Finite Group Databasebeta

← #16819910

Subgroups — #16819910

Order 3712.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#14271670C211yesmaximal, characteristic
2#14271147C211yesmaximal, characteristic, complemented
3#14271126C212yesmaximal, complemented
4#14270934C212yesmaximal, complemented
5C29×#568C211yesmaximal, characteristic, complemented
6C2×#41×C29C2211yescharacteristic
7C22×C4×D29C2211yescharacteristic
8C29×#131—24no—
9C2×M16×C29—22no—
10C2×M16×C29C2212yes—
11#11745308—21no—
12#11745300—21no—
13#11745299C412yes—
14#11745297C412yes—
15C4×C22⋊D29—22no—
16#11745288C2211yescharacteristic
17C2×#90777C2212yes—
18#11745184—21no—
19#11745181—21no—
20#11745168—24no—
21C23×D29C2×C411yescharacteristic
22#41×C29—22no—
23#41×C29—22no—
24C2×Q8×C29D411yescharacteristic
25C2×D4×C29C2×C412yes—
26C2×D4×C29D411yescharacteristic
27C22×C4×C29D412yes—
28C22×C4×C29C2311yescharacteristic
29C2×C22⋊D29—22no—
30C22×Dic29C2×C411yescharacteristic
31C2×C4×D29—21no—
32C2×C4×D29—21no—
33C2×C4×D29—22no—
34C2×C4×D29—22no—
35M16×C29—24no—
36M16×C29—22no—
37M16×C29—22no—
38C2×C8×C29—24no—
39C2×C8×C29D414yescomplemented
40C22⋊Dic29—22no—
41C22⋊Dic29C2×C412yes—
42#90781—22no—
43C4⋊Dic29—21no—
44#90779—22no—
45C4×Dic29—21no—
46#90777—22no—
47#90777—22no—
48C2×#6597—22no—
49C23×C29C22⋊C412yes—
50C23×C29C22×C411yescharacteristic
51C22×D29—21no—
52C22×D29—22no—
53C22×D29—42no—
54Q8×C29—22no—
55D4×C29—24no—
56D4×C29—22no—
57C2×C4×C29—22no—
58C2×C4×C29—22no—
59C2×C4×C29C22⋊C412yes—
60C2×C4×C29C2×D411yescharacteristic
61C2×C4×C29C2×D412yes—
62C2×C4×C29C2×D411yescharacteristic
63C2×C4×C29#4112yes—
64C22⋊D29—44no—
65C2×Dic29—22no—
66C2×Dic29—21no—
67C2×Dic29—22no—
68C4×D29—41no—
69C4×D29—42no—
70C4×D29—41no—
71C8×C29—24no—
72#6597—42no—
73#1891—291nomaximal, Sylow
74C22×C29—22no—
75C22×C29—41no—
76C22×C29—42no—
77C22×C29C2×C22⋊C411yescharacteristic
78C22×C29C4×D412yes—
79C2×D29—22no—
80C2×D29—24no—
81C2×D29—41no—
82C2×D29—42no—
83C2×D29—41no—
84C4×C29—22no—
85C4×C29#12011yescharacteristic
86C4×C29#12112yes—
87C4×C29#12311yescharacteristic
88Dic29—42no—
89Dic29—41no—
90#568D2911yescharacteristic, complemented
91#525—291no—
92#413—291no—
93#407—292no—
94#344—292no—
95C2×C29—21no—
96C2×C29—22no—
97C2×C29—42no—
98C2×C29#38611yescharacteristic
99D29—42no—
100C2×#41C2×D2911yescharacteristic
101C23×C4—291no—
102#131—24no—
103C2×M16—22no—
104C2×M16—292no—
105C2×M16C2×D2912yes—
106#123—292no—
107C22⋊Q8—581no—
108#121—581no—
109#120—292no—
110C4×D4—582no—
111#117—291no—
112#101—581no—
113#100—581no—
114C22⋊C8—584no—
115C29#189111yescharacteristic, complemented, Sylow
116C24—291no—
117#41—22no—
118#41—22no—
119C2×Q8D11611yescharacteristic
120C2×D4—582no—
121C2×D4C4×D2912yes—
122C2×D4D11611yescharacteristic
123C22×C4—291no—
124C22×C4—581no—
125C22×C4—582no—
126C22×C4—582no—
127C22×C4—581no—
128C22×C4C22⋊D2912yes—
129C22×C4C22×D2911yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—582no—
134M16—582no—
135C2×C8—24no—
136C2×C8—582no—
137C2×C8C22⋊D2914yescomplemented
138C4⋊C4—582no—
139C4⋊C4—581no—
140C22⋊C4—292no—
141C22⋊C4—582no—
142C22⋊C4—582no—
143C42—581no—
144C23—581no—
145C23—582no—
146C23—1162no—
147C23#9078112yes—
148C23C2×C4×D2911yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—1164no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—582no—
156C2×C4—581no—
157C2×C4—582no—
158C2×C4—1161no—
159C2×C4—1162no—
160C2×C4—1161no—
161C2×C4#9078112yes—
162C2×C4C2×D11611yescharacteristic
163C2×C4#9080512yes—
164C2×C4C2×C22⋊D2911yescharacteristic
165C2×C4C2×C22⋊D2912yes—
166C8—24no—
167C8—1162no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—582no—
172C22—584no—
173C22—1161no—
174C22—1162no—
175C22—1161no—
176C22C2×#9078111yescharacteristic
177C22C4×C22⋊D2912yes—
178C4—22no—
179C4—1162no—
180C4—1161no—
181C4#1174529111yescharacteristic
182C4#1174529212yes—
183C4#1174531511yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—1162no—
188C2#1427104211yescharacteristic
subgroupG/H
centerC4#11745315
commutatorC23×C29C22×C4
FrattiniC22×C4C22×D29
FittingC29×#568C2
socleC2×C29#386
prime radical#14271670C2
cosocleC22×C4×C29C23
hypercenter#568D29
hypocenterC29#1891
LCSG ▷ C23×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #14271147C2semidirect
3. #14271126C2semidirect
4. #14270934C2semidirect
5. C29×#568C2semidirect
39. C2×C8×C29D4semidirect
90. #568D29semidirect
115. C29#1891semidirect
137. C2×C8C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
#142716702
#142711472
#142711262
#142709342
C29×#5682
#189129