Finite Group Databasebeta

← #3003×C173 #36350788

Subgroups — #3003×C173 #36350788

Order 22144.

994 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 994 conjugacy classes of subgroups — too many to draw. Only its 224 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
1C23×D4×C173C211yesmaximal, characteristic, complemented
2C2×AGL(1,ℤ/8)×C173C212yesmaximal, complemented
3#525×C173C211yesmaximal, characteristic, complemented
4#450×C173C214yesmaximal, complemented
5#447×C173C214yesmaximal, complemented
6#418×C173C212yesmaximal, complemented
7#407×C173C211yesmaximal, characteristic, complemented
8C25×C173—21no—
9C2×#41×C173C2211yescharacteristic, complemented
10C22×D4×C173—28no—
11C22×D4×C173—22no—
12C22×D4×C173—21no—
13C22×D4×C173C2214yes—
14C22×D4×C173C2212yes—
15C23×C4×C173C2211yescharacteristic
16AGL(1,ℤ/8)×C173—28no—
17C2×SD16×C173C2214yescomplemented
18C2×D8×C173C2214yescomplemented
19C2×M16×C173C2212yescomplemented
20#123×C173—21no—
21C22⋊Q8×C173C2212yescomplemented
22#121×C173C2212yescomplemented
23#120×C173—21no—
24C4×D4×C173—22no—
25#117×C173C2211yescharacteristic, complemented
26#102×C173C2218yescomplemented
27C22⋊C8×C173C2214yescomplemented
28C24×C173—28no—
29C24×C173—24no—
30C24×C173—22no—
31C24×C173—48no—
32C24×C173D411yescharacteristic
33#41×C173—24no—
34C2×Q8×C173C2311yescharacteristic, complemented
35C2×D4×C173—28no—
36C2×D4×C173—216no—
37C2×D4×C173—216no—
38C2×D4×C173—21no—
39C2×D4×C173—22no—
40C2×D4×C173—24no—
41C2×D4×C173—22no—
42C2×D4×C173—24no—
43C2×D4×C173D418yescomplemented
44C2×D4×C173C2314yes—
45C2×D4×C173C2311yescharacteristic, complemented
46C22×C4×C173—24no—
47C22×C4×C173—21no—
48C22×C4×C173D412yes—
49C22×C4×C173D411yescharacteristic
50C22×C4×C173C2312yes—
51C22×C4×C173C2311yescharacteristic
52SD16×C173—28no—
53D8×C173—28no—
54M16×C173—24no—
55C2×C8×C173C2314yescomplemented
56C4⋊C4×C173—22no—
57C4⋊C4×C173C2312yescomplemented
58C22⋊C4×C173—21no—
59C22⋊C4×C173—22no—
60C22⋊C4×C173—22no—
61C42×C173—21no—
62C23×C173—22no—
63C23×C173—24no—
64C23×C173—216no—
65C23×C173—28no—
66C23×C173—24no—
67C23×C173—21no—
68C23×C173—432no—
69C23×C173—48no—
70C23×C173—48no—
71C23×C173—48no—
72C23×C173C2×D411yescharacteristic
73C23×C173C2×D412yes—
74Q8×C173—22no—
75D4×C173—216no—
76D4×C173—28no—
77D4×C173—24no—
78D4×C173—22no—
79D4×C173—42no—
80D4×C173—44no—
81D4×C173C2×D418yes—
82C2×C4×C173—22no—
83C2×C4×C173—24no—
84C2×C4×C173—24no—
85C2×C4×C173—22no—
86C2×C4×C173—21no—
87C2×C4×C173—42no—
88C2×C4×C173C2×D412yes—
89C2×C4×C173C2×D411yescharacteristic
90C2×C4×C173C2×D414yes—
91C2×C4×C173C2411yescharacteristic
92C8×C173—24no—
93C22×C173—28no—
94C22×C173—22no—
95C22×C173—22no—
96C22×C173—22no—
97C22×C173—28no—
98C22×C173—416no—
99C22×C173—432no—
100C22×C173—48no—
101C22×C173—41no—
102C22×C173—42no—
103C22×C173#12012yes—
104C22×C173AGL(1,ℤ/8)14yescomplemented
105C22×C173C22×D411yescharacteristic
106C4×C173—22no—
107C4×C173—41no—
108C4×C173—42no—
109C4×C173#12012yes—
110C4×C173C22×D412yes—
111C2×C173—22no—
112C2×C173—24no—
113C2×C173—48no—
114C2×C173—81no—
115C2×C173C2×#12011yescharacteristic
116C2×C173C2×AGL(1,ℤ/8)12yes—
117C173#300311yescharacteristic, direct product factor, Sylow
118#3003C17311yesmaximal, characteristic, direct product factor, Sylow
119C23×D4C2×C17311yescharacteristic, complemented
120C2×AGL(1,ℤ/8)C2×C17312yescomplemented
121#525C2×C17311yescharacteristic, complemented
122#450C2×C17314yescomplemented
123#447C2×C17314yescomplemented
124#418C2×C17312yescomplemented
125#407C2×C17311yescharacteristic, complemented
126C25—21no—
127C2×#41C22×C17311yescharacteristic, complemented
128C22×D4—28no—
129C22×D4—22no—
130C22×D4—21no—
131C22×D4C22×C17314yes—
132C22×D4C22×C17312yes—
133C23×C4C22×C17311yescharacteristic
134AGL(1,ℤ/8)—28no—
135C2×SD16C22×C17314yescomplemented
136C2×D8C22×C17314yescomplemented
137C2×M16C22×C17312yescomplemented
138#123—21no—
139C22⋊Q8C22×C17312yescomplemented
140#121C22×C17312yescomplemented
141#120—21no—
142C4×D4—22no—
143#117C22×C17311yescharacteristic, complemented
144#102C22×C17318yescomplemented
145C22⋊C8C22×C17314yescomplemented
146C24—28no—
147C24—24no—
148C24—22no—
149C24—48no—
150C24D4×C17311yescharacteristic
151#41—24no—
152C2×Q8C23×C17311yescharacteristic, complemented
153C2×D4—28no—
154C2×D4—216no—
155C2×D4—216no—
156C2×D4—22no—
157C2×D4—21no—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—22no—
161C2×D4D4×C17318yescomplemented
162C2×D4C23×C17314yes—
163C2×D4C23×C17311yescharacteristic, complemented
164C22×C4—24no—
165C22×C4—21no—
166C22×C4D4×C17312yes—
167C22×C4D4×C17311yescharacteristic
168C22×C4C23×C17312yes—
169C22×C4C23×C17311yescharacteristic
170SD16—28no—
171D8—28no—
172M16—24no—
173C2×C8C23×C17314yescomplemented
174C4⋊C4—22no—
175C4⋊C4C23×C17312yescomplemented
176C22⋊C4—21no—
177C22⋊C4—22no—
178C22⋊C4—22no—
179C42—21no—
180C23—24no—
181C23—22no—
182C23—216no—
183C23—24no—
184C23—28no—
185C23—21no—
186C23—432no—
187C23—48no—
188C23—48no—
189C23—48no—
190C23C2×D4×C17312yes—
191C23C2×D4×C17311yescharacteristic
192Q8—22no—
193D4—216no—
194D4—24no—
195D4—22no—
196D4—28no—
197D4—44no—
198D4—42no—
199D4C2×D4×C17318yes—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4C2×D4×C17312yes—
207C2×C4C2×D4×C17314yes—
208C2×C4C2×D4×C17311yescharacteristic
209C2×C4C24×C17311yescharacteristic
210C8—24no—
211C22—28no—
212C22—22no—
213C22—22no—
214C22—22no—
215C22—28no—
216C22—432no—
217C22—416no—
218C22—48no—
219C22—41no—
220C22—42no—
221C22#120×C17312yes—
222C22AGL(1,ℤ/8)×C17314yescomplemented
223C22C22×D4×C17311yescharacteristic
224C4—22no—
225C4—41no—
226C4—42no—
227C4#120×C17312yes—
228C4C22×D4×C17312yes—
229C2—24no—
230C2—22no—
231C2—48no—
232C2—81no—
233C2C2×#120×C17311yescharacteristic
234C2C2×AGL(1,ℤ/8)×C17312yes—
subgroupG/H
center, socleC22×C173C22×D4
commutator, Frattini, cosocleC2×C4C24×C173
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C173 ◁ C23×C4×C173 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×D4×C173C2semidirect
2. C2×AGL(1,ℤ/8)×C173C2semidirect
3. #525×C173C2semidirect
4. #450×C173C2semidirect
5. #447×C173C2semidirect
6. #418×C173C2semidirect
7. #407×C173C2semidirect
9. C2×#41×C173C22semidirect
17. C2×SD16×C173C22semidirect
18. C2×D8×C173C22semidirect
19. C2×M16×C173C22semidirect
21. C22⋊Q8×C173C22semidirect
22. #121×C173C22semidirect
25. #117×C173C22semidirect
26. #102×C173C22semidirect
27. C22⋊C8×C173C22semidirect
34. C2×Q8×C173C23semidirect
43. C2×D4×C173D4semidirect
45. C2×D4×C173C23semidirect
55. C2×C8×C173C23semidirect
57. C4⋊C4×C173C23semidirect
104. C22×C173AGL(1,ℤ/8)semidirect
117. C173#3003direct (N × H)
118. #3003C173direct (N × H)
119. C23×D4C2×C173semidirect
120. C2×AGL(1,ℤ/8)C2×C173semidirect
121. #525C2×C173semidirect
122. #450C2×C173semidirect
123. #447C2×C173semidirect
124. #418C2×C173semidirect
125. #407C2×C173semidirect
127. C2×#41C22×C173semidirect
135. C2×SD16C22×C173semidirect
136. C2×D8C22×C173semidirect
137. C2×M16C22×C173semidirect
139. C22⋊Q8C22×C173semidirect
140. #121C22×C173semidirect
143. #117C22×C173semidirect
144. #102C22×C173semidirect
145. C22⋊C8C22×C173semidirect
152. C2×Q8C23×C173semidirect
161. C2×D4D4×C173semidirect
163. C2×D4C23×C173semidirect
173. C2×C8C23×C173semidirect
175. C4⋊C4C23×C173semidirect
222. C22AGL(1,ℤ/8)×C173semidirect

Maximal subgroups

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

subgroupindex
C23×D4×C1732
C2×AGL(1,ℤ/8)×C1732
#525×C1732
#450×C1732
#447×C1732
#418×C1732
#407×C1732
#3003173