Finite Group Databasebeta

← #26130421

Subgroups — #26130421

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#20245536C211yesmaximal, characteristic
2#20245430C211yesmaximal, characteristic
3#20242982C211yesmaximal, characteristic, complemented
4#20242182C211yesmaximal, characteristic
5#20241451C211yesmaximal, characteristic, complemented
6#20241447C211yesmaximal, characteristic, complemented
7C47×#1362C211yesmaximal, characteristic, complemented
8C2×#13093967—21no—
9#15689528C2211yescharacteristic
10#15689519—21no—
11#15689475—21no—
12C4×D376C2211yescharacteristic
13#15689188—21no—
14#15688951—21no—
15C4⋊(C47⋊C16)C2211yescharacteristic
16#15688927—21no—
17#15688925—21no—
18#15688850—21no—
19C8×C47⋊C8C2211yescharacteristic
20C47×#494C2211yescharacteristic
21C47×C4⋊C16C2211yescharacteristic
22C47×#335C2211yescharacteristic
23C2×D376D411yescharacteristic
24C2×C8×D47—41no—
25#13094024—21no—
26C4×D188—21no—
27#13093967—42no—
28C2×C47⋊C16—21no—
29#13093958—21no—
30#13093958—21no—
31#13093958D411yescharacteristic
32#13093955—41no—
33#13093955—41no—
34#13093947—42no—
35#13093946—21no—
36(C2×D47)⋊C8—41no—
37C4×C47⋊C8D411yescharacteristic
38C4×C47⋊C8D411yescharacteristic
39C2×Q16×C47—21no—
40C2×Q16×C47—21no—
41C4⋊Q8×C47—21no—
42C2×C16×C47—21no—
43C4⋊C8×C47—21no—
44C4×C8×C47C2311yescharacteristic
45C2×Q8×C47—41no—
46C2×D188—21no—
47C2×C4×D47—41no—
48C2×Dic94—41no—
49Q16×C47—41no—
50Q16×C47—41no—
51Q16×C47—81no—
52C2×C8×C47—41no—
53C2×C8×C47C2×D411yescharacteristic
54C2×C8×C47#4111yescharacteristic
55C4⋊C4×C47—41no—
56C4⋊C4×C47—41no—
57C42×C47C2×D411yescharacteristic
58(C2×D47)⋊C4—41no—
59C4⋊Dic47—21no—
60C4⋊Dic47—41no—
61C4⋊Dic47—41no—
62C2×C47⋊C8—21no—
63C2×C47⋊C8D811yescharacteristic
64C2×C47⋊C8D811yescharacteristic
65D376—21no—
66D376SD1612yes—
67C8×D47—82no—
68C16×C47—41no—
69C47⋊C16—41no—
70C22×D47—41no—
71Q8×C47—81no—
72Q8×C47—81no—
73C2×C4×C47—81no—
74C2×C4×C47#12111yescharacteristic
75C2×C4×C47C2×D811yescharacteristic
76C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
77C2×Dic47—41no—
78C2×Dic47—81no—
79D188—22no—
80D188—41no—
81C4×D47—81no—
82C4×D47—81no—
83Dic94—82no—
84C8×C47—21no—
85C8×C47—81no—
86C8×C47C2×SD1611yescharacteristic
87C8×C47#13511yescharacteristic
88C47⋊C8—21no—
89C47⋊C8—41no—
90C47⋊C8SD3212yescomplemented
91#12420—471nomaximal, Sylow
92C22×C47#45911yescharacteristic
93C2×D47—42no—
94C2×D47—81no—
95C2×D47—81no—
96C4×C47—21no—
97C4×C47—161no—
98C4×C47#46511yescharacteristic
99C4×C47#47111yescharacteristic
100C4×C47C2×SD3211yescharacteristic
101C4×C47#51011yescharacteristic
102Dic47—81no—
103Dic47—161no—
104#2226—471no—
105#2208—471no—
106#1707—471no—
107C8×D8—471no—
108#1367—471no—
109#1365—471no—
110#1362D4711yescharacteristic, complemented
111C2×C47#166511yescharacteristic
112C2×C47#220811yescharacteristic
113C2×C47#221211yescharacteristic
114D47—82no—
115C2×SD32—941no—
116#500—471no—
117#498—941no—
118#494C2×D4711yescharacteristic
119#487—941no—
120C4×D8—471no—
121C8×D4—941no—
122#367—941no—
123C4⋊C16—471no—
124C4⋊C16C2×D4711yescharacteristic
125#358—941no—
126#357—941no—
127#335C2×D4711yescharacteristic
128D4⋊C8—941no—
129C82—471no—
130C47#1242011yescharacteristic, complemented, Sylow
131C2×Q16—21no—
132C2×Q16—21no—
133C2×D8—471no—
134C22×C8—1881no—
135C4⋊Q8—21no—
136C4⋊Q8—941no—
137C4×D4—941no—
138SD32—1882no—
139C2×C16—21no—
140C2×C16—941no—
141#107—471no—
142#107—941no—
143#107—941no—
144#106—1881no—
145#106—1881no—
146C4⋊C8—21no—
147#103—1882no—
148#102—941no—
149C22⋊C8—1881no—
150C4×C8—471no—
151C4×C8—471no—
152C4×C8C22×D4711yescharacteristic
153C2×Q8—41no—
154C2×Q8—1881no—
155C2×D4—941no—
156C22×C4—1881no—
157Q16—41no—
158Q16—41no—
159Q16—81no—
160D8—472no—
161D8—941no—
162C2×C8—41no—
163C2×C8—471no—
164C2×C8—471no—
165C2×C8—941no—
166C2×C8—3762no—
167C2×C8#1062535011yescharacteristic
168C2×C8D4×D4711yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—941no—
172C4⋊C4—1881no—
173C4⋊C4—1881no—
174C22⋊C4—1881no—
175C42D4×D4711yescharacteristic
176C16—41no—
177C16—1881no—
178C23—1881no—
179Q8—81no—
180Q8—81no—
181Q8—3762no—
182D4—942no—
183D4—1881no—
184C2×C4—81no—
185C2×C4—3761no—
186C2×C4—3761no—
187C2×C4—3761no—
188C2×C4—1881no—
189C2×C4#1309399711yescharacteristic
190C2×C4D8×D4711yescharacteristic
191C2×C4#1309405111yescharacteristic
192C8—21no—
193C8—81no—
194C8—472no—
195C8—941no—
196C8—1881no—
197C8SD16×D4711yescharacteristic
198C8#1309404311yescharacteristic
199C22—3761no—
200C22—3761no—
201C22—1882no—
202C22#1568931711yescharacteristic
203C4—21no—
204C4—161no—
205C4—3761no—
206C4—7521no—
207C4#1568935111yescharacteristic
208C4#1568938511yescharacteristic
209C4SD32×D4711yescharacteristic
210C4#1568956511yescharacteristic
211C2—3762no—
212C2#2024272811yescharacteristic
213C2#2024543211yescharacteristic
214C2#2024546011yescharacteristic
subgroupG/H
centerC22#15689317
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#1362C2
socleC22×C47#459
prime radicalC2×D376D4
hypercenter#1362D47
hypocenterC47#12420
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
3. #20242982C2semidirect
5. #20241451C2semidirect
6. #20241447C2semidirect
7. C47×#1362C2semidirect
90. C47⋊C8SD32semidirect
110. #1362D47semidirect
130. C47#12420semidirect

Maximal subgroups

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

subgroupindex
#202455362
#202454302
#202429822
#202421822
#202414512
#202414472
C47×#13622
#1242047