Finite Group Databasebeta

← #34053151

Subgroups — #34053151

Order 19072.

800 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 249 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×#18011956C211yesmaximal, characteristic
2#23750100C211yesmaximal, characteristic, complemented
3#23749598C214yesmaximal, complemented
4C2×#18011825C218yesmaximal, complemented
5#518×C149C211yesmaximal, characteristic, complemented
6#18011956—24no—
7#18011956—24no—
8C2×#14558736C2212yes—
9C2×#14558735C414yescomplemented
10C22×C22⋊D149C2214yes—
11C2×D4×D149C414yescomplemented
12D4×Dic149—24no—
13#18011870C2212yescomplemented
14C2×C22⋊Dic149C2214yescomplemented
15#18011825C22116yescomplemented
16C22×D4×C149C2211yescharacteristic
17C4×D4×C149—24no—
18#117×C149C2212yescomplemented
19C2×C22⋊C4×C149C2214yescomplemented
20C23×D149C2×C414yescomplemented
21#14558736—24no—
22#14558736—24no—
23#14558735—216no—
24C2×Dic298C2×C412yescomplemented
25C2×D596C2×C412yescomplemented
26C2×C22⋊D149—28no—
27C2×C22⋊D149—28no—
28C2×C22⋊D149C2×C418yescomplemented
29C2×C22⋊D149C2314yes—
30D4×D149—216no—
31C22×Dic149—24no—
32C22×Dic149C2×C414yescomplemented
33C2×C4×D149C2×C414yescomplemented
34C4⋊Dic149—22no—
35C22⋊Dic149—24no—
36C22⋊Dic149C2314yes—
37C4×Dic149—22no—
38C24×C149C2312yes—
39C2×D4×C149C2×C414yes—
40C2×D4×C149D414yes—
41C2×D4×C149D414yes—
42C22×C4×C149—24no—
43C22×C4×C149C2311yescharacteristic
44C4⋊C4×C149—22no—
45C22⋊C4×C149—24no—
46C22⋊C4×C149C2314yes—
47C42×C149—22no—
48C22×D149—28no—
49C22×D149—84no—
50C22×D149C22×C414yes—
51Dic298—24no—
52D596—24no—
53C22⋊D149—216no—
54C22⋊D149—24no—
55C22⋊D149—24no—
56C2×Dic149—28no—
57C2×Dic149—24no—
58C2×Dic149—84no—
59C2×Dic149C22×C414yes—
60C4×D149—28no—
61C23×C149—82no—
62C23×C149C22×C414yes—
63C23×C149C2×D414yes—
64C23×C149C2×D414yes—
65C23×C149C2411yescharacteristic
66D4×C149C22⋊C4116yes—
67C2×C4×C149—24no—
68C2×C4×C149—84no—
69C2×C4×C149C22×C412yes—
70C2×C4×C149C2×D412yes—
71C2×C4×C149C2×D412yes—
72C22×C149—44no—
73C22×C149—41no—
74C22×C149—44no—
75C22×C149—44no—
76C22×C149C2×C22⋊C418yes—
77C22×C149C23×C411yescharacteristic
78C22×C149C22×D411yescharacteristic
79C22×C149C22×D411yescharacteristic
80C2×D149—24no—
81C2×D149—44no—
82C2×D149—88no—
83C4×C149—44no—
84C4×C149C2×C22⋊C414yes—
85Dic149—24no—
86Dic149—44no—
87C2×C149—21no—
88C2×C149—21no—
89C2×C149—21no—
90C2×C149—28no—
91C2×C149C22×C22⋊C411yescharacteristic
92D149—44no—
93C149#288311yescharacteristic, complemented, Sylow
94#2883—1491nomaximal, Sylow
95C2×H(5,𝔽2)—1491no—
96#518—1491no—
97#518D14911yescharacteristic, complemented
98#410—1494no—
99C2×#99—1498no—
100H(5,𝔽2)—2984no—
101H(5,𝔽2)—2984no—
102C2×#41—1494no—
103C2×#41—1492no—
104C22×D4—1494no—
105C22×D4—1494no—
106C22×D4C2×D14911yescharacteristic
107C4×D4—24no—
108C4×D4—2984no—
109#117—1492no—
110#117C2×D14912yescomplemented
111C2×C22⋊C4—1494no—
112C2×C22⋊C4C2×D14914yescomplemented
113#99—14916no—
114C24—1494no—
115C24C22×D14912yes—
116#41—29816no—
117#41—2984no—
118#41—2984no—
119C2×Q8—1492no—
120C2×D4—2988no—
121C2×D4—2988no—
122C2×D4—29816no—
123C2×D4—1498no—
124C2×D4—1494no—
125C2×D4—1492no—
126C2×D4C4×D14914yes—
127C2×D4C22⋊D14914yes—
128C2×D4D59614yes—
129C22×C4—24no—
130C22×C4—2984no—
131C22×C4—1494no—
132C22×C4—1494no—
133C22×C4C22×D14911yescharacteristic
134C4⋊C4—22no—
135C4⋊C4—2982no—
136C22⋊C4—24no—
137C22⋊C4—2984no—
138C22⋊C4—1494no—
139C22⋊C4C22×D14914yes—
140C42—22no—
141C42—2982no—
142C23—82no—
143C23—2988no—
144C23—1494no—
145C23—11924no—
146C23C2×C4×D14914yes—
147C23C2×C22⋊D14914yes—
148C23C2×D59614yes—
149C23C23×D14911yescharacteristic
150Q8—2984no—
151D4—2984no—
152D4—2984no—
153D4—29816no—
154D4—2984no—
155D4#14558713116yes—
156C2×C4—24no—
157C2×C4—84no—
158C2×C4—2988no—
159C2×C4—2988no—
160C2×C4—2984no—
161C2×C4—1494no—
162C2×C4—11924no—
163C2×C4C2×C4×D14912yes—
164C2×C4C2×C22⋊D14912yes—
165C2×C4C2×D59612yes—
166C22—44no—
167C22—44no—
168C22—41no—
169C22—44no—
170C22—2984no—
171C22—5964no—
172C22—11928no—
173C22C2×#1455871318yes—
174C22C22×C4×D14911yescharacteristic
175C22C22×C22⋊D14911yescharacteristic
176C22C22×D59611yescharacteristic
177C4—44no—
178C4—2984no—
179C4—5964no—
180C4C2×#1455871314yes—
181C2—21no—
182C2—21no—
183C2—21no—
184C2—28no—
185C2—5964no—
186C2C22×#1455871311yescharacteristic
subgroupG/H
centerC2C22×#14558713
commutatorC22×C149C23×C4
FrattiniC23C23×D149
Fitting#518×C149C2
socleC2×C149C22×C22⋊C4
prime radicalC2×#18011956C2
cosocleC23×C149C24
hypercenter#518D149
hypocenterC149#2883
LCSG ▷ C22×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22×D4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #23750100C2semidirect
3. #23749598C2semidirect
4. C2×#18011825C2semidirect
5. #518×C149C2semidirect
9. C2×#14558735C4semidirect
11. C2×D4×D149C4semidirect
13. #18011870C22semidirect
14. C2×C22⋊Dic149C22semidirect
15. #18011825C22semidirect
18. #117×C149C22semidirect
19. C2×C22⋊C4×C149C22semidirect
20. C23×D149C2×C4semidirect
24. C2×Dic298C2×C4semidirect
25. C2×D596C2×C4semidirect
28. C2×C22⋊D149C2×C4semidirect
32. C22×Dic149C2×C4semidirect
33. C2×C4×D149C2×C4semidirect
93. C149#2883semidirect
97. #518D149semidirect
110. #117C2×D149semidirect
112. C2×C22⋊C4C2×D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119562
#237501002
#237495982
C2×#180118252
#518×C1492
#2883149