Finite Group Databasebeta

← #16818728

Subgroups — #16818728

Order 3712.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#14271816C211yesmaximal, characteristic
2#14271566C211yesmaximal, characteristic, complemented
3#14271444C211yesmaximal, characteristic, complemented
4#14271369C211yesmaximal, characteristic, complemented
5#14270890C211yesmaximal, characteristic, complemented
6#14270890C211yesmaximal, characteristic, complemented
7C29×#332C211yesmaximal, characteristic, complemented
8C2×#90809—21no—
9C29×C4⋊Q8C2211yescharacteristic
10#11745308—21no—
11C2×#90773—21no—
12#11745257—21no—
13#11745256—22no—
14C4⋊C4×D29—21no—
15C4×D116C2211yescharacteristic
16C4×D116C2211yescharacteristic
17#11745231C2211yescharacteristic
18C29×C4⋊C8C2211yescharacteristic
19C29×C4⋊C8C2211yescharacteristic
20#11745169—21no—
21#11745166—21no—
22#11745164—21no—
23#11745155—22no—
24C4×#6597C2211yescharacteristic
25C2×Q8×C29—21no—
26#90809—42no—
27#90809—42no—
28C2×D116—22no—
29C2×D116D411yescharacteristic
30C2×D116D411yescharacteristic
31C2×C4×D29—22no—
32C2×C4×D29—21no—
33C2×C4×D29—21no—
34C2×Dic58—21no—
35C2×C8×C29—21no—
36C2×C8×C29—21no—
37C4⋊C4×C29—21no—
38C4⋊C4×C29D412yes—
39C42×C29C2311yescharacteristic
40#90781—22no—
41#90781—21no—
42#90781—21no—
43C4⋊Dic29—21no—
44C4⋊Dic29—21no—
45C4⋊Dic29D411yescharacteristic
46C4⋊Dic29D411yescharacteristic
47#90779—22no—
48C2×#6597—21no—
49#90773—42no—
50C22×D29—21no—
51C22×D29—21no—
52Q8×C29—41no—
53Q8×C29—41no—
54C2×C4×C29—22no—
55C2×C4×C29C2×D411yescharacteristic
56C2×C4×C29C2×D411yescharacteristic
57C2×C4×C29C2×D411yescharacteristic
58C2×Dic29—21no—
59C2×Dic29—21no—
60C2×Dic29—41no—
61D116—21no—
62D116—21no—
63D116—21no—
64D116—44no—
65D116D812yes—
66C4×D29—44no—
67C4×D29—41no—
68C4×D29—41no—
69C4×D29—41no—
70C4×D29—41no—
71Dic58—41no—
72Dic58—41no—
73C8×C29—41no—
74C8×C29—41no—
75#6597—22no—
76#1647—291nomaximal, Sylow
77C22×C29#12011yescharacteristic
78C2×D29—22no—
79C2×D29—41no—
80C2×D29—41no—
81C2×D29—41no—
82C2×D29—41no—
83C2×D29—41no—
84C4×C29—21no—
85C4×C29—42no—
86C4×C29C2×D811yescharacteristic
87C4×C29#13511yescharacteristic
88C4×C29AGL(1,ℤ/8)11yescharacteristic
89C4×C29#13711yescharacteristic
90Dic29—41no—
91Dic29—41no—
92Dic29—81no—
93#547—291no—
94#500—291no—
95#474—291no—
96#461—291no—
97#332D2911yescharacteristic, complemented
98D4⋊C8—291no—
99D4⋊C8—291no—
100C2×C29#44711yescharacteristic
101C2×C29#45211yescharacteristic
102C2×C29#45611yescharacteristic
103D29—42no—
104D29—81no—
105C2×#41—581no—
106C2×SD16—581no—
107C4⋊Q8—291no—
108C4⋊Q8C2×D2911yescharacteristic
109#123—582no—
110C22⋊Q8—581no—
111#121—581no—
112C4×D4—291no—
113C4×D4—291no—
114C2×C4⋊C4—581no—
115#107—581no—
116#107—582no—
117C4⋊C8C2×D2911yescharacteristic
118C4⋊C8C2×D2911yescharacteristic
119#103—581no—
120#102—581no—
121C4×C8—291no—
122C29#164711yescharacteristic, complemented, Sylow
123#41—1162no—
124#41—1162no—
125C2×Q8—21no—
126C2×Q8—581no—
127C2×D4—291no—
128C2×D4—291no—
129C2×D4—582no—
130C22×C4—582no—
131C22×C4—581no—
132C22×C4—581no—
133SD16—1162no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—581no—
137C4⋊C4—21no—
138C4⋊C4—291no—
139C4⋊C4—291no—
140C4⋊C4—582no—
141C4⋊C4—581no—
142C4⋊C4—581no—
143C4⋊C4D11612yes—
144C22⋊C4—582no—
145C22⋊C4—581no—
146C22⋊C4—581no—
147C42C22×D2911yescharacteristic
148C23—581no—
149C23—581no—
150Q8—41no—
151Q8—41no—
152Q8—1161no—
153Q8—1161no—
154D4—292no—
155D4—581no—
156D4—581no—
157D4—581no—
158D4—1164no—
159C2×C4—22no—
160C2×C4—581no—
161C2×C4—581no—
162C2×C4—1164no—
163C2×C4—1161no—
164C2×C4—1161no—
165C2×C4—1161no—
166C2×C4—1161no—
167C2×C4—1161no—
168C2×C4C2×D11611yescharacteristic
169C2×C4D4×D2911yescharacteristic
170C2×C4D4×D2911yescharacteristic
171C8—41no—
172C8—41no—
173C8—582no—
174C22—582no—
175C22—1161no—
176C22—1161no—
177C22—1161no—
178C22—1161no—
179C22—1161no—
180C22#1174524411yescharacteristic
181C4—21no—
182C4—42no—
183C4—1161no—
184C4—1161no—
185C4—2321no—
186C4D8×D2911yescharacteristic
187C4#1174527611yescharacteristic
188C4#1174527811yescharacteristic
189C4#1174528011yescharacteristic
190C2—1162no—
191C2—2321no—
192C2#1427129811yescharacteristic
193C2#1427132811yescharacteristic
194C2#1427134811yescharacteristic
subgroupG/H
centerC22#11745244
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#332C2
socleC22×C29#120
prime radical#14271816C2
hypercenter#332D29
hypocenterC29#1647
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #14271566C2semidirect
3. #14271444C2semidirect
4. #14271369C2semidirect
5. #14270890C2semidirect
6. #14270890C2semidirect
7. C29×#332C2semidirect
97. #332D29semidirect
122. C29#1647semidirect

Maximal subgroups

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

subgroupindex
#142718162
#142715662
#142714442
#142713692
#142708902
C29×#3322
#164729