Finite Group Databasebeta

← #26130514

Subgroups — #26130514

Order 12032.

246 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 63 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#20245645C211yesmaximal, characteristic, complemented
2#20245219C211yesmaximal, characteristic, complemented
3#20242750C211yesmaximal, characteristic, complemented
4#20242744C211yesmaximal, characteristic
5#20241459C211yesmaximal, characteristic, complemented
6#20241435C211yesmaximal, characteristic, complemented
7C47×#1362C211yesmaximal, characteristic, complemented
8C2×#13093967—21no—
9C2×#13093967—21no—
10C2×#13093963—21no—
11#15689500C2211yescharacteristic
12#15689400—21no—
13#15689319—21no—
14C8×D188C2211yescharacteristic
15(C2×D47)⋊C16—21no—
16C4×C47⋊C16C2211yescharacteristic
17#15688876C2211yescharacteristic
18#15688869—21no—
19#15688866—21no—
20C47×#494C2211yescharacteristic
21C47×C4⋊C16C2211yescharacteristic
22C47×#335C2211yescharacteristic
23C2×D376—21no—
24C2×D376—21no—
25C2×D376—41no—
26C2×C8×D47—21no—
27#13094021—21no—
28C4×D188D411yescharacteristic
29#13093967—42no—
30#13093967—42no—
31#13093963—42no—
32C2×C47⋊C16D411yescharacteristic
33C2×C47⋊C16D411yescharacteristic
34#13093957—41no—
35C4⋊(C47⋊C8)—21no—
36C4⋊(C47⋊C8)D411yescharacteristic
37#13093946—41no—
38#13093944—41no—
39(C2×D47)⋊C8—21no—
40C2×Q16×C47—21no—
41C2×Q16×C47—21no—
42C4⋊Q8×C47—21no—
43C2×C16×C47—21no—
44C4⋊C8×C47—21no—
45C4×C8×C47C2311yescharacteristic
46C2×Q8×C47—41no—
47C2×D188—41no—
48C2×D188—41no—
49C2×D188—41no—
50C2×D188D811yescharacteristic
51C2×C4×D47—21no—
52Q16×C47—41no—
53Q16×C47—41no—
54Q16×C47—81no—
55C2×C8×C47—41no—
56C2×C8×C47C2×D411yescharacteristic
57C2×C8×C47#4111yescharacteristic
58C4⋊C4×C47—41no—
59C4⋊C4×C47—41no—
60C42×C47C2×D411yescharacteristic
61(C2×D47)⋊C4—21no—
62C4⋊Dic47D811yescharacteristic
63C2×C47⋊C8—21no—
64C2×C47⋊C8—41no—
65D376—41no—
66D376—41no—
67D376—81no—
68D376—82no—
69C8×D47—41no—
70C8×D47—41no—
71C16×C47—41no—
72C47⋊C16—21no—
73C47⋊C16D812yescomplemented
74C22×D47—21no—
75C22×D47—81no—
76Q8×C47—81no—
77Q8×C47—81no—
78C2×C4×C47—81no—
79C2×C4×C47#12111yescharacteristic
80C2×C4×C47C2×D811yescharacteristic
81C2×C4×C47#13511yescharacteristic
82C2×Dic47—21no—
83D188—21no—
84D188—21no—
85D188—81no—
86D188—81no—
87D188—81no—
88D188—81no—
89D188—161no—
90C4×D47—41no—
91C4×D47—41no—
92C8×C47—21no—
93C8×C47—81no—
94C8×C47C2×D811yescharacteristic
95C8×C47AGL(1,ℤ/8)11yescharacteristic
96C47⋊C8—41no—
97C47⋊C8—81no—
98#12435—471nomaximal, Sylow
99C22×C47#46611yescharacteristic
100C2×D47—41no—
101C2×D47—41no—
102C2×D47—41no—
103C2×D47—161no—
104C2×D47—161no—
105C2×D47—161no—
106C4×C47—21no—
107C4×C47—161no—
108C4×C47#45911yescharacteristic
109C4×C47#47211yescharacteristic
110C4×C47#50911yescharacteristic
111C4×C47#51011yescharacteristic
112Dic47—41no—
113#2248—471no—
114#2169—471no—
115#1669—471no—
116#1667—471no—
117#1371—471no—
118#1362D4711yescharacteristic, complemented
119#1361—471no—
120C2×C47#166711yescharacteristic
121C2×C47#222011yescharacteristic
122C2×C47#222311yescharacteristic
123D47—81no—
124D47—321no—
125C2×SD32—941no—
126C2×SD32—941no—
127C2×M32—941no—
128#494C2×D4711yescharacteristic
129#493—471no—
130#474—941no—
131#459—941no—
132C8×D4—471no—
133C4⋊C16C2×D4711yescharacteristic
134C22⋊C16—941no—
135C4×C16—471no—
136#335—471no—
137#335C2×D4711yescharacteristic
138#332—941no—
139#331—941no—
140C47#1243511yescharacteristic, complemented, Sylow
141C2×Q16—21no—
142C2×Q16—21no—
143C2×D8—941no—
144C2×D8—941no—
145C2×D8—1881no—
146C22×C8—941no—
147C4⋊Q8—21no—
148#127—941no—
149C4×D4—471no—
150SD32—1882no—
151SD32—1882no—
152M32—1882no—
153C2×C16—21no—
154C2×C16—471no—
155C2×C16—471no—
156#107—1881no—
157C4⋊C8—21no—
158C4⋊C8—471no—
159C4⋊C8—941no—
160#102—1881no—
161#102—1881no—
162C22⋊C8—941no—
163C4×C8C22×D4711yescharacteristic
164C2×Q8—41no—
165C2×D4—471no—
166C2×D4—1881no—
167C2×D4—1881no—
168C2×D4—1881no—
169C22×C4—941no—
170Q16—41no—
171Q16—41no—
172Q16—81no—
173D8—3761no—
174D8—3762no—
175D8—1881no—
176D8—1881no—
177C2×C8—41no—
178C2×C8—941no—
179C2×C8—1881no—
180C2×C8—1881no—
181C2×C8—1881no—
182C2×C8#1062535011yescharacteristic
183C2×C8D4×D4711yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—471no—
187C22⋊C4—941no—
188C42D4×D4711yescharacteristic
189C16—41no—
190C16—472no—
191C16—941no—
192C23—941no—
193C23—3761no—
194Q8—81no—
195Q8—81no—
196D4—941no—
197D4—941no—
198D4—3761no—
199D4—3761no—
200D4—3761no—
201D4—3761no—
202D4—7521no—
203C2×C4—81no—
204C2×C4—941no—
205C2×C4—1881no—
206C2×C4—1881no—
207C2×C4#1309399711yescharacteristic
208C2×C4D8×D4711yescharacteristic
209C2×C4#1309404511yescharacteristic
210C8—21no—
211C8—81no—
212C8—3761no—
213C8—1881no—
214C8D8×D4711yescharacteristic
215C8#1309405111yescharacteristic
216C22—1881no—
217C22—1881no—
218C22—1881no—
219C22—7521no—
220C22—7521no—
221C22—7521no—
222C22#1568935811yescharacteristic
223C4—21no—
224C4—161no—
225C4—1881no—
226C4#1568931711yescharacteristic
227C4#1568939211yescharacteristic
228C4#1568956011yescharacteristic
229C4#1568956511yescharacteristic
230C2—3761no—
231C2—15041no—
232C2#2024274211yescharacteristic
233C2#2024550411yescharacteristic
234C2#2024552111yescharacteristic
subgroupG/H
centerC22#15689358
commutator, cosocleC4×C8×C47C23
FrattiniC4×C8C22×D47
FittingC47×#1362C2
socleC22×C47#466
prime radical#20242744C2
hypercenter#1362D47
hypocenterC47#12435
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
1. #20245645C2semidirect
2. #20245219C2semidirect
3. #20242750C2semidirect
5. #20241459C2semidirect
6. #20241435C2semidirect
7. C47×#1362C2semidirect
73. C47⋊C16D8semidirect
118. #1362D47semidirect
140. C47#12435semidirect

Maximal subgroups

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

subgroupindex
#202456452
#202452192
#202427502
#202427442
#202414592
#202414352
C47×#13622
#1243547