Finite Group Databasebeta

← #46809813

Subgroups — #46809813

Order 34432.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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#31771204C211yesmaximal, characteristic, complemented
2#31770739C211yesmaximal, characteristic
3#31770519C211yesmaximal, characteristic, complemented
4#31770342C211yesmaximal, characteristic, complemented
5#31770340C211yesmaximal, characteristic, complemented
6#31770330C211yesmaximal, characteristic, complemented
7#450×C269C211yesmaximal, characteristic, complemented
8C2×#16864387—21no—
9#22575448—21no—
10#22575444—21no—
11#22575438—21no—
12#22575437—21no—
13#22575429—21no—
14#22575427C2211yescharacteristic
15#22575413—21no—
16C2×#16864381C2211yescharacteristic
17C2×C22⋊Dic269—21no—
18C2×C22⋊Dic269—21no—
19#22575374—21no—
20C22⋊#14524130C2211yescharacteristic
21#22575360C2211yescharacteristic
22C22×D4×C269C2211yescharacteristic
23C2×SD16×C269—21no—
24C22⋊Q8×C269C2211yescharacteristic
25#102×C269—21no—
26C22⋊C8×C269C2211yescharacteristic
27C2×D1076D411yescharacteristic
28C2×C22⋊D269—21no—
29C22×Dic269—21no—
30C22×Dic269D411yescharacteristic
31C22×Dic269D411yescharacteristic
32C2×C4×D269—21no—
33#16864387—42no—
34C2×#14524130—21no—
35C4⋊Dic269—21no—
36C4⋊Dic269—21no—
37C4⋊Dic269D411yescharacteristic
38#16864381—21no—
39#16864381—21no—
40#16864381—21no—
41C22⋊Dic269—42no—
42C22⋊Dic269—42no—
43#16864379—21no—
44#16864379—21no—
45#16864379—21no—
46#16864379—21no—
47C4×Dic269—21no—
48C24×C269—21no—
49C2×Q8×C269D411yescharacteristic
50C2×D4×C269—21no—
51C2×D4×C269—21no—
52C2×D4×C269—42no—
53C22×C4×C269C2311yescharacteristic
54SD16×C269—42no—
55C2×C8×C269—21no—
56C4⋊C4×C269—21no—
57C4⋊C4×C269D411yescharacteristic
58C22⋊C4×C269—21no—
59C22×D269—21no—
60D1076—41no—
61C22⋊D269—41no—
62C22⋊D269—41no—
63C2×Dic269—21no—
64C2×Dic269—21no—
65C2×Dic269—21no—
66C2×Dic269—21no—
67C2×Dic269—21no—
68C2×Dic269—41no—
69C2×Dic269—41no—
70C2×Dic269—41no—
71C2×Dic269—41no—
72C4×D269—81no—
73#14524130—41no—
74C23×C269—22no—
75C23×C269—41no—
76C23×C269—41no—
77C23×C269—41no—
78C23×C269—41no—
79C23×C269—41no—
80C23×C269—41no—
81C23×C269C2×D411yescharacteristic
82Q8×C269—41no—
83D4×C269—22no—
84D4×C269—22no—
85D4×C269—41no—
86D4×C269—41no—
87C2×C4×C269—21no—
88C2×C4×C269—21no—
89C2×C4×C269—41no—
90C2×C4×C269C2×D411yescharacteristic
91C2×C4×C269C2×D411yescharacteristic
92C8×C269—41no—
93C22×C269—21no—
94C22×C269—21no—
95C22×C269—42no—
96C22×C269—42no—
97C22×C269—42no—
98C22×C269—81no—
99C22×C269—81no—
100C22×C269—81no—
101C22×C269—81no—
102C22×C269#12011yescharacteristic
103C22×C269#13511yescharacteristic
104C22×C269AGL(1,ℤ/8)11yescharacteristic
105C2×D269—41no—
106C2×D269—41no—
107C2×D269—41no—
108C4×C269—21no—
109C4×C269—21no—
110C4×C269—41no—
111C4×C269—41no—
112Dic269—41no—
113Dic269—41no—
114Dic269—41no—
115Dic269—81no—
116C2×C269—21no—
117C2×C269—21no—
118C2×C269—82no—
119C2×C269#44911yescharacteristic
120C2×C269#45411yescharacteristic
121C2×C269B(4,𝔽2)11yescharacteristic
122D269—81no—
123C269#159711yescharacteristic, complemented, Sylow
124#1597—2691nomaximal, Sylow
125#539—2691no—
126#450D26911yescharacteristic, complemented
127#447—2691no—
128#394—2691no—
129#328—2691no—
130#327—2691no—
131#324—2691no—
132C22×D4C2×D26911yescharacteristic
133C2×SD16—21no—
134C2×D8—5381no—
135#126—5381no—
136#125—5381no—
137#123—5381no—
138#123—5381no—
139C22⋊Q8—5381no—
140C22⋊Q8C2×D26911yescharacteristic
141#121—2691no—
142C4×D4—5381no—
143C2×C4⋊C4—2691no—
144C2×C22⋊C4—5381no—
145C2×C22⋊C4—5381no—
146#102—21no—
147#102—5381no—
148C22⋊C8—2691no—
149C22⋊C8C2×D26911yescharacteristic
150#95—2691no—
151C24—21no—
152C2×Q8C22⋊D26911yescharacteristic
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—42no—
156C2×D4—2691no—
157C2×D4—5381no—
158C22×C4—2691no—
159C22×C4—2691no—
160C22×C4—5381no—
161C22×C4—5381no—
162C22×C4C22×D26911yescharacteristic
163SD16—42no—
164D8—10762no—
165C2×C8—21no—
166C2×C8—5381no—
167C4⋊C4—21no—
168C4⋊C4—2691no—
169C4⋊C4—5381no—
170C4⋊C4—5381no—
171C4⋊C4—5381no—
172C4⋊C4—5381no—
173C4⋊C4—5381no—
174C4⋊C4C22⋊D26911yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—5381no—
177C22⋊C4—5381no—
178C22⋊C4—5381no—
179C22⋊C4—5381no—
180C22⋊C4—10762no—
181C22⋊C4—10762no—
182C42—5381no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—5381no—
191C23D4×D26911yescharacteristic
192Q8—41no—
193D4—22no—
194D4—22no—
195D4—41no—
196D4—41no—
197D4—10761no—
198D4—10761no—
199D4—10761no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—41no—
203C2×C4—5381no—
204C2×C4—5381no—
205C2×C4—5381no—
206C2×C4—5381no—
207C2×C4—5381no—
208C2×C4—10761no—
209C2×C4—10761no—
210C2×C4—10761no—
211C2×C4—10761no—
212C2×C4—21521no—
213C2×C4D4×D26911yescharacteristic
214C2×C4C2×C22⋊D26911yescharacteristic
215C8—41no—
216C8—10761no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—10761no—
227C22—10761no—
228C22—10761no—
229C22#2257542111yescharacteristic
230C22#2257547411yescharacteristic
231C22#2257548011yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—10761no—
237C4—10761no—
238C4—10761no—
239C4—21521no—
240C2—21no—
241C2—21no—
242C2—82no—
243C2—21521no—
244C2#3177074911yescharacteristic
245C2#3177077611yescharacteristic
246C2#3177079111yescharacteristic
subgroupG/H
centerC22#22575421
commutator, cosocleC22×C4×C269C23
FrattiniC22×C4C22×D269
Fitting#450×C269C2
socleC22×C269#120
prime radical#31770739C2
hypercenter#450D269
hypocenterC269#1597
LCSG ▷ C22×C4×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #31771204C2semidirect
3. #31770519C2semidirect
4. #31770342C2semidirect
5. #31770340C2semidirect
6. #31770330C2semidirect
7. #450×C269C2semidirect
123. C269#1597semidirect
126. #450D269semidirect

Maximal subgroups

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

subgroupindex
#317712042
#317707392
#317705192
#317703422
#317703402
#317703302
#450×C2692
#1597269