Finite Group Databasebeta

← C2×#23750598 #34061271

Subgroups — C2×#23750598 #34061271

Order 19072.

792 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 792 conjugacy classes of subgroups — too many to draw. Only its 218 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×#14558745C211yesmaximal, characteristic, complemented
2C23×#11924753C211yesmaximal, characteristic, complemented
3#23750598C218yesmaximal, direct product factor
4C2×#18011968C212yesmaximal, complemented
5C2×C22⋊#11924753C212yesmaximal, complemented
6C22×#14558735C211yesmaximal, characteristic
7C2×#14558745—24no—
8C2×#14558745C2214yescomplemented
9C22×#11924753—24no—
10C22×#11924753C2212yes—
11C22×#11924753C2212yes—
12C22×#11924753C2214yescomplemented
13#18011968C2218yescomplemented
14C22⋊#11924753C2218yescomplemented
15C2×#14558735—24no—
16C2×#14558735—24no—
17C2×#14558735C414yes—
18C2×#14558735C2214yes—
19C22×Dic298C411yescharacteristic
20C22×C22⋊D149C411yescharacteristic
21C22×C22⋊D149C411yescharacteristic
22C23×Dic149C2211yescharacteristic
23C23×Dic149C2211yescharacteristic
24C22×C4×D149C2211yescharacteristic
25C22×D4×C149C411yescharacteristic
26#14558745—28no—
27C2×#11924753—24no—
28C2×#11924753—28no—
29C2×#11924753—24no—
30C2×#11924753C2314yes—
31C2×#11924753C2314yes—
32C23×D149C2×C411yescharacteristic
33#14558735—216no—
34#14558735—28no—
35#14558735—28no—
36C2×Dic298—24no—
37C2×Dic298C2×C414yes—
38C2×C22⋊D149—24no—
39C2×C22⋊D149—24no—
40C2×C22⋊D149C2×C414yes—
41C2×C22⋊D149C2×C414yes—
42C22×Dic149—22no—
43C22×Dic149—22no—
44C22×Dic149C2×C411yescharacteristic
45C22×Dic149C2×C412yes—
46C22×Dic149C2×C411yescharacteristic
47C22×Dic149C2×C412yes—
48C22×Dic149D414yes—
49C22×Dic149D414yes—
50C22×Dic149C2312yes—
51C22×Dic149C2311yescharacteristic
52C22×Dic149C2312yes—
53C2×C4×D149—22no—
54C2×C4×D149—22no—
55C2×C4×D149C2×C412yes—
56C2×C4×D149C2312yes—
57C24×C149C2×C411yescharacteristic
58C24×C149C2×C411yescharacteristic
59C2×D4×C149—24no—
60C2×D4×C149C2×C414yes—
61C22×C4×C149C2×C411yescharacteristic
62#11924753—24no—
63#11924753—24no—
64C22×D149—22no—
65C22×D149—24no—
66C22×D149C22×C412yes—
67Dic298—28no—
68C22⋊D149—28no—
69C22⋊D149—28no—
70C2×Dic149—22no—
71C2×Dic149—24no—
72C2×Dic149—22no—
73C2×Dic149—22no—
74C2×Dic149C22⋊C414yes—
75C2×Dic149C22⋊C418yes—
76C2×Dic149C22×C412yes—
77C2×Dic149C22×C411yescharacteristic
78C2×Dic149C22×C412yes—
79C2×Dic149C2×D412yes—
80C2×Dic149C2×D412yes—
81C2×Dic149C2×D414yes—
82C2×Dic149C2×D414yes—
83C2×Dic149C2411yescharacteristic
84C4×D149—24no—
85C4×D149—22no—
86C4×D149—22no—
87C23×C149—24no—
88C23×C149—22no—
89C23×C149—24no—
90C23×C149C22⋊C414yes—
91C23×C149C22×C412yes—
92C23×C149C22×C412yes—
93C23×C149C22×C411yescharacteristic
94D4×C149—28no—
95C2×C4×C149—22no—
96C2×C4×C149C22×C412yes—
97C22×C149—24no—
98C22×C149—28no—
99C22×C149—22no—
100C22×C149—24no—
101C22×C149—44no—
102C22×C149C2×C22⋊C414yes—
103C22×C149C2×C22⋊C412yes—
104C22×C149#13114yes—
105C22×C149C23×C411yescharacteristic
106C2×D149—22no—
107C2×D149—24no—
108C2×D149—44no—
109C4×C149—22no—
110Dic149—22no—
111Dic149C2×C22⋊C412yes—
112Dic149C2×C22⋊C414yes—
113Dic149C22×D411yescharacteristic
114Dic149C22×D411yescharacteristic
115C2×C149—24no—
116C2×C149—42no—
117C2×C149#40814yescomplemented
118C2×C149C22×C22⋊C411yescharacteristic
119C2×C149C2×#13112yes—
120D149—42no—
121C149C2×#40811yescharacteristic, complemented, Sylow
122C2×#408—1491nomaximal, Sylow
123C22×#41—1491no—
124C22×M16—1491no—
125C23×C8—1491no—
126#408—1498no—
127C2×C22⋊C8—1492no—
128C2×C22⋊C8—1492no—
129C2×#41—2984no—
130C2×#41—2984no—
131C2×#41—1494no—
132C2×#41—1494no—
133C22×Q8—1491no—
134C22×D4—1491no—
135C22×D4—1491no—
136C22×D4#1059701011yescharacteristic
137C23×C4—1491no—
138C23×C4—1491no—
139C23×C4—1491no—
140C2×M16—2984no—
141C2×M16—1494no—
142C22×C8—2984no—
143C22×C8—1492no—
144C22×C8—1494no—
145C22×C8—1492no—
146C22⋊C8—1498no—
147C22⋊C8—1498no—
148C24—1491no—
149C24C2×#1059701011yescharacteristic
150C24C2×#1059701011yescharacteristic
151#41—29816no—
152#41—2988no—
153#41—2988no—
154C2×Q8—2984no—
155C2×Q8—1494no—
156C2×D4—24no—
157C2×D4—2984no—
158C2×D4—2984no—
159C2×D4—1494no—
160C2×D4—1494no—
161C2×D4C2×#1059701014yes—
162C22×C4—2982no—
163C22×C4—2982no—
164C22×C4—2982no—
165C22×C4—2982no—
166C22×C4—1492no—
167C22×C4—1492no—
168C22×C4—1491no—
169C22×C4—1492no—
170C22×C4—1492no—
171C22×C4—1494no—
172C22×C4—1494no—
173C22×C4—1492no—
174C22×C4—1492no—
175C22×C4—1491no—
176C22×C4—1491no—
177C22×C4C2×#1059701011yescharacteristic
178M16—2988no—
179C2×C8—2984no—
180C2×C8—2988no—
181C2×C8—2984no—
182C2×C8—1494no—
183C2×C8—1494no—
184C23—24no—
185C23—22no—
186C23—24no—
187C23—2982no—
188C23—2984no—
189C23—1492no—
190C23C22⋊#1059701014yes—
191C23C22×#1059701012yes—
192C23C22×#1059701012yes—
193C23C22×#1059701011yescharacteristic
194Q8—2988no—
195D4—28no—
196D4—2988no—
197D4—2988no—
198C2×C4—22no—
199C2×C4—2982no—
200C2×C4—2982no—
201C2×C4—2984no—
202C2×C4—2982no—
203C2×C4—2982no—
204C2×C4—2982no—
205C2×C4—2984no—
206C2×C4—1492no—
207C2×C4—1494no—
208C2×C4—1498no—
209C2×C4—1492no—
210C2×C4—1492no—
211C2×C4—1494no—
212C2×C4—1494no—
213C2×C4—1491no—
214C2×C4—1492no—
215C2×C4—1491no—
216C2×C4C22×#1059701012yes—
217C8—2984no—
218C8—2984no—
219C22—24no—
220C22—28no—
221C22—22no—
222C22—24no—
223C22—44no—
224C22—2982no—
225C22—2984no—
226C22—5964no—
227C22C2×C22⋊#1059701014yes—
228C22C2×C22⋊#1059701012yes—
229C22#1801199814yes—
230C22C23×#1059701011yescharacteristic
231C4—22no—
232C4—2982no—
233C4—1491no—
234C4—1491no—
235C4—1492no—
236C4—1494no—
237C2—24no—
238C2—42no—
239C2—5962no—
240C2#2375059814yesdirect product factor
241C2C22×C22⋊#1059701011yescharacteristic
242C2C2×#1801199812yes—
subgroupG/H
centerC23C22×#10597010
commutatorC22×C149C23×C4
FrattiniC22C23×#10597010
FittingC22×D4×C149C4
socleC23×C149C22×C4
prime radicalC22×#14558735C2
cosocleC2×Dic149C24
hypercenterC22×D4#10597010
hypocenterC149C2×#408
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14558745C2semidirect
2. C23×#11924753C2semidirect
3. #23750598C2direct (N × H)
4. C2×#18011968C2semidirect
5. C2×C22⋊#11924753C2semidirect
8. C2×#14558745C22semidirect
12. C22×#11924753C22semidirect
13. #18011968C22semidirect
14. C22⋊#11924753C22semidirect
117. C2×C149#408semidirect
121. C149C2×#408semidirect
240. C2#23750598direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145587452
C23×#119247532
#237505982
C2×#180119682
C2×C22⋊#119247532
C22×#145587352
C2×#408149