Finite Group Databasebeta

← #44558841

Subgroups — #44558841

Order 32128.

756 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 209 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×#22548760C211yesmaximal, characteristic
2#30632453C212yesmaximal, complemented
3#30631669C212yesmaximal, complemented
4#30631668C214yesmaximal, complemented
5C2×#22548632C211yesmaximal, characteristic, complemented
6#100×D251C214yesmaximal, complemented
7C2×#100×C251C211yesmaximal, characteristic, complemented
8#22548760—24no—
9#22548760—24no—
10C2×#16860810C2211yescharacteristic
11C2×#16860810C2211yescharacteristic
12C2×#16860809C412yes—
13C2×#16860809C2212yes—
14C22×C22⋊D251C414yes—
15C2×D4×D251C412yes—
16C2×D4×D251C2212yes—
17#22548723—24no—
18#22548721—24no—
19C2×#16860794C2212yescomplemented
20M16×D251C2218yescomplemented
21#22548633C2214yescomplemented
22#22548632C2214yescomplemented
23#22548631C2214yescomplemented
24C22×D4×C251C2211yescharacteristic
25C2×M16×C251C2212yescomplemented
26#100×C251C2214yescomplemented
27C23×D251C2×C414yes—
28#16860810—22no—
29#16860810—22no—
30#16860810D414yes—
31#16860810D414yes—
32#16860809—28no—
33#16860809—28no—
34C2×Dic502C2×C411yescharacteristic
35C2×Dic502C2311yescharacteristic
36C2×D1004C2×C411yescharacteristic
37C2×D1004C2311yescharacteristic
38C2×C22⋊D251—28no—
39C2×C22⋊D251—28no—
40C2×C22⋊D251C2×C414yes—
41C2×C22⋊D251C2×C414yes—
42C2×C22⋊D251C2×C412yes—
43C2×C22⋊D251C2×C412yes—
44D4×D251—28no—
45D4×D251—28no—
46C22×Dic251C2×C414yes—
47C2×C4×D251C2×C412yes—
48C2×C4×D251C2312yes—
49#16860794—22no—
50#16860794C2314yes—
51#16860793—28no—
52C2×C251⋊C8—22no—
53C8×D251—28no—
54C24×C251C2×C412yes—
55C2×D4×C251—22no—
56C2×D4×C251—22no—
57C2×D4×C251C2×C412yes—
58C2×D4×C251C2312yes—
59C22×C4×C251C2311yescharacteristic
60M16×C251—22no—
61M16×C251C2314yes—
62C2×C8×C251—22no—
63C22×D251—28no—
64C22×D251—84no—
65C22×D251C22×C412yes—
66C22×D251C22×C412yes—
67Dic502—22no—
68Dic502C22⋊C414yes—
69D1004—22no—
70D1004C22⋊C414yes—
71C22⋊D251—22no—
72C22⋊D251—22no—
73C22⋊D251—28no—
74C22⋊D251—28no—
75C22⋊D251C22⋊C414yes—
76C22⋊D251C22⋊C414yes—
77C2×Dic251—28no—
78C2×Dic251C22×C412yes—
79C2×Dic251C22×C412yes—
80C4×D251—24no—
81C4×D251C2×D418yes—
82C251⋊C8—24no—
83C23×C251—22no—
84C23×C251—22no—
85C23×C251—82no—
86C23×C251C22×C414yes—
87C23×C251C22×C411yescharacteristic
88D4×C251—24no—
89D4×C251—24no—
90C2×C4×C251C22×C411yescharacteristic
91C2×C4×C251C2×D412yes—
92C2×C4×C251C2×D412yes—
93C2×C4×C251C2411yescharacteristic
94C8×C251—24no—
95C22×C251—24no—
96C22×C251—41no—
97C22×C251—44no—
98C22×C251—82no—
99C22×C251—82no—
100C22×C251C2×C22⋊C411yescharacteristic
101C22×C251C2×C22⋊C411yescharacteristic
102C22×C251C23×C411yescharacteristic
103C2×D251—22no—
104C2×D251—42no—
105C2×D251—42no—
106C2×D251—88no—
107C2×D251C2×C22⋊C414yes—
108C4×C251C2×C22⋊C412yes—
109C4×C251C22×D412yes—
110Dic251—22no—
111Dic251C2×C22⋊C414yes—
112C2×C251—21no—
113C2×C251—21no—
114C2×C251—21no—
115C2×C251—44no—
116C2×C251C22×C22⋊C411yescharacteristic
117D251—24no—
118D251—42no—
119C251#288811yescharacteristic, complemented, Sylow
120#2888—2511nomaximal, Sylow
121C2×H(5,𝔽2)—2511no—
122#568—2512no—
123#413—2512no—
124#413—2514no—
125C2×#100—2511no—
126C2×#100—2514no—
127C2×#100D25111yescharacteristic, complemented
128H(5,𝔽2)—5024no—
129H(5,𝔽2)—5024no—
130C2×#41—2512no—
131C2×#41—2512no—
132C2×#41—2511no—
133C2×#41—2511no—
134C22×D4—2514no—
135C22×D4—2512no—
136C22×D4—2512no—
137C22×D4C2×D25111yescharacteristic
138#131—5024no—
139#131—5024no—
140C2×M16—2512no—
141C2×M16—2518no—
142C2×M16C2×D25112yescomplemented
143#101—2514no—
144#100—2514no—
145#100—2514no—
146#100C2×D25114yescomplemented
147C24—2514no—
148C24C4×D25112yes—
149#41—5028no—
150#41—5028no—
151#41—5022no—
152#41—5022no—
153#41—2514no—
154#41—2514no—
155C2×Q8—2511no—
156C2×Q8—2511no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—5028no—
160C2×D4—5028no—
161C2×D4—5028no—
162C2×D4—5028no—
163C2×D4—2514no—
164C2×D4—2512no—
165C2×D4—2514no—
166C2×D4—2512no—
167C2×D4—2511no—
168C2×D4—2511no—
169C2×D4C4×D25112yes—
170C2×D4C22×D25112yes—
171C22×C4—2514no—
172C22×C4—2512no—
173C22×C4—2512no—
174C22×C4C22×D25111yescharacteristic
175M16—22no—
176M16—5028no—
177M16—5022no—
178M16—2514no—
179M16C22×D25114yes—
180C2×C8—22no—
181C2×C8—5028no—
182C2×C8—5022no—
183C23—22no—
184C23—22no—
185C23—82no—
186C23—20084no—
187C23—5028no—
188C23—2512no—
189C23—2512no—
190C23C2×C4×D25114yes—
191C23C2×C4×D25111yescharacteristic
192Q8—5022no—
193Q8—2514no—
194D4—24no—
195D4—24no—
196D4—5028no—
197D4—5022no—
198D4—5022no—
199D4—5028no—
200D4—5022no—
201D4—2514no—
202D4—2514no—
203D4—2514no—
204C2×C4—5028no—
205C2×C4—5024no—
206C2×C4—2512no—
207C2×C4—2512no—
208C2×C4—2518no—
209C2×C4C2×C4×D25111yescharacteristic
210C2×C4D4×D25112yes—
211C2×C4D4×D25112yes—
212C2×C4C23×D25111yescharacteristic
213C8—24no—
214C8—5024no—
215C22—24no—
216C22—41no—
217C22—44no—
218C22—82no—
219C22—82no—
220C22—20088no—
221C22—10042no—
222C22—10042no—
223C22—5022no—
224C22—2514no—
225C22C22⋊C4×D25111yescharacteristic
226C22C22⋊C4×D25111yescharacteristic
227C22C22×C4×D25111yescharacteristic
228C4—5022no—
229C4—2514no—
230C4C22⋊C4×D25112yes—
231C4C2×D4×D25112yes—
232C2—21no—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—10042no—
237C2—5024no—
238C2C2×C22⋊C4×D25111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D251
commutatorC22×C251C23×C4
FrattiniC2×C4C23×D251
FittingC2×#100×C251C2
socleC2×C251C22×C22⋊C4
prime radicalC2×#22548760C2
cosocleC2×C4×C251C24
hypercenterC2×#100D251
hypocenterC251#2888
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #30632453C2semidirect
3. #30631669C2semidirect
4. #30631668C2semidirect
5. C2×#22548632C2semidirect
6. #100×D251C2semidirect
7. C2×#100×C251C2semidirect
19. C2×#16860794C22semidirect
20. M16×D251C22semidirect
21. #22548633C22semidirect
22. #22548632C22semidirect
23. #22548631C22semidirect
25. C2×M16×C251C22semidirect
26. #100×C251C22semidirect
119. C251#2888semidirect
127. C2×#100D251semidirect
142. C2×M16C2×D251semidirect
146. #100C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487602
#306324532
#306316692
#306316682
C2×#225486322
#100×D2512
C2×#100×C2512
#2888251