Finite Group Databasebeta

← #26135364

Subgroups — #26135364

Order 12032.

370 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 101 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#20245107C212yesmaximal
2#20244259C211yesmaximal, characteristic
3#20241493C212yesmaximal
4#20241472C211yesmaximal, characteristic
5C47×#3162C211yesmaximal, characteristic
6C22×Dic188C2211yescharacteristic
7C2×#13093964C2212yes—
8C2×#13093956C2211yescharacteristic
9C2×#13093949—21no—
10#15689029—21no—
11#15688926—24no—
12C47×#533—21no—
13C47×#501—24no—
14C47×#427—21no—
15C47×#427C2212yes—
16C2×#97×C47C2211yescharacteristic
17C22×Dic94—21no—
18C2×Dic188—22no—
19C2×Dic188—22no—
20C2×Dic188—21no—
21C2×Dic188Q812yescomplemented
22C2×C4⋊Dic47—21no—
23#13093964—22no—
24#13093964—22no—
25C2×C47⋊C16—22no—
26#13093956—22no—
27#13093949—42no—
28C47⋊#95—42no—
29C22×C8×C47D411yescharacteristic
30C22×C8×C47C2311yescharacteristic
31C4⋊Q8×C47—22no—
32#125×C47—22no—
33C22⋊Q8×C47—42no—
34C4×Q8×C47—42no—
35#117×C47—22no—
36C2×C42×C47D411yescharacteristic
37#107×C47—22no—
38#107×C47D414yes—
39#106×C47—22no—
40#106×C47—22no—
41#97×C47—22no—
42C2×Q8×C47—42no—
43C22×C4×C47D812yes—
44C22×C4×C47C2×D411yescharacteristic
45C22×Dic47—41no—
46C22×Dic47—41no—
47C2×Dic94—22no—
48C2×Dic94—41no—
49C2×Dic94—41no—
50C2×Dic94—41no—
51C2×Dic94—41no—
52C2×Dic94—41no—
53C2×C8×C47—21no—
54C2×C8×C47—21no—
55C2×C8×C47SD1612yes—
56C2×C8×C47C2×D412yes—
57C2×C8×C47C2×Q811yescharacteristic
58C2×C8×C47#4112yes—
59C2×C8×C47#4111yescharacteristic
60C4⋊C4×C47—24no—
61C4⋊C4×C47—44no—
62C4⋊C4×C47—42no—
63C22⋊C4×C47—42no—
64C42×C47—21no—
65C42×C47—21no—
66C42×C47—42no—
67C4⋊Dic47—41no—
68C4⋊Dic47—41no—
69Dic188—24no—
70Dic188—42no—
71C47⋊C16—42no—
72C23×C47C2×D811yescharacteristic
73Q8×C47—84no—
74C2×C4×C47—22no—
75C2×C4×C47—42no—
76C2×C4×C47—44no—
77C2×C4×C47#12011yescharacteristic
78C2×C4×C47C22⋊Q811yescharacteristic
79C2×C4×C47#12411yescharacteristic
80C2×C4×C47C2×SD1611yescharacteristic
81C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
82C2×C4×C47#13711yescharacteristic
83C2×Dic47—42no—
84C2×Dic47—81no—
85C2×Dic47—81no—
86C2×Dic47—82no—
87C2×Dic47—81no—
88Dic94—42no—
89Dic94—42no—
90Dic94—81no—
91Dic94—81no—
92C8×C47—22no—
93C8×C47C22⋊Q812yes—
94C8×C47#12412yes—
95#13368—471nomaximal, Sylow
96C22×C47—22no—
97C22×C47#44711yescharacteristic
98C22×C47#47411yescharacteristic
99C22×C47#48611yescharacteristic
100C4×C47—42no—
101C4×C47—84no—
102C4×C47#39311yescharacteristic
103C4×C47#44811yescharacteristic
104C4×C47#47611yescharacteristic
105C4×C47#48911yescharacteristic
106Dic47—82no—
107Dic47—161no—
108#3162D4711yescharacteristic
109#2141—472no—
110#1971—471no—
111#1386—472no—
112#1377—471no—
113C2×C47—21no—
114C2×C47—21no—
115C2×C47#202311yescharacteristic
116C2×C47#219112yes—
117C22×Q16—471no—
118#533—21no—
119C2×M32—472no—
120#501—24no—
121#427—21no—
122#427C2×D4712yes—
123C2×#106—471no—
124C2×#103—941no—
125C2×#97C2×D4711yescharacteristic
126#391—941no—
127#358—944no—
128C47#1336811yescharacteristic, complemented, Sylow
129C22×Q8—941no—
130C2×Q16—472no—
131C2×Q16—942no—
132C2×Q16—941no—
133C2×Q16—942no—
134C22×C8C22⋊D4711yescharacteristic
135C22×C8C22×D4711yescharacteristic
136C4⋊Q8—22no—
137#125—22no—
138C22⋊Q8—42no—
139C4×Q8—42no—
140#117—22no—
141C2×C4⋊C4—941no—
142C2×C42C22⋊D4711yescharacteristic
143M32—942no—
144M32—942no—
145C2×C16—942no—
146#107—22no—
147#107C22⋊D4714yes—
148#106—22no—
149#106—22no—
150#106—942no—
151#103—1882no—
152#97—22no—
153#95—1882no—
154C2×Q8—42no—
155C2×Q8—942no—
156C2×Q8—1881no—
157C2×Q8—1881no—
158C2×Q8—1881no—
159C2×Q8—1881no—
160C2×Q8—1881no—
161C22×C4—1881no—
162C22×C4—1881no—
163C22×C4#1062533412yes—
164C22×C4C2×C22⋊D4711yescharacteristic
165Q16—944no—
166Q16—1882no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#1062533612yes—
170C2×C8Q8×D4711yescharacteristic
171C2×C8#1062535412yes—
172C2×C8#1062535411yescharacteristic
173C2×C8C2×C22⋊D4712yes—
174C4⋊C4—24no—
175C4⋊C4—44no—
176C4⋊C4—42no—
177C4⋊C4—1881no—
178C4⋊C4—1881no—
179C22⋊C4—42no—
180C42—21no—
181C42—21no—
182C42—42no—
183C16—1882no—
184C23C2×#1062533411yescharacteristic
185Q8—84no—
186Q8—3761no—
187Q8—3761no—
188Q8—1882no—
189Q8—1882no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—3761no—
194C2×C4—3761no—
195C2×C4—3762no—
196C2×C4—3761no—
197C2×C4—1882no—
198C2×C4#1309399311yescharacteristic
199C2×C4#1309400011yescharacteristic
200C2×C4#1309401311yescharacteristic
201C2×C4C2×#1062533611yescharacteristic
202C2×C4#1309405011yescharacteristic
203C2×C4#1309405511yescharacteristic
204C8—22no—
205C8#1309400012yes—
206C8#1309401312yes—
207C22—22no—
208C22#1568925711yescharacteristic
209C22#1568940011yescharacteristic
210C22#1568947111yescharacteristic
211C4—42no—
212C4—84no—
213C4—3762no—
214C4—7521no—
215C4#1568903411yescharacteristic
216C4#1568926211yescharacteristic
217C4#1568941011yescharacteristic
218C4#1568948511yescharacteristic
219C2—21no—
220C2—21no—
221C2#2024452111yescharacteristic
222C2#2024533912yes—
subgroupG/H
centerC22#15689257
commutator, cosocleC22×C8×C47C23
FrattiniC22×C8C22×D47
FittingC47×#3162C2
socleC22×C47#447
prime radicalC23×C47C2×D8
hypercenter#3162D47
hypocenterC47#13368
LCSG ▷ C22×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #3162

Complements

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

NH ≅ G/Ntype
21. C2×Dic188Q8semidirect
128. C47#13368semidirect

Maximal subgroups

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

subgroupindex
#202451072
#202442592
#202414932
#202414722
C47×#31622
#1336847