Finite Group Databasebeta

← #26130404

Subgroups — #26130404

Order 12032.

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

normal subgroups only

This group has 240 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#20245527C211yesmaximal, characteristic, complemented
2#20245429C211yesmaximal, characteristic
3#20242981C211yesmaximal, characteristic
4#20242194C211yesmaximal, characteristic, complemented
5#20241451C211yesmaximal, characteristic, complemented
6#20241433C211yesmaximal, characteristic
7C47×#1364C211yesmaximal, characteristic, complemented
8C2×#13093969—21no—
9#15689500C2211yescharacteristic
10#15689497—21no—
11#15689471—21no—
12#15689247—21no—
13C4×Dic188C2211yescharacteristic
14#15688950—21no—
15C4⋊(C47⋊C16)C2211yescharacteristic
16#15688926—21no—
17#15688925—21no—
18#15688853—21no—
19C8×C47⋊C8C2211yescharacteristic
20C47×#500C2211yescharacteristic
21C47×C4⋊C16C2211yescharacteristic
22C47×#335C2211yescharacteristic
23C2×Dic188D411yescharacteristic
24C2×#10625336—41no—
25C2×#10625336—41no—
26C2×D376—21no—
27C2×D376—21no—
28#13094021—21no—
29C4×Dic94—21no—
30#13093969—42no—
31C2×C47⋊C16—21no—
32#13093958D411yescharacteristic
33Dic47⋊C8—41no—
34#13093949—21no—
35#13093944—42no—
36C4×C47⋊C8D411yescharacteristic
37C4×C47⋊C8D411yescharacteristic
38C8×Dic47—41no—
39C4⋊Q8×C47—21no—
40C2×C16×C47—21no—
41#107×C47—21no—
42#107×C47—21no—
43C4⋊C8×C47—21no—
44C4×C8×C47C2311yescharacteristic
45C2×Q8×C47—41no—
46C2×D188—41no—
47C2×D188—41no—
48C2×D188—41no—
49C2×Dic94—21no—
50C2×C8×C47—41no—
51C2×C8×C47C2×D411yescharacteristic
52C2×C8×C47#4111yescharacteristic
53C4⋊C4×C47—41no—
54C4⋊C4×C47—41no—
55C42×C47C2×D411yescharacteristic
56#10625336—82no—
57#10625336—82no—
58C4⋊Dic47—21no—
59Dic47⋊C4—41no—
60C4×Dic47—41no—
61C2×C47⋊C8—21no—
62C2×C47⋊C8D811yescharacteristic
63C2×C47⋊C8D811yescharacteristic
64Dic188—21no—
65Dic188SD1612yescomplemented
66D376—41no—
67D376—41no—
68D376—81no—
69C16×C47—41no—
70C47⋊C16—41no—
71C22×D47—81no—
72Q8×C47—81no—
73Q8×C47—81no—
74C2×C4×C47—81no—
75C2×C4×C47#12111yescharacteristic
76C2×C4×C47C2×D811yescharacteristic
77C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
78C2×Dic47—41no—
79C2×Dic47—41no—
80D188—82no—
81D188—81no—
82D188—81no—
83D188—81no—
84D188—81no—
85Dic94—22no—
86Dic94—41no—
87C8×C47—21no—
88C8×C47—81no—
89C8×C47C2×SD1611yescharacteristic
90C8×C47#13511yescharacteristic
91C47⋊C8—21no—
92C47⋊C8—41no—
93C47⋊C8SD3212yescomplemented
94#12417—471nomaximal, Sylow
95C22×C47#45911yescharacteristic
96C2×D47—161no—
97C2×D47—161no—
98C2×D47—161no—
99C4×C47—21no—
100C4×C47—161no—
101C4×C47#46511yescharacteristic
102C4×C47#47111yescharacteristic
103C4×C47C2×SD3211yescharacteristic
104C4×C47#50911yescharacteristic
105Dic47—42no—
106Dic47—81no—
107#2225—471no—
108#2207—471no—
109#1706—471no—
110C8×Q16—471no—
111#1367—471no—
112#1364D4711yescharacteristic, complemented
113#1361—471no—
114C2×C47#166511yescharacteristic
115C2×C47#220711yescharacteristic
116C2×C47#221211yescharacteristic
117D47—321no—
118C2×SD32—941no—
119#500C2×D4711yescharacteristic
120#493—471no—
121#492—941no—
122#486—941no—
123C8×Q8—941no—
124C4×Q16—471no—
125#367—941no—
126C4⋊C16—471no—
127C4⋊C16C2×D4711yescharacteristic
128#358—941no—
129#357—941no—
130#335C2×D4711yescharacteristic
131Q8⋊C8—941no—
132C82—471no—
133C47#1241711yescharacteristic, complemented, Sylow
134C2×Q16—471no—
135C2×SD16—1881no—
136C2×SD16—1881no—
137C2×D8—941no—
138C2×D8—941no—
139C4⋊Q8—21no—
140#127—941no—
141C4×Q8—941no—
142SD32—1882no—
143C2×C16—21no—
144C2×C16—941no—
145#107—21no—
146#107—21no—
147#107—471no—
148C4⋊C8—21no—
149C4⋊C8—1881no—
150#103—941no—
151#102—1882no—
152C4×C8—471no—
153C4×C8—471no—
154C4×C8—1881no—
155C4×C8C22×D4711yescharacteristic
156C2×Q8—41no—
157C2×Q8—941no—
158C2×D4—1881no—
159C2×D4—1881no—
160C2×D4—1881no—
161Q16—472no—
162Q16—941no—
163SD16—3762no—
164SD16—3762no—
165D8—3761no—
166D8—1881no—
167D8—1881no—
168C2×C8—41no—
169C2×C8—471no—
170C2×C8—471no—
171C2×C8—941no—
172C2×C8#1062535011yescharacteristic
173C2×C8D4×D4711yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—941no—
177C4⋊C4—1881no—
178C42—1881no—
179C42D4×D4711yescharacteristic
180C16—41no—
181C16—1881no—
182C23—3761no—
183Q8—81no—
184Q8—81no—
185Q8—942no—
186Q8—1881no—
187D4—3762no—
188D4—3761no—
189D4—3761no—
190D4—3761no—
191D4—3761no—
192C2×C4—81no—
193C2×C4—1881no—
194C2×C4—1881no—
195C2×C4#1309399711yescharacteristic
196C2×C4D8×D4711yescharacteristic
197C2×C4#1309405111yescharacteristic
198C8—21no—
199C8—81no—
200C8—472no—
201C8—941no—
202C8—1881no—
203C8SD16×D4711yescharacteristic
204C8#1309404711yescharacteristic
205C22—7521no—
206C22—7521no—
207C22—7521no—
208C22#1568931711yescharacteristic
209C4—21no—
210C4—161no—
211C4—3761no—
212C4—1882no—
213C4#1568935511yescharacteristic
214C4#1568938911yescharacteristic
215C4SD32×D4711yescharacteristic
216C4#1568955711yescharacteristic
217C2—15041no—
218C2#2024273211yescharacteristic
219C2#2024542711yescharacteristic
220C2#2024546011yescharacteristic
subgroupG/H
centerC22#15689317
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#1364C2
socleC22×C47#459
prime radical#15689500C22
hypercenter#1364D47
hypocenterC47#12417
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1364

Complements

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

NH ≅ G/Ntype
1. #20245527C2semidirect
4. #20242194C2semidirect
5. #20241451C2semidirect
7. C47×#1364C2semidirect
65. Dic188SD16semidirect
93. C47⋊C8SD32semidirect
112. #1364D47semidirect
133. C47#12417semidirect

Maximal subgroups

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

subgroupindex
#202455272
#202454292
#202429812
#202421942
#202414512
#202414332
C47×#13642
#1241747