Finite Group Databasebeta

← #36361937

Subgroups — #36361937

Order 22144.

524 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 153 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×#19149670C211yesmaximal, characteristic, complemented
2C2×#19149668C211yesmaximal, characteristic
3#26011398C211yesmaximal, characteristic, complemented
4#26011393C211yesmaximal, characteristic
5#26011392C211yesmaximal, characteristic
6#26011383C214yesmaximal, complemented
7#26011380C214yesmaximal, complemented
8C42×Dic173C211yesmaximal, characteristic, complemented
9#391×C173C211yesmaximal, characteristic, complemented
10C22×D692C2211yescharacteristic
11#19149670—24no—
12#19149668—24no—
13C2×C4⋊Dic173C2211yescharacteristic
14C2×C4⋊Dic173C2212yes—
15C2×#14583805C2212yes—
16C2×#14583803C2214yes—
17C2×#14583803C2214yes—
18C2×#14583803C2212yes—
19C2×#14583803C2212yes—
20C2×C4×Dic173C2211yescharacteristic
21C2×C4×Dic173C2212yes—
22C2×C4×Dic173C2212yes—
23C2×C4×Dic173C2211yescharacteristic
24#19149586C2214yes—
25C22×Q8×C173C2211yescharacteristic
26C2×C4⋊C4×C173C2211yescharacteristic
27C2×C42×C173C2211yescharacteristic
28#95×C173C2214yes—
29C23×D173C2312yes—
30C2×D692—22no—
31C2×D692—22no—
32C2×D692—22no—
33C22×Dic173C2312yes—
34C22×Dic173C2311yescharacteristic
35C22×Dic173C2311yescharacteristic
36C22×Dic173C2312yes—
37C4⋊Dic173—24no—
38C4⋊Dic173—22no—
39#14583805—24no—
40#14583803—24no—
41#14583803—24no—
42#14583803—28no—
43#14583803—28no—
44C4×Dic173—24no—
45C4×Dic173—22no—
46C4×Dic173—24no—
47C4×Dic173D414yes—
48C2×Q8×C173—22no—
49C2×Q8×C173—22no—
50C2×Q8×C173—22no—
51C22×C4×C173C2312yes—
52C22×C4×C173C2312yes—
53C22×C4×C173C2312yes—
54C22×C4×C173C2311yescharacteristic
55C4⋊C4×C173—22no—
56C42×C173—22no—
57C22×D173—22no—
58C22×D173—22no—
59C22×D173—22no—
60C22×D173—24no—
61C22×D173—24no—
62D692—44no—
63C2×Dic173—22no—
64C2×Dic173—22no—
65C2×Dic173—22no—
66C2×Dic173—22no—
67C2×Dic173—22no—
68C2×Dic173—24no—
69C2×Dic173C2×D412yes—
70C2×Dic173C2×D412yes—
71C2×Dic173#4114yes—
72C23×C173C2411yescharacteristic
73Q8×C173—44no—
74C2×C4×C173—22no—
75C2×C4×C173—22no—
76C2×C4×C173—22no—
77C2×C4×C173—22no—
78C2×C4×C173—22no—
79C2×C4×C173—22no—
80C2×C4×C173—24no—
81C2×C4×C173C2×D412yes—
82C2×C4×C173#4112yes—
83C2×C4×C173#4112yes—
84C2×C4×C173#4114yes—
85C22×C173C22×D411yescharacteristic
86C22×C173C2×#4111yescharacteristic
87C22×C173C2×#4111yescharacteristic
88C22×C173C2×#4112yes—
89C22×C173C2×#4112yes—
90C2×D173—42no—
91C2×D173—44no—
92C2×D173—44no—
93C2×D173—42no—
94C2×D173—42no—
95C4×C173—24no—
96C4×C173—42no—
97C4×C173—42no—
98C4×C173#12414yes—
99Dic173—24no—
100Dic173—22no—
101Dic173—22no—
102Dic173—42no—
103C2×C173C2×#12411yescharacteristic
104C2×C173#52912yes—
105C2×C173#52912yes—
106C2×C173#53211yescharacteristic
107C2×C173#53211yescharacteristic
108D173—82no—
109C173#286111yescharacteristic, complemented, Sylow
110#2861—1731nomaximal, Sylow
111C2×#125—1731no—
112C2×#124—1731no—
113#391—1731no—
114#391D17311yescharacteristic, complemented
115#390—1731no—
116#390—1731no—
117#388—1734no—
118#388—1734no—
119C43—1731no—
120C22×Q8C2×D17311yescharacteristic
121C22×D4—1731no—
122#125—3464no—
123#124—3464no—
124C2×C4⋊C4—1732no—
125C2×C4⋊C4—1732no—
126C2×C4⋊C4—1731no—
127C2×C4⋊C4C2×D17311yescharacteristic
128C2×C22⋊C4—1732no—
129C2×C22⋊C4—1732no—
130C2×C22⋊C4—1734no—
131C2×C22⋊C4—1734no—
132C2×C42—1732no—
133C2×C42—1732no—
134C2×C42—1731no—
135C2×C42—1731no—
136C2×C42C2×D17311yescharacteristic
137#95—1734no—
138#95C2×D17314yes—
139C24—1732no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—22no—
143C2×D4—3462no—
144C2×D4—3462no—
145C2×D4—3462no—
146C22×C4—1732no—
147C22×C4—1732no—
148C22×C4—1731no—
149C22×C4—1731no—
150C22×C4C22×D17312yes—
151C22×C4C22×D17312yes—
152C22×C4C22×D17312yes—
153C22×C4C22×D17311yescharacteristic
154C4⋊C4—22no—
155C4⋊C4—3464no—
156C4⋊C4—3464no—
157C4⋊C4—3462no—
158C22⋊C4—3464no—
159C22⋊C4—3464no—
160C22⋊C4—3468no—
161C22⋊C4—3468no—
162C42—22no—
163C42—3464no—
164C42—3462no—
165C42—3464no—
166C42—1734no—
167C23—3462no—
168C23—3464no—
169C23—3462no—
170C23—3462no—
171C23—3464no—
172C23C23×D17311yescharacteristic
173Q8—44no—
174D4—6924no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—3462no—
183C2×C4—3462no—
184C2×C4—3462no—
185C2×C4—3462no—
186C2×C4—3462no—
187C2×C4—3464no—
188C2×C4—1732no—
189C2×C4—1732no—
190C2×C4—1734no—
191C2×C4D4×D17312yes—
192C2×C4#1458382512yes—
193C2×C4#1458382612yes—
194C2×C4#1458382714yes—
195C22—6922no—
196C22—6924no—
197C22—6924no—
198C22—6922no—
199C22—6922no—
200C22C2×D4×D17311yescharacteristic
201C22#41×D17312yes—
202C22C2×#1458382511yescharacteristic
203C22C2×#1458382611yescharacteristic
204C22C2×#1458382712yes—
205C4—24no—
206C4—42no—
207C4—42no—
208C4—3464no—
209C4—3462no—
210C4—3462no—
211C4—6922no—
212C4#1914966514yes—
213C2—13842no—
214C2C2×#1914966511yescharacteristic
215C2#2601202712yes—
216C2#2601202812yes—
217C2#2601204011yescharacteristic
218C2#2601204211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#391×C173C2
prime radicalC22×D692C22
hypercenter#391D173
hypocenterC173#2861
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#19149670C2semidirect
3. #26011398C2semidirect
6. #26011383C2semidirect
7. #26011380C2semidirect
8. C42×Dic173C2semidirect
9. #391×C173C2semidirect
109. C173#2861semidirect
114. #391D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496702
C2×#191496682
#260113982
#260113932
#260113922
#260113832
#260113802
C42×Dic1732
#391×C1732
#2861173