Finite Group Databasebeta

← #46822252

Subgroups — #46822252

Order 34432.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#31771139C211yesmaximal, characteristic
2#31770912C212yesmaximal, complemented
3#31770897C211yesmaximal, characteristic, complemented
4#31770892C211yesmaximal, characteristic, complemented
5#31770861C212yesmaximal, complemented
6#31770853C212yesmaximal
7#31770827C212yesmaximal, complemented
8#31770812C211yesmaximal, characteristic, complemented
9#31770807C211yesmaximal, characteristic, complemented
10#31770629C211yesmaximal, characteristic, complemented
11#533×C269C211yesmaximal, characteristic, complemented
12C2×#16864392C2212yes—
13C2×#16864389C2212yes—
14C2×#16864386C2212yes—
15#22575447—21no—
16#22575445C2211yescharacteristic
17#22575441C2211yescharacteristic
18#22575436—21no—
19#22575432C2211yescharacteristic
20#22575427C2211yescharacteristic
21C4×Dic538C2211yescharacteristic
22C4×D1076C2211yescharacteristic
23C4×C22⋊D269—21no—
24#22575410—21no—
25#22575386C2212yes—
26#22575384C2212yes—
27C4⋊#14524130C2211yescharacteristic
28C4⋊#14524130C2211yescharacteristic
29C4⋊#14524130C2212yes—
30#22575377C2212yes—
31#22575377C2212yes—
32#22575376C2212yes—
33#22575373C2212yes—
34C4⋊Q8×C269C2211yescharacteristic
35C4⋊Q8×C269C2211yescharacteristic
36#125×C269—21no—
37C22⋊Q8×C269—21no—
38C22⋊Q8×C269C2212yes—
39C4×Q8×C269—21no—
40C4×Q8×C269C2212yes—
41#117×C269—21no—
42C2×C42×C269C2211yescharacteristic
43C2×Dic538C2311yescharacteristic
44C2×D1076C2311yescharacteristic
45C2×C22⋊D269—21no—
46C2×C4×D269—21no—
47#16864392—42no—
48#16864389—42no—
49#16864386—42no—
50C2×#14524130C2312yes—
51C2×#14524130C2312yes—
52C4⋊Dic269C2311yescharacteristic
53C4⋊Dic269C2311yescharacteristic
54#16864381—21no—
55#16864381—21no—
56#16864381—21no—
57#16864381—21no—
58C22⋊Dic269—21no—
59#16864379—21no—
60#16864379—21no—
61#16864379—21no—
62#16864379—21no—
63C4×Dic269—21no—
64C2×Q8×C269—21no—
65C2×Q8×C269C2312yes—
66C22×C4×C269D411yescharacteristic
67C22×C4×C269D411yescharacteristic
68C22×C4×C269C2311yescharacteristic
69C4⋊C4×C269—21no—
70C4⋊C4×C269—21no—
71C4⋊C4×C269—21no—
72C4⋊C4×C269—22no—
73C4⋊C4×C269—22no—
74C4⋊C4×C269C2312yes—
75C22⋊C4×C269—22no—
76C42×C269—22no—
77C42×C269D411yescharacteristic
78C42×C269D411yescharacteristic
79C42×C269C2311yescharacteristic
80C42×C269C2311yescharacteristic
81C22×D269—21no—
82Dic538—21no—
83Dic538—21no—
84D1076—21no—
85D1076—21no—
86C22⋊D269—42no—
87C2×Dic269—21no—
88C2×Dic269—21no—
89C2×Dic269—21no—
90C4×D269—42no—
91#14524130—22no—
92#14524130—22no—
93C23×C269C2×D411yescharacteristic
94Q8×C269—22no—
95Q8×C269—22no—
96Q8×C269—42no—
97C2×C4×C269—22no—
98C2×C4×C269—21no—
99C2×C4×C269—21no—
100C2×C4×C269—22no—
101C2×C4×C269—22no—
102C2×C4×C269—41no—
103C2×C4×C269C2×D411yescharacteristic
104C2×C4×C269C2×D411yescharacteristic
105C2×C4×C269C2×D411yescharacteristic
106C2×C4×C269C2×D411yescharacteristic
107C2×C4×C269C2×D411yescharacteristic
108C2×C4×C269C2411yescharacteristic
109C22×C269—21no—
110C22×C269—21no—
111C22×C269—21no—
112C22×C269C22×D411yescharacteristic
113C2×D269—41no—
114C2×D269—41no—
115C2×D269—41no—
116C4×C269—21no—
117C4×C269—21no—
118C4×C269—21no—
119C4×C269—21no—
120C4×C269—42no—
121C4×C269—42no—
122C4×C269#13712yes—
123C4×C269H(5,𝔽2)11yescharacteristic
124C4×C2692−1+411yescharacteristic
125Dic269—41no—
126Dic269—41no—
127Dic269—41no—
128C2×C269—41no—
129C2×C269#53711yescharacteristic
130C2×C269C2×#13711yescharacteristic
131C2×C269#57511yescharacteristic
132D269—81no—
133C269#325211yescharacteristic, complemented, Sylow
134#3252—2691nomaximal, Sylow
135#533D26911yescharacteristic, complemented
136#529—2691no—
137#479—2692no—
138#477—2691no—
139#476—2691no—
140#470—2692no—
141#468—2692no—
142#464—2692no—
143#462—2691no—
144#461—2691no—
145#423—2691no—
146C2×Q16—2692no—
147C2×SD16—2692no—
148C2×M16—2692no—
149C4⋊Q8C2×D26911yescharacteristic
150C4⋊Q8C2×D26911yescharacteristic
151#126—5381no—
152#125—21no—
153#125—2691no—
154#124—2691no—
155#123—5381no—
156C22⋊Q8—21no—
157C22⋊Q8—2691no—
158C22⋊Q8C2×D26912yes—
159#121—2691no—
160C4×Q8—21no—
161C4×Q8—2691no—
162C4×Q8C2×D26912yes—
163C4×D4—2691no—
164C4×D4—5381no—
165#117—21no—
166#117—5381no—
167C2×C42C2×D26911yescharacteristic
168#107—2692no—
169#106—2692no—
170C4⋊C8—2691no—
171C4⋊C8—2691no—
172C4⋊C8—2692no—
173#103—2692no—
174#103—2692no—
175#103—2692no—
176#102—2692no—
177C2×Q8—21no—
178C2×Q8—2691no—
179C2×Q8C22×D26912yes—
180C2×D4—2691no—
181C2×D4—5381no—
182C22×C4—5381no—
183C22×C4C22⋊D26911yescharacteristic
184C22×C4C22⋊D26911yescharacteristic
185C22×C4C22×D26911yescharacteristic
186Q16—10762no—
187SD16—10762no—
188M16—10762no—
189C2×C8—2692no—
190C2×C8—2692no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—2691no—
197C4⋊C4—2691no—
198C4⋊C4—5381no—
199C4⋊C4—5381no—
200C4⋊C4—5381no—
201C4⋊C4—5381no—
202C4⋊C4C22×D26912yes—
203C22⋊C4—22no—
204C22⋊C4—5381no—
205C22⋊C4—5381no—
206C22⋊C4—5381no—
207C22⋊C4—5381no—
208C22⋊C4—5381no—
209C42—22no—
210C42—5381no—
211C42C22⋊D26911yescharacteristic
212C42C22⋊D26911yescharacteristic
213C42C22×D26911yescharacteristic
214C42C22×D26911yescharacteristic
215C23—5381no—
216C23C2×C22⋊D26911yescharacteristic
217Q8—22no—
218Q8—22no—
219Q8—42no—
220Q8—5381no—
221Q8—5381no—
222D4—5381no—
223D4—5381no—
224D4—10762no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—5381no—
232C2×C4—5381no—
233C2×C4—5381no—
234C2×C4—10762no—
235C2×C4C2×C22⋊D26911yescharacteristic
236C2×C4C2×C22⋊D26911yescharacteristic
237C2×C4C2×C22⋊D26911yescharacteristic
238C2×C4C2×C22⋊D26911yescharacteristic
239C2×C4C2×C22⋊D26911yescharacteristic
240C2×C4C23×D26911yescharacteristic
241C8—5382no—
242C8—5382no—
243C22—21no—
244C22—21no—
245C22—21no—
246C22—10761no—
247C22—10761no—
248C22—10761no—
249C22C22×C22⋊D26911yescharacteristic
250C4—21no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—42no—
255C4—42no—
256C4—10761no—
257C4—10761no—
258C4—10761no—
259C4#2257548412yes—
260C4#2257549911yescharacteristic
261C4#2257550011yescharacteristic
262C2—41no—
263C2—21521no—
264C2#3177118311yescharacteristic
265C2C2×#2257548411yescharacteristic
266C2#3177140311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#533×C269C2
socleC22×C269C22×D4
prime radical#22575427C22
hypercenter#533D269
hypocenterC269#3252
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #31770912C2semidirect
3. #31770897C2semidirect
4. #31770892C2semidirect
5. #31770861C2semidirect
7. #31770827C2semidirect
8. #31770812C2semidirect
9. #31770807C2semidirect
10. #31770629C2semidirect
11. #533×C269C2semidirect
133. C269#3252semidirect
135. #533D269semidirect

Maximal subgroups

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

subgroupindex
#317711392
#317709122
#317708972
#317708922
#317708612
#317708532
#317708272
#317708122
#317708072
#317706292
#533×C2692
#3252269