Finite Group Databasebeta

← C2×#30631864 #44559694

Subgroups — C2×#30631864 #44559694

Order 32128.

760 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C2×#22548746C212yesmaximal, complemented
3#30631864C218yesmaximal, direct product factor
4C2×#22548643C212yesmaximal, complemented
5C2×#22548632C211yesmaximal, characteristic, complemented
6C2×#124×C251C211yesmaximal, characteristic, complemented
7#22548760—24no—
8#22548760—22no—
9#22548760C2214yes—
10C2×#16860810C2212yes—
11C2×#16860809—22no—
12C22×C22⋊D251—22no—
13C2×D4×D251—22no—
14#22548746—24no—
15#22548746C2218yescomplemented
16C2×#16860800—22no—
17C2×#16860797—22no—
18C2×#16860794C2212yescomplemented
19#22548643C2218yescomplemented
20#22548632C2214yescomplemented
21C22×Q8×C251C2211yescharacteristic, complemented
22C22×D4×C251C2211yescharacteristic
23#124×C251—22no—
24#124×C251C2214yescomplemented
25C2×C22⋊C4×C251—22no—
26C2×C42×C251C2211yescharacteristic, complemented
27C23×D251—22no—
28#16860810—22no—
29#16860810—22no—
30#16860810D414yes—
31#16860810C2314yes—
32#16860809—28no—
33#16860809—28no—
34C2×Dic502D412yes—
35C2×D1004D412yes—
36C2×C22⋊D251—24no—
37C2×C22⋊D251—28no—
38C2×C22⋊D251—22no—
39C2×C22⋊D251—28no—
40D4×D251—28no—
41D4×D251—28no—
42C22×Dic251—22no—
43C2×C4×D251—22no—
44#16860800—28no—
45#16860797—28no—
46#16860794—22no—
47#16860794C2314yescomplemented
48C2×C251⋊C8—22no—
49C24×C251D412yes—
50C2×Q8×C251—21no—
51C2×Q8×C251—22no—
52C2×Q8×C251—22no—
53C2×Q8×C251C2312yescomplemented
54C2×D4×C251—22no—
55C2×D4×C251—22no—
56C2×D4×C251D412yes—
57C2×D4×C251C2312yes—
58C22×C4×C251—21no—
59C22×C4×C251—21no—
60C22×C4×C251C2311yescharacteristic
61C22⋊C4×C251—28no—
62C42×C251—21no—
63C42×C251C2312yescomplemented
64C22×D251—22no—
65C22×D251—28no—
66C22×D251—84no—
67Dic502—22no—
68Dic502C2×D414yes—
69D1004—22no—
70D1004C2×D414yes—
71C22⋊D251—216no—
72C22⋊D251—24no—
73C22⋊D251—24no—
74C2×Dic251—28no—
75C2×Dic251—22no—
76C4×D251—24no—
77C4×D251—24no—
78C251⋊C8—24no—
79C23×C251—22no—
80C23×C251—22no—
81C23×C251—44no—
82C23×C251C2×D411yescharacteristic
83C23×C251C2×D414yes—
84Q8×C251—24no—
85Q8×C251—42no—
86D4×C251—24no—
87D4×C251—24no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—42no—
91C2×C4×C251—41no—
92C2×C4×C251—41no—
93C2×C4×C251C2×D412yes—
94C2×C4×C251C2×D412yes—
95C2×C4×C251C2×D411yescharacteristic
96C2×C4×C251C2411yescharacteristic
97C22×C251—22no—
98C22×C251—24no—
99C22×C251—44no—
100C22×C251—44no—
101C22×C251—82no—
102C22×C251#12011yescharacteristic
103C22×C251#12011yescharacteristic
104C22×C251C22×D411yescharacteristic
105C2×D251—24no—
106C2×D251—44no—
107C2×D251—88no—
108C4×C251—42no—
109C4×C251—42no—
110C4×C251#12012yes—
111C4×C251C22×D412yes—
112Dic251—24no—
113C2×C251—21no—
114C2×C251—21no—
115C2×C251—44no—
116C2×C251#45512yescomplemented
117C2×C251C2×#12011yescharacteristic
118D251—44no—
119C251C2×#45511yescharacteristic, complemented, Sylow
120C2×#455—2511nomaximal, Sylow
121C2×H(5,𝔽2)—2511no—
122C2×#137—2512no—
123C2×#124D25111yescharacteristic, complemented
124#455—2518no—
125C2×#104—2512no—
126C2×#100—2511no—
127H(5,𝔽2)—5024no—
128H(5,𝔽2)—5022no—
129H(5,𝔽2)—2514no—
130C2×#41—5022no—
131C2×#41—2512no—
132C22×Q8C2×D25111yescharacteristic, complemented
133C22×D4—5022no—
134C22×D4—5022no—
135C22×D4C2×D25111yescharacteristic
136#137—5024no—
137#137—2518no—
138C2×Q16—5022no—
139C2×SD16—5022no—
140C2×M16—2512no—
141#124—22no—
142#124C2×D25114yescomplemented
143C2×C22⋊C4—22no—
144C2×C42C2×D25111yescharacteristic, complemented
145#104—2518no—
146#100—2514no—
147C24—5022no—
148C24C22⋊D25112yes—
149#41—5028no—
150#41—5028no—
151#41—5022no—
152#41—5022no—
153#41—2514no—
154#41—2514no—
155C2×Q8—21no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—2512no—
159C2×Q8C22×D25112yescomplemented
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—5028no—
163C2×D4—5028no—
164C2×D4—5024no—
165C2×D4—5028no—
166C2×D4—5022no—
167C2×D4—5028no—
168C2×D4—2512no—
169C2×D4C22⋊D25112yes—
170C2×D4C22×D25112yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—5022no—
174C22×C4—5022no—
175C22×C4C22×D25111yescharacteristic
176Q16—5028no—
177SD16—5028no—
178M16—5022no—
179M16—2514no—
180C2×C8—5022no—
181C22⋊C4—28no—
182C42—21no—
183C42C22×D25112yescomplemented
184C23—22no—
185C23—22no—
186C23—44no—
187C23—20084no—
188C23—5028no—
189C23—5022no—
190C23C2×C22⋊D25111yescharacteristic
191C23C2×C22⋊D25114yes—
192Q8—24no—
193Q8—42no—
194Q8—5022no—
195Q8—2514no—
196D4—24no—
197D4—24no—
198D4—50216no—
199D4—5024no—
200D4—5024no—
201D4—5022no—
202D4—2514no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—5028no—
209C2×C4—5022no—
210C2×C4—5024no—
211C2×C4—5024no—
212C2×C4D4×D25112yes—
213C2×C4D4×D25112yes—
214C2×C4C2×C22⋊D25111yescharacteristic
215C2×C4C23×D25111yescharacteristic
216C8—5024no—
217C22—22no—
218C22—24no—
219C22—44no—
220C22—44no—
221C22—82no—
222C22—20088no—
223C22—10044no—
224C22—5024no—
225C22#2254868311yescharacteristic
226C22#2254868311yescharacteristic
227C22C22×C22⋊D25111yescharacteristic
228C4—42no—
229C4—42no—
230C4—5024no—
231C4#2254868312yes—
232C4C2×D4×D25112yes—
233C2—21no—
234C2—21no—
235C2—44no—
236C2—10044no—
237C2#3063186412yesdirect product factor
238C2C2×#2254868311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#124×C251C2
socleC22×C251#120
prime radicalC2×#22548760C2
hypercenterC2×#124D251
hypocenterC251C2×#455
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#22548746C2semidirect
3. #30631864C2direct (N × H)
4. C2×#22548643C2semidirect
5. C2×#22548632C2semidirect
6. C2×#124×C251C2semidirect
15. #22548746C22semidirect
18. C2×#16860794C22semidirect
19. #22548643C22semidirect
20. #22548632C22semidirect
21. C22×Q8×C251C22semidirect
24. #124×C251C22semidirect
26. C2×C42×C251C22semidirect
47. #16860794C23semidirect
53. C2×Q8×C251C23semidirect
63. C42×C251C23semidirect
116. C2×C251#455semidirect
119. C251C2×#455semidirect
123. C2×#124D251semidirect
132. C22×Q8C2×D251semidirect
142. #124C2×D251semidirect
144. C2×C42C2×D251semidirect
159. C2×Q8C22×D251semidirect
183. C42C22×D251semidirect
237. C2#30631864direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487602
C2×#225487462
#306318642
C2×#225486432
C2×#225486322
C2×#124×C2512
C2×#455251