Finite Group Databasebeta

← #26113706

Subgroups — #26113706

Order 12032.

554 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 554 conjugacy classes of subgroups — too many to draw. Only its 115 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#20252473C211yesmaximal, characteristic, complemented
2#20248387C211yesmaximal, characteristic
3#20242057C212yesmaximal
4#20241289C212yesmaximal, complemented
5C47×#1801C211yesmaximal, characteristic, complemented
6C2×#13094021C2211yescharacteristic
7C2×#13094013—21no—
8#15689254—24no—
9#15689180—22no—
10#15689176C2212yes—
11#15689040C414yes—
12#15689025—22no—
13#15689024—21no—
14C4×C4⋊Dic47C2211yescharacteristic
15#15688866—24no—
16C2×C4×Q8×C47C2211yescharacteristic
17C47×#423C2212yes—
18C2×#101×C47—21no—
19C47×#343—21no—
20C22×D188—21no—
21C22×D188—21no—
22C22×D188C2×C412yes—
23#13094021—22no—
24#13094021—22no—
25#13094013—44no—
26C2×C4⋊Dic47—21no—
27C2×(C2×D47)⋊C4—24no—
28C2×(C2×D47)⋊C4—24no—
29C2×C4×Dic47—21no—
30C4⋊(C47⋊C8)—24no—
31C4⋊(C47⋊C8)D414yes—
32C22⋊(C47⋊C8)—22no—
33#13093935—24no—
34C4×C47⋊C8—22no—
35C47⋊#95C2×C412yes—
36C22×Q8×C47D411yescharacteristic
37C2×M16×C47—22no—
38C4×Q8×C47—22no—
39C4×Q8×C47—22no—
40C2×C4⋊C4×C47C2×C412yes—
41C2×C4⋊C4×C47D411yescharacteristic
42C2×C42×C47D412yes—
43C2×C42×C47C2311yescharacteristic
44C4⋊C8×C47—22no—
45C4⋊C8×C47—22no—
46#101×C47—42no—
47C23×D47—22no—
48C2×Q8×C47—22no—
49C2×Q8×C47—22no—
50C2×Q8×C47—21no—
51C2×Q8×C47—21no—
52C22×C4×C47C22⋊C412yes—
53C22×C4×C47C22⋊C412yes—
54C22×C4×C47C22×C411yescharacteristic
55C22×C4×C47C2×D411yescharacteristic
56C22×C4×C47C2×D411yescharacteristic
57C22×Dic47—22no—
58C2×D188—24no—
59C2×D188—44no—
60C2×D188—41no—
61C2×D188—42no—
62C2×D188—41no—
63C2×D188—42no—
64C2×D188—41no—
65C2×D188—41no—
66C2×D188—82no—
67M16×C47—42no—
68M16×C47—42no—
69C2×C8×C47—42no—
70C4⋊C4×C47—24no—
71C4⋊C4×C47—22no—
72C42×C47—22no—
73C42×C47—22no—
74C42×C47C2×D412yes—
75C42×C47#4112yes—
76(C2×D47)⋊C4—48no—
77(C2×D47)⋊C4—84no—
78C4⋊Dic47—42no—
79C4×Dic47—42no—
80C2×C47⋊C8—24no—
81C23×C47C2×C22⋊C411yescharacteristic
82C22×D47—44no—
83C22×D47—44no—
84C22×D47—42no—
85C22×D47—42no—
86C22×D47—42no—
87Q8×C47—42no—
88Q8×C47—42no—
89C2×C4×C47—22no—
90C2×C4×C47—22no—
91C2×C4×C47—22no—
92C2×C4×C47—22no—
93C2×C4×C47—22no—
94C2×C4×C47—41no—
95C2×C4×C47—42no—
96C2×C4×C47C4×D412yes—
97C2×C4×C47#12011yescharacteristic
98C2×C4×C47#12112yes—
99C2×C4×C47#12311yescharacteristic
100C2×C4×C47AGL(1,ℤ/8)12yes—
101C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
102C2×C4×C47#13711yescharacteristic
103C2×Dic47—42no—
104C2×Dic47—42no—
105C2×Dic47—42no—
106D188—84no—
107D188—82no—
108D188—82no—
109D188—82no—
110D188—82no—
111C8×C47—82no—
112C47⋊C8—44no—
113#9406—471nomaximal, Sylow
114C22×C47—21no—
115C22×C47—21no—
116C22×C47#38611yescharacteristic
117C22×C47#41711yescharacteristic
118C22×C47#41811yescharacteristic
119C2×D47—84no—
120C2×D47—84no—
121C2×D47—82no—
122C2×D47—82no—
123C2×D47—82no—
124C4×C47—21no—
125C4×C47—22no—
126C4×C47—42no—
127C4×C47—42no—
128C4×C47#45412yes—
129C4×C47#46812yes—
130C4×C47#46912yes—
131Dic47—82no—
132#2983—471no—
133#2548—471no—
134#1801D4711yescharacteristic, complemented
135#1545—472no—
136#1303—472no—
137C2×C47—21no—
138C2×C47—21no—
139C2×C47#188811yescharacteristic
140C2×C47#189511yescharacteristic
141C2×C47#191411yescharacteristic
142D47—162no—
143C2×#127—471no—
144C2×#124—941no—
145C2×C4×Q8C2×D4711yescharacteristic
146#446—944no—
147#432—942no—
148#431—472no—
149#423C2×D4712yes—
150C2×#101—21no—
151#394—474no—
152#390—941no—
153#390—942no—
154C4×C4⋊C4—471no—
155#343—21no—
156#331—944no—
157C47#940611yescharacteristic, complemented, Sylow
158C22×Q8C22⋊D4711yescharacteristic
159C22×D4—472no—
160C22×D4—941no—
161C22×D4—941no—
162C2×M16—22no—
163#127—942no—
164#127—942no—
165#124—1884no—
166C4×Q8—22no—
167C4×Q8—22no—
168C2×C4⋊C4—941no—
169C2×C4⋊C4C4×D4712yes—
170C2×C4⋊C4C22⋊D4711yescharacteristic
171C2×C22⋊C4—944no—
172C2×C22⋊C4—944no—
173C2×C42—941no—
174C2×C42D18812yes—
175C2×C42C22×D4711yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—474no—
179C4⋊C8—944no—
180#101—42no—
181C22⋊C8—942no—
182#97—944no—
183C4×C8—942no—
184#95—472no—
185C24—942no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×D4—944no—
191C2×D4—3762no—
192C2×D4—1884no—
193C2×D4—1881no—
194C2×D4—1882no—
195C2×D4—1881no—
196C2×D4—1882no—
197C2×D4—1881no—
198C2×D4—1881no—
199C22×C4—942no—
200C22×C4(C2×D47)⋊C412yes—
201C22×C4(C2×D47)⋊C412yes—
202C22×C4C2×C4×D4711yescharacteristic
203C22×C4C2×D18811yescharacteristic
204C22×C4C2×C22⋊D4711yescharacteristic
205M16—42no—
206M16—42no—
207C2×C8—42no—
208C2×C8—944no—
209C4⋊C4—24no—
210C4⋊C4—22no—
211C4⋊C4—1882no—
212C22⋊C4—3764no—
213C22⋊C4—1888no—
214C42—22no—
215C42—22no—
216C42—1882no—
217C42D4×D4712yes—
218C42#1062535212yes—
219C23—1884no—
220C23—1884no—
221C23—1882no—
222C23—1882no—
223C23—1882no—
224C23C2×(C2×D47)⋊C411yescharacteristic
225Q8—42no—
226Q8—42no—
227D4—3764no—
228D4—3762no—
229D4—3762no—
230D4—3762no—
231D4—3762no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—1882no—
240C2×C4—1882no—
241C2×C4—1882no—
242C2×C4(C22⋊D47)⋊C412yes—
243C2×C4#1309399411yescharacteristic
244C2×C4#1309399612yes—
245C2×C4#1309400811yescharacteristic
246C2×C4#1309404912yes—
247C2×C4#1309405011yescharacteristic
248C2×C4#1309405511yescharacteristic
249C8—82no—
250C8—1884no—
251C22—21no—
252C22—21no—
253C22—3764no—
254C22—3764no—
255C22—3762no—
256C22—3762no—
257C22—3762no—
258C22#1568900011yescharacteristic
259C22#1568912211yescharacteristic
260C22#1568912711yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4—42no—
265C4—3762no—
266C4#1568929812yes—
267C4#1568937112yes—
268C4#1568937412yes—
269C2—21no—
270C2—21no—
271C2—7522no—
272C2#2024378911yescharacteristic
273C2#2024382411yescharacteristic
274C2#2024393311yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1801C2
socleC22×C47#386
prime radicalC2×#13094021C22
cosocleC2×C42×C47C23
hypercenter#1801D47
hypocenterC47#9406
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
1. #20252473C2semidirect
4. #20241289C2semidirect
5. C47×#1801C2semidirect
134. #1801D47semidirect
157. C47#9406semidirect

Maximal subgroups

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

subgroupindex
#202524732
#202483872
#202420572
#202412892
C47×#18012
#940647