Finite Group Databasebeta

← #49157756

Subgroups — #49157756

Order 37504.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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#34036710C211yesmaximal, characteristic
2#34036270C212yesmaximal, complemented
3#34036049C211yesmaximal, characteristic, complemented
4#34035864C212yesmaximal, complemented
5#324×C293C211yesmaximal, characteristic, complemented
6C2×#17989802—21no—
7C22×Dic586C2211yescharacteristic
8C2×#17989791—22no—
9#22627604—22no—
10#22627599—22no—
11#22627598—21no—
12#22627598C2212yes—
13C2×#17989781—21no—
14#22627549—22no—
15#22627531—21no—
16#22627531C2211yescharacteristic
17C2×C4⋊C4×C293C2211yescharacteristic
18C22⋊C8×C293C2212yes—
19#17989802—42no—
20#17989802—42no—
21C2×Dic586—21no—
22C2×Dic586—22no—
23C2×Dic586—42no—
24C2×D1172—21no—
25C2×D1172—21no—
26C2×D1172D412yes—
27C2×C22⋊D293—22no—
28C22×Dic293—21no—
29C22×Dic293—21no—
30C2×C4×D293—22no—
31#17989791—44no—
32C4⋊Dic293—21no—
33C4⋊Dic293D412yes—
34#17989781—42no—
35#17989779—22no—
36#17989779—22no—
37C22×C4×C293D411yescharacteristic
38C22×C4×C293D411yescharacteristic
39C22×C4×C293C2311yescharacteristic
40C2×C8×C293—22no—
41C4⋊C4×C293—22no—
42C22×D293—22no—
43Dic586—24no—
44Dic586—42no—
45Dic586—42no—
46D1172—42no—
47D1172—42no—
48D1172—81no—
49C22⋊D293—42no—
50C22⋊D293—42no—
51C2×Dic293—22no—
52C2×Dic293—22no—
53C2×Dic293—22no—
54C2×Dic293—41no—
55C2×Dic293—41no—
56C2×Dic293—41no—
57C2×Dic293—41no—
58C4×D293—44no—
59C23×C293C2×D411yescharacteristic
60C2×C4×C293—21no—
61C2×C4×C293—21no—
62C2×C4×C293—21no—
63C2×C4×C293—41no—
64C2×C4×C293—41no—
65C2×C4×C293SD1612yescomplemented
66C2×C4×C293C2×D412yes—
67C8×C293—42no—
68C22×C293—21no—
69C22×C293—21no—
70C22×C293#12011yescharacteristic
71C22×C293C2×SD1611yescharacteristic
72C22×C293AGL(1,ℤ/8)11yescharacteristic
73C2×D293—42no—
74C2×D293—42no—
75C2×D293—42no—
76C4×C293—22no—
77C4×C293—22no—
78C4×C293—41no—
79Dic293—42no—
80Dic293—42no—
81Dic293—42no—
82C2×C293—21no—
83C2×C293—21no—
84C2×C293#45011yescharacteristic
85C2×C293#45611yescharacteristic
86C2×C293B(4,𝔽2)11yescharacteristic
87D293—82no—
88C293#160411yescharacteristic, complemented, Sylow
89#1604—2931nomaximal, Sylow
90#537—2931no—
91#448—2932no—
92#395—2931no—
93#327—2932no—
94#324D29311yescharacteristic, complemented
95C2×#41—5861no—
96C22×Q8—2931no—
97C2×SD16—5862no—
98C22⋊Q8—5862no—
99#121—2932no—
100#121—5862no—
101#121—5861no—
102C2×C4⋊C4—5861no—
103C2×C4⋊C4C2×D29311yescharacteristic
104#103—5862no—
105C22⋊C8C2×D29312yes—
106#95—2931no—
107#95—5861no—
108#41—11722no—
109#41—11722no—
110C2×Q8—5861no—
111C2×Q8—5862no—
112C2×Q8—11722no—
113C2×D4—2932no—
114C2×D4—5861no—
115C2×D4—5861no—
116C2×D4—5862no—
117C22×C4—5862no—
118C22×C4—5861no—
119C22×C4—5861no—
120C22×C4D117211yescharacteristic
121C22×C4D117211yescharacteristic
122C22×C4C22×D29311yescharacteristic
123SD16—11724no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—2932no—
127C4⋊C4—5861no—
128C4⋊C4—11722no—
129C22⋊C4—5862no—
130C22⋊C4—5862no—
131C23—5862no—
132C23C2×D117211yescharacteristic
133Q8—5864no—
134Q8—11722no—
135Q8—11722no—
136D4—23441no—
137D4—11722no—
138D4—11722no—
139D4—11722no—
140D4—11722no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—5862no—
147C2×C4—5862no—
148C2×C4—5862no—
149C2×C4—11724no—
150C2×C4—11721no—
151C2×C4—11721no—
152C2×C4—11721no—
153C2×C4—11721no—
154C2×C4#1798979112yescomplemented
155C2×C4D4×D29312yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—11722no—
160C22—11722no—
161C22—11722no—
162C22#2262759411yescharacteristic
163C22C2×#1798979111yescharacteristic
164C22#2262765211yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—11722no—
169C4—11722no—
170C4—11722no—
171C2—21no—
172C2—21no—
173C2—23442no—
174C2#3403628211yescharacteristic
175C2#3403631411yescharacteristic
176C2#3403631711yescharacteristic
subgroupG/H
centerC22#22627594
commutator, cosocleC22×C4×C293C23
FrattiniC22×C4C22×D293
Fitting#324×C293C2
socleC22×C293#120
prime radical#34036710C2
hypercenter#324D293
hypocenterC293#1604
LCSG ▷ C22×C4×C293 ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #34036270C2semidirect
3. #34036049C2semidirect
4. #34035864C2semidirect
5. #324×C293C2semidirect
65. C2×C4×C293SD16semidirect
88. C293#1604semidirect
94. #324D293semidirect
154. C2×C4#17989791semidirect

Maximal subgroups

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

subgroupindex
#340367102
#340362702
#340360492
#340358642
#324×C2932
#1604293