Finite Group Databasebeta

← #27142151

Subgroups — #27142151

Order 12416.

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

normal subgroups only

This group has 342 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
1#20261501C211yesmaximal, characteristic, complemented
2#20261246C211yesmaximal, characteristic
3#20260717C211yesmaximal, characteristic, complemented
4#20260699C211yesmaximal, characteristic, complemented
5#20260507C211yesmaximal, characteristic, complemented
6#20260506C211yesmaximal, characteristic, complemented
7#407×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690818—22no—
10#15690816—22no—
11C2×#13095495C2211yescharacteristic
12C2×#13095495C2211yescharacteristic
13M16×D97—22no—
14#15690793C412yes—
15#15690789—21no—
16#15690784—21no—
17#15690779C412yes—
18C4×C22⋊D97—22no—
19#15690769C2211yescharacteristic
20#15690728—21no—
21#15690726—21no—
22C22⋊#11715895—22no—
23C23×C4×C97C2211yescharacteristic
24C2×M16×C97C2211yescharacteristic
25C2×M16×C97C2211yescharacteristic
26C22⋊C8×C97—22no—
27#13095511—21no—
28#13095511—21no—
29#13095511—21no—
30#13095511—21no—
31C2×Dic194D411yescharacteristic
32C2×D388D411yescharacteristic
33C2×C22⋊D97—22no—
34C2×C22⋊D97C2×C412yes—
35C2×C4×D97D412yes—
36#13095495—21no—
37#13095495—21no—
38#13095495—21no—
39#13095495—21no—
40#13095494—24no—
41C2×#11715895—21no—
42C2×#11715895D412yescomplemented
43C8×D97—24no—
44C4⋊Dic97—21no—
45#13095490—22no—
46C22⋊Dic97—22no—
47C22⋊Dic97C2×C412yes—
48#13095488—22no—
49C4×Dic97—21no—
50C24×C97C2×C411yescharacteristic
51C22×C4×C97—21no—
52C22×C4×C97—21no—
53C22×C4×C97—21no—
54C22×C4×C97—21no—
55C22×C4×C97—21no—
56C22×C4×C97—21no—
57C22×C4×C97C2×C411yescharacteristic
58C22×C4×C97C2311yescharacteristic
59M16×C97—21no—
60M16×C97—21no—
61M16×C97—21no—
62M16×C97—21no—
63C2×C8×C97—21no—
64C2×C8×C97D412yescomplemented
65C23×C97—21no—
66C23×C97—21no—
67C23×C97—21no—
68C23×C97—42no—
69C23×C97C22×C411yescharacteristic
70C22×D97C22⋊C412yes—
71C2×C4×C97—21no—
72C2×C4×C97—21no—
73C2×C4×C97—21no—
74C2×C4×C97—41no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97#4111yescharacteristic
83C2×C4×C97#4111yescharacteristic
84C22⋊D97—22no—
85C22⋊D97—22no—
86C22⋊D97—44no—
87C2×Dic97—22no—
88C2×Dic97C22⋊C412yes—
89D388—21no—
90D388—21no—
91C4×D97—22no—
92C4×D97—22no—
93Dic194—21no—
94Dic194—21no—
95C8×C97—22no—
96C8×C97—41no—
97#11715895—22no—
98#11715895—41no—
99C22×C97—22no—
100C22×C97—22no—
101C22×C97—22no—
102C22×C97—41no—
103C22×C97—41no—
104C22×C97—41no—
105C22×C97—41no—
106C22×C97—41no—
107C22×C97C2×C22⋊C411yescharacteristic
108C22×C97C4×D411yescharacteristic
109C22×C97C4×D411yescharacteristic
110C2×D97—22no—
111C2×D97—42no—
112C4×C97—41no—
113C4×C97#12011yescharacteristic
114C4×C97#12111yescharacteristic
115C4×C97#12111yescharacteristic
116C4×C97#12311yescharacteristic
117Dic97—22no—
118Dic97—42no—
119C2×C97—21no—
120C2×C97—21no—
121C2×C97—21no—
122C2×C97—42no—
123C2×C97#38611yescharacteristic
124D97—42no—
125#1891—971nomaximal, Sylow
126C97#189111yescharacteristic, complemented, Sylow
127#568—971no—
128#525—971no—
129#413—971no—
130#407—971no—
131#407D9711yescharacteristic, complemented
132#344—971no—
133#344—971no—
134C2×#41—971no—
135C23×C4C2×D9711yescharacteristic
136#131—1942no—
137#131—1942no—
138C2×M16—971no—
139C2×M16—971no—
140C2×M16—1942no—
141C2×M16C2×D9711yescharacteristic
142C2×M16C2×D9711yescharacteristic
143#123—972no—
144C22⋊Q8—1941no—
145#121—1941no—
146#120—972no—
147C4×D4—1942no—
148#117—971no—
149#101—1941no—
150#100—1941no—
151C22⋊C8—22no—
152C22⋊C8—1942no—
153C24C4×D9711yescharacteristic
154#41—1941no—
155#41—1941no—
156#41—1941no—
157#41—1941no—
158C2×Q8—971no—
159C2×D4—972no—
160C2×D4—971no—
161C2×D4—1942no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—972no—
169C22×C4C4×D9711yescharacteristic
170C22×C4C22×D9711yescharacteristic
171M16—21no—
172M16—21no—
173M16—21no—
174M16—21no—
175M16—1944no—
176M16—1941no—
177M16—1941no—
178M16—1941no—
179M16—1941no—
180C2×C8—21no—
181C2×C8—972no—
182C2×C8—1944no—
183C2×C8—1941no—
184C2×C8C22⋊D9712yescomplemented
185C4⋊C4—1942no—
186C4⋊C4—1941no—
187C22⋊C4—972no—
188C22⋊C4—1942no—
189C22⋊C4—1942no—
190C42—1941no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23—972no—
196C23C2×C4×D9711yescharacteristic
197Q8—1941no—
198Q8—1941no—
199D4—3884no—
200D4—1942no—
201D4—1942no—
202D4—1941no—
203D4—1941no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—972no—
212C2×C4—1942no—
213C2×C4—1942no—
214C2×C4—1942no—
215C2×C4D4×D9711yescharacteristic
216C2×C4D4×D9711yescharacteristic
217C2×C4D4×D9711yescharacteristic
218C2×C4C2×C22⋊D9711yescharacteristic
219C2×C4#1309551011yescharacteristic
220C2×C4#1309551111yescharacteristic
221C8—22no—
222C8—41no—
223C8—3881no—
224C8—1942no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—3882no—
234C22—1942no—
235C22C22⋊C4×D9711yescharacteristic
236C22#1569077011yescharacteristic
237C22C4×C22⋊D9711yescharacteristic
238C4—41no—
239C4—3882no—
240C4—1942no—
241C4#1569077811yescharacteristic
242C4#1569078111yescharacteristic
243C4#1569078311yescharacteristic
244C4#1569079211yescharacteristic
245C2—21no—
246C2—21no—
247C2—21no—
248C2—42no—
249C2—3882no—
250C2#2026061111yescharacteristic
subgroupG/H
centerC4#15690778
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#407×C97C2
socleC2×C97#386
prime radical#20261246C2
cosocleC22×C4×C97C23
hypercenter#407D97
hypocenterC97#1891
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
1. #20261501C2semidirect
3. #20260717C2semidirect
4. #20260699C2semidirect
5. #20260507C2semidirect
6. #20260506C2semidirect
7. #407×C97C2semidirect
42. C2×#11715895D4semidirect
64. C2×C8×C97D4semidirect
126. C97#1891semidirect
131. #407D97semidirect
184. C2×C8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202615012
#202612462
#202607172
#202606992
#202605072
#202605062
#407×C972
#189197