22C2×#10625329—21no—
23C2×#10625329—21no—
24C2×#10625329—22no—
25C2×#10625329D412yes—
26C22×C47⋊C8—21no—
27C22×C47⋊C8D412yes—
28#13093983—21no—
29#13093960—44no—
30C4⋊(C47⋊C8)—21no—
31C4⋊(C47⋊C8)—21no—
32#13093951—42no—
33#13093939—22no—
34C22⋊(C47⋊C8)—22no—
35C22⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37C4×C47⋊C8—22no—
38C2×#41×C47D411yescharacteristic
39C2×#41×C47C2311yescharacteristic
40C2×M16×C47—22no—
41C4⋊Q8×C47C2×C411yescharacteristic
42#127×C47Q811yescharacteristic
43#124×C47C2×C411yescharacteristic
44#124×C47Q811yescharacteristic
45#121×C47—22no—
46C4×D4×C47—22no—
47C2×C42×C47D411yescharacteristic
48#101×C47C2×C414yes—
49#100×C47—22no—
50#41×C47—41no—
51#41×C47—41no—
52#41×C47—41no—
53#41×C47—41no—
54C2×Q8×C47C4⋊C411yescharacteristic
55C2×Q8×C47C22×C411yescharacteristic
56C2×D4×C47—22no—
57C2×D4×C47—21no—
58C2×D4×C47C4⋊C411yescharacteristic
59C2×D4×C47C2×Q811yescharacteristic
60C2×D4×C47#4112yes—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D412yes—
65M16×C47—24no—
66M16×C47—42no—
67C2×C8×C47—42no—
68C4⋊C4×C47—22no—
69C22⋊C4×C47—22no—
70C22⋊C4×C47—22no—
71C42×C47—21no—
72C42×C47C4⋊C411yescharacteristic
73C42×C47C4⋊C411yescharacteristic
74C22⋊Dic47—41no—
75C4⋊Dic47—41no—
76C4×Dic47—41no—
77#10625329—22no—
78#10625329—24no—
79#10625329—41no—
80#10625329—42no—
81#10625329—81no—
82C2×C47⋊C8—22no—
83C2×C47⋊C8—22no—
84C2×C47⋊C8—42no—
85C2×C47⋊C8—41no—
86C2×C47⋊C8—42no—
87C2×C47⋊C8—81no—
88C23×C47—21no—
89C23×C47C22⋊Q812yes—
90C23×C47#12311yescharacteristic
91Q8×C47—41no—
92Q8×C47—41no—
93D4×C47—41no—
94D4×C47—42no—
95D4×C47—41no—
96D4×C47—44no—
97D4×C47—81no—
98C2×C4×C47—21no—
99C2×C4×C47—22no—
100C2×C4×C47—21no—
101C2×C4×C47—22no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—42no—
106C2×C4×C47—41no—
107C2×C4×C47—41no—
108C2×C4×C47C2×C4⋊C411yescharacteristic
109C2×C4×C47C4×D412yes—
110C2×C4×C47#12011yescharacteristic
111C2×Dic47—81no—
112C8×C47—44no—
113C47⋊C8—42no—
114C47⋊C8—42no—
115C47⋊C8—42no—
116#12975—471nomaximal, Sylow
117C22×C47—21no—
118C22×C47—22no—
119C22×C47—41no—
120C22×C47—42no—
121C22×C47—42no—
122C22×C47—41no—
123C22×C47#38511yescharacteristic
124C4×C47—22no—
125C4×C47—22no—
126C4×C47—21no—
127C4×C47—41no—
128C4×C47—41no—
129C4×C47B(4,𝔽2)11yescharacteristic
130C4×C47#45811yescharacteristic
131Dic47—161no—
132#2979—471no—
133#2132D4711yescharacteristic
134#1884—471no—
135#1858—472no—
136#1395—472no—
137C2×C47—21no—
138C2×C47—41no—
139C2×C47—42no—
140C2×C47—81no—
141C2×C47#186711yescharacteristic
142#568—471no—
143#532C2×D4711yescharacteristic
144#436—942no—
145C8×D4—942no—
146#431—941no—
147#430—942no—
148#421—941no—
149#413C2×D4712yes—
150#410—471no—
151#408—472no—
152#408—941no—
153#356Dic4712yescomplemented
154#355Dic4712yescomplemented
155#341—942no—
156#339—941no—
157C47#1297511yescharacteristic, complemented, Sylow
158C2×#41D18811yescharacteristic
159C2×#41C22×D4711yescharacteristic
160#131—1882no—
161C2×M16—22no—
162C2×M16—472no—
163C2×M16—941no—
164C2×M16—941no—
165C2×M16—942no—
166C22×C8—472no—
167C22×C8—941no—
168C4⋊Q8C2×Dic4711yescharacteristic
169#127Dic9411yescharacteristic
170#124Dic9411yescharacteristic
171#124C2×Dic4711yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—941no—
175C2×C42D18811yescharacteristic
176#108—1884no—
177C4⋊C8—941no—
178C4⋊C8—941no—
179#104—1882no—
180#101C2×Dic4714yes—
181#100—22no—
182#99—942no—
183C22⋊C8—942no—
184C22⋊C8—942no—
185C22⋊C8—942no—
186C4×C8—942no—
187#41—41no—
188#41—41no—
189#41—41no—
190#41—41no—
191C2×Q8C4⋊Dic4711yescharacteristic
192C2×Q8C22×Dic4711yescharacteristic
193C2×D4—22no—
194C2×D4—21no—
195C2×D4C4⋊Dic4711yescharacteristic
196C2×D4C2×Dic9411yescharacteristic
197C2×D4#1062535212yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4C2×D18811yescharacteristic
201C22×C4D4×D4712yes—
202M16—24no—
203M16—42no—
204M16—942no—
205M16—944no—
206M16—3761no—
207M16—1881no—
208M16—1882no—
209C2×C8—42no—
210C2×C8—942no—
211C2×C8—942no—
212C2×C8—3761no—
213C2×C8—1881no—
214C2×C8—1882no—
215C2×C8—1882no—
216C4⋊C4—22no—
217C4⋊C4—1881no—
218C22⋊C4—22no—
219C22⋊C4—22no—
220C22⋊C4—1881no—
221C42—21no—
222C42—1881no—
223C42C4⋊Dic4711yescharacteristic
224C42C4⋊Dic4711yescharacteristic
225C23—21no—
226C23#1309400112yes—
227C23#1309400811yescharacteristic
228Q8—41no—
229Q8—41no—
230D4—41no—
231D4—42no—
232D4—41no—
233D4—44no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—3761no—
246C2×C4C2×C4⋊Dic4711yescharacteristic
247C2×C4D4×Dic4712yes—
248C2×C4#1309399411yescharacteristic
249C8—44no—
250C8—1882no—
251C8—1882no—
252C8—1882no—
253C22—21no—
254C22—22no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22#1568899811yescharacteristic
260C4—22no—
261C4—22no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—7521no—
266C4#1568931111yescharacteristic
267C4#1568931411yescharacteristic
268C2—21no—
269C2—41no—
270C2—42no—
271C2—81no—
272C2#2024367111yescharacteristic
subgroupG/H
centerC4#15689311
commutatorC2×Q8×C47C22×C4
FrattiniC2×#41C22×D47
FittingC47×#2132C2
socleC2×C47#1867
prime radicalC47×#532C22
cosocleC2×#41×C47C23
hypercenter#2132D47
hypocenterC47#12975
LCSG ▷ C2×Q8×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2132

Complements

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

NH ≅ G/Ntype
3. #20243610C2semidirect
4. #20241518C2semidirect
19. C47×#356C4semidirect
20. C47×#355C4semidirect
153. #356Dic47semidirect
154. #355Dic47semidirect
157. C47#12975semidirect

Maximal subgroups

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

subgroupindex
#202524552
#202437722
#202436102
#202415182
C47×#21322
#1297547