Finite Group Databasebeta

← #36356842

Subgroups — #36356842

Order 22144.

840 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 177 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×C4×D692C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D173C214yesmaximal, complemented
3#26011427C212yesmaximal, complemented
4#26011426C212yesmaximal, complemented
5#26011400C212yesmaximal, complemented
6#26011399C212yesmaximal, complemented
7#26011337C211yesmaximal, characteristic, complemented
8C2×C4×D4×C173C211yesmaximal, characteristic, complemented
9C22×D692C2211yescharacteristic
10C22×C22⋊D173C2214yes—
11C22×C4×D173C2212yes—
12C4×D692—24no—
13C4×C22⋊D173—216no—
14C2×C4⋊Dic173C2211yescharacteristic
15C2×#14583805C2214yes—
16C2×C22⋊Dic173C2214yes—
17C2×#14583803C2212yes—
18C2×#14583803C2214yes—
19C2×C4×Dic173C2212yes—
20#19149586C2212yes—
21#19149586C2212yes—
22C22×D4×C173C2211yescharacteristic
23C23×C4×C173C2212yes—
24C4×D4×C173—24no—
25C2×C4⋊C4×C173C2211yescharacteristic
26C2×C22⋊C4×C173C2212yes—
27C2×C42×C173C2211yescharacteristic
28C23×D173C2312yes—
29C2×D692—24no—
30C2×D692—22no—
31C2×C22⋊D173—28no—
32C2×C22⋊D173—28no—
33C2×C22⋊D173—28no—
34C22×Dic173C2312yes—
35C22×Dic173C2314yes—
36C2×C4×D173—28no—
37C2×C4×D173—24no—
38C4⋊Dic173—22no—
39#14583805—28no—
40C22⋊Dic173—28no—
41#14583803—28no—
42#14583803—24no—
43C4×Dic173D418yescomplemented
44C24×C173C2312yes—
45C2×D4×C173—24no—
46C2×D4×C173—22no—
47C22×C4×C173—28no—
48C22×C4×C173—24no—
49C22×C4×C173C2312yes—
50C22×C4×C173C2311yescharacteristic
51C22×C4×C173C2311yescharacteristic
52C22×C4×C173C2311yescharacteristic
53C4⋊C4×C173—22no—
54C22⋊C4×C173—24no—
55C42×C173D414yescomplemented
56C22×D173—24no—
57C22×D173—22no—
58C22×D173—24no—
59C22×D173—22no—
60C22×D173#4114yes—
61D692—28no—
62C22⋊D173—232no—
63C2×Dic173—24no—
64C2×Dic173—24no—
65C2×Dic173—22no—
66C2×Dic173—24no—
67C2×Dic173C2×D418yes—
68C4×D173—48no—
69C23×C173—24no—
70C23×C173—22no—
71C23×C173—24no—
72C23×C173—22no—
73C23×C173#4114yes—
74C23×C173C2411yescharacteristic
75D4×C173—28no—
76C2×C4×C173—22no—
77C2×C4×C173—22no—
78C2×C4×C173—22no—
79C2×C4×C173—24no—
80C2×C4×C173—48no—
81C2×C4×C173C2×D414yes—
82C2×C4×C173C2×D412yes—
83C2×C4×C173C2×D412yes—
84C2×C4×C173C2×D412yes—
85C22×C173—24no—
86C22×C173—28no—
87C22×C173—42no—
88C22×C173—44no—
89C22×C173—42no—
90C22×C173C22×D412yes—
91C22×C173C22×D411yescharacteristic
92C22×C173C2×#4112yes—
93C22×C173C2×#4112yes—
94C2×D173—28no—
95C2×D173—24no—
96C2×D173—44no—
97C2×D173—42no—
98C2×D173—42no—
99C4×C173—22no—
100C4×C173—22no—
101C4×C173—42no—
102C4×C173#12714yes—
103Dic173—28no—
104Dic173—42no—
105C2×C173—44no—
106C2×C173#52511yescharacteristic
107C2×C173#52514yes—
108C2×C173#52511yescharacteristic
109C2×C173C2×#12711yescharacteristic
110D173—44no—
111C173#239711yescharacteristic, complemented, Sylow
112#2397—1731nomaximal, Sylow
113C2×C4×D4—1734no—
114C2×C4×D4—1731no—
115C2×C4×D4D17311yescharacteristic, complemented
116#399—1732no—
117#399—1732no—
118#392—1732no—
119#392—1732no—
120#376—1731no—
121C22×D4—1734no—
122C22×D4—1731no—
123C22×D4C2×D17311yescharacteristic
124C23×C4—1732no—
125C23×C4C2×D17312yes—
126C4×D4—24no—
127C4×D4—34616no—
128C4×D4—3464no—
129C2×C4⋊C4—1734no—
130C2×C4⋊C4—1731no—
131C2×C4⋊C4C2×D17311yescharacteristic
132C2×C22⋊C4—1734no—
133C2×C22⋊C4—1734no—
134C2×C22⋊C4—1732no—
135C2×C22⋊C4C2×D17312yes—
136C2×C42—1732no—
137C2×C42C2×D17311yescharacteristic
138#95—1732no—
139#95—1732no—
140C24—1732no—
141C24C22×D17312yes—
142C2×D4—24no—
143C2×D4—22no—
144C2×D4—3464no—
145C2×D4—3468no—
146C2×D4—3468no—
147C2×D4—3468no—
148C2×D4—3462no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—3468no—
152C22×C4—3464no—
153C22×C4—1732no—
154C22×C4—1734no—
155C22×C4C22×D17312yes—
156C22×C4C22×D17311yescharacteristic
157C22×C4C22×D17311yescharacteristic
158C22×C4C22×D17311yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—3468no—
161C4⋊C4—3462no—
162C22⋊C4—24no—
163C22⋊C4—3468no—
164C22⋊C4—3468no—
165C22⋊C4—3464no—
166C42—1738no—
167C42C22⋊D17314yescomplemented
168C23—24no—
169C23—22no—
170C23—24no—
171C23—22no—
172C23—3464no—
173C23—3462no—
174C23—3462no—
175C23—3464no—
176C23—1734no—
177C23#1458382614yes—
178C23C23×D17311yescharacteristic
179D4—28no—
180D4—3468no—
181D4—34632no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—3464no—
188C2×C4—3464no—
189C2×C4—3464no—
190C2×C4—3462no—
191C2×C4—1738no—
192C2×C4—6928no—
193C2×C4D4×D17314yes—
194C2×C4C2×C22⋊D17312yes—
195C2×C4C2×C22⋊D17312yes—
196C2×C4C2×C22⋊D17312yes—
197C22—28no—
198C22—24no—
199C22—42no—
200C22—44no—
201C22—42no—
202C22—3464no—
203C22—3468no—
204C22—6924no—
205C22—6922no—
206C22—6922no—
207C22C2×D4×D17312yes—
208C22C22×C22⋊D17311yescharacteristic
209C22#41×D17312yes—
210C22C2×#1458382612yes—
211C4—22no—
212C4—22no—
213C4—42no—
214C4—3468no—
215C4—6922no—
216C4#1914967514yes—
217C2—44no—
218C2—6924no—
219C2#2601199811yescharacteristic
220C2#2601200014yes—
221C2#2601200211yescharacteristic
222C2C2×#1914967511yescharacteristic
subgroupG/H
centerC22×C4C22×D173
commutator, socle, cosocleC23×C173C24
FrattiniC23C23×D173
FittingC2×C4×D4×C173C2
hypercenterC2×C4×D4D173
hypocenterC173#2397
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D692C2semidirect
2. C2×C4×C22⋊D173C2semidirect
3. #26011427C2semidirect
4. #26011426C2semidirect
5. #26011400C2semidirect
6. #26011399C2semidirect
7. #26011337C2semidirect
8. C2×C4×D4×C173C2semidirect
43. C4×Dic173D4semidirect
55. C42×C173D4semidirect
111. C173#2397semidirect
115. C2×C4×D4D173semidirect
167. C42C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D6922
C2×C4×C22⋊D1732
#260114272
#260114262
#260114002
#260113992
#260113372
C2×C4×D4×C1732
#2397173