Finite Group Databasebeta

← #36353886

Subgroups — #36353886

Order 22144.

412 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 113 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×#19149638C211yesmaximal, characteristic, complemented
2C2×#19149600C212yesmaximal, complemented
3C2×C22⋊#13063436C211yesmaximal, characteristic, complemented
4#26011253C212yesmaximal, complemented
5C2×#121×C173C211yesmaximal, characteristic
6C22×#13063436C2212yes—
7#19149638—24no—
8C2×C4⋊Dic173C2212yes—
9C2×C22⋊Dic173—21no—
10C2×C4×Dic173—21no—
11#19149600—24no—
12C22⋊#13063436—22no—
13C22×D4×C173—21no—
14C22×D4×C173C2211yescharacteristic
15C23×C4×C173C2211yescharacteristic
16#121×C173—22no—
17#121×C173C414yescomplemented
18C2×C4⋊C4×C173C2211yescharacteristic
19C2×C22⋊C4×C173—21no—
20C22×Dic173—22no—
21C2×#13063436—22no—
22C2×#13063436—22no—
23C2×#13063436—22no—
24C4⋊Dic173—22no—
25C4⋊Dic173D414yescomplemented
26C22⋊Dic173—24no—
27C4×Dic173—24no—
28C24×C173—21no—
29C24×C173D411yescharacteristic
30C2×D4×C173—24no—
31C2×D4×C173—22no—
32C2×D4×C173—21no—
33C2×D4×C173—22no—
34C2×D4×C173—44no—
35C2×D4×C173C2×C412yes—
36C22×C4×C173—22no—
37C22×C4×C173—22no—
38C22×C4×C173—21no—
39C22×C4×C173C2×C412yes—
40C22×C4×C173D411yescharacteristic
41C22×C4×C173D412yes—
42C22×C4×C173C2311yescharacteristic
43C4⋊C4×C173—21no—
44C4⋊C4×C173C2×C412yes—
45C22⋊C4×C173—24no—
46C2×Dic173—24no—
47C2×Dic173—42no—
48C2×Dic173—42no—
49#13063436—42no—
50C23×C173—21no—
51C23×C173—21no—
52C23×C173—22no—
53C23×C173—22no—
54C23×C173—22no—
55C23×C173—41no—
56C23×C173—41no—
57C23×C173—42no—
58C23×C173—42no—
59C23×C173C22⋊C412yes—
60C23×C173C2×D411yescharacteristic
61D4×C173—24no—
62D4×C173—48no—
63D4×C173—42no—
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—24no—
67C2×C4×C173—24no—
68C2×C4×C173—22no—
69C2×C4×C173—42no—
70C2×C4×C173C22⋊C412yes—
71C2×C4×C173C22×C411yescharacteristic
72C2×C4×C173C2×D412yes—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173#4112yes—
75C22×C173—22no—
76C22×C173—22no—
77C22×C173—22no—
78C22×C173—42no—
79C22×C173—42no—
80C22×C173—42no—
81C22×C173—42no—
82C22×C173—42no—
83C22×C173—82no—
84C22×C173—82no—
85C22×C173C2×C22⋊C411yescharacteristic
86C22×C173#12011yescharacteristic
87C22×C173#12311yescharacteristic
88C22×C173#13512yes—
89C22×C173AGL(1,ℤ/8)12yes—
90C4×C173—22no—
91C4×C173—42no—
92C4×C173C4×D412yes—
93C4×C173#12112yes—
94Dic173—44no—
95C2×C173—42no—
96C2×C173—82no—
97C2×C173#38611yescharacteristic
98C2×C173#41611yescharacteristic
99C2×C173#41811yescharacteristic
100C2×C173#44912yes—
101C2×C173#48212yes—
102C173#187511yescharacteristic, complemented, Sylow
103#1875—1731nomaximal, Sylow
104C2×#121D17311yescharacteristic
105C2×#117—1731no—
106C2×#102—1732no—
107C2×C22⋊C8—1731no—
108#340—1732no—
109C22×D4—21no—
110C22×D4C2×D17311yescharacteristic
111C23×C4C2×D17311yescharacteristic
112C22×C8—1732no—
113#121—22no—
114#121Dic17314yescomplemented
115#117—3464no—
116C2×C4⋊C4—1732no—
117C2×C4⋊C4C2×D17311yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—3461no—
120C2×C42—3461no—
121#102—3464no—
122C22⋊C8—3462no—
123C24—21no—
124C24C22⋊D17311yescharacteristic
125C2×D4—24no—
126C2×D4—22no—
127C2×D4—21no—
128C2×D4—22no—
129C2×D4—44no—
130C2×D4C2×Dic17312yes—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—3462no—
135C22×C4C2×Dic17312yes—
136C22×C4C22⋊D17312yes—
137C22×C4C22⋊D17311yescharacteristic
138C22×C4C22×D17311yescharacteristic
139C2×C8—3462no—
140C2×C8—3462no—
141C2×C8—3462no—
142C4⋊C4—21no—
143C4⋊C4—3462no—
144C4⋊C4—1734no—
145C4⋊C4C2×Dic17312yes—
146C22⋊C4—24no—
147C22⋊C4—3464no—
148C42—3464no—
149C23—21no—
150C23—21no—
151C23—22no—
152C23—22no—
153C23—22no—
154C23—41no—
155C23—41no—
156C23—42no—
157C23—42no—
158C23C22⋊Dic17312yes—
159C23C2×C22⋊D17311yescharacteristic
160D4—24no—
161D4—48no—
162D4—42no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—42no—
169C2×C4—3464no—
170C2×C4—6922no—
171C2×C4—6922no—
172C2×C4C22⋊Dic17312yes—
173C2×C4C22×Dic17311yescharacteristic
174C2×C4D4×D17312yes—
175C2×C4C2×C22⋊D17311yescharacteristic
176C2×C4#1458382512yes—
177C8—6922no—
178C22—22no—
179C22—22no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—42no—
185C22—42no—
186C22—82no—
187C22—82no—
188C22C2×C22⋊Dic17311yescharacteristic
189C22#1914964711yescharacteristic
190C22#1914966011yescharacteristic
191C22#1914969912yes—
192C22#1914970512yes—
193C4—22no—
194C4—42no—
195C4—6924no—
196C4D4×Dic17312yes—
197C4#1914965112yes—
198C2—42no—
199C2—82no—
200C2#2601137111yescharacteristic
201C2#2601148711yescharacteristic
202C2#2601149611yescharacteristic
203C2#2601163412yes—
204C2#2601181212yes—
subgroupG/H
centerC23C2×C22⋊D173
commutatorC2×C4×C173C22×C4
FrattiniC22×C4C22×D173
Fitting, prime radicalC2×#121×C173C2
socleC23×C173C2×D4
cosocleC22×C4×C173C23
hypercenterC2×#121D173
hypocenterC173#1875
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#19149638C2semidirect
2. C2×#19149600C2semidirect
3. C2×C22⋊#13063436C2semidirect
4. #26011253C2semidirect
17. #121×C173C4semidirect
25. C4⋊Dic173D4semidirect
102. C173#1875semidirect
114. #121Dic173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496382
C2×#191496002
C2×C22⋊#130634362
#260112532
C2×#121×C1732
#1875173