Finite Group Databasebeta

← #27157994

Subgroups — #27157994

Order 12416.

280 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 52 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
1D97⋊#100C211yesmaximal, characteristic, complemented
2D97⋊#99C211yesmaximal, characteristic, complemented
3#20261637C212yesmaximal, complemented
4#20261636C212yesmaximal, complemented
5#20261296C211yesmaximal, characteristic
6D97⋊M16—21no—
7C2×C22⋊#87604—21no—
8#15690870—21no—
9#15690870C2212yes—
10#15690869—21no—
11#15690868C2212yes—
12C22×D388C411yescharacteristic, complemented
13C2×D4×D97C2211yescharacteristic
14#15690795—21no—
15#15690793C2211yescharacteristic
16#15690785—21no—
17#15690783—21no—
18#15690780—21no—
19#15690780C2211yescharacteristic
20#15690778C411yescharacteristic, complemented
21C22⋊C4×D97C411yescharacteristic, complemented
22#123×C97C411yescharacteristic, complemented
23C22×#87604—41no—
24#13095521—41no—
25#13095520—22no—
26#13095519—41no—
27C22⋊#87604—22no—
28C22⋊#87604—41no—
29C23×D97C2×C411yescharacteristic
30C23×D97C2×C411yescharacteristic, complemented
31C23×D97D411yescharacteristic
32C2×D388—21no—
33C2×D388—21no—
34C2×D388—21no—
35C2×D388—41no—
36C2×D388—41no—
37C2×D388D411yescharacteristic
38C2×C22⋊D97—21no—
39C2×C22⋊D97D411yescharacteristic
40C2×C22⋊D97C2311yescharacteristic
41D4×D97—41no—
42D4×D97—41no—
43C2×C4×D97—21no—
44C2×C4×D97D411yescharacteristic
45#13095490—21no—
46C22⋊Dic97C2×C411yescharacteristic, complemented
47#13095488—21no—
48#13095488—21no—
49#13095488—21no—
50#13095488—21no—
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97C2×C411yescharacteristic, complemented
53C4⋊C4×C97—21no—
54C22⋊C4×C97—21no—
55C22⋊C4×C97C2×C411yescharacteristic, complemented
56C23×C97C22⋊C411yescharacteristic
57C23×C97C22×C411yescharacteristic
58C22×D97—21no—
59C22×D97—21no—
60C22×D97—21no—
61C22×D97—21no—
62C22×D97—21no—
63C22×D97—41no—
64C22×D97—81no—
65C22×D97—81no—
66C22×D97—81no—
67C22×D97C22⋊C411yescharacteristic
68C22×D97C22⋊C411yescharacteristic
69C22×D97C2×D411yescharacteristic
70C2×#87604—42no—
71C2×#87604—82no—
72D4×C97—41no—
73C2×C4×C97—21no—
74C2×C4×C97—21no—
75C2×C4×C97—41no—
76C2×C4×C97C22⋊C411yescharacteristic
77C22⋊D97—41no—
78C22⋊D97—41no—
79C22⋊D97—81no—
80C2×Dic97—21no—
81C2×Dic97C2×D411yescharacteristic
82D388—41no—
83D388—42no—
84D388—41no—
85D388—41no—
86D388—81no—
87C4×D97—21no—
88C4×D97—21no—
89C4×D97—81no—
90#11715897—42no—
91C22×C97—21no—
92C22×C97—21no—
93C22×C97—41no—
94C22×C97—41no—
95C22×C97C2×C22⋊C411yescharacteristic
96C2×D97—21no—
97C2×D97—21no—
98C2×D97—41no—
99C2×D97—42no—
100C2×D97—41no—
101C2×D97—41no—
102C2×D97—41no—
103C2×D97—41no—
104C2×D97—41no—
105C2×D97—81no—
106C2×D97—81no—
107C2×D97—81no—
108C2×D97—82no—
109C2×D97#9911yescharacteristic
110C2×D97#9911yescharacteristic
111#87604—82no—
112C4×C97—21no—
113C4×C97—41no—
114C4×C97—41no—
115Dic97—21no—
116Dic97—41no—
117C2×C97—21no—
118C2×C97—41no—
119C2×C97—41no—
120C2×C97C2×#9911yescharacteristic
121D97—21no—
122D97—21no—
123D97—41no—
124D97—41no—
125D97—81no—
126#2121—971nomaximal, Sylow
127C97#212111yescharacteristic, complemented, Sylow
128#534—971no—
129C2×#100—971no—
130C2×#99—971no—
131#352—972no—
132AΣL(1,𝔽16)—972no—
133C22×D4—971no—
134C22×D4—971no—
135C2×M16—1941no—
136#123—971no—
137#123—1941no—
138#123#8760411yescharacteristic, complemented
139#121—1941no—
140#121—1941no—
141#120—971no—
142#120—971no—
143#120—1941no—
144C2×C22⋊C4—971no—
145C2×C22⋊C4—1941no—
146#100—972no—
147#100—1941no—
148#99—972no—
149#99—1941no—
150C24—971no—
151C24—971no—
152C24—971no—
153C2×D4—971no—
154C2×D4—971no—
155C2×D4—971no—
156C2×D4—3881no—
157C2×D4—3881no—
158C2×D4—3881no—
159C2×D4—3881no—
160C2×D4—1941no—
161C2×D4—1941no—
162C2×D4—1941no—
163C2×D4—1941no—
164C2×D4C2×#8760411yescharacteristic
165C22×C4—971no—
166C22×C4—3881no—
167C22×C4—1941no—
168C22×C4C2×#8760411yescharacteristic, complemented
169M16—3881no—
170M16—1942no—
171C2×C8—3881no—
172C4⋊C4—21no—
173C4⋊C4—1941no—
174C22⋊C4—21no—
175C22⋊C4—971no—
176C22⋊C4—3881no—
177C22⋊C4—1941no—
178C22⋊C4—1941no—
179C22⋊C4—1941no—
180C22⋊C4—1942no—
181C22⋊C4—1941no—
182C22⋊C4C2×#8760411yescharacteristic, complemented
183C23—7761no—
184C23—7761no—
185C23—7761no—
186C23—971no—
187C23—971no—
188C23—971no—
189C23—3881no—
190C23—1941no—
191C23—1941no—
192C23—1941no—
193C23—1941no—
194C23—1941no—
195C23C22⋊#8760411yescharacteristic
196C23C22×#8760411yescharacteristic
197D4—41no—
198D4—7761no—
199D4—7761no—
200D4—3881no—
201D4—3881no—
202D4—3881no—
203D4—3882no—
204D4—3881no—
205D4—3881no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—7761no—
210C2×C4—7762no—
211C2×C4—971no—
212C2×C4—3882no—
213C2×C4—1941no—
214C2×C4—1941no—
215C2×C4—1941no—
216C2×C4C22⋊#8760411yescharacteristic
217C8—3882no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—7762no—
223C22—7761no—
224C22—7761no—
225C22—7761no—
226C22—971no—
227C22—971no—
228C22—3881no—
229C22—3881no—
230C22—3881no—
231C22—3881no—
232C22—3881no—
233C22—3882no—
234C22—3881no—
235C22—1941no—
236C22—1941no—
237C22C2×C22⋊#8760411yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4—7762no—
242C4—3881no—
243C4—1941no—
244C2—21no—
245C2—41no—
246C2—41no—
247C2—7761no—
248C2—3881no—
249C2—3881no—
250C2—1941no—
251C2—1941no—
252C2D97⋊#9911yescharacteristic
subgroupG/H
centerC2D97⋊#99
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
Fitting#123×C97C4
socleC2×C97C2×#99
prime radical#20261296C2
cosocleC2×C22⋊D97C23
hypercenter#123#87604
hypocenterC97#2121
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
1. D97⋊#100C2semidirect
2. D97⋊#99C2semidirect
3. #20261637C2semidirect
4. #20261636C2semidirect
12. C22×D388C4semidirect
20. #15690778C4semidirect
21. C22⋊C4×D97C4semidirect
22. #123×C97C4semidirect
30. C23×D97C2×C4semidirect
46. C22⋊Dic97C2×C4semidirect
52. C22×C4×C97C2×C4semidirect
55. C22⋊C4×C97C2×C4semidirect
127. C97#2121semidirect
138. #123#87604semidirect
168. C22×C4C2×#87604semidirect
182. C22⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
D97⋊#1002
D97⋊#992
#202616372
#202616362
#202612962
#212197