Finite Group Databasebeta

← #34061291

Subgroups — #34061291

Order 19072.

800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 198 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
1#23750692C212yesmaximal, complemented
2#23750606C212yesmaximal, complemented
3#23750604C212yesmaximal, complemented
4C2×#18011970C214yesmaximal, complemented
5D149⋊#99C214yesmaximal, complemented
6C2×#18011957C211yesmaximal, characteristic
7D4×#10597010—28no—
8#18011990C2214yescomplemented
9C2×C22⋊#10597010C2218yescomplemented
10#18011970C2218yescomplemented
11#18011970C2214yescomplemented
12#18011969C2214yescomplemented
13#18011957—28no—
14C2×#14558737C411yescharacteristic, complemented
15C2×#14558737C2211yescharacteristic
16C2×#14558736C412yescomplemented
17C2×#14558736C2211yescharacteristic
18C22×D596C411yescharacteristic, complemented
19C22×D596C2212yes—
20C2×D4×D149C412yescomplemented
21C2×D4×D149C411yescharacteristic, complemented
22C2×D4×D149C2212yes—
23C2×D4×D149C2211yescharacteristic
24C2×#41×C149C411yescharacteristic, complemented
25C22×#10597010—28no—
26C4⋊#10597010—24no—
27C22⋊#10597010—28no—
28C22⋊#10597010C2318yes—
29C4×#10597010—24no—
30C23×D149C2×C412yescomplemented
31C23×D149C2×C412yescomplemented
32C23×D149C2312yes—
33#14558737—24no—
34#14558737—24no—
35#14558736—28no—
36#14558736—24no—
37C2×Dic298C2×C411yescharacteristic, complemented
38C2×D596—28no—
39C2×D596—24no—
40C2×D596C2×C411yescharacteristic
41C2×D596C2×C411yescharacteristic, complemented
42C2×D596C2×C412yescomplemented
43C2×D596C2×C412yescomplemented
44C2×D596C2312yes—
45C2×D596C2311yescharacteristic
46C2×C22⋊D149C2×C412yes—
47C2×C22⋊D149C2×C411yescharacteristic, complemented
48C2×C22⋊D149C2×C412yescomplemented
49C2×C22⋊D149C2311yescharacteristic
50D4×D149—28no—
51D4×D149—28no—
52D4×D149—24no—
53D4×D149D418yes—
54C2×C4×D149C2×C412yescomplemented
55C2×C4×D149C2×C411yescharacteristic, complemented
56C2×C4×D149C2×C412yescomplemented
57C2×C4×D149C2311yescharacteristic
58#41×C149—24no—
59C2×Q8×C149C2×C411yescharacteristic, complemented
60C2×D4×C149C2×C411yescharacteristic
61C2×D4×C149C2×C412yescomplemented
62C22×C4×C149C2×C411yescharacteristic, complemented
63C22×C4×C149C2×C412yescomplemented
64C2×#10597010—28no—
65C2×#10597010—88no—
66C22×D149—24no—
67C22×D149—24no—
68C22×D149—82no—
69C22×D149—82no—
70C22×D149—82no—
71C22×D149C22×C412yes—
72C22×D149C22×C411yescharacteristic
73C22×D149C22×C412yes—
74C22×D149C2×D418yes—
75C22×D149C2411yescharacteristic
76Dic298—22no—
77D596—24no—
78D596—24no—
79D596—24no—
80D596—22no—
81D596—22no—
82D596C22⋊C414yes—
83C22⋊D149—24no—
84C22⋊D149—22no—
85C22⋊D149—22no—
86C22⋊D149C22⋊C418yes—
87C2×Dic149C22×C411yescharacteristic
88C2×Dic149C22×C411yescharacteristic
89C4×D149—24no—
90C4×D149—22no—
91C4×D149—24no—
92C4×D149C2×D414yes—
93C23×C149C22×C411yescharacteristic
94C23×C149C22×C412yes—
95Q8×C149—22no—
96D4×C149—24no—
97D4×C149C22⋊C414yes—
98C2×C4×C149—24no—
99C2×C4×C149—22no—
100C2×C4×C149C22×C412yes—
101C2×C4×C149C22×C411yescharacteristic
102C2×C4×C149C22×C411yescharacteristic
103C22×C149—21no—
104C22×C149—42no—
105C22×C149—41no—
106C22×C149C2×C22⋊C414yes—
107C22×C149C23×C411yescharacteristic
108C2×D149—21no—
109C2×D149—22no—
110C2×D149—42no—
111C2×D149—41no—
112C2×D149—48no—
113C2×D149—42no—
114C2×D149—41no—
115C2×D149—84no—
116C2×D149—84no—
117C2×D149C2×C22⋊C414yes—
118C2×D149C22×D412yes—
119#10597010—48no—
120C4×C149—21no—
121C4×C149—22no—
122C4×C149C2×C22⋊C412yes—
123Dic149—21no—
124Dic149C2×C22⋊C412yes—
125C2×C149—21no—
126C2×C149—24no—
127C2×C149—41no—
128C2×C149C22×C22⋊C411yescharacteristic
129D149—22no—
130D149—24no—
131D149—42no—
132D149—41no—
133C149#288311yescharacteristic, complemented, Sylow
134#2883—1491nomaximal, Sylow
135C2×H(5,𝔽2)—1491no—
136#518—1492no—
137#410—1492no—
138#410—1492no—
139C2×#99—1494no—
140C2×#99—1494no—
141H(5,𝔽2)—2988no—
142C2×#41—1491no—
143C2×#41—1491no—
144C2×#41—1492no—
145C2×#41—1491no—
146C2×#41#1059701011yescharacteristic, complemented
147C22×D4—1491no—
148C22×D4—1492no—
149C22×D4—1492no—
150C22×D4—1491no—
151C22×D4—1491no—
152C22×D4—1492no—
153C4×D4—2988no—
154#117—1494no—
155C2×C22⋊C4—1498no—
156#99—1494no—
157#99—1498no—
158#99—1494no—
159C24—1492no—
160C24—1492no—
161C24—1492no—
162#41—24no—
163#41—2988no—
164#41—2984no—
165#41—2984no—
166#41—2984no—
167C2×Q8—1491no—
168C2×Q8C2×#1059701011yescharacteristic, complemented
169C2×D4—2988no—
170C2×D4—2984no—
171C2×D4—2988no—
172C2×D4—2988no—
173C2×D4—2984no—
174C2×D4—1492no—
175C2×D4—1491no—
176C2×D4—1491no—
177C2×D4—1492no—
178C2×D4—1498no—
179C2×D4—1492no—
180C2×D4—1492no—
181C2×D4—1491no—
182C2×D4—1492no—
183C2×D4—1491no—
184C2×D4—1491no—
185C2×D4C2×#1059701011yescharacteristic
186C2×D4C2×#1059701012yescomplemented
187C22×C4—2988no—
188C22×C4—1492no—
189C22×C4—1492no—
190C22×C4—1491no—
191C22×C4—1491no—
192C22×C4C2×#1059701012yescomplemented
193C22×C4C2×#1059701011yescharacteristic, complemented
194C4⋊C4—2984no—
195C22⋊C4—2988no—
196C22⋊C4—1498no—
197C42—2984no—
198C23—2984no—
199C23—2984no—
200C23—1492no—
201C23—1491no—
202C23—1498no—
203C23—1492no—
204C23—1491no—
205C23—11922no—
206C23—11922no—
207C23—11922no—
208C23C22×#1059701011yescharacteristic
209C23C22×#1059701012yes—
210Q8—22no—
211Q8—2982no—
212D4—24no—
213D4—2982no—
214D4—2984no—
215D4—2984no—
216D4—2984no—
217D4—2982no—
218D4—2984no—
219D4—2982no—
220D4—2982no—
221D4—1498no—
222D4—1494no—
223D4C22⋊#1059701014yes—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—2984no—
227C2×C4—2984no—
228C2×C4—2982no—
229C2×C4—2988no—
230C2×C4—1491no—
231C2×C4—1491no—
232C2×C4—1494no—
233C2×C4—11928no—
234C2×C4C22×#1059701012yes—
235C2×C4C22×#1059701011yescharacteristic
236C2×C4C22×#1059701011yescharacteristic
237C22—21no—
238C22—42no—
239C22—41no—
240C22—2982no—
241C22—2981no—
242C22—5968no—
243C22—5961no—
244C22—5962no—
245C22—5961no—
246C22—5962no—
247C22—1492no—
248C22—1494no—
249C22—11924no—
250C22—11924no—
251C22C2×C22⋊#1059701014yes—
252C22C23×#1059701011yescharacteristic
253C4—21no—
254C4—22no—
255C4—2981no—
256C4—5968no—
257C4—1492no—
258C4C2×C22⋊#1059701012yes—
259C2—21no—
260C2—24no—
261C2—41no—
262C2—2982no—
263C2—2984no—
264C2—5961no—
265C2—5962no—
266C2C22×C22⋊#1059701011yescharacteristic
subgroupG/H
centerC2C22×C22⋊#10597010
commutatorC22×C149C23×C4
FrattiniC22C23×#10597010
FittingC2×#41×C149C4
socleC2×C149C22×C22⋊C4
prime radicalC2×#18011957C2
cosocleC22×D149C24
hypercenterC2×#41#10597010
hypocenterC149#2883
LCSG ▷ C22×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #23750692C2semidirect
2. #23750606C2semidirect
3. #23750604C2semidirect
4. C2×#18011970C2semidirect
5. D149⋊#99C2semidirect
8. #18011990C22semidirect
9. C2×C22⋊#10597010C22semidirect
10. #18011970C22semidirect
11. #18011970C22semidirect
12. #18011969C22semidirect
14. C2×#14558737C4semidirect
16. C2×#14558736C4semidirect
18. C22×D596C4semidirect
20. C2×D4×D149C4semidirect
21. C2×D4×D149C4semidirect
24. C2×#41×C149C4semidirect
30. C23×D149C2×C4semidirect
31. C23×D149C2×C4semidirect
37. C2×Dic298C2×C4semidirect
41. C2×D596C2×C4semidirect
42. C2×D596C2×C4semidirect
43. C2×D596C2×C4semidirect
47. C2×C22⋊D149C2×C4semidirect
48. C2×C22⋊D149C2×C4semidirect
54. C2×C4×D149C2×C4semidirect
55. C2×C4×D149C2×C4semidirect
56. C2×C4×D149C2×C4semidirect
59. C2×Q8×C149C2×C4semidirect
61. C2×D4×C149C2×C4semidirect
62. C22×C4×C149C2×C4semidirect
63. C22×C4×C149C2×C4semidirect
133. C149#2883semidirect
146. C2×#41#10597010semidirect
168. C2×Q8C2×#10597010semidirect
186. C2×D4C2×#10597010semidirect
192. C22×C4C2×#10597010semidirect
193. C22×C4C2×#10597010semidirect

Maximal subgroups

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

subgroupindex
#237506922
#237506062
#237506042
C2×#180119702
D149⋊#992
C2×#180119572
#2883149