Finite Group Databasebeta

← #2403×C173 #36350188

Subgroups — #2403×C173 #36350188

Order 22144.

1564 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 280 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×C173C212yesmaximal, complemented
2C2×#120×C173C214yesmaximal, complemented
3#392×C173C214yesmaximal, complemented
4#386×C173C214yesmaximal, complemented
5#379×C173C211yesmaximal, characteristic, complemented
6C25×C173C2211yescharacteristic
7C25×C173C2212yescomplemented
8C22×D4×C173—28no—
9C22×D4×C173—216no—
10C22×D4×C173C2214yes—
11C22×D4×C173C2214yescomplemented
12C22×D4×C173C2214yescomplemented
13C23×C4×C173C2212yescomplemented
14#120×C173—216no—
15C2×C22⋊C4×C173C2214yescomplemented
16C2×C22⋊C4×C173C2218yescomplemented
17C2×C22⋊C4×C173C2212yescomplemented
18#95×C173C2214yescomplemented
19C24×C173—28no—
20C24×C173—24no—
21C24×C173—28no—
22C24×C173—28no—
23C24×C173—42no—
24C24×C173—44no—
25C24×C173D414yescomplemented
26C24×C173C2311yescharacteristic
27C24×C173C2314yes—
28C24×C173C2312yes—
29C2×D4×C173—264no—
30C2×D4×C173—28no—
31C2×D4×C173—28no—
32C2×D4×C173—28no—
33C2×D4×C173—28no—
34C2×D4×C173—216no—
35C2×D4×C173—28no—
36C2×D4×C173—44no—
37C2×D4×C173D4116yescomplemented
38C22×C4×C173—24no—
39C22×C4×C173—28no—
40C22×C4×C173C2314yes—
41C22×C4×C173C2314yes—
42C22⋊C4×C173—28no—
43C22⋊C4×C173—216no—
44C22⋊C4×C173—42no—
45C23×C173—24no—
46C23×C173—24no—
47C23×C173—216no—
48C23×C173—216no—
49C23×C173—28no—
50C23×C173—24no—
51C23×C173—24no—
52C23×C173—24no—
53C23×C173—22no—
54C23×C173—28no—
55C23×C173—44no—
56C23×C173—416no—
57C23×C173—432no—
58C23×C173—48no—
59C23×C173—48no—
60C23×C173C2×D412yes—
61C23×C173C2×D418yes—
62C23×C173C2×D414yes—
63C23×C173C2×D418yescomplemented
64C23×C173C2411yescharacteristic
65D4×C173—232no—
66D4×C173—232no—
67D4×C173—416no—
68C2×C4×C173—216no—
69C2×C4×C173—24no—
70C2×C4×C173—24no—
71C2×C4×C173—28no—
72C2×C4×C173—24no—
73C2×C4×C173C2×D418yescomplemented
74C22×C173—24no—
75C22×C173—24no—
76C22×C173—216no—
77C22×C173—24no—
78C22×C173—28no—
79C22×C173—22no—
80C22×C173—216no—
81C22×C173—416no—
82C22×C173—432no—
83C22×C173—44no—
84C22×C173—44no—
85C22×C173—48no—
86C22×C173—44no—
87C22×C173#12018yescomplemented
88C22×C173C22×D411yescharacteristic
89C22×C173C22×D414yes—
90C22×C173H(5,𝔽2)12yes—
91C4×C173—28no—
92C4×C173—44no—
93C2×C173—28no—
94C2×C173—42no—
95C2×C173—48no—
96C2×C173C2×#12012yes—
97C2×C173#53411yescharacteristic
98C2×C173D4214yes—
99C173#240311yescharacteristic, direct product factor, Sylow
100#2403C17311yesmaximal, characteristic, direct product factor, Sylow
101C23×D4C2×C17312yescomplemented
102C2×#120C2×C17314yescomplemented
103#392C2×C17314yescomplemented
104#386C2×C17314yescomplemented
105#379C2×C17311yescharacteristic, complemented
106C25C22×C17311yescharacteristic
107C25C22×C17312yescomplemented
108C22×D4—28no—
109C22×D4—216no—
110C22×D4C22×C17314yes—
111C22×D4C22×C17314yescomplemented
112C22×D4C22×C17314yescomplemented
113C23×C4C22×C17312yescomplemented
114#120—216no—
115C2×C22⋊C4C22×C17314yescomplemented
116C2×C22⋊C4C22×C17318yescomplemented
117C2×C22⋊C4C22×C17312yescomplemented
118#95C22×C17314yescomplemented
119C24—28no—
120C24—24no—
121C24—28no—
122C24—28no—
123C24—42no—
124C24—44no—
125C24D4×C17314yescomplemented
126C24C23×C17311yescharacteristic
127C24C23×C17314yes—
128C24C23×C17312yes—
129C2×D4—264no—
130C2×D4—28no—
131C2×D4—28no—
132C2×D4—28no—
133C2×D4—28no—
134C2×D4—216no—
135C2×D4—28no—
136C2×D4—44no—
137C2×D4D4×C173116yescomplemented
138C22×C4—24no—
139C22×C4—28no—
140C22×C4C23×C17314yes—
141C22×C4C23×C17314yes—
142C22⋊C4—28no—
143C22⋊C4—216no—
144C22⋊C4—42no—
145C23—24no—
146C23—24no—
147C23—216no—
148C23—28no—
149C23—24no—
150C23—24no—
151C23—24no—
152C23—22no—
153C23—28no—
154C23—216no—
155C23—44no—
156C23—416no—
157C23—432no—
158C23—48no—
159C23—48no—
160C23C2×D4×C17312yes—
161C23C2×D4×C17318yes—
162C23C2×D4×C17314yes—
163C23C2×D4×C17318yescomplemented
164C23C24×C17311yescharacteristic
165D4—232no—
166D4—232no—
167D4—416no—
168C2×C4—216no—
169C2×C4—24no—
170C2×C4—24no—
171C2×C4—28no—
172C2×C4—24no—
173C2×C4C2×D4×C17318yescomplemented
174C22—24no—
175C22—24no—
176C22—216no—
177C22—24no—
178C22—28no—
179C22—22no—
180C22—216no—
181C22—416no—
182C22—432no—
183C22—44no—
184C22—44no—
185C22—48no—
186C22—44no—
187C22#120×C17318yescomplemented
188C22C22×D4×C17311yescharacteristic
189C22C22×D4×C17314yes—
190C22H(5,𝔽2)×C17312yes—
191C4—28no—
192C4—44no—
193C2—28no—
194C2—42no—
195C2—48no—
196C2C2×#120×C17312yes—
197C2#534×C17311yescharacteristic
198C2D42×C17314yes—
subgroupG/H
center, socleC23×C173C24
commutator, Frattini, cosocleC23C24×C173
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C173 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×D4×C173C2semidirect
2. C2×#120×C173C2semidirect
3. #392×C173C2semidirect
4. #386×C173C2semidirect
5. #379×C173C2semidirect
7. C25×C173C22semidirect
11. C22×D4×C173C22semidirect
12. C22×D4×C173C22semidirect
13. C23×C4×C173C22semidirect
15. C2×C22⋊C4×C173C22semidirect
16. C2×C22⋊C4×C173C22semidirect
17. C2×C22⋊C4×C173C22semidirect
18. #95×C173C22semidirect
25. C24×C173D4semidirect
37. C2×D4×C173D4semidirect
63. C23×C173C2×D4semidirect
73. C2×C4×C173C2×D4semidirect
87. C22×C173#120semidirect
99. C173#2403direct (N × H)
100. #2403C173direct (N × H)
101. C23×D4C2×C173semidirect
102. C2×#120C2×C173semidirect
103. #392C2×C173semidirect
104. #386C2×C173semidirect
105. #379C2×C173semidirect
107. C25C22×C173semidirect
111. C22×D4C22×C173semidirect
112. C22×D4C22×C173semidirect
113. C23×C4C22×C173semidirect
115. C2×C22⋊C4C22×C173semidirect
116. C2×C22⋊C4C22×C173semidirect
117. C2×C22⋊C4C22×C173semidirect
118. #95C22×C173semidirect
125. C24D4×C173semidirect
137. C2×D4D4×C173semidirect
163. C23C2×D4×C173semidirect
173. C2×C4C2×D4×C173semidirect
187. C22#120×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×#120×C1732
#392×C1732
#386×C1732
#379×C1732
#2403173