Finite Group Databasebeta

← #26104061

Subgroups — #26104061

Order 12032.

286 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 59 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#20245073C211yesmaximal, characteristic
2#20244217C211yesmaximal, characteristic
3C47×#2132C211yesmaximal, characteristic
4#15689147C411yescharacteristic
5C2×#13093951—21no—
6C2×#13093951—21no—
7#15689107—21no—
8C42×Dic47C411yescharacteristic
9#15688922—21no—
10#15688920C412yescomplemented
11C47×#532C2211yescharacteristic
12C47×#413—21no—
13C47×#356—21no—
14C47×#355C412yescomplemented
15C2×#10625329—21no—
16C2×#10625329—41no—
17C2×C4×Dic47—21no—
18C2×C4×Dic47—41no—
19C4⋊(C47⋊C8)—41no—
20#13093951—42no—
21#13093951—42no—
22#13093943—22no—
23#13093941—41no—
24C2×#41×C47D411yescharacteristic
25C2×#41×C47D411yescharacteristic
26C2×M16×C47—41no—
27C4⋊Q8×C47D411yescharacteristic
28#127×C47Q811yescharacteristic
29#124×C47C2×C411yescharacteristic
30#124×C47C2×C411yescharacteristic
31#121×C47—41no—
32C4×D4×C47—41no—
33C2×C42×C47C2×C411yescharacteristic
34#101×C47—22no—
35#100×C47—41no—
36#41×C47—41no—
37#41×C47—41no—
38#41×C47—41no—
39#41×C47—41no—
40C2×Q8×C47C22⋊C411yescharacteristic
41C2×Q8×C47C22⋊C411yescharacteristic
42C2×D4×C47—21no—
43C2×D4×C47—21no—
44C2×D4×C47—41no—
45C2×D4×C47C4⋊C411yescharacteristic
46C2×D4×C47C4⋊C411yescharacteristic
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47C22⋊C411yescharacteristic
51C22×Dic47—41no—
52M16×C47—42no—
53M16×C47—81no—
54C2×C8×C47—81no—
55C4⋊C4×C47—41no—
56C22⋊C4×C47—41no—
57C22⋊C4×C47—41no—
58C42×C47—21no—
59C42×C47C4211yescharacteristic
60C42×C47C4⋊C411yescharacteristic
61C4×Dic47—22no—
62C4×Dic47—41no—
63C4×Dic47—44no—
64#10625329—42no—
65#10625329—41no—
66#10625329—41no—
67#10625329—81no—
68C2×C47⋊C8—41no—
69C2×C47⋊C8—81no—
70C23×C47—21no—
71C23×C47—21no—
72C23×C47#9911yescharacteristic
73Q8×C47—41no—
74Q8×C47—41no—
75D4×C47—41no—
76D4×C47—41no—
77D4×C47—82no—
78D4×C47—81no—
79D4×C47—81no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—21no—
86C2×C4×C47—42no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—41no—
90C2×C4×C47—41no—
91C2×C4×C47#9511yescharacteristic
92C2×C4×C47#9911yescharacteristic
93C2×Dic47—42no—
94C2×Dic47—44no—
95C8×C47—82no—
96C47⋊C8—82no—
97C47⋊C8—81no—
98#7501—471nomaximal, Sylow
99C22×C47—21no—
100C22×C47—41no—
101C22×C47—41no—
102C22×C47—41no—
103C22×C47—81no—
104C22×C47—81no—
105C22×C47#34211yescharacteristic
106C4×C47—22no—
107C4×C47—21no—
108C4×C47—41no—
109C4×C47—41no—
110C4×C47—41no—
111C4×C47#35311yescharacteristic
112C4×C47#35411yescharacteristic
113Dic47—44no—
114#2132—471no—
115#2132D4711yescharacteristic
116#1962—471no—
117C2×C47—21no—
118C2×C47—41no—
119C2×C47—81no—
120C2×C47—81no—
121C2×C47#139311yescharacteristic
122#532C2×D4711yescharacteristic
123#423—471no—
124C2×#104—941no—
125C2×#104—941no—
126#413—21no—
127#413—941no—
128C43—471no—
129#356—21no—
130#356—941no—
131#355—472no—
132#355Dic4712yescomplemented
133C47#750111yescharacteristic, complemented, Sylow
134C2×#41D18811yescharacteristic
135C2×#41C22⋊D4711yescharacteristic
136C2×M16—41no—
137C2×M16—941no—
138C2×M16—1881no—
139C4⋊Q8C22⋊D4711yescharacteristic
140#127Dic9411yescharacteristic
141#124C4×D4711yescharacteristic
142#124C2×Dic4711yescharacteristic
143#121—41no—
144C4×D4—41no—
145C2×C42—941no—
146C2×C42—1881no—
147C2×C42C4×D4711yescharacteristic
148C4⋊C8—1881no—
149#104—1882no—
150#104—1882no—
151#101—22no—
152#101—942no—
153#100—41no—
154#100—1881no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8(C2×D47)⋊C411yescharacteristic
160C2×Q8C22⋊Dic4711yescharacteristic
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—41no—
164C2×D4Dic47⋊C411yescharacteristic
165C2×D4C4⋊Dic4711yescharacteristic
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—1881no—
170C22×C4(C2×D47)⋊C411yescharacteristic
171M16—42no—
172M16—81no—
173M16—3761no—
174M16—1882no—
175M16—1881no—
176M16—1881no—
177C2×C8—81no—
178C2×C8—3761no—
179C2×C8—1881no—
180C4⋊C4—41no—
181C22⋊C4—41no—
182C22⋊C4—41no—
183C42—21no—
184C42—942no—
185C42—1881no—
186C42—1884no—
187C42C4×Dic4711yescharacteristic
188C42Dic47⋊C411yescharacteristic
189C23—21no—
190C23—21no—
191C23#1309393811yescharacteristic
192Q8—41no—
193Q8—41no—
194D4—41no—
195D4—41no—
196D4—82no—
197D4—81no—
198D4—81no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—1882no—
211C2×C4—1884no—
212C2×C4C47⋊#9511yescharacteristic
213C2×C4#1309393811yescharacteristic
214C8—82no—
215C8—3762no—
216C8—3761no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—81no—
222C22—81no—
223C22#1568888911yescharacteristic
224C4—22no—
225C4—21no—
226C4—41no—
227C4—41no—
228C4—41no—
229C4—1884no—
230C4#1568891511yescharacteristic
231C4#1568891711yescharacteristic
232C2—21no—
233C2—41no—
234C2—81no—
235C2—81no—
236C2#2024151211yescharacteristic
subgroupG/H
centerC4#15688917
commutatorC42×C47C42
Frattini#532C2×D47
FittingC47×#2132C2
socleC2×C47#1393
prime radical, cosocleC47×#532C22
hypercenter#2132D47
hypocenterC47#7501
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2132

Complements

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

NH ≅ G/Ntype
10. #15688920C4semidirect
14. C47×#355C4semidirect
132. #355Dic47semidirect
133. C47#7501semidirect

Maximal subgroups

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

subgroupindex
#202450732
#202442172
C47×#21322
#750147