Finite Group Databasebeta

← #36353883

Subgroups — #36353883

Order 22144.

412 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 105 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×#19149653C211yesmaximal, characteristic
2C2×#19149638C211yesmaximal, characteristic, complemented
3C2×#19149601C212yesmaximal, complemented
4#26011254C212yesmaximal, complemented
5C2×C22⋊C8×C173C211yesmaximal, characteristic, complemented
6C22×D692C2211yescharacteristic
7C22×C22⋊D173—21no—
8#19149653—22no—
9#19149653C414yescomplemented
10#19149638—24no—
11C2×C4⋊Dic173C2211yescharacteristic
12C2×C4⋊Dic173C2212yes—
13C2×C22⋊Dic173—21no—
14C2×#14583803—21no—
15C2×C4×Dic173—21no—
16#19149601—24no—
17C23×C4×C173C2211yescharacteristic
18C22×C8×C173C2212yes—
19C22⋊C8×C173—22no—
20C23×D173—21no—
21C2×D692—22no—
22C2×D692—21no—
23C2×D692—22no—
24C2×D692C2×C412yes—
25C2×C22⋊D173—24no—
26C2×C22⋊D173—44no—
27C22×Dic173—22no—
28C22×Dic173—21no—
29C4⋊Dic173—22no—
30C4⋊Dic173—21no—
31C4⋊Dic173C2×C412yes—
32C4⋊Dic173D414yescomplemented
33C22⋊Dic173—24no—
34#14583803—24no—
35C4×Dic173—24no—
36C24×C173D411yescharacteristic
37C22×C4×C173—22no—
38C22×C4×C173—22no—
39C22×C4×C173C2×C412yes—
40C22×C4×C173D411yescharacteristic
41C22×C4×C173D412yes—
42C22×C4×C173C2311yescharacteristic
43C2×C8×C173—22no—
44C2×C8×C173—22no—
45C2×C8×C173—22no—
46C22×D173—22no—
47C22×D173—41no—
48C22×D173—41no—
49C22×D173—42no—
50C22×D173—42no—
51D692—24no—
52D692—42no—
53C22⋊D173—48no—
54C2×Dic173—22no—
55C2×Dic173—24no—
56C2×Dic173—42no—
57C2×Dic173—42no—
58C2×Dic173—42no—
59C23×C173—21no—
60C23×C173—21no—
61C23×C173—22no—
62C23×C173—22no—
63C23×C173C22⋊C412yes—
64C23×C173C2×D411yescharacteristic
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—24no—
68C2×C4×C173—24no—
69C2×C4×C173C22⋊C412yes—
70C2×C4×C173C22×C411yescharacteristic
71C2×C4×C173C2×D412yes—
72C2×C4×C173C2×D411yescharacteristic
73C2×C4×C173#4112yes—
74C8×C173—42no—
75C22×C173—22no—
76C22×C173—22no—
77C22×C173—22no—
78C22×C173—42no—
79C22×C173—42no—
80C22×C173C2×C22⋊C411yescharacteristic
81C22×C173#12011yescharacteristic
82C22×C173#12311yescharacteristic
83C22×C173#13512yes—
84C22×C173AGL(1,ℤ/8)12yes—
85C2×D173—42no—
86C2×D173—42no—
87C2×D173—42no—
88C2×D173—82no—
89C2×D173—82no—
90C4×C173—22no—
91C4×C173C4×D412yes—
92C4×C173#12112yes—
93Dic173—42no—
94Dic173—44no—
95C2×C173—42no—
96C2×C173#38611yescharacteristic
97C2×C173#41611yescharacteristic
98C2×C173#41811yescharacteristic
99C2×C173#44912yes—
100C2×C173#48212yes—
101D173—82no—
102C173#187511yescharacteristic, complemented, Sylow
103#1875—1731nomaximal, Sylow
104C2×#121—1731no—
105C2×#117—1731no—
106C2×#102—1732no—
107C2×C22⋊C8D17311yescharacteristic, complemented
108#340—1732no—
109C22×D4—3461no—
110C22×D4—1731no—
111C23×C4C2×D17311yescharacteristic
112C22×C8C2×D17312yes—
113#121—3462no—
114#121—1734no—
115#117—3464no—
116C2×C4⋊C4—1732no—
117C2×C4⋊C4—1731no—
118C2×C22⋊C4—3461no—
119C2×C22⋊C4—3461no—
120C2×C42—3461no—
121#102—3464no—
122C22⋊C8—22no—
123C24—3461no—
124C24D69211yescharacteristic
125C2×D4—3464no—
126C2×D4—3462no—
127C2×D4—3461no—
128C2×D4—3462no—
129C2×D4—1732no—
130C2×D4—6924no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—3462no—
134C22×C4—3461no—
135C22×C4C4×D17312yes—
136C22×C4C22⋊D17312yes—
137C22×C4D69211yescharacteristic
138C22×C4C22×D17311yescharacteristic
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C4⋊C4—3462no—
143C4⋊C4—3461no—
144C4⋊C4—1732no—
145C4⋊C4—1734no—
146C22⋊C4—3464no—
147C22⋊C4—3464no—
148C42—3464no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23—3462no—
154C23—6921no—
155C23—6921no—
156C23—6922no—
157C23—6922no—
158C23#1458380312yes—
159C23C2×D69211yescharacteristic
160D4—3464no—
161D4—6928no—
162D4—6922no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—3462no—
168C2×C4—3464no—
169C2×C4—6922no—
170C2×C4—6922no—
171C2×C4—6922no—
172C2×C4#1458380312yes—
173C2×C4C2×C4×D17311yescharacteristic
174C2×C4D4×D17312yes—
175C2×C4C2×C22⋊D17311yescharacteristic
176C2×C4#1458382512yes—
177C8—42no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—13842no—
184C22—13842no—
185C22—6922no—
186C22—6922no—
187C22—6922no—
188C22C2×#1458380311yescharacteristic
189C22#1914964911yescharacteristic
190C22#1914966311yescharacteristic
191C22#1914970112yes—
192C22#1914970712yes—
193C4—22no—
194C4—6922no—
195C4—6924no—
196C4#1914963912yes—
197C4#1914965112yes—
198C2—42no—
199C2—13842no—
200C2#2601136811yescharacteristic
201C2#2601148811yescharacteristic
202C2#2601149711yescharacteristic
203C2#2601163812yes—
204C2#2601181612yes—
subgroupG/H
centerC23C2×D692
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
FittingC2×C22⋊C8×C173C2
socleC23×C173C2×D4
prime radicalC2×#19149653C2
cosocleC22×C4×C173C23
hypercenterC2×C22⋊C8D173
hypocenterC173#1875
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#19149638C2semidirect
3. C2×#19149601C2semidirect
4. #26011254C2semidirect
5. C2×C22⋊C8×C173C2semidirect
9. #19149653C4semidirect
32. C4⋊Dic173D4semidirect
102. C173#1875semidirect
107. C2×C22⋊C8D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496532
C2×#191496382
C2×#191496012
#260112542
C2×C22⋊C8×C1732
#1875173