Finite Group Databasebeta

← C2×#30632022 #44560270

Subgroups — C2×#30632022 #44560270

Order 32128.

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

normal subgroups only

This group has 460 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×#22548694C211yesmaximal, characteristic
2#30632022C218yesmaximal, direct product factor
3C2×#22548648C212yesmaximal, complemented
4C2×#22548638C212yesmaximal, complemented
5C2×C22⋊(C251⋊C8)C211yesmaximal, characteristic
6C22×C4⋊C4×C251C211yesmaximal, characteristic
7C22×Dic502C2211yescharacteristic
8C22×C251⋊C8C2212yes—
9#22548694—22no—
10#22548694C2214yes—
11C2×C4⋊Dic251C2211yescharacteristic
12C2×Dic251⋊C4—21no—
13C2×C22⋊Dic251—21no—
14#22548648C2218yescomplemented
15#22548638C2218yescomplemented
16C22⋊(C251⋊C8)C2214yes—
17C23×C4×C251—21no—
18C23×C4×C251C2211yescharacteristic
19C2×C4⋊C4×C251—21no—
20C2×C4⋊C4×C251—22no—
21C2×C4⋊C4×C251C2212yes—
22C2×C4⋊C4×C251C2214yes—
23C2×Dic502—21no—
24C2×Dic502—24no—
25C2×Dic502C2312yes—
26C22×Dic251—21no—
27C22×Dic251—21no—
28C2×C251⋊C8—24no—
29C2×C251⋊C8C2314yes—
30C4⋊Dic251—21no—
31C4⋊Dic251C2312yes—
32Dic251⋊C4—24no—
33C22⋊Dic251—24no—
34C24×C251D411yescharacteristic
35C22×C4×C251—24no—
36C22×C4×C251—24no—
37C22×C4×C251—22no—
38C22×C4×C251—44no—
39C22×C4×C251D411yescharacteristic
40C22×C4×C251D412yes—
41C22×C4×C251C2312yes—
42C22×C4×C251C2311yescharacteristic
43C4⋊C4×C251—24no—
44C4⋊C4×C251—22no—
45C4⋊C4×C251C2314yes—
46Dic502—24no—
47Dic502—42no—
48C2×Dic251—22no—
49C2×Dic251—22no—
50C2×Dic251—42no—
51C2×Dic251—42no—
52C251⋊C8—24no—
53C23×C251—22no—
54C23×C251—22no—
55C23×C251Q1614yes—
56C23×C251C2×D412yes—
57C23×C251C2×D411yescharacteristic
58C2×C4×C251—22no—
59C2×C4×C251—24no—
60C2×C4×C251—24no—
61C2×C4×C251—24no—
62C2×C4×C251—48no—
63C2×C4×C251—44no—
64C2×C4×C251C2×D412yes—
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251#4114yes—
67C2×C4×C251C2411yescharacteristic
68C22×C251—22no—
69C22×C251—22no—
70C22×C251—28no—
71C22×C251#12312yes—
72C22×C251C2×Q1614yes—
73C22×C251C2×Q1612yes—
74C22×C251AGL(1,ℤ/8)12yes—
75C22×C251C22×D411yescharacteristic
76C4×C251—22no—
77C4×C251—44no—
78C4×C251#12312yes—
79C4×C251C2×#4112yes—
80Dic251—42no—
81Dic251—42no—
82C2×C251—24no—
83C2×C251#48414yescomplemented
84C2×C251C2×#12311yescharacteristic
85C2×C251C22×Q1611yescharacteristic
86C2×C251C2×AGL(1,ℤ/8)11yescharacteristic
87C251C2×#48411yescharacteristic, complemented, Sylow
88C2×#484—2511nomaximal, Sylow
89C2×C22⋊Q8—2511no—
90C22×C4⋊C4D25111yescharacteristic
91#484—2518no—
92C2×#107—2512no—
93C2×#103—2512no—
94C2×C22⋊C8—2511no—
95C22×Q8—2511no—
96C23×C4—21no—
97C23×C4C2×D25111yescharacteristic
98C22×C8—2512no—
99C22⋊Q8—5022no—
100C22⋊Q8—2514no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—5021no—
104C2×C4⋊C4—2511no—
105C2×C4⋊C4C2×D25112yes—
106C2×C4⋊C4C2×D25114yes—
107C2×C22⋊C4—5021no—
108#107—2518no—
109#103—2518no—
110C22⋊C8—2514no—
111C24C22⋊D25111yescharacteristic
112C2×Q8—5024no—
113C2×Q8—5021no—
114C2×Q8—2512no—
115C22×C4—24no—
116C22×C4—24no—
117C22×C4—22no—
118C22×C4—44no—
119C22×C4—5021no—
120C22×C4—5021no—
121C22×C4C22⋊D25112yes—
122C22×C4C22⋊D25111yescharacteristic
123C22×C4C22×D25112yes—
124C22×C4C22×D25111yescharacteristic
125C2×C8—5024no—
126C2×C8—2514no—
127C4⋊C4—24no—
128C4⋊C4—22no—
129C4⋊C4—5024no—
130C4⋊C4—5021no—
131C4⋊C4—2512no—
132C4⋊C4C22×D25114yes—
133C22⋊C4—5024no—
134C23—22no—
135C23—22no—
136C23#1686080014yes—
137C23C2×C22⋊D25112yes—
138C23C2×C22⋊D25111yescharacteristic
139Q8—10042no—
140Q8—5024no—
141C2×C4—22no—
142C2×C4—24no—
143C2×C4—24no—
144C2×C4—24no—
145C2×C4—48no—
146C2×C4—44no—
147C2×C4—10042no—
148C2×C4—10042no—
149C2×C4—5022no—
150C2×C4—5022no—
151C2×C4C2×C22⋊D25112yes—
152C2×C4C2×C22⋊D25111yescharacteristic
153C2×C4#1686081014yes—
154C2×C4C23×D25111yescharacteristic
155C8—5024no—
156C22—22no—
157C22—28no—
158C22—22no—
159C22#2254869812yes—
160C22C2×#1686080014yes—
161C22C2×#1686080012yes—
162C22#2254874112yes—
163C22C22×C22⋊D25111yescharacteristic
164C4—22no—
165C4—44no—
166C4—10042no—
167C4—10042no—
168C4#2254869812yes—
169C4C2×#1686081012yes—
170C2—24no—
171C2#3063202214yesdirect product factor
172C2C2×#2254869811yescharacteristic
173C2C22×#1686080011yescharacteristic
174C2C2×#2254874111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC22×C4⋊C4×C251C2
socleC23×C251C2×D4
prime radicalC24×C251D4
hypercenterC22×C4⋊C4D251
hypocenterC251C2×#484
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. #30632022C2direct (N × H)
3. C2×#22548648C2semidirect
4. C2×#22548638C2semidirect
14. #22548648C22semidirect
15. #22548638C22semidirect
83. C2×C251#484semidirect
87. C251C2×#484semidirect
171. C2#30632022direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225486942
#306320222
C2×#225486482
C2×#225486382
C2×C22⋊(C251⋊C8)2
C22×C4⋊C4×C2512
C2×#484251