Finite Group Databasebeta

← #26130699

Subgroups — #26130699

Order 12032.

234 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C16×D188C211yesmaximal, characteristic, complemented
2#20242784C211yesmaximal, characteristic, complemented
3#20242740C211yesmaximal, characteristic
4#20241462C211yesmaximal, characteristic, complemented
5#20241446C211yesmaximal, characteristic, complemented
6#20241434C211yesmaximal, characteristic, complemented
7C47×#2240C211yesmaximal, characteristic, complemented
8C2×C16×D47—21no—
9#15689410—21no—
10#15689315—21no—
11C8×D188C2211yescharacteristic
12C4⋊(C47⋊C16)C2211yescharacteristic
13(C2×D47)⋊C16—21no—
14#15688876C2211yescharacteristic
15#15688876C2211yescharacteristic
16#15688872—21no—
17#15688867—21no—
18C47×#500C2211yescharacteristic
19C47×#493C2211yescharacteristic
20C47×#357—21no—
21C47×#357—21no—
22C4×C16×C47C2211yescharacteristic
23C2×#10625334—41no—
24C2×C8×D47—21no—
25C4×D188D411yescharacteristic
26C2×C47⋊C16—21no—
27C16×D47—42no—
28#13093955—41no—
29C4⋊(C47⋊C8)—21no—
30C4⋊(C47⋊C8)—21no—
31C4⋊(C47⋊C8)D411yescharacteristic
32#13093945—41no—
33#13093944—41no—
34(C2×D47)⋊C8—21no—
35C2×D8×C47—21no—
36C2×D8×C47—21no—
37C4⋊Q8×C47—21no—
38#127×C47—21no—
39C2×C16×C47D411yescharacteristic
40C2×C16×C47D411yescharacteristic
41#107×C47—21no—
42#107×C47—21no—
43C4×C8×C47C2311yescharacteristic
44C2×Q8×C47—41no—
45C2×D4×C47—41no—
46C2×D4×C47—41no—
47C2×D4×C47—41no—
48C2×D188D811yescharacteristic
49C2×C4×D47—21no—
50D8×C47—41no—
51D8×C47—41no—
52D8×C47—81no—
53C2×C8×C47C2×D411yescharacteristic
54C2×C8×C47#4111yescharacteristic
55C4⋊C4×C47—41no—
56C4⋊C4×C47—41no—
57C42×C47C2×D411yescharacteristic
58#10625334—82no—
59(C2×D47)⋊C4—21no—
60C4⋊Dic47D811yescharacteristic
61C2×C47⋊C8—21no—
62C2×C47⋊C8—41no—
63C2×C47⋊C8—41no—
64C8×D47—41no—
65C8×D47—41no—
66C16×C47—21no—
67C16×C47—21no—
68C47⋊C16—41no—
69C23×C47—81no—
70C22×D47—21no—
71Q8×C47—161no—
72D4×C47—81no—
73D4×C47—81no—
74D4×C47—81no—
75D4×C47—81no—
76D4×C47—161no—
77C2×C4×C47—81no—
78C2×C4×C47#12111yescharacteristic
79C2×C4×C47C2×D811yescharacteristic
80C2×C4×C47#13511yescharacteristic
81C2×Dic47—21no—
82D188—21no—
83D188SD3212yes—
84C4×D47—41no—
85C4×D47—41no—
86C8×C47—21no—
87C8×C47#13511yescharacteristic
88C8×C47AGL(1,ℤ/8)11yescharacteristic
89C47⋊C8—41no—
90C47⋊C8—81no—
91C47⋊C8—81no—
92#12468—471nomaximal, Sylow
93C22×C47—161no—
94C22×C47—161no—
95C22×C47—161no—
96C22×C47#46611yescharacteristic
97C2×D47—22no—
98C2×D47—41no—
99C2×D47—41no—
100C4×C47—21no—
101C4×C47—161no—
102C4×C47#46311yescharacteristic
103C4×C47#47111yescharacteristic
104C4×C47C2×SD3211yescharacteristic
105C4×C47#50811yescharacteristic
106Dic47—41no—
107#2240D4711yescharacteristic, complemented
108D4×C16—471no—
109#1675—471no—
110#1667—471no—
111#1372—471no—
112#1365—471no—
113#1361—471no—
114C2×C47—321no—
115C2×C47#168111yescharacteristic
116C2×C47#221711yescharacteristic
117C2×C47#222111yescharacteristic
118D47—42no—
119C22×C16—941no—
120#500C2×D4711yescharacteristic
121#493C2×D4711yescharacteristic
122#476—941no—
123#459—941no—
124C8×D4—471no—
125C4⋊C16—471no—
126#357—21no—
127#357—21no—
128C22⋊C16—941no—
129C4×C16C2×D4711yescharacteristic
130#335—471no—
131#335—471no—
132#333—941no—
133#331—941no—
134C47#1246811yescharacteristic, complemented, Sylow
135C2×D8—21no—
136C2×D8—21no—
137C2×D8—1881no—
138C22×C8—941no—
139C4⋊Q8—21no—
140#127—21no—
141C4×D4—471no—
142C2×C16—941no—
143C2×C16—1882no—
144C2×C16C22⋊D4711yescharacteristic
145C2×C16C22⋊D4711yescharacteristic
146#107—21no—
147#107—21no—
148#106—1881no—
149C4⋊C8—471no—
150C4⋊C8—941no—
151C4⋊C8—941no—
152#102—1881no—
153#102—1881no—
154C22⋊C8—941no—
155C4×C8C22×D4711yescharacteristic
156C2×Q8—41no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—471no—
161C22×C4—941no—
162D8—41no—
163D8—41no—
164D8—81no—
165D8—3762no—
166C2×C8—941no—
167C2×C8—1881no—
168C2×C8—1881no—
169C2×C8—1881no—
170C2×C8—1881no—
171C2×C8#1062535211yescharacteristic
172C2×C8C2×C22⋊D4711yescharacteristic
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—471no—
176C22⋊C4—941no—
177C42D4×D4711yescharacteristic
178C16—21no—
179C16—21no—
180C16—1881no—
181C23—81no—
182C23—941no—
183Q8—161no—
184D4—81no—
185D4—81no—
186D4—81no—
187D4—81no—
188D4—161no—
189D4—472no—
190D4—941no—
191C2×C4—81no—
192C2×C4—941no—
193C2×C4—1881no—
194C2×C4—1881no—
195C2×C4#1309399511yescharacteristic
196C2×C4D8×D4711yescharacteristic
197C2×C4#1309404311yescharacteristic
198C8—21no—
199C8—3761no—
200C8—3761no—
201C8—1881no—
202C8#1309404411yescharacteristic
203C8#1309404911yescharacteristic
204C22—161no—
205C22—161no—
206C22—161no—
207C22—942no—
208C22—1881no—
209C22—1881no—
210C22#1568935611yescharacteristic
211C4—21no—
212C4—161no—
213C4—1881no—
214C4#1568933511yescharacteristic
215C4#1568938311yescharacteristic
216C4SD32×D4711yescharacteristic
217C4#1568955411yescharacteristic
218C2—321no—
219C2—1882no—
220C2#2024282211yescharacteristic
221C2#2024548111yescharacteristic
222C2#2024550511yescharacteristic
subgroupG/H
centerC22#15689356
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#2240C2
socleC22×C47#466
prime radical#20242740C2
hypercenter#2240D47
hypocenterC47#12468
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. C16×D188C2semidirect
2. #20242784C2semidirect
4. #20241462C2semidirect
5. #20241446C2semidirect
6. #20241434C2semidirect
7. C47×#2240C2semidirect
107. #2240D47semidirect
134. C47#12468semidirect

Maximal subgroups

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

subgroupindex
C16×D1882
#202427842
#202427402
#202414622
#202414462
#202414342
C47×#22402
#1246847