Finite Group Databasebeta

← #26104071

Subgroups — #26104071

Order 12032.

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

normal subgroups only

This group has 260 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×#15688922C211yesmaximal, characteristic
2#20244209C211yesmaximal, characteristic
3C2×C47×#356C211yesmaximal, characteristic
4C2×#13093948—21no—
5C2×#13093948—21no—
6C2×#13093943—21no—
7C2×#13093935C411yescharacteristic
8#15688989C411yescharacteristic
9#15688922—21no—
10#15688922C412yescomplemented
11C2×C4⋊Q8×C47C2211yescharacteristic
12C2×#101×C47—21no—
13C47×#356—21no—
14C47×#356C412yescomplemented
15C2×#10625329—41no—
16C22×C47⋊C8—21no—
17C2×C4⋊Dic47—21no—
18#13093948—42no—
19#13093948—42no—
20#13093943—22no—
21#13093943—41no—
22#13093935—41no—
23C47⋊#95—41no—
24C22×Q8×C47D411yescharacteristic
25C22×Q8×C47D411yescharacteristic
26C2×M16×C47—41no—
27C4⋊Q8×C47—41no—
28C4⋊Q8×C47C2×C411yescharacteristic
29C4⋊Q8×C47C2×C411yescharacteristic
30C4⋊Q8×C47D411yescharacteristic
31C4⋊Q8×C47Q811yescharacteristic
32C2×C4⋊C4×C47—41no—
33C2×C42×C47C2×C411yescharacteristic
34#101×C47—22no—
35#101×C47—41no—
36C2×Q8×C47—21no—
37C2×Q8×C47—21no—
38C2×Q8×C47—41no—
39C2×Q8×C47—41no—
40C2×Q8×C47—41no—
41C2×Q8×C47—41no—
42C2×Q8×C47C22⋊C411yescharacteristic
43C2×Q8×C47C22⋊C411yescharacteristic
44C2×Q8×C47C4⋊C411yescharacteristic
45C2×Q8×C47C4⋊C411yescharacteristic
46C22×C4×C47—21no—
47C22×C4×C47—21no—
48C22×C4×C47—21no—
49C22×C4×C47C22⋊C411yescharacteristic
50C22×Dic47—41no—
51M16×C47—42no—
52M16×C47—81no—
53C2×C8×C47—81no—
54C4⋊C4×C47—41no—
55C4⋊C4×C47—41no—
56C4⋊C4×C47—41no—
57C4⋊C4×C47—41no—
58C42×C47—21no—
59C42×C47C4211yescharacteristic
60C42×C47C4⋊C411yescharacteristic
61C4⋊Dic47—41no—
62C4⋊Dic47—41no—
63#10625329—42no—
64#10625329—81no—
65C2×C47⋊C8—22no—
66C2×C47⋊C8—41no—
67C2×C47⋊C8—41no—
68C2×C47⋊C8—81no—
69C23×C47#9911yescharacteristic
70Q8×C47—41no—
71Q8×C47—41no—
72Q8×C47—41no—
73Q8×C47—41no—
74Q8×C47—81no—
75Q8×C47—81no—
76C2×C4×C47—21no—
77C2×C4×C47—21no—
78C2×C4×C47—21no—
79C2×C4×C47—21no—
80C2×C4×C47—21no—
81C2×C4×C47—21no—
82C2×C4×C47—21no—
83C2×C4×C47—21no—
84C2×C4×C47—42no—
85C2×C4×C47—41no—
86C2×C4×C47—41no—
87C2×C4×C47—41no—
88C2×C4×C47—41no—
89C2×C4×C47#9511yescharacteristic
90C2×C4×C47#9911yescharacteristic
91C2×Dic47—82no—
92C2×Dic47—81no—
93C8×C47—82no—
94C47⋊C8—42no—
95C47⋊C8—82no—
96#7505—471nomaximal, Sylow
97C22×C47—22no—
98C22×C47#34211yescharacteristic
99C22×C47#35311yescharacteristic
100C22×C47#35411yescharacteristic
101C4×C47—21no—
102C4×C47—21no—
103C4×C47—41no—
104C4×C47—41no—
105C4×C47—41no—
106C4×C47—41no—
107C4×C47—41no—
108C4×C47—41no—
109Dic47—161no—
110C2×#356—471no—
111C2×#356D4711yescharacteristic
112#1960—471no—
113C2×C47—21no—
114C2×C47—21no—
115C2×C47#139311yescharacteristic
116C2×C47#141312yes—
117C2×C4⋊Q8C2×D4711yescharacteristic
118C2×#103—941no—
119C2×#103—941no—
120C2×#101—21no—
121C2×#101—941no—
122C2×#97—471no—
123#382—471no—
124#356—21no—
125#356—472no—
126#356—941no—
127#356Dic4712yescomplemented
128C47#750511yescharacteristic, complemented, Sylow
129C22×Q8D18811yescharacteristic
130C22×Q8C22⋊D4711yescharacteristic
131C2×M16—41no—
132C2×M16—1881no—
133C22×C8—941no—
134C4⋊Q8—41no—
135C4⋊Q8Dic9411yescharacteristic
136C4⋊Q8C4×D4711yescharacteristic
137C4⋊Q8C2×Dic4711yescharacteristic
138C4⋊Q8C22⋊D4711yescharacteristic
139C2×C4⋊C4—41no—
140C2×C4⋊C4—941no—
141C2×C42C4×D4711yescharacteristic
142#103—1882no—
143#103—1882no—
144#101—22no—
145#101—41no—
146#101—942no—
147#101—1881no—
148#97—1881no—
149#95—1881no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—41no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8Dic47⋊C411yescharacteristic
157C2×Q8C4⋊Dic4711yescharacteristic
158C2×Q8(C2×D47)⋊C411yescharacteristic
159C2×Q8C22⋊Dic4711yescharacteristic
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—1881no—
164C22×C4(C2×D47)⋊C411yescharacteristic
165M16—42no—
166M16—81no—
167M16—3761no—
168M16—1882no—
169C2×C8—81no—
170C2×C8—942no—
171C2×C8—3761no—
172C2×C8—1881no—
173C2×C8—1881no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—1881no—
179C4⋊C4—1881no—
180C42—21no—
181C42C4×Dic4711yescharacteristic
182C42Dic47⋊C411yescharacteristic
183C23#1309393811yescharacteristic
184Q8—41no—
185Q8—41no—
186Q8—41no—
187Q8—41no—
188Q8—81no—
189Q8—81no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—3762no—
204C2×C4—3761no—
205C2×C4C47⋊#9511yescharacteristic
206C2×C4#1309393811yescharacteristic
207C8—82no—
208C8—3762no—
209C8—1882no—
210C22—22no—
211C22#1568888911yescharacteristic
212C22#1568891511yescharacteristic
213C22#1568891711yescharacteristic
214C4—21no—
215C4—21no—
216C4—41no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—7521no—
223C2—21no—
224C2—21no—
225C2#2024151211yescharacteristic
226C2#2024156012yes—
subgroupG/H
centerC22#15688917
commutatorC42×C47C42
FrattiniC2×C4⋊Q8C2×D47
FittingC2×C47×#356C2
socleC22×C47#354
prime radicalC23×C47#99
cosocleC2×C4⋊Q8×C47C22
hypercenterC2×#356D47
hypocenterC47#7505
LCSG ▷ C42×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ C2×#356

Complements

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

NH ≅ G/Ntype
10. #15688922C4semidirect
14. C47×#356C4semidirect
127. #356Dic47semidirect
128. C47#7505semidirect

Maximal subgroups

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

subgroupindex
C2×#156889222
#202442092
C2×C47×#3562
#750547