Finite Group Databasebeta

← #26132562

Subgroups — #26132562

Order 12032.

334 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 95 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#20243492C212yesmaximal
2#20241517C212yesmaximal
3#20241511C211yesmaximal, characteristic
4#20241509C211yesmaximal, characteristic
5C47×#2923C211yesmaximal, characteristic
6#15689107C2212yes—
7#15689097C2212yes—
8#15688883—22no—
9#15688879—22no—
10C4×#41×C47C2211yescharacteristic
11C47×#433—22no—
12C47×#432—22no—
13C47×#424C411yescharacteristic, complemented
14C47×#424C411yescharacteristic, complemented
15C47×#408C2211yescharacteristic
16C47×#408C2211yescharacteristic
17C4×M16×C47C411yescharacteristic, complemented
18C2×#97×C47C411yescharacteristic, complemented
19C2×#10625329—22no—
20#13093983—22no—
21#13093943—22no—
22#13093941—22no—
23#13093939—22no—
24#13093939—22no—
25C2×#41×C47C2311yescharacteristic
26C2×M16×C47C2×C411yescharacteristic
27C2×M16×C47C2×C411yescharacteristic
28C22×C8×C47C2×C411yescharacteristic
29C22×C8×C47C2×C411yescharacteristic
30C4×Q8×C47—21no—
31C4×D4×C47—21no—
32C4×D4×C47—22no—
33#117×C47C2×C411yescharacteristic
34#117×C47D412yes—
35C2×C42×C47C2×C411yescharacteristic
36C2×C42×C47D412yes—
37C4⋊C8×C47—22no—
38C4⋊C8×C47—22no—
39C22⋊C8×C47—22no—
40C22⋊C8×C47—22no—
41#97×C47—21no—
42#97×C47—21no—
43#97×C47—21no—
44C4×C8×C47—21no—
45#41×C47—41no—
46#41×C47—41no—
47C2×Q8×C47#4111yescharacteristic
48C2×D4×C47#4111yescharacteristic
49C2×D4×C47#4112yes—
50C22×C4×C47—21no—
51C22×C4×C47—22no—
52C22×C4×C47C22×C411yescharacteristic
53C22×C4×C47C2×D412yes—
54M16×C47—41no—
55M16×C47—41no—
56C2×C8×C47—21no—
57C2×C8×C47—21no—
58C2×C8×C47—21no—
59C2×C8×C47—21no—
60C2×C8×C47—41no—
61C2×C8×C47—41no—
62C4⋊C4×C47—21no—
63C4⋊C4×C47—22no—
64C22⋊C4×C47—21no—
65C22⋊C4×C47—22no—
66C42×C47—21no—
67C42×C47—22no—
68C42×C47C22⋊C412yes—
69C42×C47C22⋊C411yescharacteristic
70C42×C47C22⋊C411yescharacteristic
71C22⋊Dic47—42no—
72C4⋊Dic47—42no—
73C4×Dic47—42no—
74#10625329—42no—
75#10625329—42no—
76C2×C47⋊C8—42no—
77C23×C47#11711yescharacteristic
78C23×C47#12312yes—
79Q8×C47—41no—
80D4×C47—42no—
81D4×C47—41no—
82C2×C4×C47—22no—
83C2×C4×C47—22no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—21no—
87C2×C4×C47—22no—
88C2×C4×C47—21no—
89C2×C4×C47—81no—
90C2×C4×C47—82no—
91C2×C4×C47C2×C22⋊C411yescharacteristic
92C2×C4×C47#11711yescharacteristic
93C2×C4×C47#12312yes—
94C2×Dic47—82no—
95C8×C47—41no—
96C8×C47—41no—
97C8×C47—41no—
98C8×C47—41no—
99C47⋊C8—82no—
100#12865—471nomaximal, Sylow
101C22×C47—22no—
102C22×C47—21no—
103C22×C47—41no—
104C22×C47—42no—
105C22×C47#38111yescharacteristic
106C4×C47—22no—
107C4×C47—21no—
108C4×C47—42no—
109C4×C47—41no—
110C4×C47—41no—
111C4×C47#41011yescharacteristic
112C4×C47#41011yescharacteristic
113Dic47—162no—
114#2923D4711yescharacteristic
115#1827—472no—
116#1395—472no—
117#1392—471no—
118#1391—471no—
119C2×C47—21no—
120C2×C47—42no—
121C2×C47—41no—
122C2×C47#182011yescharacteristic
123C4×#41C2×D4711yescharacteristic
124#433—22no—
125#432—22no—
126#424Dic4711yescharacteristic, complemented
127#424Dic4711yescharacteristic, complemented
128#413—472no—
129#410—472no—
130#408C2×D4711yescharacteristic
131#408C2×D4711yescharacteristic
132C4×M16Dic4711yescharacteristic, complemented
133C2×#97Dic4711yescharacteristic, complemented
134#339—942no—
135H(3,ℤ/4)—942no—
136C47#1286511yescharacteristic, complemented, Sylow
137C2×#41C22×D4711yescharacteristic
138C2×M16—942no—
139C2×M16C2×Dic4711yescharacteristic
140C2×M16C2×Dic4711yescharacteristic
141C22×C8C2×Dic4711yescharacteristic
142C22×C8C2×Dic4711yescharacteristic
143C4×Q8—21no—
144C4×D4—21no—
145C4×D4—22no—
146#117—942no—
147#117C2×Dic4711yescharacteristic
148#117C22⋊D4712yes—
149C2×C42C2×Dic4711yescharacteristic
150C2×C42C22⋊D4712yes—
151C4⋊C8—22no—
152C4⋊C8—22no—
153#101—942no—
154#100—942no—
155#99—942no—
156#99—942no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159#97—21no—
160#97—21no—
161#97—21no—
162C4×C8—21no—
163#41—41no—
164#41—41no—
165C2×Q8#1062535011yescharacteristic
166C2×D4#1062535011yescharacteristic
167C2×D4#1062535012yes—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4C22×Dic4711yescharacteristic
171C22×C4C2×C22⋊D4712yes—
172M16—41no—
173M16—41no—
174M16—1882no—
175M16—1882no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—1882no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—1882no—
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—1882no—
189C42—21no—
190C42—22no—
191C42—1882no—
192C42C22⋊Dic4711yescharacteristic
193C42C22⋊Dic4712yes—
194C42C22⋊Dic4711yescharacteristic
195C23#1309398311yescharacteristic
196C23#1309400712yes—
197Q8—41no—
198D4—42no—
199D4—41no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—3762no—
210C2×C4C2×C22⋊Dic4711yescharacteristic
211C2×C4#1309398311yescharacteristic
212C2×C4#1309400712yes—
213C8—41no—
214C8—41no—
215C8—41no—
216C8—41no—
217C8—3762no—
218C22—22no—
219C22—21no—
220C22—41no—
221C22—42no—
222C22#1568898611yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—42no—
227C4—41no—
228C4—7522no—
229C4#1568909711yescharacteristic
230C4#1568909711yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2024346511yescharacteristic
subgroupG/H
centerC4#15689097
commutatorC22×C4×C47C22×C4
FrattiniC2×#41C22×D47
FittingC47×#2923C2
socleC2×C47#1820
prime radical, cosocleC2×#41×C47C23
hypercenter#2923D47
hypocenterC47#12865
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C42 ◁ C4×#41 ◁ #2923

Complements

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

NH ≅ G/Ntype
13. C47×#424C4semidirect
14. C47×#424C4semidirect
17. C4×M16×C47C4semidirect
18. C2×#97×C47C4semidirect
126. #424Dic47semidirect
127. #424Dic47semidirect
132. C4×M16Dic47semidirect
133. C2×#97Dic47semidirect
136. C47#12865semidirect

Maximal subgroups

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

subgroupindex
#202434922
#202415172
#202415112
#202415092
C47×#29232
#1286547