Finite Group Databasebeta

← #26125589

Subgroups — #26125589

Order 12032.

364 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 83 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#20252103C211yesmaximal, characteristic
2#20245007C212yesmaximal, complemented
3#20241338C212yesmaximal, complemented
4#20241326C211yesmaximal, characteristic, complemented
5C47×#1321C211yesmaximal, characteristic, complemented
6C2×#13094032C2211yescharacteristic
7C2×#13094031C411yescharacteristic
8#15689669C411yescharacteristic
9M32×D47—22no—
10#15689197—22no—
11#15689191—22no—
12#15689146C2211yescharacteristic
13#15689089C411yescharacteristic
14C2×#13093934C411yescharacteristic
15#15688905C2212yes—
16#15688904—22no—
17C47×#408C2211yescharacteristic
18C47×#349C2212yes—
19C2×#10625350C2×C411yescharacteristic
20#13094032—42no—
21#13094031—42no—
22C2×#10625329—21no—
23C2×#10625329C2×C411yescharacteristic
24C2×#10625324D412yes—
25C22×C47⋊C8—21no—
26C22×C47⋊C8C2×C411yescharacteristic
27C2×C8×D47D412yes—
28#13094022C811yescharacteristic
29#13094020C811yescharacteristic
30#13094013C811yescharacteristic
31#13094011C811yescharacteristic
32#13093996—22no—
33C4×C22⋊D47—22no—
34C2×C4×Dic47C2×C411yescharacteristic
35#13093964—22no—
36#13093963—42no—
37C16×D47—42no—
38Dic47⋊C8—22no—
39(C2×D47)⋊C8—22no—
40#13093934—21no—
41#13093934—21no—
42C2×#41×C47C2×C411yescharacteristic
43C2×M16×C47C2311yescharacteristic
44C22×C8×C47C2×C411yescharacteristic
45M32×C47—22no—
46C22⋊C8×C47—22no—
47#41×C47—41no—
48#41×C47—41no—
49C2×Q8×C47C2×C811yescharacteristic
50C2×D4×C47—21no—
51C2×D4×C47C2×C811yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47C22×C411yescharacteristic
54C2×C22⋊D47—22no—
55C2×C22⋊D47M1612yes—
56C22×Dic47—21no—
57#10625350—41no—
58#10625350—41no—
59C2×D188C2×C811yescharacteristic
60C2×C4×D47C22⋊C412yes—
61C2×Dic94C2×C811yescharacteristic
62M16×C47—41no—
63C2×C8×C47—21no—
64C2×C8×C47—41no—
65C2×C8×C47C2×D412yes—
66C22⋊Dic47—22no—
67(C2×D47)⋊C4—22no—
68Dic47⋊C4—22no—
69C4×Dic47—21no—
70C4×Dic47C2×C811yescharacteristic
71C4×Dic47C2×C811yescharacteristic
72#10625329—42no—
73#10625329—41no—
74C2×C47⋊C8—22no—
75C2×C47⋊C8—21no—
76C2×C47⋊C8—42no—
77C2×C47⋊C8C22⋊C411yescharacteristic
78C2×C47⋊C8C22⋊C411yescharacteristic
79#10625324—42no—
80C8×D47—22no—
81C8×D47—22no—
82C16×C47—42no—
83C47⋊C16—42no—
84C23×C47—21no—
85C23×C47C2×M1611yescharacteristic
86C22×D47C22⋊C812yes—
87Q8×C47—41no—
88D4×C47—41no—
89D4×C47—42no—
90C2×C4×C47—21no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—41no—
94C2×C4×C47—41no—
95C2×C4×C47C2×C22⋊C411yescharacteristic
96C2×C4×C47C22×C811yescharacteristic
97C22⋊D47—42no—
98C22⋊D47—82no—
99C2×Dic47—21no—
100C2×Dic47—21no—
101C2×Dic47—81no—
102C2×Dic47C22⋊C812yes—
103D188—41no—
104C4×D47—22no—
105C4×D47—22no—
106Dic94—41no—
107C8×C47—22no—
108C8×C47—41no—
109C47⋊C8—21no—
110C47⋊C8—41no—
111C47⋊C8#9912yes—
112#11486—471nomaximal, Sylow
113C22×C47—21no—
114C22×C47—41no—
115C22×C47—42no—
116C22×C47—41no—
117C22×C47C2×C22⋊C811yescharacteristic
118C2×D47—22no—
119C2×D47—42no—
120C4×C47—21no—
121C4×C47—41no—
122C4×C47C2×#9911yescharacteristic
123C4×C47#41311yescharacteristic
124Dic47—22no—
125Dic47—41no—
126Dic47—41no—
127#2932—471no—
128#2115—472no—
129#1327—472no—
130#1321—471no—
131#1321D4711yescharacteristic, complemented
132C2×C47—21no—
133C2×C47—41no—
134C2×C47—81no—
135C2×C47#145911yescharacteristic
136D47—42no—
137C2×#131—471no—
138C2×#131—471no—
139#532—471no—
140C2×M32—942no—
141#435—942no—
142#435—942no—
143#423—471no—
144#408—471no—
145#408C2×D4711yescharacteristic
146C2×#97—471no—
147#349—472no—
148#349—942no—
149#349C2×D4712yes—
150C47#1148611yescharacteristic, complemented, Sylow
151C2×#41—471no—
152C2×#41C4×D4711yescharacteristic
153#131—1882no—
154#131—1882no—
155C2×M16—472no—
156C2×M16—471no—
157C2×M16—941no—
158C2×M16C22×D4711yescharacteristic
159C22×C8—471no—
160C22×C8—472no—
161C22×C8—941no—
162C22×C8C4×D4711yescharacteristic
163C4⋊Q8—471no—
164#127—471no—
165#124—471no—
166#124—471no—
167#121—942no—
168C4×D4—942no—
169C2×C42—471no—
170M32—22no—
171M32—942no—
172M32—1882no—
173C2×C16—1882no—
174C4⋊C8—942no—
175C22⋊C8—22no—
176C22⋊C8—942no—
177#97—941no—
178#97—941no—
179#41—41no—
180#41—41no—
181#41—1881no—
182#41—1881no—
183C2×Q8—471no—
184C2×Q8C8×D4711yescharacteristic
185C2×D4—21no—
186C2×D4—472no—
187C2×D4—471no—
188C2×D4—942no—
189C2×D4C8×D4711yescharacteristic
190C22×C4—21no—
191C22×C4—472no—
192C22×C4—941no—
193C22×C4C2×C4×D4711yescharacteristic
194M16—41no—
195M16—1882no—
196M16—1882no—
197M16—1881no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—471no—
201C2×C8—471no—
202C2×C8—942no—
203C2×C8—942no—
204C2×C8—942no—
205C2×C8—941no—
206C2×C8—1882no—
207C2×C8D4×D4712yes—
208C4⋊C4—942no—
209C22⋊C4—942no—
210C22⋊C4—942no—
211C42—471no—
212C42—471no—
213C42—941no—
214C16—42no—
215C16—1882no—
216C23—21no—
217C23—472no—
218C23M16×D4711yescharacteristic
219Q8—41no—
220Q8—1881no—
221D4—41no—
222D4—42no—
223D4—3762no—
224D4—1881no—
225D4—1882no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—472no—
232C2×C4—942no—
233C2×C4—942no—
234C2×C4—941no—
235C2×C4—941no—
236C2×C4—3761no—
237C2×C4C22⋊C4×D4711yescharacteristic
238C2×C4C2×C8×D4711yescharacteristic
239C8—22no—
240C8—41no—
241C8—472no—
242C8—941no—
243C8—1881no—
244C22—21no—
245C22—41no—
246C22—42no—
247C22—41no—
248C22—942no—
249C22—1882no—
250C22C22⋊C8×D4711yescharacteristic
251C4—21no—
252C4—41no—
253C4—942no—
254C4—1881no—
255C4—1881no—
256C4#99×D4711yescharacteristic
257C4#1568910411yescharacteristic
258C2—21no—
259C2—41no—
260C2—81no—
261C2—1882no—
262C2#2024170311yescharacteristic
subgroupG/H
centerC4#99×D47
commutatorC2×C4×C47C22×C8
FrattiniC2×M16C22×D47
FittingC47×#1321C2
socleC2×C47#1459
prime radical#15689669C4
cosocleC2×M16×C47C23
hypercenter#1321D47
hypocenterC47#11486
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #20245007C2semidirect
3. #20241338C2semidirect
4. #20241326C2semidirect
5. C47×#1321C2semidirect
131. #1321D47semidirect
150. C47#11486semidirect

Maximal subgroups

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

subgroupindex
#202521032
#202450072
#202413382
#202413262
C47×#13212
#1148647