Finite Group Databasebeta

← #49158075

Subgroups — #49158075

Order 37504.

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

normal subgroups only

This group has 262 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, complemented
2#34036519C211yesmaximal, characteristic, complemented
3#34036518C212yesmaximal, complemented
4#34035868C212yesmaximal, complemented
5#329×C293C211yesmaximal, characteristic, complemented
6C2×#17989791—22no—
7C2×#17989789—21no—
8C2×D2344—22no—
9C2×#17989787—21no—
10#22627621C2211yescharacteristic, complemented
11#22627613—21no—
12#22627612C2212yes—
13#22627610—22no—
14#22627594—22no—
15#22627592—21no—
16#22627535C2211yescharacteristic, complemented
17#124×C293C2211yescharacteristic
18#97×C293C2212yescomplemented
19C23×D293—21no—
20C2×Dic586D412yes—
21C2×D1172—21no—
22C2×D1172—21no—
23C2×D1172—22no—
24C2×D1172D412yes—
25C2×C22⋊D293—22no—
26#17989791—44no—
27#17989789—42no—
28D2344—44no—
29#17989787—42no—
30C2×#14556873—21no—
31C22⋊Dic293—21no—
32#17989779—22no—
33#17989779—22no—
34#17989779—22no—
35C4×Dic293—21no—
36C2×Q8×C293D411yescharacteristic
37C2×D4×C293D411yescharacteristic
38C2×C8×C293—22no—
39C22⋊C4×C293—22no—
40C42×C293C2311yescharacteristic
41C22×D293—22no—
42C22×D293—41no—
43C22×D293—82no—
44C22×D293—81no—
45Dic586—42no—
46D1172—42no—
47D1172—41no—
48D1172—42no—
49D1172—41no—
50D1172—42no—
51D1172—81no—
52C22⋊D293—82no—
53C2×Dic293—22no—
54#14556873—22no—
55C23×C293—21no—
56Q8×C293—41no—
57D4×C293—41no—
58C2×C4×C293—21no—
59C2×C4×C293C2×D412yes—
60C2×C4×C293C2×D411yescharacteristic
61C8×C293—24no—
62C22×C293—41no—
63C22×C293—42no—
64C22×C293#12011yescharacteristic
65C2×D293—42no—
66C2×D293—42no—
67C2×D293—42no—
68C2×D293—84no—
69C2×D293—82no—
70C2×D293—81no—
71C4×C293—22no—
72C4×C293—21no—
73C4×C293—41no—
74Dic293—42no—
75C2×C293—81no—
76C2×C293#45312yes—
77C2×C293#45311yescharacteristic
78D293—82no—
79D293—161no—
80C293#165511yescharacteristic, complemented, Sylow
81#1655—2931nomaximal, Sylow
82H(3,𝔽4)—2931no—
83#496—2931no—
84#496—2932no—
85#329—2932no—
86#329D29311yescharacteristic, complemented
87C2×SD16—5862no—
88C2×SD16—5861no—
89C2×D8—5862no—
90C2×D8—5861no—
91#127—2931no—
92#124—2932no—
93#124—5862no—
94#124—5861no—
95#124C2×D29311yescharacteristic
96#120—5862no—
97#120—5861no—
98#97—2931no—
99#97C2×D29312yescomplemented
100C24—5861no—
101C2×Q8—2932no—
102C2×Q8D117211yescharacteristic
103C2×D4—2932no—
104C2×D4—5861no—
105C2×D4—5862no—
106C2×D4—5862no—
107C2×D4—5861no—
108C2×D4D117211yescharacteristic
109SD16—11724no—
110SD16—11722no—
111D8—11724no—
112D8—11722no—
113C2×C8—22no—
114C2×C8—5861no—
115C22⋊C4—22no—
116C22⋊C4—5862no—
117C22⋊C4—5861no—
118C22⋊C4—5862no—
119C22⋊C4—5862no—
120C42—5861no—
121C42C22×D29311yescharacteristic
122C23—21no—
123C23—23442no—
124C23—23441no—
125C23—5862no—
126C23—11721no—
127Q8—41no—
128Q8—11722no—
129D4—41no—
130D4—23441no—
131D4—23442no—
132D4—11722no—
133D4—11721no—
134D4—11722no—
135D4—11721no—
136D4—11722no—
137C2×C4—21no—
138C2×C4—5862no—
139C2×C4D4×D29312yes—
140C2×C4C2×D117211yescharacteristic
141C8—24no—
142C8—5862no—
143C22—42no—
144C22—41no—
145C22—23444no—
146C22—23442no—
147C22—23441no—
148C22—11722no—
149C22—11722no—
150C22—11722no—
151C22#2262759411yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—11722no—
156C2—81no—
157C2—23442no—
158C2—46881no—
159C2#3403629812yes—
160C2#3403629911yescharacteristic
subgroupG/H
centerC22#22627594
commutator, cosocleC42×C293C23
FrattiniC42C22×D293
Fitting#329×C293C2
socleC22×C293#120
hypercenter#329D293
hypocenterC293#1655
LCSG ▷ C42×C293 ▷ C22×C293 ▷ C293
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
1. #34036871C2semidirect
2. #34036519C2semidirect
3. #34036518C2semidirect
4. #34035868C2semidirect
5. #329×C293C2semidirect
10. #22627621C22semidirect
16. #22627535C22semidirect
18. #97×C293C22semidirect
80. C293#1655semidirect
86. #329D293semidirect
99. #97C2×D293semidirect

Maximal subgroups

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

subgroupindex
#340368712
#340365192
#340365182
#340358682
#329×C2932
#1655293