Finite Group Databasebeta

← C47×#12364 #26052062

Subgroups — C47×#12364 #26052062

Order 12032.

358 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C47×#3291C211yesmaximal, characteristic
2C47×#2230C212yesmaximal, complemented
3C47×#1705C211yesmaximal, characteristic, complemented
4C47×#1359C211yesmaximal, characteristic, complemented
5C47×#1335C212yesmaximal, complemented
6C47×#577—22no—
7C47×#498—21no—
8C47×#498C2211yescharacteristic, complemented
9C47×#492—21no—
10C47×#492C2211yescharacteristic
11C47×#471—22no—
12C47×#456—21no—
13C47×#453—21no—
14C47×#443C2212yes—
15C47×#365—22no—
16C47×#364C2212yescomplemented
17C47×#360—22no—
18C82×C47C2211yescharacteristic, complemented
192−1+4×C47—21no—
20H(5,𝔽2)×C47—21no—
21#137×C47—22no—
22#137×C47—42no—
23AGL(1,ℤ/8)×C47—22no—
24AGL(1,ℤ/8)×C47—42no—
25#135×C47—42no—
26#135×C47D412yes—
27C2×SD16×C47—22no—
28C2×SD16×C47—42no—
29C2×SD16×C47—42no—
30#131×C47—22no—
31C4⋊Q8×C47—21no—
32C4⋊Q8×C47D411yescharacteristic
33#127×C47D411yescharacteristic
34M32×C47—22no—
35#108×C47D412yes—
36#106×C47—22no—
37#106×C47—42no—
38#104×C47—22no—
39#101×C47—21no—
40#100×C47—21no—
41C4×C8×C47—21no—
42C4×C8×C47C2311yescharacteristic
43#41×C47—22no—
44#41×C47—42no—
45#41×C47—42no—
46#41×C47—81no—
47C2×Q8×C47—21no—
48C2×Q8×C47—41no—
49C2×Q8×C47—42no—
50C2×D4×C47—21no—
51C2×D4×C47—41no—
52C2×D4×C47—42no—
53C2×D4×C47—81no—
54Q16×C47—42no—
55Q16×C47SD1612yescomplemented
56SD16×C47—22no—
57SD16×C47—42no—
58SD16×C47—42no—
59SD16×C47—42no—
60SD16×C47—42no—
61SD16×C47—84no—
62D8×C47—42no—
63D8×C47SD1612yescomplemented
64M16×C47—22no—
65M16×C47—42no—
66C2×C8×C47—22no—
67C2×C8×C47—42no—
68C2×C8×C47C2×D412yes—
69C4⋊C4×C47—41no—
70C4⋊C4×C47—41no—
71C4⋊C4×C47—41no—
72C42×C47C2×D411yescharacteristic
73C16×C47—42no—
74C23×C47—41no—
75C23×C47—81no—
76Q8×C47—22no—
77Q8×C47—42no—
78Q8×C47—82no—
79Q8×C47—81no—
80D4×C47—22no—
81D4×C47—42no—
82D4×C47—42no—
83D4×C47—82no—
84D4×C47—82no—
85D4×C47—81no—
86C2×C4×C47—21no—
87C2×C4×C47—42no—
88C2×C4×C47—41no—
89C2×C4×C47—81no—
90C2×C4×C47—82no—
91C2×C4×C47—81no—
92C2×C4×C47#12011yescharacteristic
93C8×C47—24no—
94C8×C47—42no—
95C8×C47C2×SD1612yes—
96C8×C47AGL(1,ℤ/8)12yes—
97#12364C4711yesmaximal, characteristic, direct product factor, Sylow
98C22×C47—42no—
99C22×C47—81no—
100C22×C47—161no—
101C22×C47—321no—
102C22×C47#45311yescharacteristic
103C4×C47—22no—
104C4×C47—42no—
105C4×C47—81no—
106C4×C47—161no—
107C4×C47#45012yes—
108#3291C2×C4711yescharacteristic
109#2230C2×C4712yescomplemented
110#1705C2×C4711yescharacteristic, complemented
111#1359C2×C4711yescharacteristic, complemented
112#1335C2×C4712yescomplemented
113C2×C47—21no—
114C2×C47—82no—
115C2×C47—161no—
116C2×C47#162911yescharacteristic
117#577—22no—
118#498—21no—
119#498C22×C4711yescharacteristic, complemented
120#492—21no—
121#492C22×C4711yescharacteristic
122#471—22no—
123#456—21no—
124#453—21no—
125#443C22×C4712yes—
126#365—22no—
127#364C22×C4712yescomplemented
128#360—22no—
129C82C22×C4711yescharacteristic, complemented
130C47#1236411yescharacteristic, direct product factor, Sylow
1312−1+4—21no—
132H(5,𝔽2)—21no—
133#137—22no—
134#137—42no—
135AGL(1,ℤ/8)—22no—
136AGL(1,ℤ/8)—42no—
137#135—42no—
138#135D4×C4712yes—
139C2×SD16—22no—
140C2×SD16—42no—
141C2×SD16—42no—
142#131—22no—
143C4⋊Q8—21no—
144C4⋊Q8D4×C4711yescharacteristic
145#127D4×C4711yescharacteristic
146M32—22no—
147#108D4×C4712yes—
148#106—22no—
149#106—42no—
150#104—22no—
151#101—21no—
152#100—21no—
153C4×C8—21no—
154C4×C8C23×C4711yescharacteristic
155#41—22no—
156#41—42no—
157#41—42no—
158#41—81no—
159C2×Q8—21no—
160C2×Q8—41no—
161C2×Q8—42no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—81no—
166Q16—42no—
167Q16SD16×C4712yescomplemented
168SD16—22no—
169SD16—42no—
170SD16—42no—
171SD16—42no—
172SD16—42no—
173SD16—84no—
174D8—42no—
175D8SD16×C4712yescomplemented
176M16—22no—
177M16—42no—
178C2×C8—22no—
179C2×C8—42no—
180C2×C8C2×D4×C4712yes—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C42C2×D4×C4711yescharacteristic
185C16—42no—
186C23—41no—
187C23—81no—
188Q8—22no—
189Q8—42no—
190Q8—82no—
191Q8—81no—
192D4—22no—
193D4—42no—
194D4—42no—
195D4—82no—
196D4—82no—
197D4—81no—
198C2×C4—21no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—81no—
202C2×C4—82no—
203C2×C4—81no—
204C2×C4#120×C4711yescharacteristic
205C8—24no—
206C8—42no—
207C8C2×SD16×C4712yes—
208C8AGL(1,ℤ/8)×C4712yes—
209C22—42no—
210C22—81no—
211C22—161no—
212C22—321no—
213C22C47×#45311yescharacteristic
214C4—22no—
215C4—42no—
216C4—81no—
217C4—161no—
218C4C47×#45012yes—
219C2—21no—
220C2—82no—
221C2—161no—
222C2C47×#162911yescharacteristic
subgroupG/H
center, socleC2×C47#1629
commutator, Frattini, cosocleC4×C8C23×C47
prime radicalC47×#3291C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C47 ◁ C2×C4×C47 ◁ C4×C8×C47 ◁ G

Complements

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

NH ≅ G/Ntype
2. C47×#2230C2semidirect
3. C47×#1705C2semidirect
4. C47×#1359C2semidirect
5. C47×#1335C2semidirect
8. C47×#498C22semidirect
16. C47×#364C22semidirect
18. C82×C47C22semidirect
55. Q16×C47SD16semidirect
63. D8×C47SD16semidirect
97. #12364C47direct (N × H)
109. #2230C2×C47semidirect
110. #1705C2×C47semidirect
111. #1359C2×C47semidirect
112. #1335C2×C47semidirect
119. #498C22×C47semidirect
127. #364C22×C47semidirect
129. C82C22×C47semidirect
130. C47#12364direct (N × H)
167. Q16SD16×C47semidirect
175. D8SD16×C47semidirect

Maximal subgroups

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

subgroupindex
C47×#32912
C47×#22302
C47×#17052
C47×#13592
C47×#13352
#1236447