Finite Group Databasebeta

← #26125778

Subgroups — #26125778

Order 12032.

348 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 91 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#20245204C212yesmaximal, complemented
2#20241326C212yesmaximal, complemented
3#20241324C212yesmaximal, complemented
4C47×#2929C211yesmaximal, characteristic
5#15688944—22no—
6#15688905C2214yes—
7#15688901—22no—
8C2×#131×C47C411yescharacteristic
9C23×C8×C47C411yescharacteristic
10C47×#525C411yescharacteristic
11C47×#435—22no—
12C8×D4×C47—22no—
13C47×#424C411yescharacteristic
14C47×#408C2211yescharacteristic
15C47×#407C2211yescharacteristic
16C47×#405C2211yescharacteristic
17#13093964—24no—
18C2×#41×C47C2×C411yescharacteristic
19C23×C4×C47C2×C411yescharacteristic
20#131×C47—42no—
21C2×M16×C47—21no—
22C2×M16×C47C2×C411yescharacteristic
23C2×M16×C47C2311yescharacteristic
24C22×C8×C47—21no—
25C22×C8×C47—21no—
26C22×C8×C47—21no—
27C22×C8×C47—42no—
28C22×C8×C47C2×C411yescharacteristic
29C22×C8×C47C2×C411yescharacteristic
30#123×C47—21no—
31C22⋊Q8×C47—21no—
32#121×C47—21no—
33#120×C47—21no—
34C4×D4×C47—22no—
35#117×C47C2×C411yescharacteristic
36C4⋊C8×C47—22no—
37C22⋊C8×C47—22no—
38C22⋊C8×C47—22no—
39#97×C47D412yes—
40C4×C8×C47D412yes—
41C24×C47M1611yescharacteristic
42#41×C47—41no—
43#41×C47—41no—
44C2×Q8×C47M1611yescharacteristic
45C2×D4×C47—22no—
46C2×D4×C47—21no—
47C2×D4×C47M1611yescharacteristic
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—41no—
52C22×C4×C47—41no—
53C22×C4×C47M1611yescharacteristic
54C22×C4×C47C22×C411yescharacteristic
55M16×C47—41no—
56M16×C47—42no—
57M16×C47—41no—
58C2×C8×C47—22no—
59C2×C8×C47—21no—
60C2×C8×C47—21no—
61C2×C8×C47—42no—
62C2×C8×C47—42no—
63C2×C8×C47—42no—
64C2×C8×C47—41no—
65C2×C8×C47C22⋊C411yescharacteristic
66C2×C8×C47C22⋊C411yescharacteristic
67C2×C8×C47C2×D412yes—
68C4⋊C4×C47—22no—
69C4⋊C4×C47—21no—
70C22⋊C4×C47—22no—
71C22⋊C4×C47—22no—
72C22⋊C4×C47—21no—
73C42×C47C22⋊C412yes—
74C47⋊C16—44no—
75C23×C47—21no—
76C23×C47—21no—
77C23×C47—41no—
78C23×C47—42no—
79C23×C47C2×M1611yescharacteristic
80Q8×C47—41no—
81D4×C47—41no—
82D4×C47—44no—
83D4×C47—42no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—22no—
87C2×C4×C47—21no—
88C2×C4×C47—21no—
89C2×C4×C47—42no—
90C2×C4×C47—41no—
91C2×C4×C47—41no—
92C2×C4×C47—41no—
93C2×C4×C47—41no—
94C2×C4×C47—41no—
95C2×C4×C47C2×C22⋊C411yescharacteristic
96C2×C4×C47C2×M1611yescharacteristic
97C8×C47—22no—
98C8×C47—21no—
99C8×C47—41no—
100C8×C47—41no—
101C8×C47#10012yes—
102#11537—471nomaximal, Sylow
103C22×C47—22no—
104C22×C47—21no—
105C22×C47—41no—
106C22×C47—41no—
107C22×C47—42no—
108C22×C47—42no—
109C22×C47—42no—
110C22×C47—41no—
111C22×C47—41no—
112C22×C47#40711yescharacteristic
113C4×C47—21no—
114C4×C47—42no—
115C4×C47—41no—
116C4×C47—41no—
117C4×C47#41011yescharacteristic
118C4×C47C2×#10011yescharacteristic
119#2929D4711yescharacteristic
120#2166—472no—
121#1321—472no—
122#1320—472no—
123C2×C47—21no—
124C2×C47—42no—
125C2×C47—41no—
126C2×C47—81no—
127C2×C47#146611yescharacteristic
128C2×#131Dic4711yescharacteristic
129C23×C8Dic4711yescharacteristic
130#525Dic4711yescharacteristic
131#435—22no—
132C8×D4—22no—
133#424Dic4711yescharacteristic
134#408C2×D4711yescharacteristic
135#407C2×D4711yescharacteristic
136#405C2×D4711yescharacteristic
137#364—942no—
138#349—474no—
139#347—942no—
140C47#1153711yescharacteristic, complemented, Sylow
141C2×#41C2×Dic4711yescharacteristic
142C23×C4C2×Dic4711yescharacteristic
143#131—42no—
144C2×M16—21no—
145C2×M16C2×Dic4711yescharacteristic
146C2×M16C22×D4711yescharacteristic
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—42no—
151C22×C8C2×Dic4711yescharacteristic
152C22×C8C2×Dic4711yescharacteristic
153#123—21no—
154C22⋊Q8—21no—
155#121—21no—
156#120—21no—
157C4×D4—22no—
158#117C2×Dic4711yescharacteristic
159M32—944no—
160C4⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8—22no—
163#97C22⋊D4712yes—
164C4×C8C22⋊D4712yes—
165C24#1062532911yescharacteristic
166#41—41no—
167#41—41no—
168C2×Q8#1062532911yescharacteristic
169C2×D4—22no—
170C2×D4—21no—
171C2×D4#1062532911yescharacteristic
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4#1062532911yescharacteristic
178C22×C4C22×Dic4711yescharacteristic
179M16—41no—
180M16—42no—
181M16—41no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—42no—
188C2×C8—41no—
189C2×C8C22⋊Dic4711yescharacteristic
190C2×C8C22⋊Dic4711yescharacteristic
191C2×C8C2×C22⋊D4712yes—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C22⋊C4—22no—
195C22⋊C4—22no—
196C22⋊C4—21no—
197C42C22⋊Dic4712yes—
198C16—1884no—
199C23—21no—
200C23—21no—
201C23—41no—
202C23—42no—
203C23C2×#1062532911yescharacteristic
204Q8—41no—
205D4—41no—
206D4—44no—
207D4—42no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4C2×C22⋊Dic4711yescharacteristic
220C2×C4C2×#1062532911yescharacteristic
221C8—22no—
222C8—21no—
223C8—41no—
224C8—41no—
225C8#1309394112yes—
226C22—22no—
227C22—21no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22#1568908711yescharacteristic
236C4—21no—
237C4—42no—
238C4—41no—
239C4—41no—
240C4#1568909711yescharacteristic
241C4C2×#1309394111yescharacteristic
242C2—21no—
243C2—42no—
244C2—41no—
245C2—81no—
246C2#2024172911yescharacteristic
subgroupG/H
centerC4#15689097
commutatorC22×C4×C47C22×C4
FrattiniC2×M16C22×D47
FittingC47×#2929C2
socleC2×C47#1466
prime radicalC47×#525C4
cosocleC2×M16×C47C23
hypercenter#2929D47
hypocenterC47#11537
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ #2929

Complements

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

NH ≅ G/Ntype
1. #20245204C2semidirect
2. #20241326C2semidirect
3. #20241324C2semidirect
140. C47#11537semidirect

Maximal subgroups

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

subgroupindex
#202452042
#202413262
#202413242
C47×#29292
#1153747