Finite Group Databasebeta

← #26132555

Subgroups — #26132555

Order 12032.

342 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 79 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#20252028C211yesmaximal, characteristic
2#20243489C212yesmaximal
3#20241516C212yesmaximal
4#20241511C211yesmaximal, characteristic
5C47×#1391C211yesmaximal, characteristic
6#41×Dic47C2211yescharacteristic
7#15689181—22no—
8#15689173—22no—
9#15689151C411yescharacteristic, complemented
10#15689149C411yescharacteristic, complemented
11#15689107C2212yes—
12#15689090C2211yescharacteristic
13M16×Dic47C411yescharacteristic, complemented
14C2×C8×Dic47C411yescharacteristic, complemented
15#15688882—22no—
16#15688879—22no—
17C47×#410C2212yes—
18C47×#408C2211yescharacteristic
19C2×#10625329—22no—
20C2×#10625329C2×C411yescharacteristic
21C22×C47⋊C8C2×C411yescharacteristic
22Q8×Dic47—21no—
23D4×Dic47—21no—
24D4×Dic47—22no—
25#13093983C2×C411yescharacteristic
26#13093983D412yes—
27C2×C4×Dic47C2×C411yescharacteristic
28C2×C4×Dic47D412yes—
29Dic47⋊C8—22no—
30Dic47⋊C8—22no—
31#13093943—22no—
32#13093941—22no—
33C22⋊(C47⋊C8)—22no—
34#13093934—21no—
35C8×Dic47—21no—
36C8×Dic47—21no—
37C8×Dic47—21no—
38C2×#41×C47C2311yescharacteristic
39C2×M16×C47C2×C411yescharacteristic
40C22×C8×C47C2×C411yescharacteristic
41#117×C47—22no—
42#99×C47—22no—
43#99×C47—22no—
44C22⋊C8×C47—22no—
45#41×C47—41no—
46#41×C47—41no—
47C2×Q8×C47#4111yescharacteristic
48C2×D4×C47#4111yescharacteristic
49C2×D4×C47#4112yes—
50C22×C4×C47C22×C411yescharacteristic
51C22×C4×C47C2×D412yes—
52C22×Dic47—21no—
53C22×Dic47—22no—
54M16×C47—41no—
55C2×C8×C47—21no—
56C2×C8×C47—21no—
57C2×C8×C47—41no—
58C4⋊C4×C47—42no—
59C22⋊C4×C47—42no—
60C42×C47—42no—
61C22⋊Dic47—21no—
62C22⋊Dic47—22no—
63C4⋊Dic47—21no—
64C4⋊Dic47—22no—
65C4×Dic47—21no—
66C4×Dic47—22no—
67C4×Dic47C22⋊C411yescharacteristic
68C4×Dic47C22⋊C411yescharacteristic
69C4×Dic47C22⋊C412yes—
70#10625329—41no—
71#10625329—42no—
72#10625329—42no—
73C2×C47⋊C8—21no—
74C2×C47⋊C8—21no—
75C2×C47⋊C8—41no—
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—21no—
84C2×C4×C47—22no—
85C2×C4×C47—21no—
86C2×C4×C47—82no—
87C2×C4×C47C2×C22⋊C411yescharacteristic
88C2×C4×C47#11711yescharacteristic
89C2×C4×C47#12312yes—
90C2×Dic47—21no—
91C2×Dic47—21no—
92C2×Dic47—22no—
93C2×Dic47—42no—
94C2×Dic47—44no—
95C8×C47—41no—
96C8×C47—41no—
97C47⋊C8—41no—
98C47⋊C8—41no—
99C47⋊C8—82no—
100#12864—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—162no—
109C4×C47#41011yescharacteristic
110C4×C47#41011yescharacteristic
111Dic47—22no—
112Dic47—41no—
113Dic47—42no—
114#2922—471no—
115#1827—472no—
116#1395—472no—
117#1392—471no—
118#1391D4711yescharacteristic
119C2×C47—21no—
120C2×C47—42no—
121C2×C47—41no—
122C2×C47#182011yescharacteristic
123C4×#41—471no—
124#433—942no—
125#431—942no—
126#424—471no—
127#424—471no—
128#413—472no—
129#410C2×D4712yes—
130#408—471no—
131#408C2×D4711yescharacteristic
132C4×M16—471no—
133C2×C4×C8—471no—
134#339—942no—
135H(3,ℤ/4)—942no—
136C47#1286411yescharacteristic, complemented, Sylow
137C2×#41C22×D4711yescharacteristic
138C2×M16—471no—
139C2×M16—942no—
140C2×M16C4×D4711yescharacteristic
141C22×C8—471no—
142C22×C8C4×D4711yescharacteristic
143C4×Q8—941no—
144C4×D4—942no—
145C4×D4—941no—
146#117—22no—
147#117—471no—
148#117—472no—
149C2×C42—471no—
150C2×C42—472no—
151C4⋊C8—942no—
152C4⋊C8—942no—
153#101—942no—
154#100—942no—
155#99—22no—
156#99—22no—
157C22⋊C8—22no—
158C22⋊C8—942no—
159#97—941no—
160C4×C8—941no—
161C4×C8—941no—
162C4×C8—941no—
163#41—41no—
164#41—41no—
165C2×Q8#1062535411yescharacteristic
166C2×D4#1062535012yes—
167C2×D4#1062535211yescharacteristic
168C22×C4—941no—
169C22×C4—942no—
170C22×C4C2×C4×D4711yescharacteristic
171C22×C4D4×D4712yes—
172M16—41no—
173M16—1882no—
174M16—1882no—
175M16—1881no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—941no—
180C2×C8—941no—
181C2×C8—1882no—
182C2×C8—1881no—
183C4⋊C4—42no—
184C4⋊C4—941no—
185C4⋊C4—942no—
186C22⋊C4—42no—
187C22⋊C4—941no—
188C22⋊C4—942no—
189C42—42no—
190C42—471no—
191C42—471no—
192C42—472no—
193C42—941no—
194C42—942no—
195C23#1309398111yescharacteristic
196C23#1309400612yes—
197Q8—41no—
198D4—42no—
199D4—41no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—82no—
205C2×C4—941no—
206C2×C4—941no—
207C2×C4—942no—
208C2×C4—1882no—
209C2×C4—1884no—
210C2×C4C22⋊C4×D4711yescharacteristic
211C2×C4#1309398211yescharacteristic
212C2×C4#1309400912yes—
213C8—41no—
214C8—41no—
215C8—3762no—
216C8—1881no—
217C8—1881no—
218C22—22no—
219C22—21no—
220C22—41no—
221C22—42no—
222C22#1568898511yescharacteristic
223C4—22no—
224C4—21no—
225C4—162no—
226C4—942no—
227C4—1881no—
228C4—1882no—
229C4#1568909511yescharacteristic
230C4#1568909511yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2#2024346411yescharacteristic
subgroupG/H
centerC4#15689095
commutatorC22×C4×C47C22×C4
FrattiniC2×#41C22×D47
FittingC47×#1391C2
socleC2×C47#1820
prime radical, cosocleC2×#41×C47C23
hypercenter#1391D47
hypocenterC47#12864
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #1391

Complements

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

NH ≅ G/Ntype
9. #15689151C4semidirect
10. #15689149C4semidirect
13. M16×Dic47C4semidirect
14. C2×C8×Dic47C4semidirect
136. C47#12864semidirect

Maximal subgroups

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

subgroupindex
#202520282
#202434892
#202415162
#202415112
C47×#13912
#1286447