Finite Group Databasebeta

← #46809823

Subgroups — #46809823

Order 34432.

266 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 51 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#31771234C211yesmaximal, characteristic, complemented
2#31770756C211yesmaximal, characteristic
3#31770519C211yesmaximal, characteristic
4#31770343C211yesmaximal, characteristic, complemented
5#31770342C211yesmaximal, characteristic, complemented
6#31770330C211yesmaximal, characteristic, complemented
7#450×C269C211yesmaximal, characteristic
8C2×#16864390—21no—
9#22575452—21no—
10#22575451—21no—
11#22575443—21no—
12#22575443—21no—
13#22575432C2211yescharacteristic
14#22575430—21no—
15#22575409—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×Dic538—21no—
28C2×Dic538D411yescharacteristic
29C22×Dic269—21no—
30C22×Dic269D411yescharacteristic
31C22×Dic269D411yescharacteristic
32#16864390—42no—
33C2×#14524130—21no—
34C4⋊Dic269—21no—
35C4⋊Dic269D411yescharacteristic
36#16864381—21no—
37#16864381—21no—
38#16864381—21no—
39#16864381—21no—
40#16864381—21no—
41#16864381—21no—
42#16864381—21no—
43C22⋊Dic269—21no—
44C22⋊Dic269—42no—
45C22⋊Dic269—42no—
46C4×Dic269—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—
59Dic538—41no—
60Dic538—81no—
61C2×Dic269—21no—
62C2×Dic269—21no—
63C2×Dic269—21no—
64C2×Dic269—21no—
65C2×Dic269—21no—
66C2×Dic269—41no—
67C2×Dic269—41no—
68C2×Dic269—41no—
69C2×Dic269—41no—
70C2×Dic269SD1612yescomplemented
71#14524130—41no—
72C23×C269—22no—
73C23×C269—41no—
74C23×C269—41no—
75C23×C269—41no—
76C23×C269—41no—
77C23×C269—41no—
78C23×C269—41no—
79C23×C269C2×D411yescharacteristic
80Q8×C269—41no—
81D4×C269—22no—
82D4×C269—22no—
83D4×C269—41no—
84D4×C269—41no—
85C2×C4×C269—21no—
86C2×C4×C269—21no—
87C2×C4×C269—41no—
88C2×C4×C269C2×D411yescharacteristic
89C2×C4×C269C2×D411yescharacteristic
90C8×C269—41no—
91C22×C269—21no—
92C22×C269—21no—
93C22×C269—42no—
94C22×C269—42no—
95C22×C269—42no—
96C22×C269—81no—
97C22×C269—81no—
98C22×C269—81no—
99C22×C269—81no—
100C22×C269#12011yescharacteristic
101C22×C269C2×SD1611yescharacteristic
102C22×C269#13711yescharacteristic
103C4×C269—21no—
104C4×C269—21no—
105C4×C269—41no—
106C4×C269—41no—
107Dic269—22no—
108Dic269—41no—
109Dic269—41no—
110Dic269—41no—
111Dic269—81no—
112C2×C269—21no—
113C2×C269—21no—
114C2×C269—82no—
115C2×C269#44811yescharacteristic
116C2×C269#45411yescharacteristic
117C2×C269B(4,𝔽2)11yescharacteristic
118C269#159911yescharacteristic, complemented, Sylow
119#1599—2691nomaximal, Sylow
120#544—2691no—
121#450—2691no—
122#450D26911yescharacteristic
123#394—2691no—
124#328—2691no—
125#328—2691no—
126#324—2691no—
127C22×D4C2×D26911yescharacteristic
128C2×SD16—21no—
129C2×SD16—5381no—
130C4⋊Q8—5381no—
131C4⋊Q8—5381no—
132#125—5381no—
133#125—5381no—
134C22⋊Q8—2691no—
135C22⋊Q8—5381no—
136C22⋊Q8C2×D26911yescharacteristic
137#117—5381no—
138C2×C4⋊C4—2691no—
139C2×C22⋊C4—5381no—
140C2×C22⋊C4—5381no—
141#102—21no—
142#102—5381no—
143C22⋊C8—2691no—
144C22⋊C8C2×D26911yescharacteristic
145#95—2691no—
146C24—21no—
147C2×Q8—2691no—
148C2×Q8—5381no—
149C2×Q8C22⋊D26911yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—42no—
153C22×C4—2691no—
154C22×C4—2691no—
155C22×C4—5381no—
156C22×C4C22×D26911yescharacteristic
157SD16—42no—
158SD16—10762no—
159C2×C8—21no—
160C2×C8—5381no—
161C4⋊C4—21no—
162C4⋊C4—2691no—
163C4⋊C4—5381no—
164C4⋊C4—5381no—
165C4⋊C4—5381no—
166C4⋊C4—5381no—
167C4⋊C4—5381no—
168C4⋊C4—5381no—
169C4⋊C4—5381no—
170C4⋊C4—5381no—
171C4⋊C4C22⋊D26911yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—5381no—
174C22⋊C4—10762no—
175C22⋊C4—10762no—
176C42—5381no—
177C42—5381no—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23D4×D26911yescharacteristic
186Q8—41no—
187Q8—10761no—
188Q8—21521no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—2692no—
197C2×C4—5381no—
198C2×C4—5381no—
199C2×C4—5381no—
200C2×C4—5381no—
201C2×C4—5381no—
202C2×C4—10761no—
203C2×C4—10761no—
204C2×C4—10761no—
205C2×C4—10761no—
206C2×C4D4×D26911yescharacteristic
207C2×C4C2×C22⋊D26911yescharacteristic
208C8—41no—
209C8—10761no—
210C22—21no—
211C22—21no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—81no—
219C22#2257542111yescharacteristic
220C22SD16×D26911yescharacteristic
221C22#2257548211yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—5382no—
227C4—10761no—
228C4—10761no—
229C4—10761no—
230C4—21521no—
231C2—21no—
232C2—21no—
233C2—82no—
234C2#3177074211yescharacteristic
235C2#3177077611yescharacteristic
236C2#3177079111yescharacteristic
subgroupG/H
centerC22#22575421
commutator, cosocleC22×C4×C269C23
FrattiniC22×C4C22×D269
Fitting#450×C269C2
socleC22×C269#120
prime radicalC22×D4×C269C22
hypercenter#450D269
hypocenterC269#1599
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. #31771234C2semidirect
4. #31770343C2semidirect
5. #31770342C2semidirect
6. #31770330C2semidirect
70. C2×Dic269SD16semidirect
118. C269#1599semidirect

Maximal subgroups

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

subgroupindex
#317712342
#317707562
#317705192
#317703432
#317703422
#317703302
#450×C2692
#1599269