Finite Group Databasebeta

← #26129357

Subgroups — #26129357

Order 12032.

342 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 63 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#20244738C211yesmaximal, characteristic
2#20242361C212yesmaximal
3#20241418C212yesmaximal
4#20241332C211yesmaximal, characteristic
5C47×#2844C211yesmaximal, characteristic
6#15689754C2211yescharacteristic
7#15689279—22no—
8#15689163—21no—
9#15688892C2211yescharacteristic
10#15688883—21no—
11#15688859—22no—
12#15688849C2212yes—
13C47×#400—22no—
14C47×#400—21no—
15C47×#395—22no—
16C47×#395C2212yes—
17C47×#382—21no—
18C47×#382C2211yescharacteristic
19C2×#10625337—42no—
20C2×#10625329—21no—
21#13094023—41no—
22#13094015—41no—
23#13094003—22no—
24#13093983—21no—
25#13093957—41no—
26#13093955—41no—
27#13093948—42no—
28C22⋊(C47⋊C8)—22no—
29C22×Q8×C47D412yes—
30C2×C4⋊C4×C47—21no—
31C2×C4⋊C4×C47—21no—
32C2×C4⋊C4×C47—21no—
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47D411yescharacteristic
35C2×C4⋊C4×C47C2311yescharacteristic
36C2×C42×C47D411yescharacteristic
37#95×C47—22no—
38#95×C47—22no—
39#95×C47—21no—
40#95×C47—22no—
41#95×C47—22no—
42#95×C47D412yes—
43C2×Q8×C47—42no—
44C2×Q8×C47—82no—
45C22×C4×C47—21no—
46C22×C4×C47—22no—
47C22×C4×C47—22no—
48C22×C4×C47—21no—
49C22×C4×C47C2×D411yescharacteristic
50C22×C4×C47C2×D412yes—
51C2×Dic94—22no—
52C4⋊C4×C47—41no—
53C4⋊C4×C47—41no—
54C4⋊C4×C47—81no—
55C4⋊C4×C47—81no—
56C4⋊C4×C47—82no—
57C4⋊C4×C47—81no—
58C42×C47—21no—
59C42×C47Q1612yescomplemented
60C22⋊Dic47—41no—
61#10625337—84no—
62C4⋊Dic47—22no—
63C4⋊Dic47—21no—
64C4⋊Dic47—21no—
65Dic47⋊C4—42no—
66C4×Dic47—41no—
67#10625329—22no—
68#10625329—41no—
69C2×C47⋊C8—41no—
70C2×C47⋊C8—42no—
71C23×C47#12011yescharacteristic
72Q8×C47—44no—
73Q8×C47—82no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47—22no—
77C2×C4×C47—22no—
78C2×C4×C47—41no—
79C2×C4×C47—41no—
80C2×C4×C47—42no—
81C2×C4×C47—42no—
82C2×C4×C47—42no—
83C2×C4×C47—42no—
84C2×C4×C47—42no—
85C2×C4×C47—42no—
86C2×C4×C47—42no—
87C2×C4×C47—41no—
88C2×C4×C47—42no—
89C2×C4×C47C2×Q1611yescharacteristic
90C2×C4×C47#13711yescharacteristic
91C2×Dic47—41no—
92C2×Dic47—41no—
93Dic94—82no—
94C47⋊C8—42no—
95C47⋊C8—82no—
96#12260—471nomaximal, Sylow
97C22×C47—21no—
98C22×C47—21no—
99C22×C47#45111yescharacteristic
100C22×C47#45611yescharacteristic
101C22×C47B(4,𝔽2)11yescharacteristic
102C4×C47—21no—
103C4×C47—21no—
104C4×C47—42no—
105C4×C47—42no—
106C4×C47—82no—
107C4×C47—82no—
108C4×C47—81no—
109Dic47—81no—
110Dic47—81no—
111#2844D4711yescharacteristic
112#2061—471no—
113#1605—472no—
114#1354—472no—
115#1324—471no—
116C2×C47—21no—
117C2×C47—21no—
118C2×C47#160511yescharacteristic
119C2×C47#219911yescharacteristic
120C2×C47#220311yescharacteristic
121#544—471no—
122#451—942no—
123#428—941no—
124#400—21no—
125#400—22no—
126#395—22no—
127#395C2×D4712yes—
128#382—21no—
129#382C2×D4711yescharacteristic
130#343—471no—
131#339—941no—
132#328—942no—
133#324—472no—
134C47#1226011yescharacteristic, complemented, Sylow
135C22×Q8C22⋊D4712yes—
136C2×Q16—1882no—
137C2×M16—941no—
138C4⋊Q8—1881no—
139#125—1881no—
140C22⋊Q8—942no—
141#117—941no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4C22⋊D4711yescharacteristic
147C2×C4⋊C4C22×D4711yescharacteristic
148C2×C42C22⋊D4711yescharacteristic
149#107—1881no—
150#106—1881no—
151#103—1882no—
152C22⋊C8—942no—
153#95—22no—
154#95—22no—
155#95—21no—
156#95—22no—
157#95—22no—
158#95C22⋊D4712yes—
159C2×Q8—42no—
160C2×Q8—82no—
161C2×Q8—942no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4C2×C22⋊D4711yescharacteristic
167C22×C4C2×C22⋊D4712yes—
168Q16—3764no—
169M16—942no—
170M16—1881no—
171C2×C8—1882no—
172C2×C8—1881no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—82no—
178C4⋊C4—81no—
179C4⋊C4—942no—
180C4⋊C4—941no—
181C4⋊C4—941no—
182C4⋊C4—1882no—
183C22⋊C4—1881no—
184C42—21no—
185C42—1881no—
186C42#1062533712yescomplemented
187C23#1309399311yescharacteristic
188Q8—44no—
189Q8—82no—
190Q8—3762no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—1881no—
207C2×C4—1881no—
208C2×C4C2×#1062533711yescharacteristic
209C2×C4#1309405511yescharacteristic
210C8—3762no—
211C8—1882no—
212C22—21no—
213C22—21no—
214C22#1568927911yescharacteristic
215C22#1568930611yescharacteristic
216C22#1568931011yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—82no—
222C4—82no—
223C4—81no—
224C4—3761no—
225C4—3761no—
226C2—21no—
227C2—21no—
228C2#2024236211yescharacteristic
229C2#2024538911yescharacteristic
230C2#2024540911yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2844C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
hypercenter#2844D47
hypocenterC47#12260
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2844

Complements

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

NH ≅ G/Ntype
59. C42×C47Q16semidirect
134. C47#12260semidirect
186. C42#10625337semidirect

Maximal subgroups

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

subgroupindex
#202447382
#202423612
#202414182
#202413322
C47×#28442
#1226047