Finite Group Databasebeta

← #26129975

Subgroups — #26129975

Order 12032.

312 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 75 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#20245613C211yesmaximal, characteristic, complemented
2#20245423C211yesmaximal, characteristic
3#20242748C211yesmaximal, characteristic, complemented
4#20241446C211yesmaximal, characteristic, complemented
5Q8⋊(C47⋊C16)C211yesmaximal, characteristic, complemented
6#20241349C211yesmaximal, characteristic, complemented
7C47×#3380C211yesmaximal, characteristic, complemented
8C2×#13093965—21no—
9#15689500C2211yescharacteristic
10#15689323—21no—
11C4⋊(C47⋊C16)C2211yescharacteristic
12#15688923—22no—
13#15688923—21no—
14C4×C47⋊C16C2211yescharacteristic
15#15688876C2211yescharacteristic
16#15688866—21no—
17C47×#554—21no—
18C47×#500C2211yescharacteristic
19C47×#499—21no—
20C47×#478—22no—
21C47×#476—21no—
22C8×Q8×C47C2211yescharacteristic
23C4×D8×C47—21no—
24C4×D8×C47C2211yescharacteristic
25C2×#10625336—41no—
26C2×D376—21no—
27C2×D376—21no—
28#13094021—21no—
29#13093965—42no—
30C2×C47⋊C16—21no—
31C2×C47⋊C16—21no—
32C4⋊(C47⋊C8)—21no—
33#13093944—41no—
34C2×D8×C47D411yescharacteristic
35C4⋊Q8×C47—21no—
36#125×C47—22no—
37C22⋊Q8×C47—42no—
38C22⋊Q8×C47—41no—
39C4×Q8×C47D411yescharacteristic
40C4×D4×C47—22no—
41C4×D4×C47—21no—
42C2×C4⋊C4×C47—41no—
43#107×C47—21no—
44#107×C47D412yes—
45#107×C47D411yescharacteristic
46#106×C47—22no—
47C4⋊C8×C47—21no—
48C4⋊C8×C47D411yescharacteristic
49#102×C47—22no—
50#102×C47—21no—
51C4×C8×C47—21no—
52C4×C8×C47C2311yescharacteristic
53C2×Q8×C47—41no—
54C2×Q8×C47D811yescharacteristic
55C2×D4×C47—21no—
56C22×C4×C47—42no—
57C22×C4×C47—41no—
58C2×D188—41no—
59C2×D188—41no—
60C2×D188—41no—
61D8×C47—21no—
62D8×C47SD1612yescomplemented
63C2×C8×C47—21no—
64C2×C8×C47C2×D411yescharacteristic
65C2×C8×C47C2×D411yescharacteristic
66C4⋊C4×C47—21no—
67C4⋊C4×C47—21no—
68C4⋊C4×C47—22no—
69C4⋊C4×C47—42no—
70C4⋊C4×C47—42no—
71C4⋊C4×C47—41no—
72C4⋊C4×C47D811yescharacteristic
73C22⋊C4×C47—42no—
74C22⋊C4×C47—41no—
75C42×C47—21no—
76C42×C47C2×D411yescharacteristic
77#10625336—82no—
78C2×C47⋊C8—41no—
79D376—42no—
80D376—41no—
81D376—41no—
82C47⋊C16—41no—
83C47⋊C16—41no—
84C23×C47—41no—
85C22×D47—81no—
86Q8×C47—21no—
87Q8×C47—21no—
88Q8×C47—161no—
89D4×C47—22no—
90D4×C47—41no—
91C2×C4×C47—21no—
92C2×C4×C47—21no—
93C2×C4×C47—41no—
94C2×C4×C47—42no—
95C2×C4×C47—81no—
96C2×C4×C47—81no—
97C2×C4×C47—162no—
98C2×C4×C47#12011yescharacteristic
99C2×C4×C47C2×D811yescharacteristic
100C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
101D188—81no—
102D188—81no—
103D188—81no—
104D188—81no—
105D188—161no—
106C8×C47—21no—
107C8×C47—41no—
108C8×C47C2×SD1611yescharacteristic
109C8×C47#13711yescharacteristic
110C47⋊C8—81no—
111#12355—471nomaximal, Sylow
112C22×C47—42no—
113C22×C47—81no—
114C22×C47—81no—
115C22×C47#44711yescharacteristic
116C2×D47—161no—
117C2×D47—161no—
118C2×D47—161no—
119C4×C47—21no—
120C4×C47—41no—
121C4×C47—41no—
122C4×C47—81no—
123C4×C47—82no—
124C4×C47#44811yescharacteristic
125C4×C47#45411yescharacteristic
126C4×C47#50811yescharacteristic
127C4×C47#50911yescharacteristic
128#3380D4711yescharacteristic, complemented
129#2240—471no—
130#2206—471no—
131#1668—471no—
132#1365—471no—
133Q8⋊C16—471no—
134#1331—471no—
135C2×C47—82no—
136C2×C47#162711yescharacteristic
137C2×C47#218611yescharacteristic
138C2×C47#219211yescharacteristic
139D47—321no—
140#554—21no—
141C2×D16—941no—
142#500C2×D4711yescharacteristic
143#499—21no—
144#493—471no—
145#478—22no—
146#476—21no—
147#460—941no—
148C8×Q8C2×D4711yescharacteristic
149C4×D8—21no—
150C4×D8C2×D4711yescharacteristic
151C4⋊C16—471no—
152#357—941no—
153#357—942no—
154C4×C16—471no—
155#335—471no—
156#331—941no—
157C47#1235511yescharacteristic, complemented, Sylow
158C2×SD16—1881no—
159C2×D8—941no—
160C2×D8—941no—
161C2×D8C22⋊D4711yescharacteristic
162C4⋊Q8—21no—
163#127—941no—
164#125—22no—
165C22⋊Q8—42no—
166C22⋊Q8—41no—
167C4×Q8C22⋊D4711yescharacteristic
168C4×D4—22no—
169C4×D4—21no—
170C2×C4⋊C4—41no—
171D16—1882no—
172C2×C16—941no—
173C2×C16—941no—
174#107—21no—
175#107C22⋊D4711yescharacteristic
176#107C22⋊D4712yes—
177#106—22no—
178C4⋊C8—21no—
179C4⋊C8—941no—
180C4⋊C8C22⋊D4711yescharacteristic
181#102—22no—
182#102—21no—
183#102—1881no—
184C4×C8—21no—
185C4×C8C22×D4711yescharacteristic
186C2×Q8—41no—
187C2×Q8#1062533411yescharacteristic
188C2×D4—21no—
189C2×D4—1881no—
190C2×D4—1881no—
191C2×D4—1881no—
192C22×C4—42no—
193C22×C4—41no—
194SD16—3762no—
195D8—21no—
196D8—1882no—
197D8—1881no—
198D8—1881no—
199D8#1062533612yescomplemented
200C2×C8—21no—
201C2×C8—1881no—
202C2×C8C2×C22⋊D4711yescharacteristic
203C2×C8C2×C22⋊D4711yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—22no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4#1062533411yescharacteristic
211C22⋊C4—42no—
212C22⋊C4—41no—
213C42—21no—
214C42C2×C22⋊D4711yescharacteristic
215C16—1881no—
216C16—1881no—
217C23—41no—
218C23—3761no—
219Q8—21no—
220Q8—21no—
221Q8—161no—
222D4—22no—
223D4—41no—
224D4—3761no—
225D4—3761no—
226D4—3761no—
227D4—3761no—
228D4—7521no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—162no—
236C2×C4#1309399311yescharacteristic
237C2×C4C2×#1062533411yescharacteristic
238C2×C4#1309405011yescharacteristic
239C8—21no—
240C8—41no—
241C8—3761no—
242C8C2×#1062533611yescharacteristic
243C8#1309405511yescharacteristic
244C22—42no—
245C22—81no—
246C22—81no—
247C22—7521no—
248C22—7521no—
249C22—7521no—
250C22#1568925711yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—81no—
255C4—82no—
256C4#1568926211yescharacteristic
257C4#1568929611yescharacteristic
258C4#1568955311yescharacteristic
259C4#1568955811yescharacteristic
260C2—82no—
261C2—15041no—
262C2#2024249111yescharacteristic
263C2#2024531111yescharacteristic
264C2#2024534511yescharacteristic
subgroupG/H
centerC22#15689257
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#3380C2
socleC22×C47#447
prime radical#20245423C2
hypercenter#3380D47
hypocenterC47#12355
LCSG ▷ C4×C8×C47 ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3380

Complements

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

NH ≅ G/Ntype
1. #20245613C2semidirect
3. #20242748C2semidirect
4. #20241446C2semidirect
5. Q8⋊(C47⋊C16)C2semidirect
6. #20241349C2semidirect
7. C47×#3380C2semidirect
62. D8×C47SD16semidirect
128. #3380D47semidirect
157. C47#12355semidirect
199. D8#10625336semidirect

Maximal subgroups

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

subgroupindex
#202456132
#202454232
#202427482
#202414462
Q8⋊(C47⋊C16)2
#202413492
C47×#33802
#1235547