Finite Group Databasebeta

← #36361912

Subgroups — #36361912

Order 22144.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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×#19149677C211yesmaximal, characteristic, complemented
2C2×#19149659C211yesmaximal, characteristic
3C2×#19149655C211yesmaximal, characteristic
4C2×C42×D173C211yesmaximal, characteristic
5#26011397C211yesmaximal, characteristic, complemented
6#26011388C214yesmaximal, complemented
7#26011360C211yesmaximal, characteristic, complemented
8#26011350C214yesmaximal
9#391×C173C211yesmaximal, characteristic, complemented
10C22×Dic346C2211yescharacteristic
11C22×C4×D173C2211yescharacteristic
12C22×C4×D173C2212yes—
13#19149677—24no—
14#19149659—24no—
15#19149655—24no—
16C2×C4⋊Dic173C2211yescharacteristic
17C2×C4⋊Dic173C2212yes—
18C2×#14583805C2212yes—
19C2×#14583805C2214yes—
20C2×#14583805C2214yes—
21C2×#14583803C2212yes—
22C2×#14583803C2212yes—
23C2×C4×Dic173C2212yes—
24C2×C4×Dic173C2211yescharacteristic
25C42×D173—24no—
26#19149586C2214yes—
27C22×Q8×C173C2211yescharacteristic
28C2×C4⋊C4×C173C2211yescharacteristic
29C2×C42×C173C2211yescharacteristic
30#95×C173C2214yes—
31C23×D173C2311yescharacteristic
32C2×Dic346—22no—
33C2×Dic346—22no—
34C2×Dic346—22no—
35C22×Dic173C2312yes—
36C22×Dic173C2312yes—
37C22×Dic173C2311yescharacteristic
38C22×Dic173C2312yes—
39C2×C4×D173—24no—
40C2×C4×D173—24no—
41C2×C4×D173—22no—
42C2×C4×D173—22no—
43C2×C4×D173D414yes—
44C2×C4×D173Q818yescomplemented
45C4⋊Dic173—24no—
46C4⋊Dic173—22no—
47#14583805—24no—
48#14583805—28no—
49#14583805—28no—
50#14583803—24no—
51#14583803—24no—
52C4×Dic173—24no—
53C4×Dic173D414yes—
54C2×Q8×C173—22no—
55C2×Q8×C173—22no—
56C2×Q8×C173—22no—
57C22×C4×C173C2312yes—
58C22×C4×C173C2312yes—
59C22×C4×C173C2312yes—
60C22×C4×C173C2311yescharacteristic
61C4⋊C4×C173—22no—
62C42×C173—22no—
63C22×D173—21no—
64C22×D173—22no—
65C22×D173—21no—
66C22×D173C2×D412yes—
67C22×D173C2×Q814yes—
68Dic346—44no—
69C2×Dic173—21no—
70C2×Dic173—21no—
71C2×Dic173—22no—
72C2×Dic173—22no—
73C2×Dic173—22no—
74C2×Dic173—22no—
75C2×Dic173—24no—
76C2×Dic173C2×D412yes—
77C2×Dic173C2×D414yes—
78C2×Dic173C2×Q814yes—
79C2×Dic173#4114yes—
80C4×D173—28no—
81C4×D173—216no—
82C23×C173C2411yescharacteristic
83Q8×C173—44no—
84C2×C4×C173—22no—
85C2×C4×C173—22no—
86C2×C4×C173—22no—
87C2×C4×C173—22no—
88C2×C4×C173—22no—
89C2×C4×C173—22no—
90C2×C4×C173—24no—
91C2×C4×C173C2×D412yes—
92C2×C4×C173C2×D412yes—
93C2×C4×C173C2×Q814yes—
94C2×C4×C173#4112yes—
95C22×C173C22×D411yescharacteristic
96C22×C173C22×D411yescharacteristic
97C22×C173C22×Q812yes—
98C22×C173C2×#4111yescharacteristic
99C22×C173C2×#4112yes—
100C2×D173—22no—
101C2×D173—24no—
102C2×D173—22no—
103C2×D173—24no—
104C2×D173C4⋊Q814yescomplemented
105C4×C173—24no—
106C4×C173—42no—
107C4×C173—42no—
108C4×C173#12114yes—
109Dic173—22no—
110Dic173—42no—
111Dic173—42no—
112Dic173C22⋊Q818yescomplemented
113C2×C173C2×#12111yescharacteristic
114C2×C173C2×C22⋊Q812yes—
115C2×C173C2×C4⋊Q811yescharacteristic
116C2×C173#53211yescharacteristic
117C2×C173#53312yes—
118D173—24no—
119C173#285711yescharacteristic, complemented, Sylow
120#2857—1731nomaximal, Sylow
121C2×C4⋊Q8—1731no—
122C2×C22⋊Q8—1731no—
123C2×C22⋊Q8—1731no—
124C22×C42—1731no—
125#391—1731no—
126#391D17311yescharacteristic, complemented
127#389—1734no—
128#384—1731no—
129#380—1734no—
130C22×Q8—1731no—
131C22×Q8C2×D17311yescharacteristic
132C23×C4—1731no—
133C23×C4—1732no—
134C4⋊Q8—3464no—
135C22⋊Q8—3464no—
136C22⋊Q8—3464no—
137C2×C4⋊C4—1732no—
138C2×C4⋊C4—1731no—
139C2×C4⋊C4—1732no—
140C2×C4⋊C4—1734no—
141C2×C4⋊C4—1734no—
142C2×C4⋊C4C2×D17311yescharacteristic
143C2×C22⋊C4—1732no—
144C2×C22⋊C4—1732no—
145C2×C42—3464no—
146C2×C42—1732no—
147C2×C42—1731no—
148C2×C42C2×D17311yescharacteristic
149#95—1734no—
150#95C2×D17314yes—
151C24—1731no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—22no—
155C2×Q8—3462no—
156C2×Q8—3462no—
157C2×Q8—3462no—
158C22×C4—3464no—
159C22×C4—3464no—
160C22×C4—3462no—
161C22×C4—3462no—
162C22×C4—1732no—
163C22×C4—1732no—
164C22×C4—1731no—
165C22×C4—1734no—
166C22×C4—1738no—
167C22×C4—1732no—
168C22×C4C22×D17312yes—
169C22×C4C22×D17312yes—
170C22×C4C22×D17312yes—
171C22×C4C22×D17311yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—3464no—
174C4⋊C4—3464no—
175C4⋊C4—3468no—
176C4⋊C4—3468no—
177C4⋊C4—3462no—
178C22⋊C4—3464no—
179C22⋊C4—3464no—
180C42—22no—
181C42—3464no—
182C42—1734no—
183C23—3461no—
184C23—3462no—
185C23—3461no—
186C23—1732no—
187C23—1734no—
188C23C23×D17311yescharacteristic
189Q8—44no—
190Q8—6924no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—22no—
198C2×C4—34616no—
199C2×C4—3461no—
200C2×C4—3461no—
201C2×C4—3468no—
202C2×C4—3462no—
203C2×C4—3462no—
204C2×C4—3462no—
205C2×C4—3464no—
206C2×C4—3462no—
207C2×C4—1732no—
208C2×C4—1734no—
209C2×C4—1734no—
210C2×C4—1734no—
211C2×C4D4×D17312yes—
212C2×C4D4×D17312yes—
213C2×C4Q8×D17314yes—
214C2×C4#1458382612yes—
215C22—3462no—
216C22—3464no—
217C22—3462no—
218C22—3464no—
219C22—1734no—
220C22C2×D4×D17311yescharacteristic
221C22C2×D4×D17311yescharacteristic
222C22C2×Q8×D17312yes—
223C22#41×D17312yes—
224C22C2×#1458382611yescharacteristic
225C4—24no—
226C4—42no—
227C4—42no—
228C4—3462no—
229C4—1738no—
230C4—6922no—
231C4—6922no—
232C4#1914965214yes—
233C2—3464no—
234C2C2×#1914965211yescharacteristic
235C2C22⋊Q8×D17312yes—
236C2C4⋊Q8×D17311yescharacteristic
237C2#2601204011yescharacteristic
238C2#2601204512yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#391×C173C2
prime radicalC23×D173C23
hypercenter#391D173
hypocenterC173#2857
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#19149677C2semidirect
5. #26011397C2semidirect
6. #26011388C2semidirect
7. #26011360C2semidirect
9. #391×C173C2semidirect
44. C2×C4×D173Q8semidirect
104. C2×D173C4⋊Q8semidirect
112. Dic173C22⋊Q8semidirect
119. C173#2857semidirect
126. #391D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496772
C2×#191496592
C2×#191496552
C2×C42×D1732
#260113972
#260113882
#260113602
#260113502
#391×C1732
#2857173