Finite Group Databasebeta

← #26136610

Subgroups — #26136610

Order 12032.

408 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 79 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#20245073C212yesmaximal, complemented
2#20241557C212yesmaximal, complemented
3#20241555C212yesmaximal, complemented
4C47×#3147C211yesmaximal, characteristic
5#15689107—22no—
6#15688922C2214yes—
7#15688920—22no—
8C2×#135×C47—22no—
9C47×#532C411yescharacteristic
10C47×#532C2211yescharacteristic
11C47×#495—22no—
12C47×#494C2211yescharacteristic
13C47×#493C411yescharacteristic
14C47×#492C411yescharacteristic
15C47×#492C2211yescharacteristic
16C2×C4×C8×C47C411yescharacteristic
17C2×#10625329—42no—
18#13093943—24no—
19#13093941—42no—
20C2×#41×C47—21no—
21C2×#41×C47D412yes—
22#135×C47—44no—
23#135×C47—44no—
24C2×Q16×C47—22no—
25C2×SD16×C47—22no—
26C2×SD16×C47—22no—
27C2×D8×C47—22no—
28C22×C8×C47—21no—
29C4⋊Q8×C47C2×C411yescharacteristic
30C4⋊Q8×C47C2311yescharacteristic
31#127×C47C2×C411yescharacteristic
32#127×C47C2×C411yescharacteristic
33#124×C47—21no—
34#124×C47D412yes—
35#121×C47—41no—
36#121×C47—41no—
37C4×D4×C47—41no—
38C4×D4×C47—41no—
39C2×C42×C47C2×C411yescharacteristic
40C4×C8×C47—21no—
41C4×C8×C47C2×C411yescharacteristic
42C4×C8×C47C2×C411yescharacteristic
43#41×C47—42no—
44#41×C47—42no—
45#41×C47—42no—
46#41×C47—42no—
47C2×Q8×C47—21no—
48C2×Q8×C47C2×D412yes—
49C2×D4×C47—22no—
50C2×D4×C47—21no—
51C2×D4×C47—41no—
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47C22⋊C412yes—
55C22×C4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—41no—
58C22×C4×C47C22⋊C411yescharacteristic
59Q16×C47—44no—
60SD16×C47—44no—
61SD16×C47—44no—
62D8×C47—44no—
63C2×C8×C47—22no—
64C2×C8×C47—22no—
65C2×C8×C47—21no—
66C2×C8×C47—21no—
67C4⋊C4×C47—41no—
68C4⋊C4×C47—41no—
69C22⋊C4×C47—42no—
70C22⋊C4×C47—42no—
71C42×C47—21no—
72C42×C47C22⋊C411yescharacteristic
73C42×C47C22×C411yescharacteristic
74#10625329—44no—
75#10625329—82no—
76C2×C47⋊C8—82no—
77C23×C47—22no—
78C23×C47—41no—
79C23×C47#9911yescharacteristic
80Q8×C47—42no—
81Q8×C47—42no—
82D4×C47—42no—
83D4×C47—42no—
84D4×C47—82no—
85D4×C47—82no—
86D4×C47—82no—
87D4×C47—82no—
88C2×C4×C47—21no—
89C2×C4×C47—21no—
90C2×C4×C47—21no—
91C2×C4×C47—22no—
92C2×C4×C47—21no—
93C2×C4×C47—42no—
94C2×C4×C47—42no—
95C2×C4×C47—42no—
96C2×C4×C47—41no—
97C2×C4×C47—81no—
98C2×C4×C47—81no—
99C2×C4×C47#9911yescharacteristic
100C2×C4×C47C2×C22⋊C411yescharacteristic
101C8×C47—22no—
102C8×C47—22no—
103C47⋊C8—84no—
104#13589—471nomaximal, Sylow
105C22×C47—21no—
106C22×C47—41no—
107C22×C47—42no—
108C22×C47—82no—
109C22×C47—82no—
110C22×C47—81no—
111C22×C47C2×#9911yescharacteristic
112C4×C47—22no—
113C4×C47—21no—
114C4×C47—41no—
115C4×C47—42no—
116C4×C47—81no—
117C4×C47#35311yescharacteristic
118C4×C47#35311yescharacteristic
119#3147D4711yescharacteristic
120#2132—472no—
121#1411—472no—
122#1410—472no—
123C2×C47—21no—
124C2×C47—41no—
125C2×C47—82no—
126C2×C47—161no—
127C2×C47C2×#35311yescharacteristic
128C2×#135—22no—
129#532Dic4711yescharacteristic
130#532C2×D4711yescharacteristic
131#495—22no—
132#494C2×D4711yescharacteristic
133#493Dic4711yescharacteristic
134#492Dic4711yescharacteristic
135#492C2×D4711yescharacteristic
136#413—942no—
137C2×C4×C8Dic4711yescharacteristic
138#356—474no—
139#355—942no—
140C47#1358911yescharacteristic, complemented, Sylow
141C2×#41—21no—
142C2×#41C22⋊D4712yes—
143#135—44no—
144#135—44no—
145C2×Q16—22no—
146C2×SD16—22no—
147C2×SD16—22no—
148C2×D8—22no—
149C2×M16—1882no—
150C22×C8—21no—
151C4⋊Q8C2×Dic4711yescharacteristic
152C4⋊Q8C22×D4711yescharacteristic
153#127C2×Dic4711yescharacteristic
154#127C2×Dic4711yescharacteristic
155#124—21no—
156#124C22⋊D4712yes—
157#121—41no—
158#121—41no—
159C4×D4—41no—
160C4×D4—41no—
161C2×C42C2×Dic4711yescharacteristic
162#101—944no—
163#100—1882no—
164C4×C8—21no—
165C4×C8C2×Dic4711yescharacteristic
166C4×C8C2×Dic4711yescharacteristic
167#41—42no—
168#41—42no—
169#41—42no—
170#41—42no—
171C2×Q8—21no—
172C2×Q8C2×C22⋊D4712yes—
173C2×D4—22no—
174C2×D4—21no—
175C2×D4—41no—
176C2×D4—41no—
177C2×D4—41no—
178C2×D4C22⋊Dic4712yes—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—41no—
182C22×C4C22⋊Dic4711yescharacteristic
183Q16—44no—
184SD16—44no—
185SD16—44no—
186D8—44no—
187M16—3762no—
188M16—1884no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—3762no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C22⋊C4—42no—
197C22⋊C4—42no—
198C42—21no—
199C42C22⋊Dic4711yescharacteristic
200C42C22×Dic4711yescharacteristic
201C23—22no—
202C23—41no—
203C23#1309393911yescharacteristic
204Q8—42no—
205Q8—42no—
206D4—42no—
207D4—42no—
208D4—82no—
209D4—82no—
210D4—82no—
211D4—82no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4#1309393911yescharacteristic
224C2×C4C2×C22⋊Dic4711yescharacteristic
225C8—22no—
226C8—22no—
227C8—3764no—
228C22—21no—
229C22—41no—
230C22—42no—
231C22—82no—
232C22—82no—
233C22—81no—
234C22C2×#1309393911yescharacteristic
235C4—22no—
236C4—21no—
237C4—41no—
238C4—42no—
239C4—81no—
240C4#1568891611yescharacteristic
241C4#1568891611yescharacteristic
242C2—21no—
243C2—41no—
244C2—82no—
245C2—161no—
246C2C2×#1568891611yescharacteristic
subgroupG/H
centerC4#15688916
commutatorC42×C47C22×C4
FrattiniC4⋊Q8C22×D47
Fitting, prime radicalC47×#3147C2
socleC2×C47C2×#353
cosocleC4⋊Q8×C47C23
hypercenter#3147D47
hypocenterC47#13589
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #20245073C2semidirect
2. #20241557C2semidirect
3. #20241555C2semidirect
140. C47#13589semidirect

Maximal subgroups

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

subgroupindex
#202450732
#202415572
#202415552
C47×#31472
#1358947