Finite Group Databasebeta

← #16818738

Subgroups — #16818738

Order 3712.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#14271828C211yesmaximal, characteristic
2#14271440C211yesmaximal, characteristic
3#14271370C211yesmaximal, characteristic, complemented
4#14270909C211yesmaximal, characteristic, complemented
5#14270895C211yesmaximal, characteristic, complemented
6#14270892C211yesmaximal, characteristic
7C29×#500C211yesmaximal, characteristic, complemented
8C2×Q8×D29—21no—
9C29×C4⋊Q8C2211yescharacteristic
10C29×C4⋊Q8C2211yescharacteristic
11#11745308—21no—
12C2×#90785—21no—
13#11745258—22no—
14#11745255—21no—
15#11745250—21no—
16C4×D116C2211yescharacteristic
17C4×Dic58C2211yescharacteristic
18C29×#107—22no—
19C4×C8×C29C2211yescharacteristic
20#11745183—21no—
21#11745157—21no—
22#11745155—21no—
23C4⋊#6597C2211yescharacteristic
24C4⋊#6597C2211yescharacteristic
25C2×Q8×C29—21no—
26C2×Q8×C29—21no—
27Q8×D29—42no—
28Q8×D29—42no—
29C2×D116D411yescharacteristic
30C2×C4×D29—22no—
31C2×C4×D29—21no—
32C2×Dic58—22no—
33C2×Dic58D411yescharacteristic
34C2×C8×C29—21no—
35C4⋊C4×C29—21no—
36C4⋊C4×C29—21no—
37C4⋊C4×C29—21no—
38C4⋊C4×C29D412yes—
39C42×C29C2311yescharacteristic
40#90785—42no—
41#90781—22no—
42#90781—21no—
43C4⋊Dic29D411yescharacteristic
44C4⋊Dic29D411yescharacteristic
45#90779—22no—
46#90779—21no—
47C4×Dic29—21no—
48C2×#6597—21no—
49C2×#6597—21no—
50C22×D29—21no—
51Q8×C29—41no—
52Q8×C29—41no—
53Q8×C29—41no—
54Q8×C29—41no—
55C2×C4×C29—22no—
56C2×C4×C29—41no—
57C2×C4×C29C2×D411yescharacteristic
58C2×C4×C29C2×D411yescharacteristic
59C2×C4×C29C2×D411yescharacteristic
60C2×Dic29—21no—
61C2×Dic29—21no—
62C2×Dic29—21no—
63D116—21no—
64D116Q1612yescomplemented
65C4×D29—44no—
66C4×D29—41no—
67C4×D29—41no—
68Dic58—21no—
69Dic58—44no—
70Dic58D812yes—
71C8×C29—22no—
72#6597—41no—
73#6597—41no—
74#1649—291nomaximal, Sylow
75C22×C29#12011yescharacteristic
76C2×D29—22no—
77C2×D29—41no—
78C2×D29—41no—
79C4×C29—21no—
80C4×C29—42no—
81C4×C29—81no—
82C4×C29C2×D811yescharacteristic
83C4×C29C2×Q1611yescharacteristic
84C4×C29AGL(1,ℤ/8)11yescharacteristic
85C4×C29#13711yescharacteristic
86Dic29—22no—
87Dic29—41no—
88Dic29—41no—
89D4×Q8—291no—
90#500D2911yescharacteristic, complemented
91#474—291no—
92#462—291no—
93#332—291no—
94Q8⋊C8—291no—
95D4⋊C8—291no—
96C2×C29#44711yescharacteristic
97C2×C29#45111yescharacteristic
98C2×C29#45611yescharacteristic
99D29—42no—
100C22×Q8—581no—
101C2×Q16—581no—
102C4⋊Q8—581no—
103C4⋊Q8C2×D2911yescharacteristic
104C4⋊Q8C2×D2911yescharacteristic
105C22⋊Q8—581no—
106C22⋊Q8—582no—
107C4×Q8—291no—
108C4×D4—291no—
109C4×D4—581no—
110#107—22no—
111#107—581no—
112C4⋊C8—291no—
113C4⋊C8—291no—
114#103—581no—
115#102—581no—
116C4×C8C2×D2911yescharacteristic
117C29#164911yescharacteristic, complemented, Sylow
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—291no—
121C2×Q8—582no—
122C2×Q8—1162no—
123C2×Q8—1162no—
124C2×D4—291no—
125C22×C4—582no—
126C22×C4—581no—
127Q16—1162no—
128C2×C8—21no—
129C2×C8—581no—
130C2×C8—581no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—291no—
135C4⋊C4—291no—
136C4⋊C4—582no—
137C4⋊C4—581no—
138C4⋊C4C22⋊D2912yes—
139C22⋊C4—582no—
140C22⋊C4—581no—
141C42—581no—
142C42C22×D2911yescharacteristic
143C23—581no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—292no—
149Q8—581no—
150Q8—1164no—
151D4—292no—
152D4—581no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—581no—
156C2×C4—581no—
157C2×C4—581no—
158C2×C4—1164no—
159C2×C4—1161no—
160C2×C4—1161no—
161C2×C4D4×D2911yescharacteristic
162C2×C4D4×D2911yescharacteristic
163C2×C4C2×C22⋊D2911yescharacteristic
164C8—22no—
165C8—1161no—
166C8—1161no—
167C22—582no—
168C22—1161no—
169C22—1161no—
170C22#1174529911yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—582no—
175C4—1161no—
176C4—1161no—
177C4D8×D2911yescharacteristic
178C4#1174527311yescharacteristic
179C4Q16×D2911yescharacteristic
180C4#1174528011yescharacteristic
181C2—1162no—
182C2#1427129511yescharacteristic
183C2#1427131711yescharacteristic
184C2#1427134511yescharacteristic
subgroupG/H
centerC22#11745299
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#500C2
socleC22×C29#120
prime radicalC2×D116D4
hypercenter#500D29
hypocenterC29#1649
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #14271370C2semidirect
4. #14270909C2semidirect
5. #14270895C2semidirect
7. C29×#500C2semidirect
64. D116Q16semidirect
90. #500D29semidirect
117. C29#1649semidirect

Maximal subgroups

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

subgroupindex
#142718282
#142714402
#142713702
#142709092
#142708952
#142708922
C29×#5002
#164929