Finite Group Databasebeta

← #49160798

Subgroups — #49160798

Order 37504.

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

normal subgroups only

This group has 226 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#34036318C212yesmaximal
2#34036306C211yesmaximal, characteristic
3#34035924C211yesmaximal, characteristic, complemented
4#34035919C212yesmaximal, complemented
5#563×C293C211yesmaximal, characteristic, complemented
6#22627669C2211yescharacteristic
7#22627655—21no—
8#22627607—22no—
9#22627552—21no—
10#22627541C2211yescharacteristic
11#22627539C2212yes—
12#22627538—22no—
13C4⋊Q8×C293—21no—
14#126×C293—22no—
15#125×C293—21no—
16#124×C293C2211yescharacteristic
17#123×C293C2212yes—
18C22⋊Q8×C293—22no—
19#17989802—21no—
20#17989801—41no—
21C2×C22⋊D293—22no—
22D4×D293—41no—
23#17989792—41no—
24#17989789—41no—
25#17989786—21no—
26#17989781—42no—
27C22⋊Dic293—22no—
28#17989779—42no—
29C2×Q8×C293—21no—
30C2×Q8×C293D411yescharacteristic
31C2×D4×C293C2311yescharacteristic
32C22×C4×C293D412yes—
33C4⋊C4×C293—21no—
34C4⋊C4×C293—22no—
35C4⋊C4×C293—22no—
36C4⋊C4×C293—21no—
37C4⋊C4×C293—22no—
38C22⋊C4×C293—22no—
39C22⋊C4×C293—22no—
40C22⋊C4×C293D412yes—
41C42×C293—21no—
42C42×C293D411yescharacteristic
43C22×D293—22no—
44Dic586—21no—
45D1172—21no—
46C22⋊D293—42no—
47C22⋊D293—41no—
48C2×Dic293—22no—
49C2×Dic293—42no—
50C4×D293—41no—
51#14556873—41no—
52C23×C293C2×D412yes—
53Q8×C293—41no—
54Q8×C293—81no—
55D4×C293—41no—
56C2×C4×C293—21no—
57C2×C4×C293—22no—
58C2×C4×C293—21no—
59C2×C4×C293—22no—
60C2×C4×C293—42no—
61C2×C4×C293C2×D411yescharacteristic
62C22×C293—22no—
63C22×C293—42no—
64C22×C293#12011yescharacteristic
65C2×D293—41no—
66C2×D293—82no—
67C4×C293—21no—
68C4×C293—42no—
69C4×C293—41no—
70C4×C293—42no—
71C4×C293—41no—
72Dic293—41no—
73Dic293—82no—
74C2×C293—21no—
75C2×C293—42no—
76C2×C293B(4,𝔽2)11yescharacteristic
77D293—81no—
78C293#220111yescharacteristic, complemented, Sylow
79#2201—2931nomaximal, Sylow
80#563D29311yescharacteristic, complemented
81#458—2932no—
82#455—2931no—
83#354—2931no—
84#352—2932no—
85H(5,𝔽2)—2931no—
86#137—5861no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D29311yescharacteristic
91#123—5862no—
92#123C2×D29312yes—
93C22⋊Q8—22no—
94#104—5861no—
95#100—2931no—
96#99—2932no—
97#99—5862no—
98#41—5861no—
99#41—11721no—
100C2×Q8—21no—
101C2×Q8C22⋊D29311yescharacteristic
102C2×D4—5862no—
103C2×D4—11721no—
104C2×D4C22×D29311yescharacteristic
105C22×C4C22⋊D29312yes—
106Q16—11721no—
107SD16—11721no—
108M16—5861no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—11722no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—5862no—
118C22⋊C4—11722no—
119C22⋊C4C22⋊D29312yes—
120C42—21no—
121C42C22⋊D29311yescharacteristic
122C23—5862no—
123C23C2×C22⋊D29312yes—
124Q8—41no—
125Q8—81no—
126Q8—5861no—
127D4—41no—
128D4—5861no—
129D4—11722no—
130D4—11721no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—5862no—
137C2×C4—11721no—
138C2×C4—11722no—
139C2×C4C2×C22⋊D29311yescharacteristic
140C8—11721no—
141C22—22no—
142C22—42no—
143C22—23442no—
144C22—11721no—
145C22#2262759311yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—23442no—
152C4—11721no—
153C2—21no—
154C2—42no—
155C2—23441no—
156C2#3403631611yescharacteristic
subgroupG/H
centerC2#34036316
commutator, cosocleC2×D4×C293C23
FrattiniC2×D4C22×D293
Fitting#563×C293C2
socleC2×C293B(4,𝔽2)
prime radical#22627669C22
hypercenter#563D293
hypocenterC293#2201
LCSG ▷ C2×D4×C293 ▷ C22×C293 ▷ C2×C293 ▷ C293
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #34035924C2semidirect
4. #34035919C2semidirect
5. #563×C293C2semidirect
78. C293#2201semidirect
80. #563D293semidirect

Maximal subgroups

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

subgroupindex
#340363182
#340363062
#340359242
#340359192
#563×C2932
#2201293