Finite Group Databasebeta

← #26128236

Subgroups — #26128236

Order 12032.

234 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 71 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#20245262C211yesmaximal, characteristic
2#20242303C211yesmaximal, characteristic, complemented
3#20241453C211yesmaximal, characteristic, complemented
4#20241355C211yesmaximal, characteristic, complemented
5#20241354C211yesmaximal, characteristic, complemented
6#20241309C211yesmaximal, characteristic, complemented
7C47×#1580C211yesmaximal, characteristic, complemented
8C2×#13093965C411yescharacteristic
9C4×D376C2211yescharacteristic
10#15689202—21no—
11#15688950C411yescharacteristic
12#15688924C411yescharacteristic
13#15688923C411yescharacteristic
14#15688899C2211yescharacteristic
15#15688899C2211yescharacteristic
16#15688874C2211yescharacteristic
17#15688852—21no—
18C4×D8×C47C2211yescharacteristic
19C8×D4×C47—21no—
20C47×#334C2211yescharacteristic
21C47×D4⋊C8—21no—
22C47×#322C2211yescharacteristic
23C2×D376C2×C411yescharacteristic
24C2×#10625324—41no—
25C4×D188—21no—
26#13093965—22no—
27C2×C47⋊C16—21no—
28C2×C47⋊C16C2×C411yescharacteristic
29C2×C47⋊C16C2×C411yescharacteristic
30#13093958C2×C411yescharacteristic
31C4⋊(C47⋊C8)—21no—
32#13093946—21no—
33(C2×D47)⋊C8—41no—
34C2×D8×C47C2×C411yescharacteristic
35C22×C8×C47—41no—
36C4×D4×C47—21no—
37#107×C47C2×C411yescharacteristic
38C4⋊C8×C47—21no—
39#102×C47—21no—
40C22⋊C8×C47—41no—
41C4×C8×C47D411yescharacteristic
42C4×C8×C47D411yescharacteristic
43C4×C8×C47C2311yescharacteristic
44C2×D4×C47—21no—
45C22×C4×C47—41no—
46C2×D188—21no—
47C2×C4×D47—41no—
48D8×C47—21no—
49D8×C47M1612yescomplemented
50C2×C8×C47—21no—
51C2×C8×C47—21no—
52C2×C8×C47—41no—
53C2×C8×C47—82no—
54C2×C8×C47C22×C411yescharacteristic
55C2×C8×C47#4111yescharacteristic
56C4⋊C4×C47—21no—
57C22⋊C4×C47—41no—
58C42×C47C2×D411yescharacteristic
59(C2×D47)⋊C4—41no—
60C4⋊Dic47—21no—
61C2×C47⋊C8—41no—
62D376—21no—
63D376—21no—
64#10625324—82no—
65C47⋊C16—22no—
66C47⋊C16—21no—
67C47⋊C16—21no—
68C23×C47—41no—
69C22×D47—41no—
70D4×C47—22no—
71D4×C47—41no—
72C2×C4×C47—41no—
73C2×C4×C47—81no—
74C2×C4×C47—81no—
75C2×C4×C47C4×D411yescharacteristic
76C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
78C2×Dic47—41no—
79D188—41no—
80D188—41no—
81C4×D47—81no—
82C4×D47—81no—
83C8×C47—21no—
84C8×C47—41no—
85C8×C47—41no—
86C8×C47—81no—
87C8×C47C2×M1611yescharacteristic
88C8×C47#13111yescharacteristic
89C47⋊C8—81no—
90#12078—471nomaximal, Sylow
91C22×C47—42no—
92C22×C47—81no—
93C22×C47—81no—
94C22×C47#44211yescharacteristic
95C2×D47—81no—
96C2×D47—81no—
97C2×D47—81no—
98C4×C47—21no—
99C4×C47—21no—
100C4×C47—81no—
101C4×C47#43511yescharacteristic
102C4×C47#44311yescharacteristic
103Dic47—81no—
104#2177—471no—
105#1594—471no—
106#1580D4711yescharacteristic, complemented
107#1368—471no—
108#1333—471no—
109#1333—471no—
110#1313—471no—
111C2×C47—82no—
112C2×C47#158611yescharacteristic
113C2×C47AGL(1,ℤ/16)11yescharacteristic
114C2×C47AGL(1,ℤ/16)11yescharacteristic
115D47—161no—
116C2×D16—471no—
117C4×D8—471no—
118C4×D8C2×D4711yescharacteristic
119#436—941no—
120C8×D4—21no—
121#367—471no—
122#357—471no—
123#357—471no—
124#346—471no—
125#346—471no—
126#334—471no—
127#334C2×D4711yescharacteristic
128D4⋊C8—21no—
129D4⋊C8—941no—
130#322C2×D4711yescharacteristic
131C47#1207811yescharacteristic, complemented, Sylow
132C2×D8—471no—
133C2×D8C4×D4711yescharacteristic
134C2×M16—1881no—
135C22×C8—41no—
136C4×D4—21no—
137C4×D4—941no—
138D16—942no—
139C2×C16—471no—
140C2×C16—471no—
141C2×C16—941no—
142#107—471no—
143#107C4×D4711yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—941no—
146#102—21no—
147#102—941no—
148C22⋊C8—41no—
149C22⋊C8—1881no—
150C4×C8C22⋊D4711yescharacteristic
151C4×C8C22⋊D4711yescharacteristic
152C4×C8C22×D4711yescharacteristic
153C2×D4—21no—
154C2×D4—941no—
155C22×C4—41no—
156C22×C4—1881no—
157D8—21no—
158D8—941no—
159D8—941no—
160D8#1062532412yescomplemented
161M16—3762no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—82no—
166C2×C8—1881no—
167C2×C8C2×C4×D4711yescharacteristic
168C2×C8#1062535011yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—941no—
171C22⋊C4—41no—
172C22⋊C4—1881no—
173C42C2×C22⋊D4711yescharacteristic
174C16—942no—
175C16—941no—
176C16—941no—
177C23—41no—
178C23—1881no—
179D4—22no—
180D4—41no—
181D4—1881no—
182D4—1881no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—3761no—
187C2×C4—3761no—
188C2×C4—1881no—
189C2×C4C4×C22⋊D4711yescharacteristic
190C2×C4#1309404911yescharacteristic
191C2×C4#1309405011yescharacteristic
192C8—21no—
193C8—41no—
194C8—41no—
195C8—81no—
196C8—3761no—
197C8C2×#1062532411yescharacteristic
198C8#1309403011yescharacteristic
199C22—42no—
200C22—81no—
201C22—81no—
202C22—3761no—
203C22—3761no—
204C22—3761no—
205C22#1568923311yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—3761no—
210C4#1568919411yescharacteristic
211C4#1568923811yescharacteristic
212C2—82no—
213C2—7521no—
214C2#2024225211yescharacteristic
215C2#2024528411yescharacteristic
216C2#2024528611yescharacteristic
subgroupG/H
centerC22#15689233
commutatorC2×C8×C47C22×C4
FrattiniC4×C8C22×D47
FittingC47×#1580C2
socleC22×C47#442
prime radicalC2×#13093965C4
cosocleC4×C8×C47C23
hypercenter#1580D47
hypocenterC47#12078
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ #322 ◁ #1580

Complements

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

NH ≅ G/Ntype
2. #20242303C2semidirect
3. #20241453C2semidirect
4. #20241355C2semidirect
5. #20241354C2semidirect
6. #20241309C2semidirect
7. C47×#1580C2semidirect
49. D8×C47M16semidirect
106. #1580D47semidirect
131. C47#12078semidirect
160. D8#10625324semidirect

Maximal subgroups

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

subgroupindex
#202452622
#202423032
#202414532
#202413552
#202413542
#202413092
C47×#15802
#1207847