Finite Group Databasebeta

← #26136629

Subgroups — #26136629

Order 12032.

388 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 79 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
2#20241559C214yesmaximal, complemented
3C47×#3138C211yesmaximal, characteristic
4C2×#13093939—22no—
5#15688916—22no—
6#15688916C2214yes—
7C47×#533C411yescharacteristic, complemented
8C2×#127×C47C2211yescharacteristic
9C47×#490C412yescomplemented
10C47×#490C2212yes—
11C47×#486—22no—
12C47×#418—22no—
13C4×M16×C47C411yescharacteristic, complemented
14C2×C22⋊Dic47—42no—
15#13093939—24no—
16#13093939—42no—
17C22×D4×C47—41no—
18C22×D4×C47D412yes—
19C2×M16×C47—21no—
20C4⋊Q8×C47—21no—
21#127×C47—41no—
22#127×C47C2×C411yescharacteristic
23#127×C47D412yes—
24#127×C47C2311yescharacteristic
25#125×C47C2×C412yescomplemented
26C22⋊Q8×C47—41no—
27C4×Q8×C47—41no—
28#117×C47—21no—
29C2×C42×C47C2×C411yescharacteristic
30#102×C47—24no—
31#102×C47—24no—
32#97×C47C2×C412yescomplemented
33C4×C8×C47—21no—
34C24×C47—22no—
35C2×Q8×C47—41no—
36C2×D4×C47—22no—
37C2×D4×C47—42no—
38C2×D4×C47—42no—
39C2×D4×C47—41no—
40C2×D4×C47—42no—
41C2×D4×C47—41no—
42C2×D4×C47—84no—
43C2×D4×C47C22⋊C412yes—
44C2×D4×C47C2×D412yes—
45C22×C4×C47—21no—
46C22×C4×C47C22⋊C411yescharacteristic
47C22×Dic47—82no—
48M16×C47—42no—
49C2×C8×C47—22no—
50C4⋊C4×C47—22no—
51C4⋊C4×C47—41no—
52C4⋊C4×C47—42no—
53C22⋊C4×C47—41no—
54C42×C47—21no—
55C42×C47—41no—
56C42×C47C22⋊C411yescharacteristic
57C42×C47C22×C411yescharacteristic
58C22⋊Dic47—44no—
59C22⋊Dic47—82no—
60C23×C47—22no—
61C23×C47—22no—
62C23×C47—42no—
63C23×C47—42no—
64C23×C47—84no—
65C23×C47#9911yescharacteristic
66Q8×C47—82no—
67D4×C47—42no—
68D4×C47—42no—
69D4×C47—82no—
70D4×C47—84no—
71D4×C47—82no—
72D4×C47—82no—
73C2×C4×C47—22no—
74C2×C4×C47—21no—
75C2×C4×C47—21no—
76C2×C4×C47—42no—
77C2×C4×C47—42no—
78C2×C4×C47#9911yescharacteristic
79C2×C4×C47C2×C22⋊C411yescharacteristic
80C2×Dic47—84no—
81C2×Dic47—164no—
82C8×C47—42no—
83#13595—471nomaximal, Sylow
84C22×C47—41no—
85C22×C47—42no—
86C22×C47—42no—
87C22×C47—82no—
88C22×C47—82no—
89C22×C47—84no—
90C22×C47—162no—
91C22×C47#35412yes—
92C22×C47C2×#9911yescharacteristic
93C4×C47—22no—
94C4×C47—21no—
95C4×C47—21no—
96C4×C47—41no—
97C4×C47—82no—
98Dic47—164no—
99#3138D4711yescharacteristic
100C2×#353—472no—
101#1412—474no—
102C2×C47—21no—
103C2×C47—22no—
104C2×C47—82no—
105C2×C47—82no—
106C2×C47C2×#35411yescharacteristic
107#533Dic4711yescharacteristic, complemented
108C2×#127C2×D4711yescharacteristic
109#490Dic4712yescomplemented
110#490C2×D4712yes—
111#486—22no—
112#418—22no—
113C2×#99—942no—
114C4×M16Dic4711yescharacteristic, complemented
115#353—474no—
116#353—942no—
117C47#1359511yescharacteristic, complemented, Sylow
118C22×D4—41no—
119C22×D4C22⋊D4712yes—
120C2×M16—21no—
121C4⋊Q8—21no—
122#127—41no—
123#127C2×Dic4711yescharacteristic
124#127C22⋊D4712yes—
125#127C22×D4711yescharacteristic
126#125C2×Dic4712yescomplemented
127C22⋊Q8—41no—
128C4×Q8—41no—
129#117—21no—
130C2×C22⋊C4—1882no—
131C2×C42C2×Dic4711yescharacteristic
132#102—24no—
133#102—24no—
134#99—944no—
135#99—1882no—
136#97C2×Dic4712yescomplemented
137C4×C8—21no—
138C24—22no—
139C2×Q8—41no—
140C2×D4—22no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—41no—
144C2×D4—42no—
145C2×D4—41no—
146C2×D4—84no—
147C2×D4C22⋊Dic4712yes—
148C2×D4C2×C22⋊D4712yes—
149C22×C4—21no—
150C22×C4—3762no—
151C22×C4C22⋊Dic4711yescharacteristic
152M16—42no—
153C2×C8—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C22⋊C4—41no—
158C22⋊C4—3762no—
159C22⋊C4—1884no—
160C42—21no—
161C42—41no—
162C42C22⋊Dic4711yescharacteristic
163C42C22×Dic4711yescharacteristic
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23—84no—
169C23#1309393911yescharacteristic
170Q8—82no—
171D4—42no—
172D4—42no—
173D4—82no—
174D4—84no—
175D4—82no—
176D4—82no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—3764no—
183C2×C4—7524no—
184C2×C4#1309393911yescharacteristic
185C2×C4C2×C22⋊Dic4711yescharacteristic
186C8—42no—
187C22—41no—
188C22—42no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—84no—
193C22—162no—
194C22#1568891812yes—
195C22C2×#1309393911yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—82no—
201C4—7524no—
202C2—21no—
203C2—22no—
204C2—82no—
205C2—82no—
206C2C2×#1568891811yescharacteristic
subgroupG/H
centerC2C2×#15688918
commutatorC42×C47C22×C4
Frattini#127C22×D47
FittingC47×#3138C2
socleC2×C47C2×#354
prime radicalC2×#127×C47C22
cosocle#127×C47C23
hypercenter#3138D47
hypocenterC47#13595
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ #3138

Complements

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

NH ≅ G/Ntype
2. #20241559C2semidirect
7. C47×#533C4semidirect
9. C47×#490C4semidirect
13. C4×M16×C47C4semidirect
25. #125×C47C2×C4semidirect
32. #97×C47C2×C4semidirect
107. #533Dic47semidirect
109. #490Dic47semidirect
114. C4×M16Dic47semidirect
117. C47#13595semidirect
126. #125C2×Dic47semidirect
136. #97C2×Dic47semidirect

Maximal subgroups

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

subgroupindex
C2×#156889162
#202415592
C47×#31382
#1359547