Finite Group Databasebeta

← #26104013

Subgroups — #26104013

Order 12032.

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

normal subgroups only

This group has 266 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#20243772C211yesmaximal, characteristic
2#20242023C211yesmaximal, characteristic
3C47×#2132C211yesmaximal, characteristic
4#15689890C411yescharacteristic
5#15689180—21no—
6#15689140—21no—
7#15689097C411yescharacteristic
8#15688883—21no—
9#15688863—21no—
10#15688863C412yescomplemented
11C47×#532C2211yescharacteristic
12C47×#413—21no—
13C47×#356C412yescomplemented
14C47×#355—21no—
15#13094031—42no—
16C2×#10625329—21no—
17C2×#10625329—41no—
18#13093983—21no—
19C4⋊(C47⋊C8)—41no—
20#13093951—42no—
21#13093939—41no—
22C22⋊(C47⋊C8)—41no—
23#13093935—22no—
24C2×#41×C47C2×C411yescharacteristic
25C2×#41×C47D411yescharacteristic
26C2×M16×C47—41no—
27C4⋊Q8×C47C2×C411yescharacteristic
28#127×C47Q811yescharacteristic
29#124×C47C2×C411yescharacteristic
30#124×C47D411yescharacteristic
31#121×C47—41no—
32C4×D4×C47—41no—
33C2×C42×C47D411yescharacteristic
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×C47C4211yescharacteristic
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
51M16×C47—42no—
52M16×C47—81no—
53C2×C8×C47—81no—
54C4⋊C4×C47—41no—
55C22⋊C4×C47—41no—
56C22⋊C4×C47—41no—
57C42×C47—21no—
58C42×C47C22⋊C411yescharacteristic
59C42×C47C4⋊C411yescharacteristic
60C22⋊Dic47—41no—
61C4⋊Dic47—41no—
62C4×Dic47—41no—
63#10625329—22no—
64#10625329—42no—
65#10625329—41no—
66C2×C47⋊C8—41no—
67C2×C47⋊C8—42no—
68C2×C47⋊C8—42no—
69C23×C47—21no—
70C23×C47—21no—
71C23×C47#9911yescharacteristic
72Q8×C47—41no—
73Q8×C47—41no—
74D4×C47—41no—
75D4×C47—41no—
76D4×C47—42no—
77D4×C47—81no—
78D4×C47—82no—
79C2×C4×C47—21no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—21no—
85C2×C4×C47—41no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47—81no—
90C2×C4×C47#9511yescharacteristic
91C2×C4×C47#9911yescharacteristic
92C2×Dic47—81no—
93C8×C47—82no—
94C47⋊C8—42no—
95C47⋊C8—82no—
96#7484—471nomaximal, Sylow
97C22×C47—21no—
98C22×C47—41no—
99C22×C47—41no—
100C22×C47—41no—
101C22×C47—81no—
102C22×C47—81no—
103C22×C47#34211yescharacteristic
104C4×C47—21no—
105C4×C47—41no—
106C4×C47—41no—
107C4×C47—41no—
108C4×C47—41no—
109C4×C47AΣL(1,𝔽16)11yescharacteristic
110C4×C47#35211yescharacteristic
111Dic47—161no—
112#2132D4711yescharacteristic
113#1884—471no—
114#1536—471no—
115C2×C47—21no—
116C2×C47—41no—
117C2×C47—81no—
118C2×C47—81no—
119C2×C47#139011yescharacteristic
120#568—471no—
121#532C2×D4711yescharacteristic
122#432—941no—
123#421—941no—
124#413—21no—
125#410—471no—
126#356Dic4712yescomplemented
127#355—21no—
128#339—941no—
129#329—472no—
130#329—941no—
131C47#748411yescharacteristic, complemented, Sylow
132C2×#41C4×D4711yescharacteristic
133C2×#41C22⋊D4711yescharacteristic
134#131—1882no—
135C2×M16—41no—
136C2×M16—941no—
137C2×M16—1881no—
138C4⋊Q8C2×Dic4711yescharacteristic
139#127Dic9411yescharacteristic
140#124C4×D4711yescharacteristic
141#124C22⋊D4711yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—941no—
145C2×C42D18811yescharacteristic
146C4⋊C8—1881no—
147#104—1882no—
148#101—22no—
149#100—41no—
150#99—1881no—
151C22⋊C8—1881no—
152#97—942no—
153#41—41no—
154#41—41no—
155#41—41no—
156#41—41no—
157C2×Q8C4×Dic4711yescharacteristic
158C2×Q8C22⋊Dic4711yescharacteristic
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—41no—
162C2×D4Dic47⋊C411yescharacteristic
163C2×D4Dic47⋊C411yescharacteristic
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4(C2×D47)⋊C411yescharacteristic
168M16—42no—
169M16—81no—
170M16—942no—
171M16—1882no—
172M16—1881no—
173C2×C8—81no—
174C2×C8—1882no—
175C2×C8—1881no—
176C2×C8—1882no—
177C4⋊C4—41no—
178C4⋊C4—1881no—
179C22⋊C4—41no—
180C22⋊C4—41no—
181C22⋊C4—1881no—
182C42—21no—
183C42—1881no—
184C42C4⋊Dic4711yescharacteristic
185C42(C2×D47)⋊C411yescharacteristic
186C23—21no—
187C23—21no—
188C23#1309393811yescharacteristic
189Q8—41no—
190Q8—41no—
191D4—41no—
192D4—41no—
193D4—42no—
194D4—81no—
195D4—82no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4—3761no—
208C2×C4C47⋊#9511yescharacteristic
209C2×C4#1309393811yescharacteristic
210C8—82no—
211C8—3762no—
212C8—1882no—
213C22—21no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—81no—
218C22—81no—
219C22#1568888911yescharacteristic
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—7521no—
226C4#1568890911yescharacteristic
227C4#1568891211yescharacteristic
228C2—21no—
229C2—41no—
230C2—81no—
231C2—81no—
232C2#2024150511yescharacteristic
subgroupG/H
centerC2#20241505
commutatorC2×Q8×C47C42
Frattini#532C2×D47
FittingC47×#2132C2
socleC2×C47#1390
prime radical, cosocleC47×#532C22
hypercenter#2132D47
hypocenterC47#7484
LCSG ▷ C2×Q8×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #2132

Complements

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

NH ≅ G/Ntype
10. #15688863C4semidirect
13. C47×#356C4semidirect
126. #356Dic47semidirect
131. C47#7484semidirect

Maximal subgroups

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

subgroupindex
#202437722
#202420232
C47×#21322
#748447