Finite Group Databasebeta

← #26133092

Subgroups — #26133092

Order 12032.

424 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 95 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#20245053C211yesmaximal, characteristic
2#20243772C211yesmaximal, characteristic
3#20243673C212yesmaximal, complemented
4#20241509C212yesmaximal, complemented
5C47×#2979C211yesmaximal, characteristic
6#15689890C2211yescharacteristic
7#15689149—22no—
8#15689140—21no—
9#15689097C2211yescharacteristic
10#15689097C2212yes—
11#15688918C412yescomplemented
12#15688916C412yescomplemented
13#15688895—22no—
14#15688883—21no—
15C47×#532C2211yescharacteristic
16C47×#436—22no—
17C8×D4×C47—22no—
18C47×#431—21no—
19C47×#408—21no—
20C47×#408C2212yes—
21#13094031—42no—
22C2×#10625329—22no—
23C2×#10625329—21no—
24#13093983—22no—
25#13093983—21no—
26Dic47⋊C8—44no—
27#13093951—42no—
28#13093939—22no—
29#13093939—22no—
30#13093939C2×C414yes—
31C2×#41×C47D411yescharacteristic
32C2×#41×C47C2311yescharacteristic
33C2×M16×C47—21no—
34C2×M16×C47D412yes—
35C22×C8×C47—21no—
36C22×C8×C47D412yes—
37C4⋊Q8×C47Q811yescharacteristic
38#127×C47C2×C411yescharacteristic
39#124×C47C2×C411yescharacteristic
40#124×C47Q811yescharacteristic
41#121×C47—22no—
42C4×D4×C47—22no—
43C2×C42×C47D411yescharacteristic
44C4⋊C8×C47—21no—
45C4⋊C8×C47—21no—
46C22⋊C8×C47—22no—
47C22⋊C8×C47—22no—
48C22⋊C8×C47—22no—
49C4×C8×C47—22no—
50#41×C47—41no—
51#41×C47—41no—
52#41×C47—41no—
53#41×C47—41no—
54C2×Q8×C47C4⋊C411yescharacteristic
55C2×Q8×C47C2×Q811yescharacteristic
56C2×D4×C47—22no—
57C2×D4×C47—21no—
58C2×D4×C47C4⋊C411yescharacteristic
59C2×D4×C47C22×C411yescharacteristic
60C2×D4×C47#4112yes—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47C2×D411yescharacteristic
64C22×C4×C47C2×D412yes—
65M16×C47—42no—
66M16×C47—81no—
67C2×C8×C47—22no—
68C2×C8×C47—22no—
69C2×C8×C47—42no—
70C2×C8×C47—81no—
71C4⋊C4×C47—22no—
72C22⋊C4×C47—22no—
73C22⋊C4×C47—22no—
74C42×C47—21no—
75C42×C47C4⋊C411yescharacteristic
76C42×C47C4⋊C411yescharacteristic
77C22⋊Dic47—24no—
78C22⋊Dic47—41no—
79C4⋊Dic47—42no—
80C4⋊Dic47—41no—
81C4×Dic47—42no—
82C4×Dic47—41no—
83#10625329—22no—
84#10625329—42no—
85#10625329—41no—
86C2×C47⋊C8—24no—
87C2×C47⋊C8—41no—
88C23×C47—21no—
89C23×C47C4×D412yes—
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×C47#12011yescharacteristic
110C2×C4×C47C22⋊Q812yes—
111C2×Dic47—44no—
112C2×Dic47—81no—
113C8×C47—42no—
114C8×C47—42no—
115C47⋊C8—42no—
116#12968—471nomaximal, Sylow
117C22×C47—22no—
118C22×C47—21no—
119C22×C47—41no—
120C22×C47—42no—
121C22×C47—42no—
122C22×C47—41no—
123C22×C47#38511yescharacteristic
124C4×C47—22no—
125C4×C47—21no—
126C4×C47—22no—
127C4×C47—41no—
128C4×C47—41no—
129C4×C47B(4,𝔽2)11yescharacteristic
130C4×C47#45811yescharacteristic
131Dic47—84no—
132Dic47—161no—
133#2979D4711yescharacteristic
134#2126—471no—
135#1884—471no—
136#1868—472no—
137#1391—472no—
138C2×C47—21no—
139C2×C47—42no—
140C2×C47—41no—
141C2×C47—81no—
142C2×C47#186711yescharacteristic
143#568—471no—
144#532C2×D4711yescharacteristic
145#436—22no—
146C8×D4—22no—
147#431—21no—
148#424—942no—
149#421—941no—
150#410—472no—
151#410—471no—
152#408—21no—
153#408C2×D4712yes—
154#354—472no—
155#353—472no—
156#344—942no—
157#339—941no—
158C47#1296811yescharacteristic, complemented, Sylow
159C2×#41C22⋊D4711yescharacteristic
160C2×#41C22×D4711yescharacteristic
161#131—1882no—
162C2×M16—21no—
163C2×M16—942no—
164C2×M16—941no—
165C2×M16C22⋊D4712yes—
166C22×C8—21no—
167C22×C8C22⋊D4712yes—
168C4⋊Q8Dic9411yescharacteristic
169#127C4×D4711yescharacteristic
170#124Dic9411yescharacteristic
171#124C4×D4711yescharacteristic
172#121—22no—
173C4×D4—22no—
174#117—942no—
175#117—941no—
176C2×C42C22⋊D4711yescharacteristic
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—1884no—
180#104—1882no—
181#99—474no—
182#99—942no—
183#99—942no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C4×C8—22no—
188#41—41no—
189#41—41no—
190#41—41no—
191#41—41no—
192C2×Q8Dic47⋊C411yescharacteristic
193C2×Q8C2×Dic9411yescharacteristic
194C2×D4—22no—
195C2×D4—21no—
196C2×D4Dic47⋊C411yescharacteristic
197C2×D4C2×C4×D4711yescharacteristic
198C2×D4#1062535012yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4C2×C22⋊D4711yescharacteristic
202C22×C4C2×C22⋊D4712yes—
203M16—42no—
204M16—81no—
205M16—942no—
206M16—1882no—
207M16—1881no—
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—42no—
211C2×C8—81no—
212C2×C8—944no—
213C2×C8—1881no—
214C4⋊C4—22no—
215C4⋊C4—1881no—
216C4⋊C4—1882no—
217C22⋊C4—22no—
218C22⋊C4—22no—
219C22⋊C4—944no—
220C22⋊C4—1881no—
221C42—21no—
222C42—1881no—
223C42—1882no—
224C42Dic47⋊C411yescharacteristic
225C42Dic47⋊C411yescharacteristic
226C23—21no—
227C23C4×C22⋊D4712yes—
228C23#1309400711yescharacteristic
229Q8—41no—
230Q8—41no—
231D4—41no—
232D4—42no—
233D4—41no—
234D4—44no—
235D4—81no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—3761no—
247C2×C4—1884no—
248C2×C4C2×Dic47⋊C411yescharacteristic
249C2×C4#1309399311yescharacteristic
250C2×C4#1309400312yes—
251C8—42no—
252C8—42no—
253C8—1882no—
254C22—21no—
255C22—22no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22#1568899611yescharacteristic
261C4—22no—
262C4—21no—
263C4—22no—
264C4—41no—
265C4—41no—
266C4—3764no—
267C4—7521no—
268C4#1568931011yescharacteristic
269C4#1568931311yescharacteristic
270C2—21no—
271C2—42no—
272C2—41no—
273C2—81no—
274C2#2024366911yescharacteristic
subgroupG/H
centerC4#15689310
commutatorC2×D4×C47C22×C4
FrattiniC2×#41C22×D47
FittingC47×#2979C2
socleC2×C47#1867
prime radicalC47×#532C22
cosocleC2×#41×C47C23
hypercenter#2979D47
hypocenterC47#12968
LCSG ▷ C2×D4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2979

Complements

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

NH ≅ G/Ntype
3. #20243673C2semidirect
4. #20241509C2semidirect
11. #15688918C4semidirect
12. #15688916C4semidirect
158. C47#12968semidirect

Maximal subgroups

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

subgroupindex
#202450532
#202437722
#202436732
#202415092
C47×#29792
#1296847