Finite Group Databasebeta

← C2×#30632005 #44560264

Subgroups — C2×#30632005 #44560264

Order 32128.

512 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 165 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×#22548689C211yesmaximal, characteristic
2#30632005C218yesmaximal, direct product factor
3C2×#22548646C212yesmaximal, complemented
4C2×#22548635C212yesmaximal, complemented
5C2×C22⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
6C22×C4⋊C4×C251C211yesmaximal, characteristic, complemented
7C22×D1004C2211yescharacteristic
8C22×C22⋊D251—21no—
9C22×C251⋊C8C2212yes—
10#22548689—22no—
11#22548689C2214yes—
12C2×C4⋊Dic251C2211yescharacteristic
13C2×(C2×D251)⋊C4—21no—
14#22548646C2218yescomplemented
15#22548635C2218yescomplemented
16C22⋊(C251⋊C8)C2214yescomplemented
17C23×C4×C251—21no—
18C23×C4×C251C2211yescharacteristic
19C2×C4⋊C4×C251—22no—
20C2×C4⋊C4×C251—21no—
21C2×C4⋊C4×C251C2212yes—
22C2×C4⋊C4×C251C2214yescomplemented
23C23×D251—21no—
24C2×D1004—21no—
25C2×D1004—24no—
26C2×D1004C2312yes—
27C2×C22⋊D251—24no—
28C2×C22⋊D251—44no—
29C22×Dic251—21no—
30C2×C251⋊C8—24no—
31C2×C251⋊C8C2314yes—
32C4⋊Dic251—21no—
33C4⋊Dic251C2312yes—
34(C2×D251)⋊C4—24no—
35C24×C251D411yescharacteristic
36C22×C4×C251—24no—
37C22×C4×C251—22no—
38C22×C4×C251—24no—
39C22×C4×C251—44no—
40C22×C4×C251D411yescharacteristic
41C22×C4×C251D412yes—
42C22×C4×C251C2312yes—
43C22×C4×C251C2311yescharacteristic
44C4⋊C4×C251—22no—
45C4⋊C4×C251—24no—
46C4⋊C4×C251C2314yes—
47C22×D251—22no—
48C22×D251—42no—
49C22×D251—42no—
50C22×D251—42no—
51D1004—24no—
52D1004—42no—
53C22⋊D251—48no—
54C2×Dic251—22no—
55C2×Dic251—42no—
56C251⋊C8—24no—
57C23×C251—22no—
58C23×C251—22no—
59C23×C251SD1614yes—
60C23×C251C2×D412yes—
61C23×C251C2×D411yescharacteristic
62C2×C4×C251—22no—
63C2×C4×C251—24no—
64C2×C4×C251—24no—
65C2×C4×C251—24no—
66C2×C4×C251—48no—
67C2×C4×C251—44no—
68C2×C4×C251C2×D412yes—
69C2×C4×C251C2×D411yescharacteristic
70C2×C4×C251#4114yes—
71C2×C4×C251C2411yescharacteristic
72C22×C251—22no—
73C22×C251—22no—
74C22×C251—28no—
75C22×C251#12312yes—
76C22×C251C2×SD1614yes—
77C22×C251C2×SD1612yes—
78C22×C251AGL(1,ℤ/8)12yes—
79C22×C251C22×D411yescharacteristic
80C2×D251—42no—
81C2×D251—42no—
82C2×D251—42no—
83C2×D251—84no—
84C4×C251—22no—
85C4×C251—44no—
86C4×C251#12312yes—
87C4×C251C2×#4112yes—
88Dic251—42no—
89C2×C251—24no—
90C2×C251#48114yescomplemented
91C2×C251C2×#12311yescharacteristic
92C2×C251C22×SD1611yescharacteristic
93C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
94D251—82no—
95C251C2×#48111yescharacteristic, complemented, Sylow
96C2×#481—2511nomaximal, Sylow
97C2×#121—2511no—
98C22×C4⋊C4D25111yescharacteristic, complemented
99#481—2518no—
100C2×#106—2512no—
101C2×#102—2512no—
102C2×C22⋊C8—2511no—
103C22×D4—5021no—
104C22×D4—2511no—
105C23×C4—21no—
106C23×C4C2×D25111yescharacteristic
107C22×C8—2512no—
108#121—5022no—
109#121—2514no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—2511no—
113C2×C4⋊C4C2×D25112yes—
114C2×C4⋊C4C2×D25114yescomplemented
115C2×C22⋊C4—5021no—
116#106—2518no—
117#102—2518no—
118C22⋊C8—2514no—
119C24—5021no—
120C24C22⋊D25111yescharacteristic
121C2×D4—10044no—
122C2×D4—5024no—
123C2×D4—5021no—
124C2×D4—5024no—
125C2×D4—2512no—
126C22×C4—24no—
127C22×C4—24no—
128C22×C4—22no—
129C22×C4—44no—
130C22×C4—5021no—
131C22×C4C22⋊D25112yes—
132C22×C4C22⋊D25111yescharacteristic
133C22×C4C22×D25112yes—
134C22×C4C22×D25111yescharacteristic
135C2×C8—5024no—
136C2×C8—2514no—
137C4⋊C4—22no—
138C4⋊C4—24no—
139C4⋊C4—5021no—
140C4⋊C4—2512no—
141C4⋊C4C22×D25114yes—
142C22⋊C4—5024no—
143C23—22no—
144C23—22no—
145C23—10042no—
146C23—10042no—
147C23—10042no—
148C23—5022no—
149C23#1686079714yes—
150C23C2×C22⋊D25112yes—
151C23C2×C22⋊D25111yescharacteristic
152D4—10048no—
153D4—10042no—
154D4—5024no—
155C2×C4—22no—
156C2×C4—24no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—48no—
160C2×C4—44no—
161C2×C4—10042no—
162C2×C4—5022no—
163C2×C4C2×C22⋊D25112yes—
164C2×C4C2×C22⋊D25111yescharacteristic
165C2×C4#1686081014yes—
166C2×C4C23×D25111yescharacteristic
167C8—5024no—
168C22—22no—
169C22—28no—
170C22—22no—
171C22—20084no—
172C22—10042no—
173C22—10042no—
174C22—10042no—
175C22#2254869812yes—
176C22C2×#1686079714yes—
177C22C2×#1686079712yes—
178C22#2254874112yes—
179C22C22×C22⋊D25111yescharacteristic
180C4—22no—
181C4—44no—
182C4—10042no—
183C4#2254869812yes—
184C4C2×#1686081012yes—
185C2—24no—
186C2—20082no—
187C2#3063200514yesdirect product factor
188C2C2×#2254869811yescharacteristic
189C2C22×#1686079711yescharacteristic
190C2C2×#2254874111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×C4⋊C4×C251C2
socleC23×C251C2×D4
prime radicalC2×#22548689C2
hypercenterC22×C4⋊C4D251
hypocenterC251C2×#481
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #30632005C2direct (N × H)
3. C2×#22548646C2semidirect
4. C2×#22548635C2semidirect
5. C2×C22⋊(C251⋊C8)C2semidirect
6. C22×C4⋊C4×C251C2semidirect
14. #22548646C22semidirect
15. #22548635C22semidirect
16. C22⋊(C251⋊C8)C22semidirect
22. C2×C4⋊C4×C251C22semidirect
90. C2×C251#481semidirect
95. C251C2×#481semidirect
98. C22×C4⋊C4D251semidirect
114. C2×C4⋊C4C2×D251semidirect
187. C2#30632005direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225486892
#306320052
C2×#225486462
C2×#225486352
C2×C22⋊(C251⋊C8)2
C22×C4⋊C4×C2512
C2×#481251