Finite Group Databasebeta

← #26119372

Subgroups — #26119372

Order 12032.

416 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 97 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#20251766C211yesmaximal, characteristic
2#20244365C211yesmaximal, characteristic
3#20241953C212yesmaximal
4#20241247C212yesmaximal
5C47×#1276C211yesmaximal, characteristic
6#15689680C2211yescharacteristic
7C2×#13094024—21no—
8#15689147C2211yescharacteristic
9#15689029—24no—
10#15688989—21no—
11#15688989C2212yes—
12#15688883—21no—
13#15688883—21no—
14#15688861—24no—
15C47×#431C2212yes—
16C43×C47C2211yescharacteristic
17C22×Dic94—22no—
18C2×#10625329—21no—
19#13094024—42no—
20#13094024—42no—
21#13094024Q812yescomplemented
22#13094016—21no—
23#13094003—22no—
24C4×Dic94—22no—
25#13093983—21no—
26C2×C4⋊Dic47—22no—
27C2×C4⋊Dic47—22no—
28C4⋊(C47⋊C8)—22no—
29C47⋊#95—24no—
30C47⋊#95—22no—
31C47⋊#95D414yes—
32C2×C42×C47—22no—
33C2×C42×C47D411yescharacteristic
34C2×C42×C47D411yescharacteristic
35C2×C42×C47C2311yescharacteristic
36C4⋊C8×C47—22no—
37C22⋊C8×C47—22no—
38C4×C8×C47—22no—
39C22×C4×C47—21no—
40C22×C4×C47—21no—
41C22×C4×C47C2×D412yes—
42C22×C4×C47C2×D411yescharacteristic
43C22×Dic47—24no—
44C2×Dic94—22no—
45C2×Dic94—44no—
46C2×Dic94—48no—
47C2×C8×C47—42no—
48C42×C47—21no—
49C42×C47—21no—
50C42×C47—44no—
51C42×C47SD1612yes—
52C42×C47Q1612yes—
53C42×C47C2×Q811yescharacteristic
54C42×C47#4112yes—
55C42×C47#4111yescharacteristic
56C22⋊Dic47—41no—
57C4⋊Dic47—22no—
58C4⋊Dic47—22no—
59C4⋊Dic47—44no—
60C4⋊Dic47—42no—
61C4⋊Dic47—42no—
62Dic47⋊C4—42no—
63C4×Dic47—41no—
64#10625329—41no—
65#10625329—41no—
66C2×C47⋊C8—41no—
67C23×C47#12011yescharacteristic
68C2×C4×C47—24no—
69C2×C4×C47—22no—
70C2×C4×C47—22no—
71C2×C4×C47—42no—
72C2×C4×C47—42no—
73C2×C4×C47C22⋊Q812yes—
74C2×C4×C47C22⋊Q811yescharacteristic
75C2×C4×C47#12412yes—
76C2×C4×C47#12411yescharacteristic
77C2×C4×C47C2×SD1611yescharacteristic
78C2×C4×C47C2×Q1611yescharacteristic
79C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
80C2×C4×C47#13711yescharacteristic
81C2×Dic47—48no—
82C2×Dic47—42no—
83C2×Dic47—44no—
84Dic94—88no—
85Dic94—82no—
86C8×C47—82no—
87C47⋊C8—81no—
88#10363—471nomaximal, Sylow
89C22×C47—22no—
90C22×C47#39311yescharacteristic
91C22×C47#45011yescharacteristic
92C22×C47#45111yescharacteristic
93C4×C47—24no—
94C4×C47—22no—
95C4×C47—22no—
96C4×C47—42no—
97C4×C47#47511yescharacteristic
98C4×C47#47711yescharacteristic
99C4×C47#48711yescharacteristic
100C4×C47#48911yescharacteristic
101Dic47—84no—
102Dic47—82no—
103#2862—471no—
104#1994—471no—
105#1518—472no—
106#1286—472no—
107#1276D4711yescharacteristic
108C2×C47—21no—
109C2×C47—21no—
110C2×C47#202411yescharacteristic
111C2×C47#203112yes—
112#533—471no—
113C2×C4⋊Q8—941no—
114#431C2×D4712yes—
115#423—471no—
116#391—944no—
117#382—472no—
118#382—941no—
119C43C2×D4711yescharacteristic
120#339—941no—
121#339—941no—
122#328—944no—
123C47#1036311yescharacteristic, complemented, Sylow
124C22×Q8—942no—
125C2×M16—941no—
126C4⋊Q8—472no—
127C4⋊Q8—1882no—
128C4⋊Q8—1882no—
129#125—941no—
130C22⋊Q8—942no—
131C4×Q8—942no—
132#117—941no—
133C2×C4⋊C4—942no—
134C2×C4⋊C4—942no—
135C2×C42—22no—
136C2×C42D18811yescharacteristic
137C2×C42D18811yescharacteristic
138C2×C42C22×D4711yescharacteristic
139C4⋊C8—22no—
140C4⋊C8—942no—
141C22⋊C8—22no—
142C4×C8—22no—
143#95—474no—
144#95—944no—
145#95—942no—
146C2×Q8—942no—
147C2×Q8—1884no—
148C2×Q8—1888no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—944no—
152C22×C4C2×D18811yescharacteristic
153C22×C4D4×D4712yes—
154M16—1881no—
155M16—1881no—
156C2×C8—42no—
157C2×C8—1881no—
158C4⋊C4—942no—
159C4⋊C4—942no—
160C4⋊C4—1884no—
161C4⋊C4—1882no—
162C4⋊C4—1882no—
163C4⋊C4—1882no—
164C22⋊C4—1881no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—1881no—
169C42#1062532512yes—
170C42Dic18812yes—
171C42#1062535012yes—
172C42#1062535211yescharacteristic
173C42Q8×D4711yescharacteristic
174C23#1309399411yescharacteristic
175Q8—3762no—
176Q8—3768no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—1888no—
183C2×C4—1884no—
184C2×C4—1882no—
185C2×C4#1309400212yes—
186C2×C4#1309400411yescharacteristic
187C2×C4#1309401012yes—
188C2×C4#1309401211yescharacteristic
189C2×C4C2×#1062532511yescharacteristic
190C2×C4C2×Dic18811yescharacteristic
191C2×C4#1309405211yescharacteristic
192C2×C4#1309405711yescharacteristic
193C8—82no—
194C8—3761no—
195C22—22no—
196C22#1568903611yescharacteristic
197C22#1568927611yescharacteristic
198C22#1568928111yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—3764no—
204C4—3762no—
205C4#1568940911yescharacteristic
206C4#1568941911yescharacteristic
207C4#1568947411yescharacteristic
208C4#1568948311yescharacteristic
209C2—21no—
210C2—21no—
211C2#2024453011yescharacteristic
212C2#2024456112yes—
subgroupG/H
centerC22#15689036
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1276C2
socleC22×C47#393
prime radicalC23×C47#120
hypercenter#1276D47
hypocenterC47#10363
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1276

Complements

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

NH ≅ G/Ntype
21. #13094024Q8semidirect
123. C47#10363semidirect

Maximal subgroups

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

subgroupindex
#202517662
#202443652
#202419532
#202412472
C47×#12762
#1036347