Finite Group Databasebeta

← #324×D173 #36351871

Subgroups — #324×D173 #36351871

Order 22144.

328 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 87 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×C4⋊C4×D173C211yesmaximal, characteristic
2C22⋊C8×D173C212yesmaximal
3#26011217C211yesmaximal, characteristic, complemented
4#26011216C212yesmaximal, complemented
5#324×C173C211yesmaximal, characteristic, complemented
6C22×C4×D173C411yescharacteristic
7C22×C4×D173C411yescharacteristic
8C22×C4×D173C2211yescharacteristic
9C2×C8×D173—22no—
10C2×C4⋊Dic173C411yescharacteristic
11C2×#14583805C411yescharacteristic
12C2×#14583805C2211yescharacteristic
13C4⋊C4×D173—22no—
14C4⋊C4×D173—22no—
15C22⋊#13063436C2212yes—
16#19149591—22no—
17C2×C4⋊C4×C173C2211yescharacteristic
18C22⋊C8×C173C2212yes—
19C23×D173C2×C411yescharacteristic
20C22×Dic173C2×C411yescharacteristic
21C22×Dic173C2×C411yescharacteristic
22C22×Dic173C2×C411yescharacteristic
23C2×C4×D173—21no—
24C2×C4×D173—21no—
25C2×C4×D173—22no—
26C2×C4×D173—21no—
27C2×C4×D173—21no—
28C2×C4×D173—42no—
29C2×C4×D173—42no—
30C2×C4×D173C814yescomplemented
31C2×C4×D173D412yes—
32C2×C4×D173D412yes—
33C2×#13063436—22no—
34C8×D173—44no—
35C4⋊Dic173—22no—
36#14583805—22no—
37#14583805—22no—
38C22×C4×C173C2×C411yescharacteristic
39C22×C4×C173C2×C411yescharacteristic
40C22×C4×C173C2311yescharacteristic
41C2×C8×C173—22no—
42C4⋊C4×C173—22no—
43C22×D173—22no—
44C22×D173—22no—
45C22×D173C22⋊C411yescharacteristic
46C22×D173C22⋊C411yescharacteristic
47C22×D173C2×C812yes—
48C22×D173M1612yes—
49C2×Dic173—21no—
50C2×Dic173—21no—
51C2×Dic173—21no—
52C2×Dic173—41no—
53C2×Dic173—41no—
54C2×Dic173C22⋊C412yes—
55C2×Dic173C2×C812yescomplemented
56C4×D173—24no—
57C4×D173—24no—
58C4×D173—24no—
59C4×D173—42no—
60C4×D173—42no—
61C4×D173—42no—
62#13063436—42no—
63C23×C173C22×C411yescharacteristic
64C2×C4×C173—21no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—41no—
68C2×C4×C173—41no—
69C2×C4×C173C2×C812yescomplemented
70C2×C4×C173C2×D412yes—
71C8×C173—42no—
72C22×C173—21no—
73C22×C173—21no—
74C22×C173C2×C22⋊C411yescharacteristic
75C22×C173C22×C811yescharacteristic
76C22×C173C2×M1611yescharacteristic
77C2×D173—21no—
78C2×D173—21no—
79C2×D173—24no—
80C2×D173—24no—
81C2×D173C22⋊C812yes—
82C2×D173C22⋊C812yes—
83C2×D173#9912yes—
84C2×D173#10112yes—
85C4×C173—22no—
86C4×C173—22no—
87C4×C173—41no—
88Dic173—22no—
89Dic173—22no—
90Dic173—41no—
91C2×C173—21no—
92C2×C173—21no—
93C2×C173C2×C22⋊C811yescharacteristic
94C2×C173C2×#9911yescharacteristic
95C2×C173C2×#10111yescharacteristic
96D173—22no—
97D173#32414yesdirect product factor
98C173C2×#32411yescharacteristic, complemented, Sylow
99C2×#324—1731nomaximal, Sylow
100C22×C4⋊C4—1731no—
101C2×C22⋊C8—1732no—
102#324—1731no—
103#324—1732no—
104#324D17311yescharacteristic, direct product factor
105C23×C4—1731no—
106C23×C4—1731no—
107C23×C4—1731no—
108C22×C8—3462no—
109C2×C4⋊C4—3462no—
110C2×C4⋊C4—3462no—
111C2×C4⋊C4—1731no—
112C2×C4⋊C4—1731no—
113C2×C4⋊C4—1731no—
114C2×C4⋊C4C2×D17311yescharacteristic
115C22⋊C8—3462no—
116C22⋊C8—1732no—
117C22⋊C8C2×D17312yes—
118C24—1731no—
119C22×C4—3461no—
120C22×C4—3461no—
121C22×C4—3462no—
122C22×C4—3461no—
123C22×C4—3461no—
124C22×C4—1731no—
125C22×C4—1732no—
126C22×C4—1734no—
127C22×C4—1731no—
128C22×C4—1731no—
129C22×C4—1732no—
130C22×C4—6922no—
131C22×C4—6922no—
132C22×C4C4×D17311yescharacteristic
133C22×C4C4×D17311yescharacteristic
134C22×C4C22×D17311yescharacteristic
135C2×C8—22no—
136C2×C8—3462no—
137C2×C8—6924no—
138C4⋊C4—22no—
139C4⋊C4—3462no—
140C4⋊C4—3462no—
141C4⋊C4—3462no—
142C23—3462no—
143C23—3462no—
144C23—1731no—
145C23—1732no—
146C23—1732no—
147C23—1731no—
148C23C2×C4×D17311yescharacteristic
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—3464no—
155C2×C4—3464no—
156C2×C4—3464no—
157C2×C4—3461no—
158C2×C4—3461no—
159C2×C4—3461no—
160C2×C4—1732no—
161C2×C4—1732no—
162C2×C4—6922no—
163C2×C4—6922no—
164C2×C4—6922no—
165C2×C4—6921no—
166C2×C4—6921no—
167C2×C4C8×D17312yescomplemented
168C2×C4D4×D17312yes—
169C8—42no—
170C8—6922no—
171C22—21no—
172C22—21no—
173C22—3464no—
174C22—3464no—
175C22—3461no—
176C22—3461no—
177C22—1732no—
178C22—1732no—
179C22—1732no—
180C22—1732no—
181C22C22⋊C4×D17311yescharacteristic
182C22C2×C8×D17311yescharacteristic
183C22M16×D17311yescharacteristic
184C4—22no—
185C4—22no—
186C4—41no—
187C4—3462no—
188C4—3462no—
189C4—6921no—
190C2—21no—
191C2—21no—
192C2—3462no—
193C2—1734no—
194C2C22⋊C8×D17311yescharacteristic
195C2#99×D17311yescharacteristic
196C2#101×D17311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D173
commutatorC22×C173C22×C8
FrattiniC22×C4C22×D173
Fitting#324×C173C2
socleC22×C173C2×C22⋊C4
prime radicalC23×D173C2×C4
cosocleC22×C4×C173C23
hypercenter#324D173
hypocenterC173C2×#324
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
3. #26011217C2semidirect
4. #26011216C2semidirect
5. #324×C173C2semidirect
30. C2×C4×D173C8semidirect
55. C2×Dic173C2×C8semidirect
69. C2×C4×C173C2×C8semidirect
97. D173#324direct (N × H)
98. C173C2×#324semidirect
104. #324D173direct (N × H)
167. C2×C4C8×D173semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D1732
C22⋊C8×D1732
#260112172
#260112162
#324×C1732
C2×#324173