Finite Group Databasebeta

← #26119368

Subgroups — #26119368

Order 12032.

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

normal subgroups only

This group has 416 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#20244365C211yesmaximal, characteristic
2#20241951C212yesmaximal
3#20241246C212yesmaximal
4#20241221C211yesmaximal, characteristic
5C47×#2862C211yesmaximal, characteristic
6#15689680C2211yescharacteristic
7#15689176C2212yes—
8#15689147C2211yescharacteristic
9#15688883—21no—
10#15688883—21no—
11#15688859—24no—
12C2×C4⋊Q8×C47—21no—
13C47×#391—24no—
14C47×#382—21no—
15C47×#382C2212yes—
16C43×C47C2211yescharacteristic
17C2×#10625329—21no—
18#13094024Q812yescomplemented
19#13094016—21no—
20#13094003—22no—
21C4×Dic94—22no—
22#13093983—21no—
23C4⋊(C47⋊C8)—22no—
24C4⋊(C47⋊C8)—22no—
25C22⋊(C47⋊C8)—22no—
26C4×C47⋊C8—22no—
27C22×Q8×C47—22no—
28C4⋊Q8×C47—42no—
29C4⋊Q8×C47—42no—
30C2×C4⋊C4×C47—22no—
31C2×C4⋊C4×C47—22no—
32C2×C42×C47—22no—
33C2×C42×C47D411yescharacteristic
34C2×C42×C47D411yescharacteristic
35C2×C42×C47C2311yescharacteristic
36#95×C47—24no—
37#95×C47—22no—
38#95×C47D414yes—
39C2×Q8×C47—44no—
40C2×Q8×C47—48no—
41C22×C4×C47—21no—
42C22×C4×C47—21no—
43C22×C4×C47—24no—
44C22×C4×C47C2×D412yes—
45C22×C4×C47C2×D411yescharacteristic
46C2×Dic94—22no—
47C4⋊C4×C47—44no—
48C4⋊C4×C47—42no—
49C4⋊C4×C47—42no—
50C42×C47—21no—
51C42×C47—21no—
52C42×C47—44no—
53C42×C47SD1612yes—
54C42×C47Q1612yes—
55C42×C47C2×Q811yescharacteristic
56C42×C47#4111yescharacteristic
57C42×C47#4112yes—
58C22⋊Dic47—41no—
59C4⋊Dic47—22no—
60C4⋊Dic47—22no—
61Dic47⋊C4—42no—
62C4×Dic47—41no—
63#10625329—41no—
64#10625329—41no—
65C2×C47⋊C8—42no—
66C2×C47⋊C8—41no—
67C23×C47#12011yescharacteristic
68Q8×C47—88no—
69C2×C4×C47—24no—
70C2×C4×C47—22no—
71C2×C4×C47—22no—
72C2×C4×C47—42no—
73C2×C4×C47—42no—
74C2×C4×C47—48no—
75C2×C4×C47—44no—
76C2×C4×C47C22⋊Q812yes—
77C2×C4×C47C22⋊Q811yescharacteristic
78C2×C4×C47#12412yes—
79C2×C4×C47#12411yescharacteristic
80C2×C4×C47C2×SD1611yescharacteristic
81C2×C4×C47C2×Q1611yescharacteristic
82C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
83C2×C4×C47#13711yescharacteristic
84C2×Dic47—42no—
85Dic94—82no—
86C47⋊C8—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—84no—
98C4×C47#47511yescharacteristic
99C4×C47#47711yescharacteristic
100C4×C47#48711yescharacteristic
101C4×C47#48911yescharacteristic
102Dic47—82no—
103#2862D4711yescharacteristic
104#1994—471no—
105#1518—472no—
106#1286—472no—
107#1276—471no—
108C2×C47—21no—
109C2×C47—21no—
110C2×C47#202411yescharacteristic
111C2×C47#203112yes—
112#533—471no—
113C2×C4⋊Q8—21no—
114#431—472no—
115#423—471no—
116#391—24no—
117#382—21no—
118#382C2×D4712yes—
119C43C2×D4711yescharacteristic
120#339—941no—
121#339—941no—
122#328—944no—
123C47#1036311yescharacteristic, complemented, Sylow
124C22×Q8—22no—
125C2×M16—941no—
126C4⋊Q8—42no—
127C4⋊Q8—42no—
128C4⋊Q8—472no—
129#125—941no—
130C22⋊Q8—942no—
131C4×Q8—942no—
132#117—941no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C42—22no—
136C2×C42C22⋊D4711yescharacteristic
137C2×C42C22⋊D4711yescharacteristic
138C2×C42C22×D4711yescharacteristic
139C4⋊C8—942no—
140C4⋊C8—942no—
141C22⋊C8—942no—
142C4×C8—942no—
143#95—24no—
144#95—22no—
145#95C22⋊D4714yes—
146C2×Q8—44no—
147C2×Q8—48no—
148C2×Q8—942no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—24no—
152C22×C4C2×C22⋊D4712yes—
153C22×C4C2×C22⋊D4711yescharacteristic
154M16—1881no—
155M16—1881no—
156C2×C8—1882no—
157C2×C8—1881no—
158C4⋊C4—44no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—942no—
162C4⋊C4—942no—
163C4⋊C4—1882no—
164C22⋊C4—1881no—
165C42—21no—
166C42—21no—
167C42—44no—
168C42—1881no—
169C42#1062533512yes—
170C42#1062533712yes—
171C42Q8×D4711yescharacteristic
172C42#1062535411yescharacteristic
173C42#1062535412yes—
174C23#1309399311yescharacteristic
175Q8—88no—
176Q8—3762no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—48no—
183C2×C4—44no—
184C2×C4—1882no—
185C2×C4#1309400012yes—
186C2×C4#1309400011yescharacteristic
187C2×C4#1309401312yes—
188C2×C4#1309401311yescharacteristic
189C2×C4C2×#1062533511yescharacteristic
190C2×C4C2×#1062533711yescharacteristic
191C2×C4#1309405011yescharacteristic
192C2×C4#1309405511yescharacteristic
193C8—3762no—
194C8—3761no—
195C22—22no—
196C22#1568903411yescharacteristic
197C22#1568927411yescharacteristic
198C22#1568927911yescharacteristic
199C4—24no—
200C4—22no—
201C4—22no—
202C4—42no—
203C4—84no—
204C4—3762no—
205C4#1568940511yescharacteristic
206C4#1568941511yescharacteristic
207C4#1568947511yescharacteristic
208C4#1568948511yescharacteristic
209C2—21no—
210C2—21no—
211C2#2024452611yescharacteristic
212C2#2024455712yes—
subgroupG/H
centerC22#15689034
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2862C2
socleC22×C47#393
prime radicalC23×C47#120
hypercenter#2862D47
hypocenterC47#10363
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2862

Complements

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

NH ≅ G/Ntype
18. #13094024Q8semidirect
123. C47#10363semidirect

Maximal subgroups

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

subgroupindex
#202443652
#202419512
#202412462
#202412212
C47×#28622
#1036347