Finite Group Databasebeta

← #46812530

Subgroups — #46812530

Order 34432.

280 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 55 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#31771166C211yesmaximal, characteristic
2#100×D269C211yesmaximal, characteristic, complemented
3#99×D269C211yesmaximal, characteristic, complemented
4#31770395C211yesmaximal, characteristic, complemented
5#31770393C211yesmaximal, characteristic, complemented
6#31770391C211yesmaximal, characteristic, complemented
7#352×C269C211yesmaximal, characteristic, complemented
8C22×D1076C411yescharacteristic, complemented
9C2×D4×D269C2211yescharacteristic
10M16×D269—21no—
11#22575438—21no—
12#22575436C411yescharacteristic, complemented
13#22575428—21no—
14#22575426—21no—
15#22575423—21no—
16#22575423C411yescharacteristic, complemented
17#22575421C2211yescharacteristic
18C22⋊C4×D269—21no—
19C22⋊C4×D269C411yescharacteristic, complemented
20#22575370C2211yescharacteristic
21#22575369—21no—
22#22575368C2211yescharacteristic
23#22575367—21no—
24#123×C269C2211yescharacteristic
25#100×C269C2211yescharacteristic
26#99×C269C2211yescharacteristic
27C23×D269C2×C411yescharacteristic
28C23×D269C2×C411yescharacteristic, complemented
29C23×D269D411yescharacteristic
30C2×D1076—21no—
31C2×D1076—21no—
32C2×D1076—21no—
33C2×D1076—41no—
34C2×D1076—41no—
35C2×D1076D411yescharacteristic
36C2×C22⋊D269—21no—
37C2×C22⋊D269C2×C411yescharacteristic
38C2×C22⋊D269D411yescharacteristic
39D4×D269—41no—
40D4×D269—41no—
41C2×C4×D269—21no—
42C2×C4×D269—41no—
43C2×C4×D269D411yescharacteristic
44#16864386—21no—
45#16864385—41no—
46C8×D269—41no—
47#16864381—21no—
48C22⋊Dic269—21no—
49C22⋊Dic269C2×C411yescharacteristic, complemented
50#16864379—21no—
51#16864379—21no—
52#16864379—21no—
53#16864379—21no—
54#16864379—41no—
55C2×D4×C269C2311yescharacteristic
56C22×C4×C269C2×C411yescharacteristic, complemented
57M16×C269—21no—
58C4⋊C4×C269—21no—
59C22⋊C4×C269—21no—
60C22⋊C4×C269—21no—
61C22⋊C4×C269C2×C411yescharacteristic, complemented
62C22×D269—21no—
63C22×D269—21no—
64C22×D269—21no—
65C22×D269—21no—
66C22×D269—21no—
67C22×D269—41no—
68C22×D269—81no—
69C22×D269—81no—
70C22×D269—81no—
71C22×D269C22⋊C411yescharacteristic
72C22×D269C22⋊C411yescharacteristic
73C22×D269C22⋊C411yescharacteristic
74D1076—41no—
75D1076—42no—
76D1076—41no—
77D1076—41no—
78D1076—81no—
79C22⋊D269—41no—
80C22⋊D269—41no—
81C22⋊D269—81no—
82C2×Dic269—21no—
83C2×Dic269—41no—
84C2×Dic269C22⋊C411yescharacteristic
85C4×D269—21no—
86C4×D269—21no—
87C4×D269—81no—
88C4×D269—82no—
89#14524130—41no—
90C23×C269C22×C411yescharacteristic
91C23×C269C2×D411yescharacteristic
92D4×C269—41no—
93C2×C4×C269—21no—
94C2×C4×C269—21no—
95C2×C4×C269—41no—
96C2×C4×C269—41no—
97C2×C4×C269C2×D411yescharacteristic
98C8×C269—41no—
99C22×C269—21no—
100C22×C269—21no—
101C22×C269—41no—
102C22×C269—41no—
103C22×C269C2×C22⋊C411yescharacteristic
104C2×D269—21no—
105C2×D269—21no—
106C2×D269—41no—
107C2×D269—42no—
108C2×D269—41no—
109C2×D269—41no—
110C2×D269—41no—
111C2×D269—41no—
112C2×D269—41no—
113C2×D269—81no—
114C2×D269—81no—
115C2×D269—81no—
116C2×D269—82no—
117C2×D269#9911yescharacteristic
118C2×D269#9911yescharacteristic
119C4×C269—21no—
120C4×C269—41no—
121C4×C269—41no—
122C4×C269—81no—
123Dic269—21no—
124Dic269—41no—
125Dic269—81no—
126C2×C269—21no—
127C2×C269—41no—
128C2×C269—41no—
129C2×C269C2×#9911yescharacteristic
130D269—21no—
131D269—21no—
132D269—41no—
133D269—41no—
134D269—81no—
135C269#212111yescharacteristic, complemented, Sylow
136#2121—2691nomaximal, Sylow
137#534—2691no—
138C2×#100—2691no—
139C2×#99—2691no—
140#352—2691no—
141#352D26911yescharacteristic, complemented
142AΣL(1,𝔽16)—2691no—
143AΣL(1,𝔽16)—2691no—
144C22×D4—2691no—
145C22×D4—2691no—
146C2×M16—5381no—
147#123—2691no—
148#123—5381no—
149#123C2×D26911yescharacteristic
150#121—5381no—
151#121—5381no—
152#120—2691no—
153#120—2691no—
154#120—5381no—
155C2×C22⋊C4—2691no—
156C2×C22⋊C4—5381no—
157#100—2691no—
158#100—5381no—
159#100C2×D26911yescharacteristic
160#99—2691no—
161#99—5381no—
162#99C2×D26911yescharacteristic
163C24—2691no—
164C24—2691no—
165C24—2691no—
166C2×D4—2691no—
167C2×D4—2691no—
168C2×D4—2691no—
169C2×D4—5381no—
170C2×D4—5381no—
171C2×D4—5381no—
172C2×D4—5381no—
173C2×D4—10761no—
174C2×D4—10761no—
175C2×D4—10761no—
176C2×D4—10761no—
177C2×D4C22×D26911yescharacteristic
178C22×C4—2691no—
179C22×C4—5381no—
180C22×C4—10761no—
181C22×C4C4×D26911yescharacteristic, complemented
182M16—21no—
183M16—5381no—
184M16—10761no—
185C2×C8—10761no—
186C4⋊C4—21no—
187C4⋊C4—5381no—
188C22⋊C4—21no—
189C22⋊C4—21no—
190C22⋊C4—2691no—
191C22⋊C4—5381no—
192C22⋊C4—5381no—
193C22⋊C4—5381no—
194C22⋊C4—5381no—
195C22⋊C4—5381no—
196C22⋊C4—10761no—
197C22⋊C4C4×D26911yescharacteristic, complemented
198C23—2691no—
199C23—2691no—
200C23—2691no—
201C23—5381no—
202C23—5381no—
203C23—5381no—
204C23—5381no—
205C23—5381no—
206C23—10761no—
207C23—21521no—
208C23—21521no—
209C23—21521no—
210C23C2×C4×D26911yescharacteristic
211C23D4×D26911yescharacteristic
212D4—41no—
213D4—10761no—
214D4—10761no—
215D4—10761no—
216D4—10762no—
217D4—10761no—
218D4—10761no—
219D4—21521no—
220D4—21521no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—2691no—
226C2×C4—5381no—
227C2×C4—5381no—
228C2×C4—5381no—
229C2×C4—10761no—
230C2×C4—21521no—
231C2×C4—21522no—
232C2×C4D4×D26911yescharacteristic
233C8—41no—
234C8—10761no—
235C22—21no—
236C22—21no—
237C22—41no—
238C22—41no—
239C22—2691no—
240C22—2691no—
241C22—5381no—
242C22—5381no—
243C22—10761no—
244C22—10761no—
245C22—10761no—
246C22—10761no—
247C22—10761no—
248C22—10762no—
249C22—10761no—
250C22—21521no—
251C22—21521no—
252C22—21521no—
253C22—21522no—
254C22C22⋊C4×D26911yescharacteristic
255C4—21no—
256C4—41no—
257C4—41no—
258C4—81no—
259C4—5381no—
260C4—10761no—
261C4—21521no—
262C2—21no—
263C2—41no—
264C2—41no—
265C2—5381no—
266C2—5381no—
267C2—10761no—
268C2—10761no—
269C2—21521no—
270C2#99×D26911yescharacteristic
subgroupG/H
centerC2#99×D269
commutatorC23×C269C22×C4
FrattiniC2×D4C22×D269
Fitting#352×C269C2
socleC2×C269C2×#99
prime radical#31771166C2
cosocleC2×D4×C269C23
hypercenter#352D269
hypocenterC269#2121
LCSG ▷ C23×C269 ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
2. #100×D269C2semidirect
3. #99×D269C2semidirect
4. #31770395C2semidirect
5. #31770393C2semidirect
6. #31770391C2semidirect
7. #352×C269C2semidirect
8. C22×D1076C4semidirect
12. #22575436C4semidirect
16. #22575423C4semidirect
19. C22⋊C4×D269C4semidirect
28. C23×D269C2×C4semidirect
49. C22⋊Dic269C2×C4semidirect
56. C22×C4×C269C2×C4semidirect
61. C22⋊C4×C269C2×C4semidirect
135. C269#2121semidirect
141. #352D269semidirect
181. C22×C4C4×D269semidirect
197. C22⋊C4C4×D269semidirect

Maximal subgroups

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

subgroupindex
#317711662
#100×D2692
#99×D2692
#317703952
#317703932
#317703912
#352×C2692
#2121269