Finite Group Databasebeta

← #36356828

Subgroups — #36356828

Order 22144.

560 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 147 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×#19149639C211yesmaximal, characteristic
2C2×#19149637C211yesmaximal, characteristic, complemented
3#26011429C211yesmaximal, characteristic, complemented
4#26011423C211yesmaximal, characteristic, complemented
5#26011417C212yesmaximal, complemented
6#26011416C211yesmaximal, characteristic, complemented
7#26011407C212yesmaximal
8C4⋊C4×Dic173C212yesmaximal, complemented
9C22⋊C4×Dic173C212yesmaximal, complemented
10#26011338C211yesmaximal, characteristic, complemented
11#399×C173C211yesmaximal, characteristic, complemented
12C22×C22⋊D173C2211yescharacteristic
13C23×Dic173C2211yescharacteristic
14C22×C4×D173C2211yescharacteristic
15#19149639—24no—
16#19149637—24no—
17C2×C4⋊Dic173C2212yes—
18C2×C4⋊Dic173C2211yescharacteristic
19C2×#14583805C2211yescharacteristic
20C2×C22⋊Dic173C2212yes—
21C2×#14583803C2212yes—
22C2×#14583803C2212yes—
23C2×#14583803C2211yescharacteristic
24C2×C4×Dic173C2212yes—
25C2×C4×Dic173C2211yescharacteristic
26C2×C4×Dic173C2212yes—
27#19149586C2212yes—
28#19149586C2212yes—
29#19149586C2211yescharacteristic
30#19149586C2212yes—
31#19149586C2212yes—
32C2×C4⋊C4×C173C2212yes—
33C2×C4⋊C4×C173C2211yescharacteristic
34C2×C22⋊C4×C173C2212yes—
35C2×C22⋊C4×C173C2211yescharacteristic
36#95×C173C2211yescharacteristic
37C23×D173C2311yescharacteristic
38C2×C22⋊D173—22no—
39C2×C22⋊D173—22no—
40C2×C22⋊D173—22no—
41C22×Dic173—22no—
42C22×Dic173—24no—
43C22×Dic173C2312yes—
44C22×Dic173C2311yescharacteristic
45C22×Dic173C2312yes—
46C22×Dic173C2311yescharacteristic
47C22×Dic173C2311yescharacteristic
48C2×C4×D173—24no—
49C2×C4×D173D414yes—
50C4⋊Dic173—24no—
51C4⋊Dic173—22no—
52#14583805—22no—
53C22⋊Dic173—24no—
54#14583803—24no—
55#14583803—22no—
56#14583803—24no—
57C4×Dic173—24no—
58C4×Dic173—22no—
59C4×Dic173—24no—
60C24×C173C2311yescharacteristic
61C22×C4×C173C2311yescharacteristic
62C22×C4×C173C2312yes—
63C22×C4×C173C2311yescharacteristic
64C22×C4×C173C2312yes—
65C4⋊C4×C173—22no—
66C4⋊C4×C173—24no—
67C22⋊C4×C173—22no—
68C22⋊C4×C173—24no—
69C22×D173—22no—
70C22×D173—21no—
71C22×D173—21no—
72C22×D173—22no—
73C22×D173C2×D412yes—
74C22⋊D173—28no—
75C2×Dic173—21no—
76C2×Dic173—21no—
77C2×Dic173—21no—
78C2×Dic173—22no—
79C2×Dic173—22no—
80C2×Dic173—22no—
81C2×Dic173—22no—
82C2×Dic173—22no—
83C2×Dic173—22no—
84C2×Dic173—22no—
85C2×Dic173—44no—
86C2×Dic173C2×D412yes—
87C2×Dic173#4112yes—
88C2×Dic173#4114yes—
89C4×D173—28no—
90C23×C173—21no—
91C23×C173—22no—
92C23×C173—21no—
93C23×C173—22no—
94C23×C173#4112yes—
95C23×C173C2411yescharacteristic
96C2×C4×C173—22no—
97C2×C4×C173—22no—
98C2×C4×C173—22no—
99C2×C4×C173—21no—
100C2×C4×C173—22no—
101C2×C4×C173—22no—
102C2×C4×C173—22no—
103C2×C4×C173—22no—
104C2×C4×C173C2×D412yes—
105C2×C4×C173#4114yes—
106C22×C173—22no—
107C22×C173—24no—
108C22×C173—41no—
109C22×C173—42no—
110C22×C173—41no—
111C22×C173C22×D411yescharacteristic
112C22×C173C2×#4112yes—
113C22×C173C2×#4111yescharacteristic
114C22×C173C2×#4111yescharacteristic
115C22×C173C2×#4112yes—
116C2×D173—22no—
117C2×D173—22no—
118C2×D173—22no—
119C2×D173—42no—
120C2×D173—42no—
121C4×C173—22no—
122C4×C173—24no—
123C4×C173—41no—
124C4×C173—42no—
125Dic173—22no—
126Dic173—41no—
127Dic173—42no—
128Dic173—42no—
129Dic173#12314yes—
130C2×C173—42no—
131C2×C173C2×#12311yescharacteristic
132C2×C173#52511yescharacteristic
133C2×C173#52912yes—
134C2×C173#52912yes—
135C2×C173#53211yescharacteristic
136D173—42no—
137C173#239511yescharacteristic, complemented, Sylow
138#2395—1731nomaximal, Sylow
139C2×C4×D4—1731no—
140C2×#117—1731no—
141#400—1731no—
142#399D17311yescharacteristic, complemented
143#398—1731no—
144#397—1732no—
145#397—1731no—
146#394—1732no—
147C4×C4⋊C4—1732no—
148C4×C22⋊C4—1732no—
149C4×C22⋊C4—1731no—
150C22×D4—1731no—
151C23×C4—1731no—
152C23×C4—1731no—
153C4×D4—3464no—
154#117—3464no—
155C2×C4⋊C4—1731no—
156C2×C4⋊C4—1731no—
157C2×C4⋊C4—1732no—
158C2×C4⋊C4C2×D17311yescharacteristic
159C2×C4⋊C4C2×D17312yes—
160C2×C22⋊C4—1732no—
161C2×C22⋊C4—1732no—
162C2×C22⋊C4—1731no—
163C2×C22⋊C4—1732no—
164C2×C22⋊C4C2×D17311yescharacteristic
165C2×C22⋊C4C2×D17312yes—
166C2×C42—1732no—
167C2×C42—1731no—
168C2×C42—1732no—
169#95—1732no—
170#95—1731no—
171#95—1732no—
172#95—1732no—
173#95—1732no—
174#95C2×D17311yescharacteristic
175C24—1731no—
176C24C22×D17311yescharacteristic
177C2×D4—3462no—
178C2×D4—3462no—
179C2×D4—3462no—
180C22×C4—3464no—
181C22×C4—3462no—
182C22×C4—3464no—
183C22×C4—1732no—
184C22×C4—1731no—
185C22×C4—1732no—
186C22×C4—1731no—
187C22×C4—1734no—
188C22×C4—1731no—
189C22×C4C22×D17311yescharacteristic
190C22×C4C22×D17312yes—
191C22×C4C22×D17311yescharacteristic
192C22×C4C22×D17312yes—
193C4⋊C4—22no—
194C4⋊C4—24no—
195C4⋊C4—3464no—
196C4⋊C4—3462no—
197C4⋊C4—3462no—
198C22⋊C4—22no—
199C22⋊C4—24no—
200C22⋊C4—3464no—
201C22⋊C4—3462no—
202C22⋊C4—3464no—
203C22⋊C4—3464no—
204C42—3464no—
205C42—3462no—
206C42—3464no—
207C23—21no—
208C23—22no—
209C23—21no—
210C23—22no—
211C23—3462no—
212C23—3461no—
213C23—3462no—
214C23—3461no—
215C23—1732no—
216C23#1458382712yes—
217C23C23×D17311yescharacteristic
218D4—3468no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—3468no—
228C2×C4—3461no—
229C2×C4—3461no—
230C2×C4—3462no—
231C2×C4—3461no—
232C2×C4—3462no—
233C2×C4—3462no—
234C2×C4—3462no—
235C2×C4—3462no—
236C2×C4—3462no—
237C2×C4—3462no—
238C2×C4—1732no—
239C2×C4—1732no—
240C2×C4—1734no—
241C2×C4—6924no—
242C2×C4D4×D17312yes—
243C2×C4#1458382514yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—3462no—
250C22—3462no—
251C22—3462no—
252C22—6922no—
253C22—6922no—
254C22C2×D4×D17311yescharacteristic
255C22#41×D17312yes—
256C22#41×D17311yescharacteristic
257C22C2×#1458382512yes—
258C22C2×#1458382711yescharacteristic
259C4—22no—
260C4—24no—
261C4—41no—
262C4—42no—
263C4—3462no—
264C4—1734no—
265C4—6921no—
266C4—6922no—
267C4—6922no—
268C2—42no—
269C2—6922no—
270C2#123×D17311yescharacteristic
271C2#2601200411yescharacteristic
272C2#2601201812yes—
273C2#2601202712yes—
274C2#2601203911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#399×C173C2
prime radicalC22×C22⋊D173C22
hypercenter#399D173
hypocenterC173#2395
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
2. C2×#19149637C2semidirect
3. #26011429C2semidirect
4. #26011423C2semidirect
5. #26011417C2semidirect
6. #26011416C2semidirect
8. C4⋊C4×Dic173C2semidirect
9. C22⋊C4×Dic173C2semidirect
10. #26011338C2semidirect
11. #399×C173C2semidirect
137. C173#2395semidirect
142. #399D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496392
C2×#191496372
#260114292
#260114232
#260114172
#260114162
#260114072
C4⋊C4×Dic1732
C22⋊C4×Dic1732
#260113382
#399×C1732
#2395173