Finite Group Databasebeta

← #34060115

Subgroups — #34060115

Order 19072.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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#23750648C211yesmaximal, characteristic
2#23750570C211yesmaximal, characteristic
3#23750479C214yesmaximal
4#23750180C211yesmaximal, characteristic
5#18011978—21no—
6#18011968C2212yes—
7#18011966—21no—
8(C2×#10597010)⋊C4—21no—
9(C2×#10597010)⋊C4—21no—
10(C2×#10597010)⋊C4C2212yes—
11#18011911C411yescharacteristic, complemented
12#18011902—21no—
13#18011891—21no—
14#18011887C2212yes—
15Q8×Dic149C412yescomplemented
16#18011876—21no—
17#18011869—21no—
18C42×D149C2211yescharacteristic
19C4⋊Q8×C149C411yescharacteristic, complemented
20C22×#10597010—22no—
21C2×#11924753—22no—
22C2×Dic298—21no—
23C2×C4×D149D411yescharacteristic
24C2×C4×D149D411yescharacteristic
25C2×C4×D149C2311yescharacteristic
26C4⋊Dic149—22no—
27C4⋊Dic149—21no—
28C4⋊Dic149C2×C412yes—
29#14558715—22no—
30#14558713—22no—
31C4×Dic149—22no—
32C4×Dic149C2×C411yescharacteristic
33C4×Dic149D411yescharacteristic
34C4×Dic149D411yescharacteristic
35C2×Q8×C149C2×C412yes—
36C4⋊C4×C149—21no—
37C4⋊C4×C149—21no—
38C42×C149C2×C411yescharacteristic
39C2×#10597010—42no—
40C2×#10597010—42no—
41C2×#10597010—42no—
42#11924753—42no—
43C22×D149C2×D411yescharacteristic
44Dic298—42no—
45C2×Dic149—22no—
46C2×Dic149C22⋊C411yescharacteristic
47C2×Dic149C22⋊C411yescharacteristic
48C2×Dic149C2×D411yescharacteristic
49C4×D149—22no—
50C4×D149—41no—
51C4×D149—41no—
52Q8×C149—42no—
53C2×C4×C149—22no—
54C2×C4×C149C22⋊C411yescharacteristic
55C2×C4×C149C22⋊C411yescharacteristic
56C2×C4×C149C22×C411yescharacteristic
57C22×C149C2×C22⋊C411yescharacteristic
58C2×D149—21no—
59C2×D149—21no—
60C2×D149#13712yes—
61#10597010—82no—
62C4×C149—21no—
63C4×C149—21no—
64C4×C149—21no—
65C4×C149—42no—
66Dic149—21no—
67Dic149—21no—
68Dic149—21no—
69Dic149—42no—
70C2×C149#41011yescharacteristic
71C2×C149#41911yescharacteristic
72C2×C149#42111yescharacteristic
73D149—22no—
74C149#151511yescharacteristic, complemented, Sylow
75#1515—1491nomaximal, Sylow
76#533—1491no—
77#432—1491no—
78#383—1491no—
79#328—1494no—
80C4⋊Q8—1491no—
81C4⋊Q8#1059701011yescharacteristic, complemented
82#125—2981no—
83C22⋊Q8—2981no—
84C22⋊Q8—1492no—
85C4×Q8—2981no—
86C4×Q8—1492no—
87#117—2981no—
88C2×C42—1491no—
89C4⋊C8—2981no—
90C22⋊C8—1492no—
91#97—2981no—
92#95—2981no—
93#95—2981no—
94#95—1492no—
95C2×Q8—2981no—
96C2×Q8C2×#1059701012yes—
97C22×C4—2982no—
98C22×C4—1491no—
99C22×C4—1491no—
100C22×C4—1491no—
101C2×C8—2982no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—2982no—
105C4⋊C4—2982no—
106C4⋊C4—2981no—
107C4⋊C4—1492no—
108C22⋊C4—2982no—
109C42—2982no—
110C42—1491no—
111C42—1491no—
112C42—1491no—
113C42C2×#1059701011yescharacteristic
114C23—1491no—
115Q8—42no—
116Q8—5962no—
117C2×C4—22no—
118C2×C4—2982no—
119C2×C4—2982no—
120C2×C4—5961no—
121C2×C4—5961no—
122C2×C4—5962no—
123C2×C4—5962no—
124C2×C4—5962no—
125C2×C4—1491no—
126C2×C4—1491no—
127C2×C4—1491no—
128C2×C4C22⋊#1059701011yescharacteristic
129C2×C4C22⋊#1059701011yescharacteristic
130C2×C4C22×#1059701011yescharacteristic
131C8—5962no—
132C22—2981no—
133C22—2981no—
134C22—1492no—
135C22C2×C22⋊#1059701011yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—42no—
140C4—2981no—
141C4—2981no—
142C4—2981no—
143C4—5962no—
144C4—11922no—
145C2—2982no—
146C2#2375060511yescharacteristic
147C2#2375061811yescharacteristic
148C2#2375062411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10597010
commutatorC2×C4×C149C22×C4
FrattiniC2×C4C22×#10597010
FittingC4⋊Q8×C149C4
socleC22×C149C2×C22⋊C4
prime radicalC22×D149C2×D4
cosocleC2×C4×D149C23
hypercenterC4⋊Q8#10597010
hypocenterC149#1515
LCSG ▷ C2×C4×C149 ▷ C22×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #18011911C4semidirect
15. Q8×Dic149C4semidirect
19. C4⋊Q8×C149C4semidirect
74. C149#1515semidirect
81. C4⋊Q8#10597010semidirect

Maximal subgroups

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

subgroupindex
#237506482
#237505702
#237504792
#237501802
#1515149