Finite Group Databasebeta

← #16818451

Subgroups — #16818451

Order 3712.

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

normal subgroups only

This group has 256 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#14271789C211yesmaximal, characteristic
2#14271321C212yesmaximal, complemented
3#14271075C211yesmaximal, characteristic
4#14270901C212yesmaximal, complemented
5C29×#323C211yesmaximal, characteristic
6C22×Dic58C2211yescharacteristic
7C2×C22⋊C4×C29C2211yescharacteristic
8C2×C22⋊Dic29—21no—
9C2×C22⋊Dic29—21no—
10#11745286—21no—
11#11745286C2212yes—
12C2×Dic116—22no—
13#11745240—22no—
14#11745240—22no—
15C29×C22⋊C8C2212yes—
16#11745179—21no—
17#11745179C2211yescharacteristic
18#11745164—22no—
19C24×C29D411yescharacteristic
20C22×C4×C29D411yescharacteristic
21C22×C4×C29C2311yescharacteristic
22C22×Dic29—21no—
23C22×Dic29—21no—
24C22×Dic29—21no—
25C22×Dic29—21no—
26C2×Dic58—22no—
27C2×Dic58—21no—
28C2×Dic58—42no—
29C2×Dic58D412yes—
30C2×C8×C29—22no—
31C22⋊C4×C29—22no—
32C22⋊Dic29—22no—
33C22⋊Dic29—22no—
34C22⋊Dic29—42no—
35C4⋊Dic29—21no—
36C4⋊Dic29D412yes—
37#90779—22no—
38#90779—22no—
39Dic116—44no—
40C23×C29—21no—
41C23×C29—21no—
42C23×C29—41no—
43C23×C29—41no—
44C23×C29Q1612yescomplemented
45C23×C29C2×D411yescharacteristic
46C2×C4×C29—21no—
47C2×C4×C29—41no—
48C2×C4×C29—41no—
49C2×C4×C29C2×D412yes—
50C2×Dic29—22no—
51C2×Dic29—22no—
52C2×Dic29—22no—
53C2×Dic29—41no—
54C2×Dic29—41no—
55C2×Dic29—41no—
56C2×Dic29—41no—
57C2×Dic29—41no—
58C2×Dic29—81no—
59C2×Dic29—81no—
60Dic58—24no—
61Dic58—42no—
62Dic58—42no—
63Dic58—81no—
64C8×C29—42no—
65#1602—291nomaximal, Sylow
66C22×C29—21no—
67C22×C29—21no—
68C22×C29—22no—
69C22×C29—42no—
70C22×C29—41no—
71C22×C29—41no—
72C22×C29—81no—
73C22×C29#12011yescharacteristic
74C22×C29C2×Q1611yescharacteristic
75C22×C29#13711yescharacteristic
76C4×C29—22no—
77C4×C29—41no—
78Dic29—42no—
79Dic29—42no—
80Dic29—42no—
81Dic29—81no—
82#543—291no—
83#451—292no—
84#393—291no—
85#328—292no—
86#323D2911yescharacteristic
87C2×C29—21no—
88C2×C29—21no—
89C2×C29—42no—
90C2×C29#45111yescharacteristic
91C2×C29#45511yescharacteristic
92C2×C29B(4,𝔽2)11yescharacteristic
93C22×Q8—291no—
94C2×Q16—582no—
95C22⋊Q8—292no—
96C22⋊Q8—582no—
97C22⋊Q8—582no—
98C22⋊Q8—581no—
99C2×C22⋊C4—581no—
100C2×C22⋊C4—581no—
101C2×C22⋊C4C2×D2911yescharacteristic
102#103—582no—
103C22⋊C8C2×D2912yes—
104#95—291no—
105#95—581no—
106C29#160211yescharacteristic, complemented, Sylow
107C24D11611yescharacteristic
108C2×Q8—292no—
109C2×Q8—581no—
110C2×Q8—582no—
111C2×Q8—1162no—
112C22×C4—581no—
113C22×C4—581no—
114C22×C4—581no—
115C22×C4—581no—
116C22×C4D11611yescharacteristic
117C22×C4C22×D2911yescharacteristic
118Q16—1164no—
119C2×C8—22no—
120C4⋊C4—292no—
121C4⋊C4—582no—
122C4⋊C4—581no—
123C4⋊C4—582no—
124C22⋊C4—22no—
125C22⋊C4—582no—
126C22⋊C4—582no—
127C22⋊C4—1162no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic11612yescomplemented
133C23C2×D11611yescharacteristic
134Q8—584no—
135Q8—1162no—
136Q8—1162no—
137Q8—2321no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—582no—
142C2×C4—582no—
143C2×C4—582no—
144C2×C4—1161no—
145C2×C4—1161no—
146C2×C4—1161no—
147C2×C4—1161no—
148C2×C4—1161no—
149C2×C4—2321no—
150C2×C4—2321no—
151C2×C4D4×D2912yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1174524411yescharacteristic
161C22C2×Dic11611yescharacteristic
162C22#1174527111yescharacteristic
163C4—22no—
164C4—41no—
165C4—1162no—
166C4—1162no—
167C4—1162no—
168C4—2321no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#1427132111yescharacteristic
173C2#1427134311yescharacteristic
174C2#1427135111yescharacteristic
subgroupG/H
centerC22#11745244
commutator, cosocleC22×C4×C29C23
FrattiniC22×C4C22×D29
FittingC29×#323C2
socleC22×C29#120
prime radicalC24×C29D4
hypercenter#323D29
hypocenterC29#1602
LCSG ▷ C22×C4×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #14271321C2semidirect
4. #14270901C2semidirect
44. C23×C29Q16semidirect
106. C29#1602semidirect
132. C23Dic116semidirect

Maximal subgroups

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

subgroupindex
#142717892
#142713212
#142710752
#142709012
C29×#3232
#160229