Finite Group Databasebeta

← #36354105

Subgroups — #36354105

Order 22144.

380 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 89 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
1#26012050C211yesmaximal, characteristic
2C2×#19149594C211yesmaximal, characteristic, complemented
3C2×#19149593C212yesmaximal, complemented
4#26011257C212yesmaximal, complemented
5#518×C173C211yesmaximal, characteristic, complemented
6C22×C22⋊D173C2211yescharacteristic
7#19149662—21no—
8#19149660C412yescomplemented
9#19149651—22no—
10#19149648—21no—
11#19149647C412yescomplemented
12C2×C22⋊Dic173C2212yes—
13C2×C22⋊Dic173C2211yescharacteristic
14#19149594—22no—
15#19149593—24no—
16C22×D4×C173C2211yescharacteristic
17C4×D4×C173—24no—
18#117×C173—21no—
19C2×C22⋊C4×C173—21no—
20C2×C22⋊C4×C173C2212yes—
21C23×D173C2×C411yescharacteristic, complemented
22C2×C22⋊D173—22no—
23C2×C22⋊D173—22no—
24C2×C22⋊D173—21no—
25C2×C22⋊D173—21no—
26C2×C22⋊D173—22no—
27C22×Dic173—21no—
28C22×Dic173—22no—
29C22×Dic173C2×C411yescharacteristic, complemented
30#14583805—22no—
31C22⋊Dic173—22no—
32C22⋊Dic173—21no—
33C22⋊Dic173—22no—
34C22⋊Dic173—22no—
35C22⋊Dic173C2×C412yescomplemented
36#14583803—22no—
37C24×C173D411yescharacteristic
38C24×C173C2311yescharacteristic
39C2×D4×C173—22no—
40C2×D4×C173—22no—
41C2×D4×C173C2×C412yes—
42C2×D4×C173D412yes—
43C22×C4×C173—22no—
44C22×C4×C173—22no—
45C22×C4×C173D411yescharacteristic
46C4⋊C4×C173—22no—
47C22⋊C4×C173—22no—
48C22⋊C4×C173—22no—
49C22⋊C4×C173D414yescomplemented
50C42×C173—22no—
51C22×D173—21no—
52C22×D173—22no—
53C22×D173—81no—
54C22⋊D173—41no—
55C22⋊D173—42no—
56C22⋊D173—41no—
57C22⋊D173—82no—
58C2×Dic173—22no—
59C2×Dic173—21no—
60C2×Dic173—22no—
61C2×Dic173—42no—
62C2×Dic173—81no—
63C2×Dic173—82no—
64C23×C173—21no—
65C23×C173—21no—
66C23×C173—81no—
67C23×C173—81no—
68C23×C173C22⋊C412yes—
69C23×C173C22×C411yescharacteristic
70C23×C173C2×D411yescharacteristic
71C23×C173C2×D412yes—
72C23×C173C2×D411yescharacteristic
73C23×C173#4112yes—
74D4×C173—24no—
75D4×C173—24no—
76C2×C4×C173—21no—
77C2×C4×C173—21no—
78C2×C4×C173—24no—
79C2×C4×C173—82no—
80C2×C4×C173—82no—
81C2×C4×C173C22⋊C412yes—
82C22×C173—22no—
83C22×C173—42no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173—41no—
87C22×C173—42no—
88C22×C173—42no—
89C22×C173—42no—
90C22×C173—81no—
91C22×C173C2×C22⋊C411yescharacteristic
92C22×C173C4×D412yes—
93C22×C173#12011yescharacteristic
94C22×C173#12112yes—
95C22×C173#12311yescharacteristic
96C2×D173—42no—
97C2×D173—41no—
98C2×D173—82no—
99C4×C173—22no—
100C4×C173—44no—
101Dic173—42no—
102Dic173—41no—
103Dic173—82no—
104C2×C173—21no—
105C2×C173—22no—
106C2×C173—21no—
107C2×C173—22no—
108C2×C173—21no—
109C2×C173—42no—
110C2×C173#38611yescharacteristic
111D173—81no—
112C173#191311yescharacteristic, complemented, Sylow
113#1913—1731nomaximal, Sylow
114#534—1731no—
115#518D17311yescharacteristic, complemented
116C2×#99—1731no—
117C2×#99—1732no—
118#342—1732no—
119C22×D4—1731no—
120C22×D4C2×D17311yescharacteristic
121#123—3461no—
122#123—1732no—
123#121—3462no—
124#120—3461no—
125#120—1732no—
126C4×D4—24no—
127#117—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—1732no—
130C2×C22⋊C4—1731no—
131C2×C22⋊C4C2×D17312yes—
132#99—3462no—
133#99—3464no—
134C24—1731no—
135C24C22⋊D17311yescharacteristic
136C24C22×D17311yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—3462no—
140C2×D4—3462no—
141C2×D4—3461no—
142C2×D4—3461no—
143C2×D4—3462no—
144C2×D4C4×D17312yes—
145C2×D4D69212yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—3461no—
149C22×C4—3462no—
150C22×C4—1731no—
151C22×C4C22⋊D17311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—3462no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—3461no—
157C22⋊C4—3462no—
158C22⋊C4—3462no—
159C22⋊C4—3462no—
160C22⋊C4—3462no—
161C22⋊C4—1732no—
162C22⋊C4C22⋊D17314yescomplemented
163C42—22no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—3462no—
169C23—3461no—
170C23—13841no—
171C23#1458380312yes—
172C23C2×C4×D17311yescharacteristic
173C23C2×C22⋊D17311yescharacteristic
174C23C2×C22⋊D17312yes—
175C23C2×D69211yescharacteristic
176C23#1458382612yes—
177D4—24no—
178D4—24no—
179D4—13842no—
180D4—6921no—
181D4—6922no—
182D4—6921no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—3462no—
189C2×C4—3461no—
190C2×C4—3462no—
191C2×C4—13842no—
192C2×C4—13841no—
193C2×C4—6922no—
194C2×C4#1458380312yes—
195C22—22no—
196C22—42no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—13842no—
205C22—6922no—
206C22—6921no—
207C22C2×#1458380311yescharacteristic
208C22C4×C22⋊D17312yes—
209C22#1914964811yescharacteristic
210C22#1914965312yes—
211C22#1914966211yescharacteristic
212C4—22no—
213C4—44no—
214C4—13842no—
215C4—6922no—
216C4—6921no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—13841no—
224C2#2601137011yescharacteristic
subgroupG/H
centerC2#26011370
commutatorC23×C173C22×C4
FrattiniC24C22×D173
Fitting#518×C173C2
socleC2×C173#386
prime radical#26012050C2
cosocleC24×C173C23
hypercenter#518D173
hypocenterC173#1913
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#19149594C2semidirect
3. C2×#19149593C2semidirect
4. #26011257C2semidirect
5. #518×C173C2semidirect
8. #19149660C4semidirect
11. #19149647C4semidirect
21. C23×D173C2×C4semidirect
29. C22×Dic173C2×C4semidirect
35. C22⋊Dic173C2×C4semidirect
49. C22⋊C4×C173D4semidirect
112. C173#1913semidirect
115. #518D173semidirect
162. C22⋊C4C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
#260120502
C2×#191495942
C2×#191495932
#260112572
#518×C1732
#1913173