Finite Group Databasebeta

← C2×#323×C173 #36349241

Subgroups — C2×#323×C173 #36349241

Order 22144.

456 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C22×C22⋊C4×C173C211yesmaximal, characteristic
2C2×C22⋊C8×C173C212yesmaximal, complemented
3#323×C173C214yesmaximal, direct product factor
4C25×C173C411yescharacteristic
5C23×C4×C173C411yescharacteristic
6C23×C4×C173C2211yescharacteristic
7C22×C8×C173—22no—
8C2×C22⋊C4×C173—22no—
9C2×C22⋊C4×C173—22no—
10C2×C22⋊C4×C173C412yes—
11C2×C22⋊C4×C173C2212yes—
12C22⋊C8×C173—22no—
13C22⋊C8×C173C2214yescomplemented
14C24×C173—21no—
15C24×C173—21no—
16C24×C173—22no—
17C24×C173—42no—
18C24×C173—42no—
19C24×C173C814yescomplemented
20C24×C173C2×C411yescharacteristic
21C24×C173C2×C412yes—
22C22×C4×C173—21no—
23C22×C4×C173—21no—
24C22×C4×C173—42no—
25C22×C4×C173—42no—
26C22×C4×C173C2×C412yes—
27C22×C4×C173C2×C411yescharacteristic
28C22×C4×C173D412yes—
29C22×C4×C173D412yes—
30C22×C4×C173C2311yescharacteristic
31C2×C8×C173—24no—
32C2×C8×C173—44no—
33C22⋊C4×C173—24no—
34C22⋊C4×C173—24no—
35C23×C173—22no—
36C23×C173—22no—
37C23×C173—24no—
38C23×C173—24no—
39C23×C173—22no—
40C23×C173—22no—
41C23×C173—48no—
42C23×C173—42no—
43C23×C173—42no—
44C23×C173—48no—
45C23×C173—42no—
46C23×C173—42no—
47C23×C173—42no—
48C23×C173—42no—
49C23×C173C22⋊C411yescharacteristic
50C23×C173C22⋊C411yescharacteristic
51C23×C173C2×C812yes—
52C23×C173C2×C814yescomplemented
53C23×C173M1612yes—
54C23×C173C22×C411yescharacteristic
55C2×C4×C173—24no—
56C2×C4×C173—22no—
57C2×C4×C173—41no—
58C2×C4×C173—42no—
59C2×C4×C173—41no—
60C2×C4×C173—42no—
61C2×C4×C173—42no—
62C2×C4×C173—42no—
63C2×C4×C173C22⋊C412yes—
64C2×C4×C173C2×D412yes—
65C8×C173—44no—
66C22×C173—24no—
67C22×C173—24no—
68C22×C173—21no—
69C22×C173—21no—
70C22×C173—21no—
71C22×C173—21no—
72C22×C173—24no—
73C22×C173—48no—
74C22×C173—44no—
75C22×C173—44no—
76C22×C173—48no—
77C22×C173—42no—
78C22×C173—42no—
79C22×C173C22⋊C812yes—
80C22×C173C22⋊C812yes—
81C22×C173#9912yes—
82C22×C173#10012yes—
83C22×C173C2×C22⋊C411yescharacteristic
84C22×C173C22×C811yescharacteristic
85C22×C173C2×M1611yescharacteristic
86C4×C173—22no—
87C4×C173—22no—
88C4×C173—42no—
89C2×C173—22no—
90C2×C173—21no—
91C2×C173—21no—
92C2×C173—44no—
93C2×C173#32314yesdirect product factor
94C2×C173C2×C22⋊C811yescharacteristic
95C2×C173C2×#9911yescharacteristic
96C2×C173C2×#10011yescharacteristic
97C173C2×#32311yescharacteristic, direct product factor, Sylow
98C2×#323C17311yesmaximal, characteristic, direct product factor, Sylow
99C22×C22⋊C4C2×C17311yescharacteristic
100C2×C22⋊C8C2×C17312yescomplemented
101#323C2×C17314yesdirect product factor
102C25C4×C17311yescharacteristic
103C23×C4C4×C17311yescharacteristic
104C23×C4C22×C17311yescharacteristic
105C22×C8—22no—
106C2×C22⋊C4—22no—
107C2×C22⋊C4—22no—
108C2×C22⋊C4C4×C17312yes—
109C2×C22⋊C4C22×C17312yes—
110C22⋊C8—22no—
111C22⋊C8C22×C17314yescomplemented
112C24—21no—
113C24—21no—
114C24—22no—
115C24—42no—
116C24—42no—
117C24C8×C17314yescomplemented
118C24C2×C4×C17311yescharacteristic
119C24C2×C4×C17312yes—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4C2×C4×C17312yes—
125C22×C4C2×C4×C17311yescharacteristic
126C22×C4D4×C17312yes—
127C22×C4D4×C17312yes—
128C22×C4C23×C17311yescharacteristic
129C2×C8—24no—
130C2×C8—44no—
131C22⋊C4—24no—
132C22⋊C4—24no—
133C23—22no—
134C23—22no—
135C23—22no—
136C23—24no—
137C23—24no—
138C23—22no—
139C23—48no—
140C23—42no—
141C23—42no—
142C23—48no—
143C23—42no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23C22⋊C4×C17311yescharacteristic
148C23C22⋊C4×C17311yescharacteristic
149C23C2×C8×C17312yes—
150C23C2×C8×C17314yescomplemented
151C23M16×C17312yes—
152C23C22×C4×C17311yescharacteristic
153C2×C4—24no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—42no—
161C2×C4C22⋊C4×C17312yes—
162C2×C4C2×D4×C17312yes—
163C8—44no—
164C22—24no—
165C22—24no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—24no—
171C22—44no—
172C22—44no—
173C22—48no—
174C22—48no—
175C22—42no—
176C22—42no—
177C22C22⋊C8×C17312yes—
178C22C22⋊C8×C17312yes—
179C22#99×C17312yes—
180C22#100×C17312yes—
181C22C2×C22⋊C4×C17311yescharacteristic
182C22C22×C8×C17311yescharacteristic
183C22C2×M16×C17311yescharacteristic
184C4—22no—
185C4—22no—
186C4—42no—
187C2—21no—
188C2—21no—
189C2—22no—
190C2—44no—
191C2#323×C17314yesdirect product factor
192C2C2×C22⋊C8×C17311yescharacteristic
193C2C2×#99×C17311yescharacteristic
194C2C2×#100×C17311yescharacteristic
subgroupG/H
center, socleC23×C173C22⋊C4
commutatorC22C22×C8×C173
Frattini, cosocleC22×C4C23×C173
prime radicalC25×C173C4
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23×C173 ◁ C23×C4×C173 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊C8×C173C2semidirect
3. #323×C173C2direct (N × H)
13. C22⋊C8×C173C22semidirect
19. C24×C173C8semidirect
52. C23×C173C2×C8semidirect
93. C2×C173#323direct (N × H)
97. C173C2×#323direct (N × H)
98. C2×#323C173direct (N × H)
100. C2×C22⋊C8C2×C173semidirect
101. #323C2×C173direct (N × H)
111. C22⋊C8C22×C173semidirect
117. C24C8×C173semidirect
150. C23C2×C8×C173semidirect
191. C2#323×C173direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊C4×C1732
C2×C22⋊C8×C1732
#323×C1732
C2×#323173