Finite Group Databasebeta

← C23×C22⋊#3624 #17979435

Subgroups — C23×C22⋊#3624 #17979435

Order 4352.

7646 conjugacy classes of subgroups; only the 2232 normal subgroups are shown.

normal subgroups only

This group has 7646 conjugacy classes of subgroups — too many to draw. Only its 2232 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
1C24×#3624C212yesmaximal, complemented
2C22×C22⋊#3624C2128yesmaximal, direct product factor
3C25×Dic17C211yesmaximal, characteristic
4C23×#3624—216no—
5C23×#3624C22128yescomplemented
6C2×C22⋊#3624C221112yesdirect product factor
7C24×Dic17—216no—
8C24×Dic17C411yescharacteristic
9C24×Dic17C4114yes—
10C24×Dic17C22114yes—
11C24×Dic17C2211yescharacteristic
12C26×C17C411yescharacteristic
13C25×C17—28no—
14C25×C17—28no—
15C25×C17C8116yescomplemented
16C25×C17C2×C4114yes—
17C25×C17C2×C411yescharacteristic
18C23×Dic17—28no—
19C23×Dic17—2112no—
20C23×Dic17—2112no—
21C23×Dic17C2×C4114yes—
22C23×Dic17C2×C4184yes—
23C23×Dic17C2×C417yes—
24C23×Dic17D4116yes—
25C23×Dic17C23128yes—
26C23×Dic17C2317yes—
27C22×#3624—2112no—
28C22×#3624C23156yescomplemented
29C22⋊#3624C23164yesdirect product factor
30C24×C17—2128no—
31C24×C17—256no—
32C24×C17—256no—
33C24×C17C22⋊C418yes—
34C24×C17C2×C818yes—
35C24×C17C2×C81112yescomplemented
36C24×C17M1618yes—
37C24×C17C22×C4128yes—
38C24×C17C22×C417yes—
39C22×Dic17—256no—
40C22×Dic17—2336no—
41C22×Dic17—2112no—
42C22×Dic17C22⋊C4156yes—
43C22×Dic17C22×C4128yes—
44C22×Dic17C22×C4156yes—
45C22×Dic17C22×C4121yes—
46C22×Dic17C2×D4156yes—
47C22×Dic17C2418yes—
48C22×Dic17C2417yes—
49C2×#3624—2112no—
50C2×#3624C24116yescomplemented
51C23×C22⋊C8—171nomaximal, Sylow
52C23×C17—256no—
53C23×C17—256no—
54C23×C17—2448no—
55C23×C17C22⋊C8164yescomplemented
56C23×C17C2×C22⋊C4128yes—
57C23×C17C22×C8128yes—
58C23×C17C22×C81112yescomplemented
59C23×C17C2×M16128yes—
60C23×C17C23×C418yes—
61C23×C17C23×C417yes—
62C2×Dic17—256no—
63C2×Dic17—2112no—
64C2×Dic17—216no—
65C2×Dic17C2×C22⋊C4184yes—
66C2×Dic17C23×C418yes—
67C2×Dic17C23×C417yes—
68C2×Dic17C22×D4128yes—
69C2×Dic17C2511yescharacteristic
70#3624—216no—
71C25×C4—171no—
72C24×C8—172no—
73C22×C22⋊C8—1728no—
74C22×C17—2224no—
75C22×C17—28no—
76C22×C17—28no—
77C22×C17C2×C22⋊C81112yescomplemented
78C22×C17C22×C22⋊C4114yes—
79C22×C17C23×C8114yes—
80C22×C17C23×C8116yescomplemented
81C22×C17C22×M16114yes—
82C22×C17C24×C411yescharacteristic
83Dic17—28no—
84Dic17C22×C22⋊C4114yes—
85Dic17C23×D412yes—
86C26#59511yescharacteristic
87C24×C4—171no—
88C24×C4—1714no—
89C24×C4—1714no—
90C24×C4—171no—
91C24×C4—3416no—
92C23×C8—1728no—
93C23×C8—3416no—
94C2×C22⋊C8—17112no—
95C2×C17—216no—
96C2×C17C22×C22⋊C8128yescomplemented
97C2×C17C23×C22⋊C411yescharacteristic
98C2×C17C24×C811yescharacteristic
99C2×C17C23×M1611yescharacteristic
100C25—28no—
101C25—28no—
102C25#3624116yescomplemented
103C25C2×#595114yes—
104C25C2×#59511yescharacteristic
105C23×C4—1716no—
106C23×C4—177no—
107C23×C4—177no—
108C23×C4—1784no—
109C23×C4—1714no—
110C23×C4—1728no—
111C23×C4—34112no—
112C23×C4—34112no—
113C23×C4—348no—
114C22×C8—1756no—
115C22×C8—34112no—
116C22⋊C8—1764no—
117C17C23×C22⋊C811yescharacteristic, complemented, Sylow
118C24—256no—
119C24—256no—
120C24—2128no—
121C24C2×#362418yes—
122C24C2×#36241112yescomplemented
123C24#6324718yes—
124C24C22⋊#59518yes—
125C24C22×#595128yes—
126C24C22×#59517yes—
127C22×C4—1756no—
128C22×C4—1756no—
129C22×C4—1721no—
130C22×C4—177no—
131C22×C4—1756no—
132C22×C4—1728no—
133C22×C4—178no—
134C22×C4—34112no—
135C22×C4—34336no—
136C22×C4—3456no—
137C2×C8—1716no—
138C2×C8—34112no—
139C23—2448no—
140C23—256no—
141C23—256no—
142C23C22⋊#3624164yesdirect product factor
143C23C22×#3624128yes—
144C23C22×#36241112yescomplemented
145C23C2×#63247128yes—
146C23C2×C22⋊#595128yes—
147C23C23×#59518yes—
148C23C23×#59517yes—
149C2×C4—1728no—
150C2×C4—1784no—
151C2×C4—171no—
152C2×C4—177no—
153C2×C4—178no—
154C2×C4—3416no—
155C2×C4—34112no—
156C2×C4—3456no—
157C8—3416no—
158C22—28no—
159C22—28no—
160C22—2224no—
161C22C2×C22⋊#36241112yesdirect product factor
162C22C22×C22⋊#595114yes—
163C22C23×#3624114yes—
164C22C23×#3624116yescomplemented
165C22C22×#63247114yes—
166C22C24×#59511yescharacteristic
167C4—172no—
168C4—1714no—
169C4—348no—
170C2—216no—
171C2C22×C22⋊#3624128yesdirect product factor
172C2C23×C22⋊#59511yescharacteristic
173C2C24×#362411yescharacteristic
174C2C23×#6324711yescharacteristic
subgroupG/H
centerC25C2×#595
commutatorC2×C17C24×C8
FrattiniC22C24×#595
Fitting, prime radicalC26×C17C4
socleC25×C17C2×C4
cosocleC2×Dic17C25
hypercenterC26#595
hypocenterC17C23×C22⋊C8
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C25 ◁ C26

Complements

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

NH ≅ G/Ntype
1. C24×#3624C2semidirect
2. C22×C22⋊#3624C2direct (N × H)
5. C23×#3624C22semidirect
6. C2×C22⋊#3624C22direct (N × H)
15. C25×C17C8semidirect
28. C22×#3624C23semidirect
29. C22⋊#3624C23direct (N × H)
35. C24×C17C2×C8semidirect
50. C2×#3624C24semidirect
55. C23×C17C22⋊C8semidirect
58. C23×C17C22×C8semidirect
77. C22×C17C2×C22⋊C8semidirect
80. C22×C17C23×C8semidirect
96. C2×C17C22×C22⋊C8semidirect
102. C25#3624semidirect
117. C17C23×C22⋊C8semidirect
122. C24C2×#3624semidirect
142. C23C22⋊#3624direct (N × H)
144. C23C22×#3624semidirect
161. C22C2×C22⋊#3624direct (N × H)
164. C22C23×#3624semidirect
171. C2C22×C22⋊#3624direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#36242
C22×C22⋊#36242
C25×Dic172
C23×C22⋊C817