Finite Group Databasebeta

← #26120447

Subgroups — #26120447

Order 12032.

402 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 85 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#20243559C211yesmaximal, characteristic, complemented
2#20242123C212yesmaximal, complemented
3#20241839C212yesmaximal, complemented
4C47⋊#1309C211yesmaximal, characteristic, complemented
5C47×#3253C211yesmaximal, characteristic
6C2×#13093959—22no—
7#15689147C2211yescharacteristic
8#15689147C2212yes—
9C2×C4×C47⋊C8C2211yescharacteristic
10#15688872—22no—
11#15688867—22no—
12D4⋊(C47⋊C8)—24no—
13C47×#532C2212yes—
14C47×#476—22no—
15C47×#469—22no—
16C47×#459—22no—
17C47×#423C2211yescharacteristic
18C2×#10625329—21no—
19C2×#10625329—22no—
20C22×C47⋊C8—21no—
21#13093959—44no—
22C4⋊(C47⋊C8)—22no—
23C4⋊(C47⋊C8)—22no—
24C4⋊(C47⋊C8)—22no—
25C4×C47⋊C8—21no—
26C4×C47⋊C8D412yes—
27C2×#41×C47—22no—
28C2×D8×C47—22no—
29C2×M16×C47D412yes—
30C4⋊Q8×C47D412yes—
31#127×C47D412yes—
32#124×C47—22no—
33#121×C47—22no—
34#121×C47—22no—
35C4×D4×C47—22no—
36C4×D4×C47—22no—
37C2×C42×C47C2311yescharacteristic
38#106×C47—22no—
39C4⋊C8×C47—22no—
40#102×C47—24no—
41#41×C47—82no—
42#41×C47—82no—
43C2×Q8×C47—22no—
44C2×D4×C47—22no—
45C2×D4×C47—22no—
46C2×D4×C47—22no—
47C2×D4×C47—42no—
48C22×C4×C47—42no—
49C22×C4×C47C2×D411yescharacteristic
50C22×C4×C47#4112yes—
51D8×C47—82no—
52M16×C47—42no—
53C2×C8×C47—22no—
54C4⋊C4×C47—22no—
55C4⋊C4×C47—22no—
56C22⋊C4×C47—42no—
57C22⋊C4×C47—42no—
58C42×C47C2×D412yes—
59C42×C47C2×D411yescharacteristic
60C42×C47#4111yescharacteristic
61#10625329—42no—
62#10625329—82no—
63C2×C47⋊C8—22no—
64C2×C47⋊C8—22no—
65C2×C47⋊C8—22no—
66C2×C47⋊C8—41no—
67C2×C47⋊C8—42no—
68C23×C47—42no—
69C23×C47#12411yescharacteristic
70Q8×C47—82no—
71D4×C47—42no—
72D4×C47—42no—
73D4×C47—42no—
74D4×C47—42no—
75D4×C47—82no—
76D4×C47—84no—
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—22no—
80C2×C4×C47—22no—
81C2×C4×C47—42no—
82C2×C4×C47—82no—
83C2×C4×C47—82no—
84C2×C4×C47#12011yescharacteristic
85C2×C4×C47#12111yescharacteristic
86C2×C4×C47#12112yes—
87C2×C4×C47#12312yes—
88C8×C47—42no—
89C47⋊C8—24no—
90C47⋊C8—81no—
91C47⋊C8—82no—
92#10530—471nomaximal, Sylow
93C22×C47—22no—
94C22×C47—21no—
95C22×C47—82no—
96C22×C47—82no—
97C22×C47—82no—
98C22×C47#39411yescharacteristic
99C4×C47—22no—
100C4×C47—21no—
101C4×C47—82no—
102C4×C47#45311yescharacteristic
103C4×C47#45611yescharacteristic
104C4×C47#47112yes—
105C4×C47#47212yes—
106#3253D4711yescharacteristic
107#1848—471no—
108#1560—472no—
109#1492—472no—
110#1309—471no—
111C2×C47—41no—
112C2×C47—162no—
113C2×C47#203911yescharacteristic
114C2×C47#205112yes—
115#532C2×D4712yes—
116#476—22no—
117#469—22no—
118#459—22no—
119C2×#108—942no—
120#423—471no—
121#423—472no—
122#423C2×D4711yescharacteristic
123C2×C4×C8—471no—
124#333—942no—
125#331—942no—
126D4⋊C8—944no—
127C47#1053011yescharacteristic, complemented, Sylow
128C2×#41—22no—
129C2×D8—22no—
130C2×M16—942no—
131C2×M16—941no—
132C2×M16C22⋊D4712yes—
133C22×C8—941no—
134C4⋊Q8C22⋊D4712yes—
135#127C22⋊D4712yes—
136#124—22no—
137#121—22no—
138#121—22no—
139C4×D4—22no—
140C4×D4—22no—
141C2×C42C22×D4711yescharacteristic
142#108—1884no—
143#106—22no—
144C4⋊C8—22no—
145C4⋊C8—942no—
146C4⋊C8—942no—
147C4⋊C8—942no—
148#102—24no—
149C4×C8—472no—
150C4×C8—941no—
151#41—82no—
152#41—82no—
153C2×Q8—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—42no—
158C22×C4—42no—
159C22×C4#1062535212yes—
160C22×C4C2×C22⋊D4711yescharacteristic
161D8—82no—
162M16—42no—
163M16—3762no—
164M16—1882no—
165C2×C8—22no—
166C2×C8—942no—
167C2×C8—942no—
168C2×C8—942no—
169C2×C8—1881no—
170C2×C8—1882no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C42D4×D4711yescharacteristic
176C42#1062535211yescharacteristic
177C42C2×C22⋊D4712yes—
178C23—42no—
179C23#1309401111yescharacteristic
180Q8—82no—
181D4—42no—
182D4—42no—
183D4—42no—
184D4—42no—
185D4—82no—
186D4—84no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—82no—
193C2×C4—82no—
194C2×C4#1309399311yescharacteristic
195C2×C4#1309399511yescharacteristic
196C2×C4#1309399612yes—
197C2×C4#1309400512yes—
198C8—42no—
199C8—944no—
200C8—3761no—
201C8—3762no—
202C22—21no—
203C22—22no—
204C22—82no—
205C22—82no—
206C22—82no—
207C22#1568903711yescharacteristic
208C4—21no—
209C4—22no—
210C4—82no—
211C4#1568929111yescharacteristic
212C4#1568930611yescharacteristic
213C4#1568938412yes—
214C4#1568939112yes—
215C2—41no—
216C2—162no—
217C2#2024460911yescharacteristic
218C2#2024467712yes—
subgroupG/H
centerC22#15689037
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
Fitting, prime radicalC47×#3253C2
socleC22×C47#394
hypercenter#3253D47
hypocenterC47#10530
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. #20243559C2semidirect
2. #20242123C2semidirect
3. #20241839C2semidirect
4. C47⋊#1309C2semidirect
127. C47#10530semidirect

Maximal subgroups

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

subgroupindex
#202435592
#202421232
#202418392
C47⋊#13092
C47×#32532
#1053047