Finite Group Databasebeta

← #26104070

Subgroups — #26104070

Order 12032.

336 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C2×#15688916C212yesmaximal
2C47×#1961C211yesmaximal, characteristic
3C2×#13093939—22no—
4#15688916—22no—
5#15688916C414yescomplemented
6C2×#127×C47C2211yescharacteristic
7C2×#102×C47—22no—
8C2×#97×C47C411yescharacteristic, complemented
9C47×#382C411yescharacteristic, complemented
10C2×C22⋊Dic47—42no—
11#13093939—24no—
12#13093939—42no—
13C22×D4×C47—41no—
14C22×D4×C47D412yes—
15C22×C8×C47—21no—
16#127×C47—41no—
17#127×C47C2×C412yes—
18#127×C47D411yescharacteristic
19#127×C47Q811yescharacteristic
20C2×C4⋊C4×C47—21no—
21C2×C42×C47C2×C411yescharacteristic
22#102×C47—44no—
23#97×C47—41no—
24#95×C47—41no—
25C24×C47—22no—
26C2×D4×C47—22no—
27C2×D4×C47—42no—
28C2×D4×C47—42no—
29C2×D4×C47—42no—
30C2×D4×C47—41no—
31C2×D4×C47—41no—
32C2×D4×C47—84no—
33C2×D4×C47C22⋊C412yes—
34C2×D4×C47C4⋊C412yes—
35C22×C4×C47—21no—
36C22×C4×C47—41no—
37C22×C4×C47C22⋊C411yescharacteristic
38C22×Dic47—82no—
39C2×C8×C47—22no—
40C2×C8×C47—41no—
41C2×C8×C47—41no—
42C4⋊C4×C47—42no—
43C42×C47—21no—
44C42×C47C4211yescharacteristic
45C42×C47C4⋊C411yescharacteristic
46C22⋊Dic47—44no—
47C22⋊Dic47—82no—
48C23×C47—22no—
49C23×C47—22no—
50C23×C47—42no—
51C23×C47—42no—
52C23×C47—84no—
53C23×C47#9911yescharacteristic
54D4×C47—42no—
55D4×C47—42no—
56D4×C47—82no—
57D4×C47—84no—
58D4×C47—82no—
59D4×C47—82no—
60C2×C4×C47—21no—
61C2×C4×C47—21no—
62C2×C4×C47—21no—
63C2×C4×C47—21no—
64C2×C4×C47—42no—
65C2×C4×C47—82no—
66C2×C4×C47—81no—
67C2×C4×C47#9511yescharacteristic
68C2×C4×C47#9911yescharacteristic
69C2×Dic47—84no—
70C2×Dic47—164no—
71C8×C47—42no—
72#7504—471nomaximal, Sylow
73C22×C47—22no—
74C22×C47—42no—
75C22×C47—42no—
76C22×C47—82no—
77C22×C47—82no—
78C22×C47—84no—
79C22×C47—162no—
80C22×C47#34211yescharacteristic
81C22×C47#35311yescharacteristic
82C22×C47#35411yescharacteristic
83C4×C47—21no—
84C4×C47—21no—
85C4×C47—41no—
86C4×C47—41no—
87C4×C47—161no—
88Dic47—164no—
89C2×#353—472no—
90#1961D4711yescharacteristic
91C2×C47—21no—
92C2×C47—21no—
93C2×C47—82no—
94C2×C47—82no—
95C2×C47#139311yescharacteristic
96C2×C47#141212yes—
97C2×#127C2×D4711yescharacteristic
98C2×#102—22no—
99C2×#99—942no—
100C2×#97Dic4711yescharacteristic, complemented
101#382Dic4711yescharacteristic, complemented
102#353—474no—
103#353—942no—
104C47#750411yescharacteristic, complemented, Sylow
105C22×D4—41no—
106C22×D4C22⋊D4712yes—
107C22×C8—21no—
108#127—41no—
109#127Dic9411yescharacteristic
110#127C4×D4712yes—
111#127D18811yescharacteristic
112C2×C4⋊C4—21no—
113C2×C22⋊C4—1882no—
114C2×C42C2×Dic4711yescharacteristic
115#102—44no—
116#99—944no—
117#99—1882no—
118#97—41no—
119#95—41no—
120C24—22no—
121C2×D4—22no—
122C2×D4—42no—
123C2×D4—42no—
124C2×D4—41no—
125C2×D4—42no—
126C2×D4—41no—
127C2×D4—84no—
128C2×D4Dic47⋊C412yes—
129C2×D4(C2×D47)⋊C412yes—
130C22×C4—21no—
131C22×C4—41no—
132C22×C4—3762no—
133C22×C4C22⋊Dic4711yescharacteristic
134C2×C8—22no—
135C2×C8—41no—
136C2×C8—41no—
137C4⋊C4—42no—
138C22⋊C4—3762no—
139C22⋊C4—1884no—
140C42—21no—
141C42C4×Dic4711yescharacteristic
142C42C4⋊Dic4711yescharacteristic
143C23—22no—
144C23—22no—
145C23—42no—
146C23—42no—
147C23—84no—
148C23#1309393911yescharacteristic
149D4—42no—
150D4—42no—
151D4—82no—
152D4—82no—
153D4—84no—
154D4—82no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—42no—
160C2×C4—82no—
161C2×C4—81no—
162C2×C4—3764no—
163C2×C4—7524no—
164C2×C4C47⋊#9511yescharacteristic
165C2×C4#1309393911yescharacteristic
166C8—42no—
167C22—22no—
168C22—42no—
169C22—42no—
170C22—82no—
171C22—82no—
172C22—84no—
173C22—162no—
174C22#1568889011yescharacteristic
175C22#1568891611yescharacteristic
176C22#1568891811yescharacteristic
177C4—21no—
178C4—21no—
179C4—41no—
180C4—41no—
181C4—161no—
182C4—7524no—
183C2—21no—
184C2—21no—
185C2—82no—
186C2—82no—
187C2#2024151311yescharacteristic
188C2#2024155912yes—
subgroupG/H
centerC22#15688918
commutatorC42×C47C42
FrattiniC2×#127C2×D47
FittingC47×#1961C2
socleC22×C47#354
prime radical, cosocleC2×#127×C47C22
hypercenter#1961D47
hypocenterC47#7504
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ #1961

Complements

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

NH ≅ G/Ntype
5. #15688916C4semidirect
8. C2×#97×C47C4semidirect
9. C47×#382C4semidirect
100. C2×#97Dic47semidirect
101. #382Dic47semidirect
104. C47#7504semidirect

Maximal subgroups

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

subgroupindex
C2×#156889162
C47×#19612
#750447