Finite Group Databasebeta

← #26128235

Subgroups — #26128235

Order 12032.

234 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 67 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#20245265C211yesmaximal, characteristic
2#20242301C211yesmaximal, characteristic, complemented
3#20242211C211yesmaximal, characteristic, complemented
4#20241354C211yesmaximal, characteristic, complemented
5#20241354C211yesmaximal, characteristic, complemented
6#20241310C211yesmaximal, characteristic, complemented
7C47×#1368C211yesmaximal, characteristic, complemented
8C2×D752C411yescharacteristic
9C4×D376C2211yescharacteristic
10C4×D376C2211yescharacteristic
11#15689200—21no—
12#15689188—21no—
13#15688951C411yescharacteristic
14#15688925C411yescharacteristic
15#15688925C411yescharacteristic
16#15688899C2211yescharacteristic
17#15688850—21no—
18#15688850—21no—
19#15688844C2211yescharacteristic
20C47×#346C2211yescharacteristic
21C47×#334C2211yescharacteristic
22C47×#334C2211yescharacteristic
23C2×D376C2×C411yescharacteristic
24C2×D376C2×C411yescharacteristic
25C2×#10625324—41no—
26C2×C8×D47—41no—
27C4×D188—21no—
28C4×D188—21no—
29D752—22no—
30C2×C47⋊C16—21no—
31#13093958C2×C411yescharacteristic
32#13093958C2×C411yescharacteristic
33#13093946—21no—
34#13093946—21no—
35(C2×D47)⋊C8—41no—
36(C2×D47)⋊C8—41no—
37C4×C47⋊C8D411yescharacteristic
38C4×C47⋊C8D411yescharacteristic
39C2×C16×C47C2×C411yescharacteristic
40C2×C16×C47C2×C411yescharacteristic
41C4⋊C8×C47—21no—
42C4⋊C8×C47—21no—
43C4×C8×C47C2311yescharacteristic
44C2×D188—21no—
45C2×D188—21no—
46C2×C4×D47—41no—
47C2×C4×D47—41no—
48C2×C8×C47—41no—
49C2×C8×C47—41no—
50C2×C8×C47C22×C411yescharacteristic
51C2×C8×C47#4111yescharacteristic
52C42×C47C2×D411yescharacteristic
53(C2×D47)⋊C4—41no—
54(C2×D47)⋊C4—41no—
55C4⋊Dic47—21no—
56C4⋊Dic47—21no—
57C2×C47⋊C8—21no—
58C2×C47⋊C8—21no—
59D376—21no—
60D376—21no—
61D376—21no—
62D376M1612yescomplemented
63#10625324—82no—
64C8×D47—82no—
65C16×C47—21no—
66C16×C47—21no—
67C47⋊C16—22no—
68C22×D47—41no—
69C22×D47—41no—
70C2×C4×C47C4×D411yescharacteristic
71C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
72C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
73C2×Dic47—41no—
74C2×Dic47—41no—
75D188—22no—
76D188—41no—
77D188—41no—
78D188—41no—
79C4×D47—81no—
80C4×D47—81no—
81C4×D47—81no—
82C4×D47—81no—
83C8×C47—21no—
84C8×C47—81no—
85C8×C47—81no—
86C8×C47C2×M1611yescharacteristic
87C8×C47#13111yescharacteristic
88C47⋊C8—41no—
89C47⋊C8—41no—
90#12078—471nomaximal, Sylow
91C22×C47#44211yescharacteristic
92C2×D47—42no—
93C2×D47—81no—
94C2×D47—81no—
95C2×D47—81no—
96C2×D47—81no—
97C2×D47—81no—
98C4×C47—21no—
99C4×C47—21no—
100C4×C47#43511yescharacteristic
101C4×C47#44311yescharacteristic
102Dic47—81no—
103Dic47—81no—
104#2177—471no—
105#1594—471no—
106#1580—471no—
107#1368D4711yescharacteristic, complemented
108#1333—471no—
109#1333—471no—
110#1313—471no—
111C2×C47#158611yescharacteristic
112C2×C47AGL(1,ℤ/16)11yescharacteristic
113C2×C47AGL(1,ℤ/16)11yescharacteristic
114D47—82no—
115D47—161no—
116C2×D16—471no—
117C4×D8—471no—
118C4×D8—471no—
119#436—941no—
120C8×D4—941no—
121#367—471no—
122#357—471no—
123#357—471no—
124#346—471no—
125#346C2×D4711yescharacteristic
126#334C2×D4711yescharacteristic
127#334C2×D4711yescharacteristic
128D4⋊C8—941no—
129D4⋊C8—941no—
130#322—471no—
131C47#1207811yescharacteristic, complemented, Sylow
132C2×D8—471no—
133C2×D8—471no—
134C2×M16—1881no—
135C22×C8—1881no—
136C4×D4—941no—
137C4×D4—941no—
138D16—942no—
139C2×C16—941no—
140C2×C16C4×D4711yescharacteristic
141C2×C16C4×D4711yescharacteristic
142#107—471no—
143#107—471no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146#102—941no—
147#102—941no—
148C22⋊C8—1881no—
149C22⋊C8—1881no—
150C4×C8—471no—
151C4×C8—471no—
152C4×C8C22×D4711yescharacteristic
153C2×D4—941no—
154C2×D4—941no—
155C22×C4—1881no—
156C22×C4—1881no—
157D8—472no—
158D8—941no—
159D8—941no—
160D8—941no—
161M16—3762no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—941no—
165C2×C8—941no—
166C2×C8—3762no—
167C2×C8C2×C4×D4711yescharacteristic
168C2×C8#1062535411yescharacteristic
169C4⋊C4—941no—
170C4⋊C4—941no—
171C22⋊C4—1881no—
172C22⋊C4—1881no—
173C42D4×D4711yescharacteristic
174C16—21no—
175C16—21no—
176C16—942no—
177C23—1881no—
178C23—1881no—
179D4—942no—
180D4—1881no—
181D4—1881no—
182D4—1881no—
183C2×C4—3761no—
184C2×C4—3761no—
185C2×C4—3761no—
186C2×C4—3761no—
187C2×C4—1881no—
188C2×C4—1881no—
189C2×C4#1309398611yescharacteristic
190C2×C4#1309405111yescharacteristic
191C2×C4#1309405111yescharacteristic
192C8—21no—
193C8—81no—
194C8—81no—
195C8—1881no—
196C8—1881no—
197C8M16×D4711yescharacteristic
198C8#1309403011yescharacteristic
199C22—3761no—
200C22—3761no—
201C22—3761no—
202C22—3761no—
203C22—3761no—
204C22—1882no—
205C22#1568923411yescharacteristic
206C4—21no—
207C4—21no—
208C4—3761no—
209C4—3761no—
210C4#1568919311yescharacteristic
211C4#1568923911yescharacteristic
212C2—3762no—
213C2—7521no—
214C2#2024225111yescharacteristic
215C2#2024528511yescharacteristic
216C2#2024528511yescharacteristic
subgroupG/H
centerC22#15689234
commutatorC2×C8×C47C22×C4
FrattiniC4×C8C22×D47
FittingC47×#1368C2
socleC22×C47#442
prime radicalC2×D752C4
cosocleC4×C8×C47C23
hypercenter#1368D47
hypocenterC47#12078
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1368

Complements

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

NH ≅ G/Ntype
2. #20242301C2semidirect
3. #20242211C2semidirect
4. #20241354C2semidirect
5. #20241354C2semidirect
6. #20241310C2semidirect
7. C47×#1368C2semidirect
62. D376M16semidirect
107. #1368D47semidirect
131. C47#12078semidirect

Maximal subgroups

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

subgroupindex
#202452652
#202423012
#202422112
#202413542
#202413102
C47×#13682
#1207847