Finite Group Databasebeta

← #352×D97 #27143370

Subgroups — #352×D97 #27143370

Order 12416.

278 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 57 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#123×D97C211yesmaximal, characteristic
2#100×D97C211yesmaximal, characteristic
3#99×D97C211yesmaximal, characteristic
4#20260526C211yesmaximal, characteristic, complemented
5#20260525C211yesmaximal, characteristic, complemented
6#20260524C211yesmaximal, characteristic, complemented
7#352×C97C211yesmaximal, characteristic, complemented
8C2×D4×D97C2211yescharacteristic
9C22×C4×D97C411yescharacteristic, complemented
10M16×D97—21no—
11#15690795—21no—
12#15690794C411yescharacteristic, complemented
13#15690793C411yescharacteristic, complemented
14#15690792—21no—
15#15690791C2211yescharacteristic
16C4⋊C4×D97—21no—
17C22⋊C4×D97—21no—
18C22⋊C4×D97—21no—
19C22⋊C4×D97C411yescharacteristic, complemented
20#15690727C2211yescharacteristic
21#15690726—21no—
22#15690725C2211yescharacteristic
23#15690724—21no—
24#123×C97C2211yescharacteristic
25#100×C97C2211yescharacteristic
26#99×C97C2211yescharacteristic
27C23×D97C2×C411yescharacteristic
28C23×D97D411yescharacteristic
29C2×D388D411yescharacteristic
30C2×C22⋊D97C2×C411yescharacteristic
31C2×C22⋊D97D411yescharacteristic
32D4×D97—41no—
33D4×D97—41no—
34C22×Dic97C2×C411yescharacteristic, complemented
35C2×C4×D97—21no—
36C2×C4×D97—21no—
37C2×C4×D97—21no—
38C2×C4×D97—41no—
39C2×C4×D97—41no—
40C2×C4×D97—41no—
41C2×C4×D97D411yescharacteristic
42#13095495—21no—
43#13095494—41no—
44C8×D97—41no—
45C4⋊Dic97—21no—
46#13095490—21no—
47#13095490—21no—
48C22⋊Dic97—21no—
49C22⋊Dic97—21no—
50C22⋊Dic97C2×C411yescharacteristic, complemented
51#13095488—21no—
52#13095488—21no—
53#13095488—21no—
54#13095488—41no—
55C2×D4×C97C2311yescharacteristic
56C22×C4×C97C2×C411yescharacteristic, complemented
57M16×C97—21no—
58C4⋊C4×C97—21no—
59C22⋊C4×C97—21no—
60C22⋊C4×C97—21no—
61C22⋊C4×C97C2×C411yescharacteristic, complemented
62C23×C97C22×C411yescharacteristic
63C23×C97C2×D411yescharacteristic
64C22×D97—21no—
65C22×D97—21no—
66C22×D97—21no—
67C22×D97—21no—
68C22×D97—42no—
69C22×D97—42no—
70C22×D97C22⋊C411yescharacteristic
71C22×D97C22⋊C411yescharacteristic
72C22×D97C22⋊C411yescharacteristic
73D4×C97—41no—
74C2×C4×C97—21no—
75C2×C4×C97—21no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97C2×D411yescharacteristic
79C22⋊D97—41no—
80C22⋊D97—41no—
81C2×Dic97—21no—
82C2×Dic97—21no—
83C2×Dic97—41no—
84C2×Dic97—41no—
85C2×Dic97C22⋊C411yescharacteristic
86D388—41no—
87C4×D97—21no—
88C4×D97—21no—
89C4×D97—42no—
90C4×D97—42no—
91C4×D97—41no—
92C4×D97—41no—
93C4×D97—82no—
94C8×C97—41no—
95#11715895—41no—
96C22×C97—21no—
97C22×C97—21no—
98C22×C97—41no—
99C22×C97—41no—
100C22×C97C2×C22⋊C411yescharacteristic
101C2×D97—22no—
102C2×D97—21no—
103C2×D97—21no—
104C2×D97—42no—
105C2×D97—42no—
106C2×D97—42no—
107C2×D97—41no—
108C2×D97—41no—
109C2×D97—81no—
110C2×D97#9911yescharacteristic
111C2×D97#9911yescharacteristic
112C4×C97—21no—
113C4×C97—41no—
114C4×C97—41no—
115C4×C97—81no—
116Dic97—21no—
117Dic97—41no—
118Dic97—41no—
119Dic97—81no—
120C2×C97—21no—
121C2×C97—41no—
122C2×C97—41no—
123C2×C97C2×#9911yescharacteristic
124D97—21no—
125D97—41no—
126D97—41no—
127D97#35212yesdirect product factor
128C2×#352—971nomaximal, Sylow
129C97C2×#35211yescharacteristic, complemented, Sylow
130C2×#123—971no—
131C2×#100—971no—
132C2×#99—971no—
133#352—971no—
134#352—971no—
135#352—971no—
136#352D9711yescharacteristic, direct product factor
137C22×D4—971no—
138C23×C4—971no—
139C2×M16—1941no—
140#123—971no—
141#123—971no—
142#123—971no—
143#123—1941no—
144#123—1941no—
145#123C2×D9711yescharacteristic
146C2×C4⋊C4—1941no—
147C2×C22⋊C4—971no—
148C2×C22⋊C4—1941no—
149C2×C22⋊C4—1941no—
150#100—971no—
151#100—1941no—
152#100C2×D9711yescharacteristic
153#99—971no—
154#99—1941no—
155#99C2×D9711yescharacteristic
156C24—971no—
157C24—971no—
158C2×D4—971no—
159C2×D4—971no—
160C2×D4—971no—
161C2×D4—3881no—
162C2×D4—3881no—
163C2×D4C22×D9711yescharacteristic
164C22×C4—971no—
165C22×C4—971no—
166C22×C4—3881no—
167C22×C4—3881no—
168C22×C4—3881no—
169C22×C4—1941no—
170C22×C4—1941no—
171C22×C4—1941no—
172C22×C4C4×D9711yescharacteristic, complemented
173M16—21no—
174M16—3881no—
175M16—1941no—
176C2×C8—3881no—
177C4⋊C4—21no—
178C4⋊C4—1941no—
179C4⋊C4—1941no—
180C4⋊C4—1941no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—971no—
184C22⋊C4—3881no—
185C22⋊C4—1941no—
186C22⋊C4—1941no—
187C22⋊C4—1941no—
188C22⋊C4—1941no—
189C22⋊C4—1941no—
190C22⋊C4C4×D9711yescharacteristic, complemented
191C23—971no—
192C23—971no—
193C23—971no—
194C23—3882no—
195C23—3882no—
196C23—1941no—
197C23—1941no—
198C23—1941no—
199C23—1941no—
200C23C2×C4×D9711yescharacteristic
201C23D4×D9711yescharacteristic
202D4—41no—
203D4—3881no—
204D4—3881no—
205D4—3881no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—7762no—
211C2×C4—971no—
212C2×C4—3881no—
213C2×C4—3881no—
214C2×C4—3882no—
215C2×C4—3882no—
216C2×C4—3881no—
217C2×C4—3881no—
218C2×C4—1941no—
219C2×C4—1941no—
220C2×C4—1941no—
221C2×C4—1941no—
222C2×C4D4×D9711yescharacteristic
223C8—41no—
224C8—3881no—
225C22—21no—
226C22—21no—
227C22—41no—
228C22—41no—
229C22—7761no—
230C22—971no—
231C22—971no—
232C22—3881no—
233C22—3881no—
234C22—3882no—
235C22—3882no—
236C22—3882no—
237C22—1942no—
238C22—1941no—
239C22—1941no—
240C22C22⋊C4×D9711yescharacteristic
241C4—21no—
242C4—41no—
243C4—41no—
244C4—81no—
245C4—7761no—
246C4—3881no—
247C4—3881no—
248C4—1941no—
249C2—21no—
250C2—41no—
251C2—41no—
252C2—972no—
253C2—3881no—
254C2—3881no—
255C2—1941no—
256C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC23×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#352×C97C2
socleC2×C97C2×#99
prime radicalC2×D4×D97C22
cosocleC2×D4×C97C23
hypercenter#352D97
hypocenterC97C2×#352
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #20260526C2semidirect
5. #20260525C2semidirect
6. #20260524C2semidirect
7. #352×C97C2semidirect
9. C22×C4×D97C4semidirect
12. #15690794C4semidirect
13. #15690793C4semidirect
19. C22⋊C4×D97C4semidirect
34. C22×Dic97C2×C4semidirect
50. C22⋊Dic97C2×C4semidirect
56. C22×C4×C97C2×C4semidirect
61. C22⋊C4×C97C2×C4semidirect
127. D97#352direct (N × H)
129. C97C2×#352semidirect
136. #352D97direct (N × H)
172. C22×C4C4×D97semidirect
190. C22⋊C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
#123×D972
#100×D972
#99×D972
#202605262
#202605252
#202605242
#352×C972
C2×#35297