Finite Group Databasebeta

← #49158092

Subgroups — #49158092

Order 37504.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#34036871C211yesmaximal, characteristic
2#34036524C211yesmaximal, characteristic, complemented
3#34036489C212yesmaximal, complemented
4#34035868C212yesmaximal, complemented
5#329×C293C211yesmaximal, characteristic, complemented
6C2×#17989792—21no—
7C2×#17989790—21no—
8#22627624C2211yescharacteristic, complemented
9#22627613—21no—
10#22627612C2212yes—
11#22627610—22no—
12#22627594—22no—
13#22627592—21no—
14#22627549—22no—
15#22627546—22no—
16#22627535C2211yescharacteristic, complemented
17#124×C293C2211yescharacteristic
18#97×C293C2212yescomplemented
19C23×D293—21no—
20C2×Dic586—21no—
21C2×Dic586D412yes—
22C2×D1172—21no—
23C2×D1172D412yes—
24C2×C22⋊D293—22no—
25#17989792—42no—
26#17989790—42no—
27C2×#14556873—21no—
28C4⋊Dic293—22no—
29C22⋊Dic293—21no—
30#17989779—22no—
31#17989779—22no—
32#17989779—22no—
33C4×Dic293—21no—
34C2×Q8×C293D411yescharacteristic
35C2×D4×C293D411yescharacteristic
36C2×C8×C293—22no—
37C22⋊C4×C293—22no—
38C42×C293C2311yescharacteristic
39C22×D293—22no—
40C22×D293—81no—
41C22×D293—82no—
42Dic586—42no—
43Dic586—41no—
44Dic586—41no—
45D1172—42no—
46D1172—81no—
47C22⋊D293—82no—
48C2×Dic293—22no—
49C2×Dic293—41no—
50#14556873—22no—
51C23×C293—21no—
52Q8×C293—41no—
53D4×C293—41no—
54C2×C4×C293—21no—
55C2×C4×C293C2×D412yes—
56C2×C4×C293C2×D411yescharacteristic
57C8×C293—42no—
58C22×C293—41no—
59C22×C293—42no—
60C22×C293#12011yescharacteristic
61C2×D293—42no—
62C2×D293—42no—
63C2×D293—42no—
64C2×D293—84no—
65C4×C293—21no—
66C4×C293—22no—
67C4×C293—41no—
68Dic293—42no—
69Dic293—81no—
70C2×C293—81no—
71C2×C293#45311yescharacteristic
72C2×C293#45512yes—
73D293—82no—
74C293#165911yescharacteristic, complemented, Sylow
75#1659—2931nomaximal, Sylow
76H(3,𝔽4)—2931no—
77#497—2931no—
78#489—2932no—
79#329—2932no—
80#329D29311yescharacteristic, complemented
81C2×Q16—5861no—
82C2×SD16—5861no—
83C4⋊Q8—2931no—
84#124—2932no—
85#124—5862no—
86#124—5861no—
87#124C2×D29311yescharacteristic
88#120—5862no—
89#120—5861no—
90#103—5862no—
91#102—5862no—
92#97—2931no—
93#97C2×D29312yescomplemented
94C24—5861no—
95C2×Q8—2932no—
96C2×Q8—5861no—
97C2×Q8D117211yescharacteristic
98C2×D4—2932no—
99C2×D4—5861no—
100C2×D4—5862no—
101C2×D4D117211yescharacteristic
102Q16—11722no—
103SD16—11722no—
104C2×C8—22no—
105C2×C8—5861no—
106C4⋊C4—5862no—
107C22⋊C4—22no—
108C22⋊C4—5862no—
109C22⋊C4—5861no—
110C22⋊C4—5862no—
111C22⋊C4—5862no—
112C42—5861no—
113C42C22×D29311yescharacteristic
114C23—21no—
115C23—23441no—
116C23—23442no—
117C23—5862no—
118Q8—41no—
119Q8—11721no—
120Q8—11721no—
121Q8—11722no—
122D4—41no—
123D4—23441no—
124D4—23442no—
125D4—11722no—
126C2×C4—21no—
127C2×C4—5862no—
128C2×C4—11721no—
129C2×C4D4×D29312yes—
130C2×C4C2×D117211yescharacteristic
131C8—42no—
132C8—5862no—
133C22—41no—
134C22—42no—
135C22—23444no—
136C22—11722no—
137C22—11722no—
138C22—11722no—
139C22#2262759411yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—23441no—
144C4—11722no—
145C2—81no—
146C2—23442no—
147C2#3403629911yescharacteristic
148C2#3403630812yes—
subgroupG/H
centerC22#22627594
commutator, cosocleC42×C293C23
FrattiniC42C22×D293
Fitting#329×C293C2
socleC22×C293#120
prime radical#34036871C2
hypercenter#329D293
hypocenterC293#1659
LCSG ▷ C42×C293 ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #34036524C2semidirect
3. #34036489C2semidirect
4. #34035868C2semidirect
5. #329×C293C2semidirect
8. #22627624C22semidirect
16. #22627535C22semidirect
18. #97×C293C22semidirect
74. C293#1659semidirect
80. #329D293semidirect
93. #97C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340368712
#340365242
#340364892
#340358682
#329×C2932
#1659293