Finite Group Databasebeta

← #16818677

Subgroups — #16818677

Order 3712.

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

normal subgroups only

This group has 212 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#14271898C211yesmaximal, characteristic
2#14271511C211yesmaximal, characteristic
3#14271370C211yesmaximal, characteristic, complemented
4#14270906C211yesmaximal, characteristic
5Q8⋊#6597C211yesmaximal, characteristic, complemented
6#14270893C211yesmaximal, characteristic, complemented
7C29×#475C211yesmaximal, characteristic, complemented
8C29×C4⋊Q8C2211yescharacteristic
9C4×Q8×C29C2211yescharacteristic
10#11745309—21no—
11C2×#90785—21no—
12#11745256—22no—
13#11745255—21no—
14#11745237—21no—
15#11745236C2211yescharacteristic
16C4×D116—21no—
17C4×Dic58C2211yescharacteristic
18C29×#106—21no—
19C29×C4⋊C8C2211yescharacteristic
20C29×#103—21no—
21#11745183—21no—
22#11745157—22no—
23C4⋊#6597C2211yescharacteristic
24C4×#6597C2211yescharacteristic
25C2×Q8×C29—21no—
26C2×Q8×C29D411yescharacteristic
27C2×D116—21no—
28C2×D116—21no—
29C2×D116D412yes—
30C2×C4×D29—22no—
31C2×Dic58D411yescharacteristic
32C2×C8×C29—21no—
33C4⋊C4×C29—21no—
34C4⋊C4×C29—21no—
35C4⋊C4×C29—21no—
36C4⋊C4×C29D411yescharacteristic
37C42×C29—21no—
38C42×C29C2311yescharacteristic
39#90785—42no—
40#90781—22no—
41#90781—22no—
42#90781—22no—
43C4⋊Dic29D411yescharacteristic
44#90779—21no—
45C4×Dic29—21no—
46C2×#6597—21no—
47C2×#6597—21no—
48C22×D29—22no—
49Q8×C29—21no—
50Q8×C29—21no—
51Q8×C29—41no—
52Q8×C29—41no—
53C2×C4×C29—21no—
54C2×C4×C29—21no—
55C2×C4×C29—41no—
56C2×C4×C29C2×D411yescharacteristic
57C2×C4×C29C2×D411yescharacteristic
58C2×C4×C29C2×D411yescharacteristic
59C2×Dic29—21no—
60C2×Dic29—21no—
61D116—42no—
62D116—41no—
63D116—41no—
64D116—41no—
65D116—41no—
66C4×D29—82no—
67Dic58—21no—
68Dic58—21no—
69C8×C29—41no—
70#6597—41no—
71#6597—41no—
72#1640—291nomaximal, Sylow
73C22×C29#12011yescharacteristic
74C2×D29—42no—
75C2×D29—42no—
76C2×D29—42no—
77C4×C29—21no—
78C4×C29—41no—
79C4×C29—41no—
80C4×C29—81no—
81C4×C29#13511yescharacteristic
82C4×C29#13511yescharacteristic
83C4×C29AGL(1,ℤ/8)11yescharacteristic
84C4×C29#13711yescharacteristic
85Dic29—41no—
86Dic29—41no—
87#559—291no—
88#486—291no—
89#475D2911yescharacteristic, complemented
90#462—291no—
91#331—291no—
92Q8⋊C8—291no—
93Q8⋊C8—291no—
94C2×C29#44911yescharacteristic
95C2×C29#45211yescharacteristic
96C2×C29#45511yescharacteristic
97D29—82no—
98C2×Q16—581no—
99C4⋊Q8C2×D2911yescharacteristic
100#127—291no—
101#124—581no—
102#124—581no—
103#123—582no—
104C4×Q8—291no—
105C4×Q8C2×D2911yescharacteristic
106C4×D4—581no—
107C4×D4—581no—
108#106—21no—
109C4⋊C8—291no—
110C4⋊C8C2×D2911yescharacteristic
111#103—21no—
112#103—581no—
113#102—582no—
114C4×C8—291no—
115C29#164011yescharacteristic, complemented, Sylow
116C2×Q8—21no—
117C2×Q8—291no—
118C2×Q8C22⋊D2911yescharacteristic
119C2×D4—292no—
120C2×D4—581no—
121C2×D4—581no—
122C22×C4—582no—
123Q16—1162no—
124C2×C8—21no—
125C2×C8—581no—
126C2×C8—581no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—291no—
131C4⋊C4—581no—
132C4⋊C4C22⋊D2911yescharacteristic
133C22⋊C4—582no—
134C22⋊C4—582no—
135C22⋊C4—582no—
136C42—21no—
137C42—581no—
138C42C22×D2911yescharacteristic
139C23—582no—
140Q8—21no—
141Q8—21no—
142Q8—41no—
143Q8—41no—
144Q8—581no—
145Q8—581no—
146D4—1162no—
147D4—1161no—
148D4—1161no—
149D4—1161no—
150D4—1161no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—41no—
154C2×C4—581no—
155C2×C4—581no—
156C2×C4—2322no—
157C2×C4D4×D2911yescharacteristic
158C2×C4D4×D2911yescharacteristic
159C2×C4C2×C22⋊D2911yescharacteristic
160C8—41no—
161C8—1161no—
162C8—1161no—
163C22—1162no—
164C22—1162no—
165C22—1162no—
166C22#1174529911yescharacteristic
167C4—21no—
168C4—41no—
169C4—41no—
170C4—81no—
171C4—1161no—
172C4—1161no—
173C4#1174527611yescharacteristic
174C4#1174527811yescharacteristic
175C4#1174530411yescharacteristic
176C4#1174531211yescharacteristic
177C2—2322no—
178C2#1427130711yescharacteristic
179C2#1427132311yescharacteristic
180C2#1427133911yescharacteristic
subgroupG/H
centerC22#11745299
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#475C2
socleC22×C29#120
prime radical#11745236C22
hypercenter#475D29
hypocenterC29#1640
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
3. #14271370C2semidirect
5. Q8⋊#6597C2semidirect
6. #14270893C2semidirect
7. C29×#475C2semidirect
89. #475D29semidirect
115. C29#1640semidirect

Maximal subgroups

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

subgroupindex
#142718982
#142715112
#142713702
#142709062
Q8⋊#65972
#142708932
C29×#4752
#164029