Finite Group Databasebeta

← #26130708

Subgroups — #26130708

Order 12032.

218 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 63 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#20245615C211yesmaximal, characteristic, complemented
2#20245207C211yesmaximal, characteristic
3#20242788C211yesmaximal, characteristic
4#20242750C211yesmaximal, characteristic
5#20241459C211yesmaximal, characteristic, complemented
6#20241445C211yesmaximal, characteristic, complemented
7C47×#1362C211yesmaximal, characteristic, complemented
8C2×C16×D47—21no—
9#15689528C2211yescharacteristic
10#15689414—21no—
11#15689400—21no—
12C8×D188C2211yescharacteristic
13#15688927—21no—
14#15688927—21no—
15(C2×D47)⋊C16—21no—
16C4×C47⋊C16C2211yescharacteristic
17#15688876C2211yescharacteristic
18#15688871—21no—
19#15688869—21no—
20C47×#494C2211yescharacteristic
21C47×C4⋊C16C2211yescharacteristic
22C47×#335C2211yescharacteristic
23C2×C8×D47—21no—
24#13094024—21no—
25C4×D188D411yescharacteristic
26C2×C47⋊C16D411yescharacteristic
27C2×C47⋊C16D411yescharacteristic
28C16×D47—42no—
29#13093958—21no—
30#13093958—21no—
31#13093957—41no—
32#13093956—41no—
33C4⋊(C47⋊C8)—21no—
34C4⋊(C47⋊C8)D411yescharacteristic
35#13093946—41no—
36#13093944—41no—
37(C2×D47)⋊C8—21no—
38C2×Q16×C47—21no—
39C2×Q16×C47—21no—
40C4⋊Q8×C47—21no—
41C2×C16×C47—21no—
42C4⋊C8×C47—21no—
43C4×C8×C47C2311yescharacteristic
44C2×Q8×C47—41no—
45C2×D188D811yescharacteristic
46C2×C4×D47—21no—
47C2×Dic94—41no—
48Q16×C47—41no—
49Q16×C47—41no—
50Q16×C47—81no—
51C2×C8×C47—41no—
52C2×C8×C47C2×D411yescharacteristic
53C2×C8×C47#4111yescharacteristic
54C4⋊C4×C47—41no—
55C4⋊C4×C47—41no—
56C42×C47C2×D411yescharacteristic
57(C2×D47)⋊C4—21no—
58C4⋊Dic47—41no—
59C4⋊Dic47—41no—
60C4⋊Dic47D811yescharacteristic
61C2×C47⋊C8—21no—
62C2×C47⋊C8—41no—
63C8×D47—41no—
64C8×D47—41no—
65C16×C47—41no—
66C47⋊C16—21no—
67C47⋊C16—21no—
68C22×D47—21no—
69Q8×C47—81no—
70Q8×C47—81no—
71C2×C4×C47—81no—
72C2×C4×C47#12111yescharacteristic
73C2×C4×C47C2×D811yescharacteristic
74C2×C4×C47#13511yescharacteristic
75C2×Dic47—21no—
76C2×Dic47—81no—
77D188—21no—
78D188Q3212yes—
79C4×D47—41no—
80C4×D47—41no—
81Dic94—161no—
82C8×C47—21no—
83C8×C47—81no—
84C8×C47#13511yescharacteristic
85C8×C47AGL(1,ℤ/8)11yescharacteristic
86C47⋊C8—41no—
87C47⋊C8—81no—
88#12469—471nomaximal, Sylow
89C22×C47#46611yescharacteristic
90C2×D47—22no—
91C2×D47—41no—
92C2×D47—41no—
93C4×C47—21no—
94C4×C47—161no—
95C4×C47#46311yescharacteristic
96C4×C47#47111yescharacteristic
97C4×C47C2×Q3211yescharacteristic
98C4×C47#50811yescharacteristic
99Dic47—41no—
100Dic47—161no—
101#2241—471no—
102D4×C16—471no—
103#1675—471no—
104#1669—471no—
105#1371—471no—
106#1365—471no—
107#1362D4711yescharacteristic, complemented
108C2×C47#168111yescharacteristic
109C2×C47#221611yescharacteristic
110C2×C47#222311yescharacteristic
111D47—42no—
112C22×C16—941no—
113#500—471no—
114#494C2×D4711yescharacteristic
115#476—941no—
116#474—941no—
117C8×D4—471no—
118C4⋊C16C2×D4711yescharacteristic
119#358—941no—
120#358—941no—
121C22⋊C16—941no—
122C4×C16—471no—
123#335—471no—
124#335C2×D4711yescharacteristic
125#333—941no—
126#332—941no—
127C47#1246911yescharacteristic, complemented, Sylow
128C2×Q16—21no—
129C2×Q16—21no—
130C22×C8—941no—
131C4⋊Q8—21no—
132C4⋊Q8—941no—
133C4×D4—471no—
134C2×C16—21no—
135C2×C16—471no—
136C2×C16—471no—
137C2×C16—1882no—
138#107—941no—
139#107—941no—
140#107—1881no—
141#106—1881no—
142C4⋊C8—21no—
143C4⋊C8—471no—
144C4⋊C8—941no—
145#102—1881no—
146#102—1881no—
147C22⋊C8—941no—
148C4×C8C22×D4711yescharacteristic
149C2×Q8—41no—
150C2×Q8—1881no—
151C2×D4—471no—
152C22×C4—941no—
153Q16—41no—
154Q16—41no—
155Q16—81no—
156C2×C8—41no—
157C2×C8—941no—
158C2×C8—1881no—
159C2×C8—1881no—
160C2×C8—1881no—
161C2×C8#1062535011yescharacteristic
162C2×C8D4×D4711yescharacteristic
163C4⋊C4—41no—
164C4⋊C4—41no—
165C4⋊C4—471no—
166C4⋊C4—1881no—
167C4⋊C4—1881no—
168C22⋊C4—941no—
169C42D4×D4711yescharacteristic
170C16—41no—
171C16—941no—
172C16—941no—
173C23—941no—
174Q8—81no—
175Q8—81no—
176Q8—7521no—
177D4—472no—
178D4—941no—
179C2×C4—81no—
180C2×C4—941no—
181C2×C4—3761no—
182C2×C4—1881no—
183C2×C4—1881no—
184C2×C4#1309399711yescharacteristic
185C2×C4D8×D4711yescharacteristic
186C2×C4#1309404511yescharacteristic
187C8—21no—
188C8—81no—
189C8—3761no—
190C8—1881no—
191C8#1309404511yescharacteristic
192C8#1309404811yescharacteristic
193C22—942no—
194C22—1881no—
195C22—1881no—
196C22#1568935811yescharacteristic
197C4—21no—
198C4—161no—
199C4—1881no—
200C4—7521no—
201C4#1568933711yescharacteristic
202C4#1568938511yescharacteristic
203C4Q32×D4711yescharacteristic
204C4#1568955411yescharacteristic
205C2—1882no—
206C2#2024282411yescharacteristic
207C2#2024547811yescharacteristic
208C2#2024552111yescharacteristic
subgroupG/H
centerC22#15689358
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#1362C2
socleC22×C47#466
prime radicalC2×D188D8
hypercenter#1362D47
hypocenterC47#12469
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
1. #20245615C2semidirect
5. #20241459C2semidirect
6. #20241445C2semidirect
7. C47×#1362C2semidirect
107. #1362D47semidirect
127. C47#12469semidirect

Maximal subgroups

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

subgroupindex
#202456152
#202452072
#202427882
#202427502
#202414592
#202414452
C47×#13622
#1246947