Finite Group Databasebeta

← #46782700

Subgroups — #46782700

Order 33664.

620 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#22554515C211yesmaximal, characteristic
2C2×#22554398C212yesmaximal, complemented
3#30681937C212yesmaximal, complemented
4C2×#22554387C211yesmaximal, characteristic, complemented
5#390×C263C211yesmaximal, characteristic, complemented
6#22554515—24no—
7#22554515—22no—
8#22554515C414yescomplemented
9C2×#16862376C2212yes—
10C2×#16862375—22no—
11C22×C22⋊D263—22no—
12C2×D4×D263—22no—
13C2×#16862360C2212yescomplemented
14#22554398—24no—
15#22554393—22no—
16#22554390—22no—
17#22554387—22no—
18C22×D4×C263C2211yescharacteristic
19C2×C4⋊C4×C263C2211yescharacteristic, complemented
20C2×C22⋊C4×C263—22no—
21C2×C42×C263C2211yescharacteristic, complemented
22C23×D263—22no—
23#16862376—22no—
24#16862376—22no—
25#16862376C2×C414yes—
26#16862376D414yes—
27#16862375—28no—
28#16862375—28no—
29C2×Dic526D412yes—
30C2×D1052D412yes—
31C2×C22⋊D263—24no—
32C2×C22⋊D263—28no—
33C2×C22⋊D263—28no—
34C2×C22⋊D263—22no—
35D4×D263—28no—
36D4×D263—28no—
37C22×Dic263—22no—
38C2×C4×D263—22no—
39#16862360—22no—
40#16862360—22no—
41C2×C263⋊C8—22no—
42C24×C263D412yes—
43C2×D4×C263—22no—
44C2×D4×C263—22no—
45C2×D4×C263C2×C412yes—
46C2×D4×C263D412yes—
47C22×C4×C263—21no—
48C22×C4×C263—21no—
49C22×C4×C263C2311yescharacteristic
50C4⋊C4×C263—22no—
51C22⋊C4×C263—44no—
52C42×C263—22no—
53C22×D263—22no—
54C22×D263—28no—
55C22×D263—82no—
56C22×D263—82no—
57Dic526—22no—
58Dic526C22⋊C414yescomplemented
59D1052—22no—
60D1052C22⋊C414yescomplemented
61C22⋊D263—216no—
62C22⋊D263—24no—
63C22⋊D263—24no—
64C2×Dic263—28no—
65C2×Dic263—22no—
66C4×D263—24no—
67C4×D263—24no—
68C263⋊C8—42no—
69C23×C263—22no—
70C23×C263—22no—
71C23×C263—42no—
72C23×C263—42no—
73C23×C263C22⋊C414yes—
74C23×C263C2×D411yescharacteristic
75D4×C263—24no—
76D4×C263—24no—
77C2×C4×C263—22no—
78C2×C4×C263—41no—
79C2×C4×C263—41no—
80C2×C4×C263—42no—
81C2×C4×C263—41no—
82C2×C4×C263C22×C411yescharacteristic
83C2×C4×C263C2×D412yes—
84C2×C4×C263C2×D412yes—
85C2×C4×C263C2×D411yescharacteristic
86C22×C263—21no—
87C22×C263—21no—
88C22×C263—24no—
89C22×C263—44no—
90C22×C263—42no—
91C22×C263—42no—
92C22×C263—82no—
93C22×C263C2×C22⋊C411yescharacteristic
94C22×C263#12011yescharacteristic
95C22×C263#12011yescharacteristic
96C2×D263—24no—
97C2×D263—42no—
98C2×D263—42no—
99C2×D263—88no—
100C4×C263—42no—
101C4×C263—81no—
102C4×C263C2×C22⋊C412yes—
103C4×C263#12012yes—
104Dic263—24no—
105C2×C263—21no—
106C2×C263—21no—
107C2×C263—44no—
108C2×C263#37911yescharacteristic
109C2×C263#45311yescharacteristic
110C2×C263#45511yescharacteristic
111D263—44no—
112C263#179211yescharacteristic, complemented, Sylow
113#1792—2631nomaximal, Sylow
114C2×H(5,𝔽2)—2631no—
115C2×#104—2632no—
116#417—2632no—
117C2×#100—2631no—
118#390D26311yescharacteristic, complemented
119H(5,𝔽2)—2634no—
120H(5,𝔽2)—5264no—
121H(5,𝔽2)—5262no—
122C2×#41—2632no—
123C2×#41—5262no—
124C22×D4—5262no—
125C22×D4—5262no—
126C22×D4C2×D26311yescharacteristic
127C2×M16—2632no—
128C2×C4⋊C4C2×D26311yescharacteristic, complemented
129C2×C22⋊C4—22no—
130C2×C42C2×D26311yescharacteristic, complemented
131#104—5264no—
132#103—5262no—
133#102—5262no—
134#100—5262no—
135C24—5262no—
136C24C22⋊D26312yes—
137#41—2634no—
138#41—2634no—
139#41—5268no—
140#41—5268no—
141#41—5262no—
142#41—5262no—
143C2×Q8—2632no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—2632no—
147C2×D4—5268no—
148C2×D4—5268no—
149C2×D4—5264no—
150C2×D4—5268no—
151C2×D4—5268no—
152C2×D4—5262no—
153C2×D4C4×D26312yes—
154C2×D4D105212yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—5262no—
158C22×C4—5262no—
159C22×C4C22×D26311yescharacteristic
160M16—5262no—
161M16—5262no—
162C2×C8—5262no—
163C4⋊C4—22no—
164C22⋊C4—44no—
165C42—22no—
166C23—22no—
167C23—22no—
168C23—42no—
169C23—42no—
170C23—5268no—
171C23—5262no—
172C23—21042no—
173C23—21042no—
174C23(C2×D263)⋊C414yes—
175C23C2×C22⋊D26311yescharacteristic
176Q8—2634no—
177Q8—5262no—
178D4—24no—
179D4—24no—
180D4—2634no—
181D4—52616no—
182D4—5264no—
183D4—5264no—
184D4—5262no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—5268no—
191C2×C4—5262no—
192C2×C4—5264no—
193C2×C4—5264no—
194C2×C4C2×C4×D26311yescharacteristic
195C2×C4D4×D26312yes—
196C2×C4D4×D26312yes—
197C2×C4C2×D105211yescharacteristic
198C8—10522no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—82no—
206C22—5264no—
207C22—10522no—
208C22—10522no—
209C22—21048no—
210C22C2×(C2×D263)⋊C411yescharacteristic
211C22#2255443811yescharacteristic
212C22#2255443811yescharacteristic
213C4—42no—
214C4—81no—
215C4—5264no—
216C4C22⋊C4×D26312yes—
217C4#2255444012yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—10524no—
222C2#3068179311yescharacteristic
223C2#3068210511yescharacteristic
224C2#3068211511yescharacteristic
subgroupG/H
centerC22#22554438
commutatorC2×C4×C263C22×C4
FrattiniC22×C4C22×D263
Fitting#390×C263C2
socleC22×C263#120
prime radicalC2×#22554515C2
cosocleC22×C4×C263C23
hypercenter#390D263
hypocenterC263#1792
LCSG ▷ C2×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#22554398C2semidirect
3. #30681937C2semidirect
4. C2×#22554387C2semidirect
5. #390×C263C2semidirect
8. #22554515C4semidirect
13. C2×#16862360C22semidirect
19. C2×C4⋊C4×C263C22semidirect
21. C2×C42×C263C22semidirect
58. Dic526C22⋊C4semidirect
60. D1052C22⋊C4semidirect
112. C263#1792semidirect
118. #390D263semidirect
128. C2×C4⋊C4C2×D263semidirect
130. C2×C42C2×D263semidirect

Maximal subgroups

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

subgroupindex
C2×#225545152
C2×#225543982
#306819372
C2×#225543872
#390×C2632
#1792263