Finite Group Databasebeta

← #36361884

Subgroups — #36361884

Order 22144.

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

normal subgroups only

This group has 582 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×#19149660C211yesmaximal, characteristic
2C22×C4×Dic173C211yesmaximal, characteristic, complemented
3#26011385C212yesmaximal, complemented
4#26011384C212yesmaximal, complemented
5#26011374C212yesmaximal, complemented
6#26011373C212yesmaximal, complemented
7#26011371C211yesmaximal, characteristic
8#26011366C211yesmaximal, characteristic, complemented
9#26011364C211yesmaximal, characteristic, complemented
10#26011352C211yesmaximal, characteristic, complemented
11#386×C173C211yesmaximal, characteristic
12C23×Dic173C2211yescharacteristic
13C23×Dic173C2211yescharacteristic
14#19149660—24no—
15C2×C4⋊Dic173C2212yes—
16C2×#14583805C2211yescharacteristic
17C2×#14583805C2211yescharacteristic
18C2×#14583805C2212yes—
19C2×C22⋊Dic173C2211yescharacteristic
20C2×C22⋊Dic173C2211yescharacteristic
21C2×C22⋊Dic173C2211yescharacteristic
22C2×C22⋊Dic173C2211yescharacteristic
23C2×C22⋊Dic173C2212yes—
24C2×C4×Dic173—24no—
25C2×C4×Dic173C2212yes—
26C2×C4×Dic173C2212yes—
27#19149586C2212yes—
28#19149586C2212yes—
29#19149586C2212yes—
30#19149586C2212yes—
31#19149586C2212yes—
32C22×D4×C173C2211yescharacteristic
33C23×C4×C173C2211yescharacteristic
34C2×C22⋊C4×C173C2211yescharacteristic
35C2×C22⋊C4×C173C2212yes—
36#95×C173C2212yes—
37C22×Dic173—24no—
38C22×Dic173—22no—
39C22×Dic173—22no—
40C22×Dic173—22no—
41C22×Dic173D414yes—
42C22×Dic173C2312yes—
43C22×Dic173C2311yescharacteristic
44C22×Dic173C2311yescharacteristic
45C22×Dic173C2311yescharacteristic
46C22×Dic173C2311yescharacteristic
47C22×Dic173C2312yes—
48C4⋊Dic173—24no—
49#14583805—22no—
50#14583805—22no—
51#14583805—24no—
52C22⋊Dic173—22no—
53C22⋊Dic173—22no—
54C22⋊Dic173—24no—
55C22⋊Dic173—22no—
56C22⋊Dic173—22no—
57C4×Dic173—24no—
58C4×Dic173—24no—
59C24×C173C2311yescharacteristic
60C24×C173C2311yescharacteristic
61C2×D4×C173—22no—
62C2×D4×C173—22no—
63C2×D4×C173—22no—
64C22×C4×C173—22no—
65C22×C4×C173—24no—
66C22×C4×C173C2311yescharacteristic
67C22×C4×C173C2312yes—
68C22×C4×C173C2312yes—
69C22⋊C4×C173—22no—
70C22⋊C4×C173—24no—
71C2×Dic173—28no—
72C2×Dic173—28no—
73C2×Dic173—22no—
74C2×Dic173—22no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—21no—
79C2×Dic173—21no—
80C2×Dic173—21no—
81C2×Dic173—22no—
82C2×Dic173—22no—
83C2×Dic173—22no—
84C2×Dic173—22no—
85C2×Dic173—22no—
86C2×Dic173C2×D412yes—
87C2×Dic173C2×D412yes—
88C2×Dic173#4114yes—
89C23×C173—22no—
90C23×C173—22no—
91C23×C173—21no—
92C23×C173—22no—
93C23×C173—21no—
94C23×C173—21no—
95C23×C173—22no—
96C23×C173C2×D412yes—
97C23×C173#4112yes—
98C23×C173#4112yes—
99C23×C173C2411yescharacteristic
100D4×C173—44no—
101C2×C4×C173—28no—
102C2×C4×C173—22no—
103C2×C4×C173—22no—
104C2×C4×C173—21no—
105C2×C4×C173—22no—
106C2×C4×C173—22no—
107C2×C4×C173—21no—
108C2×C4×C173—21no—
109C2×C4×C173—22no—
110C2×C4×C173#4114yes—
111C22×C173—22no—
112C22×C173—22no—
113C22×C173—24no—
114C22×C173—24no—
115C22×C173—41no—
116C22×C173—41no—
117C22×C173—41no—
118C22×C173—42no—
119C22×C173—42no—
120C22×C173#12314yes—
121C22×C173C22×D411yescharacteristic
122C22×C173C2×#4111yescharacteristic
123C22×C173C2×#4111yescharacteristic
124C22×C173C2×#4112yes—
125C22×C173C2×#4112yes—
126C4×C173—24no—
127C4×C173—41no—
128C4×C173—42no—
129Dic173—22no—
130Dic173—22no—
131Dic173—24no—
132Dic173—41no—
133Dic173—41no—
134Dic173—42no—
135C2×C173—24no—
136C2×C173—81no—
137C2×C173C2×#12311yescharacteristic
138C2×C173#52511yescharacteristic
139C2×C173#52511yescharacteristic
140C2×C173#52912yes—
141C2×C173#52912yes—
142C173#285311yescharacteristic, complemented, Sylow
143#2853—1731nomaximal, Sylow
144C2×#123—1731no—
145C22×C42—1731no—
146#388—1732no—
147#388—1732no—
148#387—1732no—
149#387—1732no—
150#386—1731no—
151#386D17311yescharacteristic
152#385—1731no—
153#385—1731no—
154#381—1731no—
155C22×D4C2×D17311yescharacteristic
156C23×C4—1731no—
157C23×C4—1731no—
158C23×C4C2×D17311yescharacteristic
159#123—3464no—
160C2×C4⋊C4—1731no—
161C2×C4⋊C4—1731no—
162C2×C4⋊C4—1732no—
163C2×C4⋊C4—1732no—
164C2×C22⋊C4—1731no—
165C2×C22⋊C4—1731no—
166C2×C22⋊C4—1731no—
167C2×C22⋊C4—1731no—
168C2×C22⋊C4—1732no—
169C2×C22⋊C4C2×D17311yescharacteristic
170C2×C22⋊C4C2×D17312yes—
171C2×C42—3464no—
172C2×C42—1732no—
173C2×C42—1732no—
174#95—1732no—
175#95—1732no—
176#95—1732no—
177#95—1732no—
178#95—1732no—
179#95C2×D17312yes—
180C24C22×D17311yescharacteristic
181C24C22×D17311yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—3462no—
188C22×C4—3462no—
189C22×C4—3462no—
190C22×C4—3464no—
191C22×C4—1732no—
192C22×C4—1731no—
193C22×C4—1731no—
194C22×C4—1734no—
195C22×C4—1732no—
196C22×C4—1731no—
197C22×C4—1731no—
198C22×C4C22×D17312yes—
199C22×C4C22×D17311yescharacteristic
200C22×C4C22×D17312yes—
201C4⋊C4—3462no—
202C4⋊C4—3462no—
203C4⋊C4—3464no—
204C4⋊C4—3464no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—3462no—
208C22⋊C4—3462no—
209C22⋊C4—3462no—
210C22⋊C4—3462no—
211C22⋊C4—3464no—
212C42—3464no—
213C42—3464no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23D4×D17312yes—
222C23#1458382512yes—
223C23#1458382612yes—
224C23C23×D17311yescharacteristic
225D4—44no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—3468no—
236C2×C4—3468no—
237C2×C4—3462no—
238C2×C4—3462no—
239C2×C4—3461no—
240C2×C4—3461no—
241C2×C4—3461no—
242C2×C4—3461no—
243C2×C4—3461no—
244C2×C4—3461no—
245C2×C4—3462no—
246C2×C4—3462no—
247C2×C4—3462no—
248C2×C4—3462no—
249C2×C4—3462no—
250C2×C4—1732no—
251C2×C4—1732no—
252C2×C4—1734no—
253C2×C4#1458382514yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22#1914966114yes—
264C22C2×D4×D17311yescharacteristic
265C22#41×D17312yes—
266C22C2×#1458382511yescharacteristic
267C22C2×#1458382512yes—
268C22C2×#1458382611yescharacteristic
269C4—24no—
270C4—41no—
271C4—42no—
272C4—3462no—
273C4—3462no—
274C4—3464no—
275C4—6921no—
276C4—6921no—
277C4—6922no—
278C2—24no—
279C2—81no—
280C2C2×#1914966111yescharacteristic
281C2#2601199711yescharacteristic
282C2#2601199911yescharacteristic
283C2#2601202112yes—
284C2#2601202212yes—
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#386×C173C2
prime radicalC22×D4×C173C22
hypercenter#386D173
hypocenterC173#2853
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic173C2semidirect
3. #26011385C2semidirect
4. #26011384C2semidirect
5. #26011374C2semidirect
6. #26011373C2semidirect
8. #26011366C2semidirect
9. #26011364C2semidirect
10. #26011352C2semidirect
142. C173#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#191496602
C22×C4×Dic1732
#260113852
#260113842
#260113742
#260113732
#260113712
#260113662
#260113642
#260113522
#386×C1732
#2853173