Finite Group Databasebeta

← #26129356

Subgroups — #26129356

Order 12032.

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

normal subgroups only

This group has 342 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#20251660C211yesmaximal, characteristic
2#20242362C212yesmaximal
3#20241417C212yesmaximal
4#20241332C211yesmaximal, characteristic
5C47×#2061C211yesmaximal, characteristic
6#15689278—22no—
7#15689065—22no—
8#15689062—21no—
9#15689044—22no—
10#15689043C2212yes—
11#15688989—21no—
12#15688860—22no—
13#15688849C2212yes—
14C47×#544C2211yescharacteristic
15C47×#428—21no—
16C47×#382C2211yescharacteristic
17C47×#343C2211yescharacteristic
18C47×#339—21no—
19C22×Dic94D412yes—
20C2×#10625337—42no—
21C2×C4⋊Dic47—21no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×Dic47⋊C4—22no—
25#13093947—42no—
26C22⋊(C47⋊C8)—22no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C47⋊#95—21no—
30C47⋊#95—22no—
31C47⋊#95D412yes—
32C2×M16×C47—21no—
33C4⋊Q8×C47—41no—
34#125×C47—41no—
35C22⋊Q8×C47—22no—
36#117×C47—21no—
37C2×C4⋊C4×C47D411yescharacteristic
38C2×C4⋊C4×C47C2311yescharacteristic
39C2×C42×C47D411yescharacteristic
40#107×C47—41no—
41#106×C47—41no—
42#95×C47—22no—
43C2×Q8×C47—22no—
44C22×C4×C47—21no—
45C22×C4×C47—21no—
46C22×C4×C47C2×D411yescharacteristic
47C22×C4×C47C2×D412yes—
48C22×Dic47—22no—
49C22×Dic47—22no—
50C2×Dic94—42no—
51C2×Dic94—82no—
52M16×C47—22no—
53M16×C47—41no—
54C2×C8×C47—41no—
55C4⋊C4×C47—21no—
56C4⋊C4×C47—21no—
57C4⋊C4×C47—22no—
58C4⋊C4×C47—41no—
59C4⋊C4×C47—42no—
60C4⋊C4×C47—41no—
61C22⋊C4×C47—41no—
62C42×C47—21no—
63C42×C47—41no—
64C42×C47Q1612yescomplemented
65#10625337—84no—
66C4⋊Dic47—81no—
67C4⋊Dic47—81no—
68C4⋊Dic47—81no—
69Dic47⋊C4—82no—
70C2×C47⋊C8—42no—
71C23×C47#12011yescharacteristic
72Q8×C47—82no—
73C2×C4×C47—21no—
74C2×C4×C47—21no—
75C2×C4×C47—22no—
76C2×C4×C47—22no—
77C2×C4×C47—41no—
78C2×C4×C47—41no—
79C2×C4×C47—42no—
80C2×C4×C47—42no—
81C2×C4×C47—41no—
82C2×C4×C47—41no—
83C2×C4×C47—41no—
84C2×C4×C47C2×Q1611yescharacteristic
85C2×C4×C47#13711yescharacteristic
86C2×Dic47—42no—
87C2×Dic47—42no—
88C2×Dic47—42no—
89C2×Dic47—42no—
90C2×Dic47—42no—
91C2×Dic47—42no—
92Dic94—44no—
93Dic94—82no—
94C8×C47—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—81no—
107C4×C47—81no—
108C4×C47—81no—
109Dic47—82no—
110Dic47—82no—
111#2844—471no—
112#2061D4711yescharacteristic
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#544C2×D4711yescharacteristic
122#451—942no—
123#428—21no—
124#400—942no—
125#400—941no—
126#395—472no—
127#395—942no—
128#382—941no—
129#382C2×D4711yescharacteristic
130#343C2×D4711yescharacteristic
131#339—21no—
132#328—942no—
133#324—472no—
134C47#1226011yescharacteristic, complemented, Sylow
135C22×Q8—472no—
136C2×Q16—1882no—
137C2×M16—21no—
138C4⋊Q8—41no—
139#125—41no—
140C22⋊Q8—22no—
141#117—21no—
142C2×C4⋊C4—941no—
143C2×C4⋊C4—941no—
144C2×C4⋊C4—941no—
145C2×C4⋊C4—942no—
146C2×C4⋊C4C22⋊D4711yescharacteristic
147C2×C4⋊C4C22×D4711yescharacteristic
148C2×C42C22⋊D4711yescharacteristic
149#107—41no—
150#106—41no—
151#103—1882no—
152C22⋊C8—942no—
153#95—22no—
154#95—472no—
155#95—942no—
156#95—942no—
157#95—941no—
158#95—942no—
159C2×Q8—22no—
160C2×Q8—3762no—
161C2×Q8—1882no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—942no—
165C22×C4—942no—
166C22×C4D4×D4712yes—
167C22×C4C2×C22⋊D4711yescharacteristic
168Q16—3764no—
169M16—22no—
170M16—41no—
171C2×C8—41no—
172C2×C8—1882no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—3762no—
180C4⋊C4—3761no—
181C4⋊C4—3761no—
182C4⋊C4—3761no—
183C22⋊C4—41no—
184C42—21no—
185C42—41no—
186C42#1062533712yescomplemented
187C23#1309399211yescharacteristic
188Q8—82no—
189Q8—3762no—
190Q8—1884no—
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—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—1882no—
203C2×C4—1882no—
204C2×C4—1882no—
205C2×C4—1882no—
206C2×C4—1882no—
207C2×C4—1882no—
208C2×C4C2×#1062533711yescharacteristic
209C2×C4#1309405411yescharacteristic
210C8—42no—
211C8—3762no—
212C22—21no—
213C22—21no—
214C22#1568927811yescharacteristic
215C22#1568930511yescharacteristic
216C22#1568930911yescharacteristic
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—81no—
222C4—81no—
223C4—81no—
224C4—3762no—
225C4—3762no—
226C2—21no—
227C2—21no—
228C2#2024236111yescharacteristic
229C2#2024538811yescharacteristic
230C2#2024540811yescharacteristic
subgroupG/H
centerC22#15689309
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2061C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
hypercenter#2061D47
hypocenterC47#12260
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
64. 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
#202516602
#202423622
#202414172
#202413322
C47×#20612
#1226047