Finite Group Databasebeta

← #36361900

Subgroups — #36361900

Order 22144.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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×#19149656C211yesmaximal, characteristic
2C22×C4×Dic173C211yesmaximal, characteristic
3#26011395C212yesmaximal, complemented
4#26011388C212yesmaximal, complemented
5#26011378C212yesmaximal, complemented
6#26011374C212yesmaximal, complemented
7#26011366C211yesmaximal, characteristic
8#26011363C211yesmaximal, characteristic
9#26011354C211yesmaximal, characteristic
10#26011349C211yesmaximal, characteristic
11#381×C173C211yesmaximal, characteristic
12C22×Dic346C2211yescharacteristic
13C23×Dic173C2211yescharacteristic
14C23×Dic173C2211yescharacteristic
15#19149656—24no—
16C2×C4⋊Dic173C2211yescharacteristic
17C2×C4⋊Dic173C2212yes—
18C2×#14583805C2212yes—
19C2×#14583805C2211yescharacteristic
20C2×#14583805C2211yescharacteristic
21C2×#14583805C2212yes—
22C2×C22⋊Dic173C2211yescharacteristic
23C2×C22⋊Dic173C2211yescharacteristic
24C2×C4×Dic173—24no—
25C2×C4×Dic173C2212yes—
26C2×C4×Dic173C2212yes—
27#19149586C2212yes—
28#19149586C2212yes—
29#19149586C2212yes—
30#19149586C2212yes—
31#19149586C2212yes—
32C23×C4×C173C2211yescharacteristic
33C2×C22⋊C4×C173C2211yescharacteristic
34C2×C22⋊C4×C173C2211yescharacteristic
35#95×C173C2212yes—
36#95×C173C2212yes—
37C2×Dic346—22no—
38C2×Dic346—24no—
39C22×Dic173—24no—
40C22×Dic173—24no—
41C22×Dic173—22no—
42C22×Dic173D414yes—
43C22×Dic173C2312yes—
44C22×Dic173C2311yescharacteristic
45C22×Dic173C2311yescharacteristic
46C22×Dic173C2311yescharacteristic
47C22×Dic173C2312yes—
48C22×Dic173C2311yescharacteristic
49C4⋊Dic173—22no—
50C4⋊Dic173—24no—
51#14583805—22no—
52#14583805—22no—
53#14583805—24no—
54#14583805—24no—
55C22⋊Dic173—22no—
56C22⋊Dic173—22no—
57C4×Dic173—24no—
58C4×Dic173—24no—
59C24×C173C2311yescharacteristic
60C22×C4×C173—22no—
61C22×C4×C173—22no—
62C22×C4×C173Q814yescomplemented
63C22×C4×C173C2312yes—
64C22×C4×C173C2311yescharacteristic
65C22×C4×C173C2311yescharacteristic
66C22×C4×C173C2312yes—
67C22⋊C4×C173—22no—
68C22⋊C4×C173—22no—
69Dic346—44no—
70C2×Dic173—28no—
71C2×Dic173—22no—
72C2×Dic173—22no—
73C2×Dic173—22no—
74C2×Dic173—28no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—22no—
78C2×Dic173—22no—
79C2×Dic173—22no—
80C2×Dic173—22no—
81C2×Dic173—22no—
82C2×Dic173C2×D412yes—
83C2×Dic173C2×D412yes—
84C2×Dic173#4114yes—
85C2×Dic173#4114yes—
86C23×C173—22no—
87C23×C173—22no—
88C23×C173C2×D412yes—
89C23×C173C2×Q812yes—
90C23×C173#4112yes—
91C23×C173C2411yescharacteristic
92C2×C4×C173—28no—
93C2×C4×C173—22no—
94C2×C4×C173—22no—
95C2×C4×C173—21no—
96C2×C4×C173—21no—
97C2×C4×C173—22no—
98C2×C4×C173—22no—
99C2×C4×C173—22no—
100C2×C4×C173—22no—
101C2×C4×C173—22no—
102C2×C4×C173C2×Q814yes—
103C22×C173—22no—
104C22×C173—22no—
105C22×C173—24no—
106C22×C173—24no—
107C22×C173C22⋊Q814yes—
108C22×C173C22×D411yescharacteristic
109C22×C173C22×Q811yescharacteristic
110C22×C173C2×#4111yescharacteristic
111C22×C173C2×#4112yes—
112C22×C173C2×#4112yes—
113C4×C173—24no—
114C4×C173—41no—
115C4×C173—41no—
116C4×C173—42no—
117Dic173—22no—
118Dic173—24no—
119Dic173—41no—
120Dic173—41no—
121Dic173—42no—
122Dic173#12314yes—
123C2×C173—24no—
124C2×C173C2×C22⋊Q811yescharacteristic
125C2×C173C2×#12311yescharacteristic
126C2×C173#52511yescharacteristic
127C2×C173#52912yes—
128C2×C173#53312yes—
129C173#285511yescharacteristic, complemented, Sylow
130#2855—1731nomaximal, Sylow
131C2×C22⋊Q8—1731no—
132C22×C42—1731no—
133#391—1732no—
134#389—1732no—
135#387—1732no—
136#387—1732no—
137#385—1731no—
138#385—1731no—
139#381—1731no—
140#381D17311yescharacteristic
141#380—1731no—
142C22×Q8—1731no—
143C23×C4—1731no—
144C23×C4—1731no—
145C23×C4C2×D17311yescharacteristic
146C22⋊Q8—3464no—
147C2×C4⋊C4—1731no—
148C2×C4⋊C4—1731no—
149C2×C4⋊C4—1731no—
150C2×C4⋊C4—1732no—
151C2×C4⋊C4—1732no—
152C2×C4⋊C4—1732no—
153C2×C22⋊C4—1731no—
154C2×C22⋊C4—1731no—
155C2×C22⋊C4C2×D17311yescharacteristic
156C2×C22⋊C4C2×D17311yescharacteristic
157C2×C42—3464no—
158C2×C42—1732no—
159C2×C42—1732no—
160#95—1732no—
161#95—1732no—
162#95—1732no—
163#95—1732no—
164#95—1732no—
165#95C2×D17312yes—
166#95C2×D17312yes—
167C24C22×D17311yescharacteristic
168C2×Q8—3462no—
169C2×Q8—3464no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—3462no—
173C22×C4—3464no—
174C22×C4—3464no—
175C22×C4—1732no—
176C22×C4—1731no—
177C22×C4—1731no—
178C22×C4—1734no—
179C22×C4—1732no—
180C22×C4—1731no—
181C22×C4—1731no—
182C22×C4Dic34614yescomplemented
183C22×C4C22×D17312yes—
184C22×C4C22×D17311yescharacteristic
185C22×C4C22×D17311yescharacteristic
186C22×C4C22×D17312yes—
187C4⋊C4—3462no—
188C4⋊C4—3462no—
189C4⋊C4—3462no—
190C4⋊C4—3464no—
191C4⋊C4—3464no—
192C4⋊C4—3464no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—3462no—
196C22⋊C4—3462no—
197C42—3464no—
198C42—3464no—
199C23—22no—
200C23—22no—
201C23D4×D17312yes—
202C23C2×Dic34612yes—
203C23#1458382512yes—
204C23C23×D17311yescharacteristic
205Q8—6924no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—3468no—
217C2×C4—3462no—
218C2×C4—3462no—
219C2×C4—3462no—
220C2×C4—3468no—
221C2×C4—3461no—
222C2×C4—3461no—
223C2×C4—3462no—
224C2×C4—3462no—
225C2×C4—3462no—
226C2×C4—3462no—
227C2×C4—3462no—
228C2×C4—1732no—
229C2×C4—1734no—
230C2×C4—1732no—
231C2×C4—1734no—
232C2×C4C2×Dic34614yes—
233C22—22no—
234C22—22no—
235C22—24no—
236C22—24no—
237C22#1914965614yes—
238C22C2×D4×D17311yescharacteristic
239C22C22×Dic34611yescharacteristic
240C22#41×D17312yes—
241C22#41×D17312yes—
242C22C2×#1458382511yescharacteristic
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—3462no—
248C4—3464no—
249C4—1734no—
250C4—6921no—
251C4—6921no—
252C4—6922no—
253C2—24no—
254C2C2×#1914965611yescharacteristic
255C2#123×D17311yescharacteristic
256C2#2601199911yescharacteristic
257C2#2601201812yes—
258C2#2601204412yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#381×C173C2
prime radicalC24×C173C23
hypercenter#381D173
hypocenterC173#2855
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #26011395C2semidirect
4. #26011388C2semidirect
5. #26011378C2semidirect
6. #26011374C2semidirect
62. C22×C4×C173Q8semidirect
129. C173#2855semidirect
182. C22×C4Dic346semidirect

Maximal subgroups

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

subgroupindex
C2×#191496562
C22×C4×Dic1732
#260113952
#260113882
#260113782
#260113742
#260113662
#260113632
#260113542
#260113492
#381×C1732
#2855173