Finite Group Databasebeta

← #16818691

Subgroups — #16818691

Order 3712.

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

normal subgroups only

This group has 250 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#14271835C211yesmaximal, characteristic
2#14271511C211yesmaximal, characteristic, complemented
3#14271365C211yesmaximal, characteristic, complemented
4#14270906C211yesmaximal, characteristic, complemented
5Q8⋊#6597C211yesmaximal, characteristic, complemented
6#14270890C211yesmaximal, characteristic, complemented
7C29×#475C211yesmaximal, characteristic, complemented
8C2×#90809—21no—
9C29×C4⋊Q8C2211yescharacteristic
10C4×Q8×C29C2211yescharacteristic
11C2×#90784—21no—
12#11745257—22no—
13#11745257—21no—
14#11745236—21no—
15#11745236C2211yescharacteristic
16C4×D116—21no—
17C4×D116C2211yescharacteristic
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
27#90809—42no—
28#90809—42no—
29C2×D116—22no—
30C2×D116—21no—
31C2×D116—21no—
32C2×D116—22no—
33C2×D116D411yescharacteristic
34C2×D116D412yes—
35C2×C4×D29—22no—
36C2×C4×D29—21no—
37C2×C8×C29—21no—
38C4⋊C4×C29—21no—
39C4⋊C4×C29—21no—
40C4⋊C4×C29—21no—
41C4⋊C4×C29D411yescharacteristic
42C42×C29—21no—
43C42×C29C2311yescharacteristic
44#90784—42no—
45#90781—22no—
46#90781—21no—
47C4⋊Dic29D411yescharacteristic
48C2×#6597—21no—
49C2×#6597—21no—
50C22×D29—21no—
51C22×D29—22no—
52Q8×C29—21no—
53Q8×C29—41no—
54Q8×C29—41no—
55Q8×C29SD1612yescomplemented
56C2×C4×C29—21no—
57C2×C4×C29—21no—
58C2×C4×C29—41no—
59C2×C4×C29C2×D411yescharacteristic
60C2×C4×C29C2×D411yescharacteristic
61C2×C4×C29C2×D411yescharacteristic
62C2×Dic29—21no—
63D116—21no—
64D116—21no—
65D116—42no—
66D116—44no—
67D116—41no—
68D116—41no—
69D116—41no—
70D116—41no—
71D116—82no—
72C4×D29—44no—
73C4×D29—41no—
74C4×D29—41no—
75C8×C29—41no—
76#6597—41no—
77#6597—41no—
78#1642—291nomaximal, Sylow
79C22×C29#12011yescharacteristic
80C2×D29—41no—
81C2×D29—42no—
82C2×D29—42no—
83C2×D29—41no—
84C2×D29—42no—
85C2×D29—41no—
86C4×C29—22no—
87C4×C29—21no—
88C4×C29—41no—
89C4×C29—81no—
90C4×C29C2×SD1611yescharacteristic
91C4×C29#13511yescharacteristic
92C4×C29AGL(1,ℤ/8)11yescharacteristic
93C4×C29AGL(1,ℤ/8)11yescharacteristic
94Dic29—41no—
95#550—291no—
96#486—291no—
97#475D2911yescharacteristic, complemented
98#461—291no—
99#331—291no—
100Q8⋊C8—291no—
101D4⋊C8—291no—
102C2×C29#44911yescharacteristic
103C2×C29#45011yescharacteristic
104C2×C29#45511yescharacteristic
105D29—81no—
106D29—82no—
107C2×#41—581no—
108C2×SD16—581no—
109C4⋊Q8C2×D2911yescharacteristic
110#127—291no—
111#127—581no—
112#121—582no—
113#121—581no—
114C4×Q8C2×D2911yescharacteristic
115C4×D4—291no—
116C4×D4—581no—
117#106—21no—
118C4⋊C8—291no—
119C4⋊C8C2×D2911yescharacteristic
120#103—21no—
121#103—581no—
122#102—582no—
123C4×C8—291no—
124C29#164211yescharacteristic, complemented, Sylow
125#41—1162no—
126#41—1162no—
127C2×Q8—21no—
128C2×Q8C22⋊D2911yescharacteristic
129C2×D4—291no—
130C2×D4—292no—
131C2×D4—582no—
132C2×D4—582no—
133C2×D4—581no—
134C2×D4—581no—
135C22×C4—582no—
136C22×C4—581no—
137SD16—1162no—
138C2×C8—21no—
139C2×C8—581no—
140C2×C8—581no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—291no—
145C4⋊C4C22⋊D2911yescharacteristic
146C22⋊C4—582no—
147C22⋊C4—581no—
148C42—21no—
149C42C22×D2911yescharacteristic
150C23—581no—
151C23—582no—
152Q8—21no—
153Q8—41no—
154Q8—41no—
155Q8#9078412yescomplemented
156D4—581no—
157D4—581no—
158D4—1162no—
159D4—1164no—
160D4—1161no—
161D4—1161no—
162D4—1161no—
163D4—1161no—
164D4—2322no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—41no—
168C2×C4—581no—
169C2×C4—1164no—
170C2×C4—1161no—
171C2×C4—1161no—
172C2×C4D4×D2911yescharacteristic
173C2×C4D4×D2911yescharacteristic
174C2×C4C2×C22⋊D2911yescharacteristic
175C8—41no—
176C8—1161no—
177C8—1161no—
178C22—1161no—
179C22—1162no—
180C22—1162no—
181C22—1161no—
182C22—1162no—
183C22—1161no—
184C22#1174529911yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—81no—
189C4—1161no—
190C4#1174527611yescharacteristic
191C4#1174527811yescharacteristic
192C4C2×#9078411yescharacteristic
193C4#1174531111yescharacteristic
194C2—2321no—
195C2—2322no—
196C2#1427130711yescharacteristic
197C2#1427131311yescharacteristic
198C2#1427133911yescharacteristic
subgroupG/H
centerC22#11745299
commutator, cosocleC42×C29C23
FrattiniC42C22×D29
FittingC29×#475C2
socleC22×C29#120
prime radical#14271835C2
hypercenter#475D29
hypocenterC29#1642
LCSG ▷ C42×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #14271511C2semidirect
3. #14271365C2semidirect
4. #14270906C2semidirect
5. Q8⋊#6597C2semidirect
6. #14270890C2semidirect
7. C29×#475C2semidirect
55. Q8×C29SD16semidirect
97. #475D29semidirect
124. C29#1642semidirect
155. Q8#90784semidirect

Maximal subgroups

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

subgroupindex
#142718352
#142715112
#142713652
#142709062
Q8⋊#65972
#142708902
C29×#4752
#164229