Finite Group Databasebeta

← #36363041

Subgroups — #36363041

Order 22144.

584 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 145 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×#19149704C211yesmaximal, characteristic, complemented
2C2×#19149675C211yesmaximal, characteristic, complemented
3#41×Dic173C211yesmaximal, characteristic, complemented
4#26011689C214yesmaximal, complemented
5#26011684C214yesmaximal, complemented
6#26011514C211yesmaximal, characteristic, complemented
7#26011497C211yesmaximal, characteristic, complemented
8#26011496C211yesmaximal, characteristic, complemented
9C2×AGL(1,ℤ/8)×C173C211yesmaximal, characteristic, complemented
10C22×D692C2211yescharacteristic
11C22×C22⋊D173—22no—
12#19149704—24no—
13C2×#14583813C2212yescomplemented
14C2×#14583811C2212yescomplemented
15C2×#14583810C2211yescharacteristic
16#19149675—21no—
17#19149675—21no—
18#19149675C2214yes—
19Q8×Dic173C2212yescomplemented
20D4×Dic173—22no—
21D4×Dic173C2212yescomplemented
22#19149638—21no—
23#19149638C2211yescharacteristic
24C2×C4×Dic173—21no—
25C2×C4×Dic173C2211yescharacteristic
26#19149608C2214yescomplemented
27#19149601C2214yescomplemented
28#19149600C2214yescomplemented
29C2×#41×C173C2211yescharacteristic
30C22×D4×C173C2211yescharacteristic
31AGL(1,ℤ/8)×C173—24no—
32C2×SD16×C173C2212yescomplemented
33C2×D8×C173C2212yescomplemented
34C2×M16×C173C2211yescharacteristic
35C23×D173—21no—
36C2×D692—22no—
37C2×D692—21no—
38C2×D692—22no—
39C2×D692C2312yes—
40C2×C22⋊D173—24no—
41C2×C22⋊D173—24no—
42C2×C22⋊D173—48no—
43C22×Dic173—22no—
44C22×Dic173D412yes—
45#14583813—24no—
46#14583811—24no—
47#14583810—22no—
48C2×#13063436C2312yes—
49C4⋊Dic173—22no—
50C4⋊Dic173C2312yes—
51C22⋊Dic173—22no—
52C22⋊Dic173—21no—
53C4×Dic173—22no—
54C4×Dic173—21no—
55C4×Dic173D411yescharacteristic
56C4×Dic173D411yescharacteristic
57C4×Dic173C2312yes—
58C24×C173—21no—
59#41×C173—22no—
60#41×C173D414yescomplemented
61C2×Q8×C173C2311yescharacteristic
62C2×D4×C173—21no—
63C2×D4×C173—22no—
64C2×D4×C173—21no—
65C2×D4×C173—22no—
66C2×D4×C173C2312yes—
67C2×D4×C173C2311yescharacteristic
68C22×C4×C173—21no—
69C22×C4×C173C2311yescharacteristic
70SD16×C173—24no—
71D8×C173—24no—
72M16×C173—22no—
73C2×C8×C173C2312yes—
74C22×D173—22no—
75C22×D173—44no—
76C22×D173—42no—
77D692—22no—
78D692—22no—
79D692—42no—
80C22⋊D173—48no—
81C22⋊D173—48no—
82C2×Dic173—24no—
83C2×Dic173—22no—
84C2×Dic173—44no—
85C2×Dic173C2×D412yes—
86C2×Dic173C2×D412yes—
87#13063436—22no—
88C23×C173—21no—
89C23×C173—22no—
90C23×C173—44no—
91C23×C173—42no—
92C23×C173C2×D411yescharacteristic
93Q8×C173—21no—
94Q8×C173C2×D412yescomplemented
95D4×C173—24no—
96D4×C173—22no—
97D4×C173—21no—
98D4×C173—22no—
99D4×C173—42no—
100D4×C173C2×D412yescomplemented
101C2×C4×C173—21no—
102C2×C4×C173—24no—
103C2×C4×C173C2×D411yescharacteristic
104C2×C4×C173C2×D412yes—
105C2×C4×C173#4112yes—
106C2×C4×C173C2411yescharacteristic
107C8×C173—22no—
108C22×C173—22no—
109C22×C173—22no—
110C22×C173—42no—
111C22×C173—42no—
112C22×C173—44no—
113C22×C173—84no—
114C22×C173#12112yes—
115C22×C173C22×D411yescharacteristic
116C2×D173—42no—
117C2×D173—44no—
118C2×D173—84no—
119C4×C173—22no—
120C4×C173#12112yes—
121C4×C173C22×D411yescharacteristic
122C4×C173C2×#4111yescharacteristic
123Dic173—22no—
124Dic173—42no—
125Dic173AGL(1,ℤ/8)14yescomplemented
126C2×C173—22no—
127C2×C173—42no—
128C2×C173—82no—
129C2×C173C2×#12111yescharacteristic
130C2×C173C2×AGL(1,ℤ/8)12yes—
131D173—82no—
132C173#303811yescharacteristic, complemented, Sylow
133#3038—1731nomaximal, Sylow
134C2×AGL(1,ℤ/8)—1731no—
135C2×AGL(1,ℤ/8)D17311yescharacteristic, complemented
136C2×#127—1731no—
137C4×#41—1731no—
138#460—1734no—
139#459—1734no—
140#423—1731no—
141#418—1731no—
142#418—1731no—
143C2×#41C2×D17311yescharacteristic
144C22×D4—3462no—
145C22×D4—1731no—
146C22×D4C2×D17311yescharacteristic
147AGL(1,ℤ/8)—24no—
148AGL(1,ℤ/8)—3464no—
149C2×SD16—1732no—
150C2×SD16C2×D17312yescomplemented
151C2×D8—1732no—
152C2×D8C2×D17312yescomplemented
153C2×M16—1731no—
154C2×M16C2×D17311yescharacteristic
155#127—3461no—
156#127—3461no—
157#127—1734no—
158C4×Q8—1732no—
159C4×D4—3462no—
160C4×D4—1732no—
161#117—3461no—
162#117—1731no—
163C2×C42—3461no—
164C2×C42—1731no—
165C4⋊C8—1734no—
166#102—1734no—
167#102—1734no—
168C24—21no—
169C24—3461no—
170#41—22no—
171#41C22⋊D17314yescomplemented
172C2×Q8C22×D17311yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—3464no—
178C2×D4—3464no—
179C2×D4—3461no—
180C2×D4—3462no—
181C2×D4—3462no—
182C2×D4—1732no—
183C2×D4—6928no—
184C2×D4C22×D17312yes—
185C2×D4C22×D17311yescharacteristic
186C22×C4—21no—
187C22×C4—3462no—
188C22×C4—1732no—
189C22×C4C22×D17311yescharacteristic
190SD16—24no—
191SD16—3464no—
192D8—24no—
193D8—3464no—
194M16—22no—
195M16—3462no—
196C2×C8—1732no—
197C2×C8C22×D17312yes—
198C4⋊C4—3462no—
199C4⋊C4—1732no—
200C22⋊C4—3461no—
201C22⋊C4—3462no—
202C42—3462no—
203C42—3461no—
204C42—1731no—
205C42—1731no—
206C42—1732no—
207C23—21no—
208C23—22no—
209C23—44no—
210C23—42no—
211C23—3462no—
212C23—6924no—
213C23—6922no—
214C23D4×D17311yescharacteristic
215Q8—21no—
216Q8C2×C22⋊D17312yescomplemented
217D4—24no—
218D4—22no—
219D4—21no—
220D4—22no—
221D4—42no—
222D4—3462no—
223D4—3462no—
224D4—6928no—
225D4—6928no—
226D4—6922no—
227D4C2×C22⋊D17312yescomplemented
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—3464no—
231C2×C4—3462no—
232C2×C4—1732no—
233C2×C4—1732no—
234C2×C4—6924no—
235C2×C4D4×D17311yescharacteristic
236C2×C4C2×C22⋊D17312yes—
237C2×C4#1458382512yes—
238C2×C4C23×D17311yescharacteristic
239C8—22no—
240C8—3462no—
241C22—22no—
242C22—22no—
243C22—42no—
244C22—42no—
245C22—44no—
246C22—84no—
247C22—13844no—
248C22—6922no—
249C22—6924no—
250C22#1914965112yes—
251C22C2×D4×D17311yescharacteristic
252C4—22no—
253C4—3462no—
254C4—1734no—
255C4—6922no—
256C4#1914965112yes—
257C4C22×C22⋊D17311yescharacteristic
258C4C2×#1458382511yescharacteristic
259C2—22no—
260C2—42no—
261C2—82no—
262C2—13842no—
263C2C2×#1914965111yescharacteristic
264C2AGL(1,ℤ/8)×D17312yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×AGL(1,ℤ/8)×C173C2
socleC22×C173C22×D4
hypercenterC2×AGL(1,ℤ/8)D173
hypocenterC173#3038
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19149704C2semidirect
2. C2×#19149675C2semidirect
3. #41×Dic173C2semidirect
4. #26011689C2semidirect
5. #26011684C2semidirect
6. #26011514C2semidirect
7. #26011497C2semidirect
8. #26011496C2semidirect
9. C2×AGL(1,ℤ/8)×C173C2semidirect
13. C2×#14583813C22semidirect
14. C2×#14583811C22semidirect
19. Q8×Dic173C22semidirect
21. D4×Dic173C22semidirect
26. #19149608C22semidirect
27. #19149601C22semidirect
28. #19149600C22semidirect
32. C2×SD16×C173C22semidirect
33. C2×D8×C173C22semidirect
60. #41×C173D4semidirect
94. Q8×C173C2×D4semidirect
100. D4×C173C2×D4semidirect
125. Dic173AGL(1,ℤ/8)semidirect
132. C173#3038semidirect
135. C2×AGL(1,ℤ/8)D173semidirect
150. C2×SD16C2×D173semidirect
152. C2×D8C2×D173semidirect
171. #41C22⋊D173semidirect
216. Q8C2×C22⋊D173semidirect
227. D4C2×C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497042
C2×#191496752
#41×Dic1732
#260116892
#260116842
#260115142
#260114972
#260114962
C2×AGL(1,ℤ/8)×C1732
#3038173