Finite Group Databasebeta

← #26130703

Subgroups — #26130703

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
1#20245612C211yesmaximal, characteristic, complemented
2C16×D188C211yesmaximal, characteristic, complemented
3#20242788C211yesmaximal, characteristic, complemented
4#20242744C211yesmaximal, characteristic
5#20241447C211yesmaximal, characteristic, complemented
6#20241433C211yesmaximal, characteristic, complemented
7C47×#1372C211yesmaximal, characteristic, complemented
8C2×C16×D47—21no—
9#15689528C2211yescharacteristic
10#15689500C2211yescharacteristic
11#15689414—21no—
12#15689319—21no—
13C8×D188C2211yescharacteristic
14C4⋊(C47⋊C16)C2211yescharacteristic
15#15688925—21no—
16#15688925—21no—
17(C2×D47)⋊C16—21no—
18#15688871—21no—
19#15688866—21no—
20C4×C16×C47C2211yescharacteristic
21C47×#335C2211yescharacteristic
22C47×#335C2211yescharacteristic
23C2×D376—21no—
24C2×D376—21no—
25C2×D376—41no—
26C2×C8×D47—21no—
27#13094024—21no—
28#13094021—21no—
29C4×D188D411yescharacteristic
30C2×C47⋊C16—21no—
31C16×D47—42no—
32#13093958—21no—
33#13093958—21no—
34#13093956—41no—
35C4⋊(C47⋊C8)D411yescharacteristic
36#13093946—41no—
37#13093946—41no—
38(C2×D47)⋊C8—21no—
39C2×C16×C47D411yescharacteristic
40C2×C16×C47D411yescharacteristic
41C4⋊C8×C47—21no—
42C4⋊C8×C47—21no—
43C4×C8×C47C2311yescharacteristic
44C2×D188—41no—
45C2×D188—41no—
46C2×D188—41no—
47C2×D188D811yescharacteristic
48C2×C4×D47—21no—
49C2×Dic94—41no—
50C2×C8×C47—41no—
51C2×C8×C47—41no—
52C2×C8×C47C2×D411yescharacteristic
53C2×C8×C47#4111yescharacteristic
54C42×C47C2×D411yescharacteristic
55(C2×D47)⋊C4—21no—
56C4⋊Dic47—41no—
57C4⋊Dic47—41no—
58C4⋊Dic47D811yescharacteristic
59C2×C47⋊C8—21no—
60D376—41no—
61D376—41no—
62D376—81no—
63D376—82no—
64C8×D47—41no—
65C8×D47—41no—
66C16×C47—21no—
67C16×C47—21no—
68C47⋊C16—41no—
69C22×D47—21no—
70C22×D47—81no—
71C2×C4×C47#12111yescharacteristic
72C2×C4×C47C2×D811yescharacteristic
73C2×C4×C47#13511yescharacteristic
74C2×Dic47—21no—
75C2×Dic47—81no—
76D188—21no—
77D188—81no—
78D188—81no—
79D188—81no—
80D188—81no—
81D188—161no—
82D188SD3212yes—
83C4×D47—41no—
84C4×D47—41no—
85Dic94—161no—
86C8×C47—21no—
87C8×C47—81no—
88C8×C47—81no—
89C8×C47#13511yescharacteristic
90C8×C47AGL(1,ℤ/8)11yescharacteristic
91C47⋊C8—41no—
92#12468—471nomaximal, Sylow
93C22×C47#46611yescharacteristic
94C2×D47—22no—
95C2×D47—41no—
96C2×D47—41no—
97C2×D47—161no—
98C2×D47—161no—
99C2×D47—161no—
100C4×C47—21no—
101C4×C47#46311yescharacteristic
102C4×C47#47111yescharacteristic
103C4×C47C2×SD3211yescharacteristic
104C4×C47#50811yescharacteristic
105Dic47—41no—
106Dic47—161no—
107#2240—471no—
108D4×C16—471no—
109#1675—471no—
110#1667—471no—
111#1372D4711yescharacteristic, complemented
112#1365—471no—
113#1361—471no—
114C2×C47#168111yescharacteristic
115C2×C47#221711yescharacteristic
116C2×C47#222111yescharacteristic
117D47—42no—
118D47—321no—
119C22×C16—941no—
120#500—471no—
121#493—471no—
122#476—941no—
123#459—941no—
124C8×D4—471no—
125C4⋊C16—471no—
126#357—941no—
127#357—941no—
128C22⋊C16—941no—
129C4×C16C2×D4711yescharacteristic
130#335C2×D4711yescharacteristic
131#335C2×D4711yescharacteristic
132#333—941no—
133#331—941no—
134C47#1246811yescharacteristic, complemented, Sylow
135C2×D8—941no—
136C2×D8—941no—
137C2×D8—1881no—
138C22×C8—941no—
139C4⋊Q8—941no—
140#127—941no—
141C4×D4—471no—
142C2×C16—941no—
143C2×C16—1882no—
144C2×C16D18811yescharacteristic
145C2×C16D18811yescharacteristic
146#107—941no—
147#107—941no—
148#106—1881no—
149C4⋊C8—21no—
150C4⋊C8—21no—
151C4⋊C8—471no—
152#102—1881no—
153#102—1881no—
154C22⋊C8—941no—
155C4×C8C22×D4711yescharacteristic
156C2×Q8—1881no—
157C2×D4—471no—
158C2×D4—1881no—
159C2×D4—1881no—
160C2×D4—1881no—
161C22×C4—941no—
162D8—3761no—
163D8—3762no—
164D8—1881no—
165D8—1881no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—941no—
169C2×C8—1881no—
170C2×C8—1881no—
171C2×C8C2×D18811yescharacteristic
172C2×C8#1062535411yescharacteristic
173C4⋊C4—471no—
174C4⋊C4—1881no—
175C4⋊C4—1881no—
176C22⋊C4—941no—
177C42D4×D4711yescharacteristic
178C16—21no—
179C16—21no—
180C16—1881no—
181C23—941no—
182C23—3761no—
183Q8—7521no—
184D4—472no—
185D4—941no—
186D4—3761no—
187D4—3761no—
188D4—3761no—
189D4—3761no—
190D4—7521no—
191C2×C4—941no—
192C2×C4—3761no—
193C2×C4—1881no—
194C2×C4—1881no—
195C2×C4#1309399911yescharacteristic
196C2×C4D8×D4711yescharacteristic
197C2×C4#1309404711yescharacteristic
198C8—21no—
199C8—81no—
200C8—81no—
201C8—1881no—
202C8#1309404611yescharacteristic
203C8#1309405211yescharacteristic
204C22—942no—
205C22—1881no—
206C22—1881no—
207C22—7521no—
208C22—7521no—
209C22—7521no—
210C22#1568936011yescharacteristic
211C4—21no—
212C4—1881no—
213C4—7521no—
214C4#1568933911yescharacteristic
215C4#1568938711yescharacteristic
216C4SD32×D4711yescharacteristic
217C4#1568955411yescharacteristic
218C2—1882no—
219C2—15041no—
220C2#2024282611yescharacteristic
221C2#2024548511yescharacteristic
222C2#2024550911yescharacteristic
subgroupG/H
centerC22#15689360
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#1372C2
socleC22×C47#466
prime radical#20242744C2
hypercenter#1372D47
hypocenterC47#12468
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1372

Complements

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

NH ≅ G/Ntype
1. #20245612C2semidirect
2. C16×D188C2semidirect
3. #20242788C2semidirect
5. #20241447C2semidirect
6. #20241433C2semidirect
7. C47×#1372C2semidirect
111. #1372D47semidirect
134. C47#12468semidirect

Maximal subgroups

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

subgroupindex
#202456122
C16×D1882
#202427882
#202427442
#202414472
#202414332
C47×#13722
#1246847