Finite Group Databasebeta

← C2×#30631859 #44559700

Subgroups — C2×#30631859 #44559700

Order 32128.

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

normal subgroups only

This group has 728 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×#22548763C211yesmaximal, characteristic, complemented
2C2×#22548741C212yesmaximal, complemented
3#30631859C218yesmaximal, direct product factor
4C2×#22548643C212yesmaximal, complemented
5C2×#22548634C211yesmaximal, characteristic, complemented
6C2×#124×C251C211yesmaximal, characteristic, complemented
7#22548763—24no—
8#22548763—22no—
9#22548763C2214yescomplemented
10C2×#16860811—22no—
11C2×#16860810—22no—
12C2×#16860810C2212yes—
13C2×Q8×D251—22no—
14#22548741—24no—
15#22548741C2218yescomplemented
16C2×#16860798—22no—
17C2×#16860795—22no—
18C2×#16860794C2212yes—
19#22548643C2218yescomplemented
20#22548634C2214yescomplemented
21C22×Q8×C251C2211yescharacteristic
22C22×D4×C251C2211yescharacteristic
23#124×C251—22no—
24#124×C251C2214yescomplemented
25C2×C22⋊C4×C251—22no—
26C2×C42×C251C2211yescharacteristic
27#16860811—28no—
28#16860811—28no—
29#16860810—28no—
30#16860810—28no—
31#16860810—22no—
32#16860810—22no—
33#16860810D414yes—
34#16860810C2314yes—
35C2×Dic502—22no—
36C2×Dic502D412yes—
37Q8×D251—28no—
38Q8×D251—28no—
39C2×D1004—22no—
40C2×D1004D412yes—
41C2×C22⋊D251—22no—
42C2×C4×D251—24no—
43C2×C4×D251—22no—
44#16860798—28no—
45#16860795—28no—
46#16860794—22no—
47#16860794C2314yes—
48C2×C251⋊C8—22no—
49C24×C251—21no—
50C2×Q8×C251—22no—
51C2×Q8×C251—22no—
52C2×Q8×C251D412yes—
53C2×Q8×C251C2312yes—
54C2×D4×C251—22no—
55C2×D4×C251—22no—
56C2×D4×C251—21no—
57C2×D4×C251C2312yes—
58C22×C4×C251—21no—
59C22×C4×C251D412yes—
60C22×C4×C251C2311yescharacteristic
61C22⋊C4×C251—28no—
62C42×C251—21no—
63C42×C251C2312yes—
64C22×D251—22no—
65Dic502—28no—
66Dic502—22no—
67Dic502C2×D414yes—
68D1004—28no—
69D1004—22no—
70D1004C2×D414yes—
71C22⋊D251—24no—
72C22⋊D251—24no—
73C2×Dic251—22no—
74C4×D251—216no—
75C4×D251—24no—
76C4×D251—24no—
77C251⋊C8—24no—
78C23×C251—22no—
79C23×C251—44no—
80C23×C251—41no—
81C23×C251—41no—
82C23×C251C2×D411yescharacteristic
83Q8×C251—24no—
84Q8×C251—24no—
85D4×C251—24no—
86D4×C251—42no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251—42no—
91C2×C4×C251C2×D412yes—
92C2×C4×C251C2×D412yes—
93C2×C4×C251C2×D414yes—
94C2×C4×C251C2×D411yescharacteristic
95C2×C4×C251C2411yescharacteristic
96C22×C251—22no—
97C22×C251—44no—
98C22×C251—42no—
99C22×C251—82no—
100C22×C251—82no—
101C22×C251#12011yescharacteristic
102C22×C251#12011yescharacteristic
103C22×C251C22×D411yescharacteristic
104C2×D251—24no—
105C2×D251—44no—
106C4×C251—24no—
107C4×C251—42no—
108C4×C251#12012yes—
109C4×C251C22×D412yes—
110Dic251—24no—
111C2×C251—21no—
112C2×C251—21no—
113C2×C251—82no—
114C2×C251#45412yescomplemented
115C2×C251C2×#12011yescharacteristic
116D251—44no—
117C251C2×#45411yescharacteristic, complemented, Sylow
118C2×#454—2511nomaximal, Sylow
119C2×2−1+4—2511no—
120C2×AGL(1,ℤ/8)—2512no—
121C2×#124D25111yescharacteristic, complemented
122#454—2518no—
123C2×#104—2512no—
124C2×#101—2511no—
1252−1+4—5024no—
1262−1+4—5022no—
1272−1+4—2514no—
128C2×#41—5022no—
129C2×#41—5022no—
130C2×#41—2512no—
131C22×Q8—5022no—
132C22×Q8C2×D25111yescharacteristic
133C22×D4C2×D25111yescharacteristic
134AGL(1,ℤ/8)—5024no—
135AGL(1,ℤ/8)—2518no—
136C2×SD16—5022no—
137C2×D8—5022no—
138C2×M16—2512no—
139#124—22no—
140#124C2×D25114yescomplemented
141C2×C22⋊C4—22no—
142C2×C42C2×D25111yescharacteristic
143#104—2518no—
144#101—2514no—
145C24—21no—
146#41—5028no—
147#41—5028no—
148#41—5028no—
149#41—5022no—
150#41—5028no—
151#41—5022no—
152#41—2514no—
153#41—2514no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—5022no—
157C2×Q8—5028no—
158C2×Q8—5028no—
159C2×Q8—2512no—
160C2×Q8C22⋊D25112yes—
161C2×Q8C22×D25112yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—5022no—
166C2×D4—5022no—
167C2×D4—2512no—
168C2×D4C22×D25112yes—
169C22×C4—21no—
170C22×C4—5024no—
171C22×C4—5022no—
172C22×C4C22⋊D25112yes—
173C22×C4C22×D25111yescharacteristic
174SD16—5028no—
175D8—5028no—
176M16—5022no—
177M16—2514no—
178C2×C8—5022no—
179C22⋊C4—28no—
180C42—21no—
181C42C22×D25112yes—
182C23—22no—
183C23—44no—
184C23—41no—
185C23—41no—
186C23—5022no—
187C23C2×C22⋊D25111yescharacteristic
188Q8—24no—
189Q8—24no—
190Q8—5028no—
191Q8—5022no—
192Q8—2514no—
193D4—24no—
194D4—42no—
195D4—5028no—
196D4—5024no—
197D4—5024no—
198D4—5022no—
199D4—2514no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—50216no—
205C2×C4—5022no—
206C2×C4—5024no—
207C2×C4—5024no—
208C2×C4D4×D25112yes—
209C2×C4D4×D25112yes—
210C2×C4C2×C22⋊D25114yes—
211C2×C4C2×C22⋊D25111yescharacteristic
212C2×C4C23×D25111yescharacteristic
213C8—5024no—
214C22—22no—
215C22—44no—
216C22—42no—
217C22—82no—
218C22—82no—
219C22—10044no—
220C22—5024no—
221C22#2254868311yescharacteristic
222C22#2254868311yescharacteristic
223C22C22×C22⋊D25111yescharacteristic
224C4—24no—
225C4—42no—
226C4—5024no—
227C4#2254868312yes—
228C4C2×D4×D25112yes—
229C2—21no—
230C2—21no—
231C2—82no—
232C2—10044no—
233C2#3063185912yesdirect product factor
234C2C2×#2254868311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#124×C251C2
socleC22×C251#120
hypercenterC2×#124D251
hypocenterC251C2×#454
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
1. C2×#22548763C2semidirect
2. C2×#22548741C2semidirect
3. #30631859C2direct (N × H)
4. C2×#22548643C2semidirect
5. C2×#22548634C2semidirect
6. C2×#124×C251C2semidirect
9. #22548763C22semidirect
15. #22548741C22semidirect
19. #22548643C22semidirect
20. #22548634C22semidirect
24. #124×C251C22semidirect
114. C2×C251#454semidirect
117. C251C2×#454semidirect
121. C2×#124D251semidirect
140. #124C2×D251semidirect
233. C2#30631859direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487632
C2×#225487412
#306318592
C2×#225486432
C2×#225486342
C2×#124×C2512
C2×#454251