Finite Group Databasebeta

← #16818415

Subgroups — #16818415

Order 3712.

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

normal subgroups only

This group has 316 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#14271313C212yesmaximal, complemented
2#14271072C211yesmaximal, characteristic
3#14270900C212yesmaximal, complemented
4#14270887C211yesmaximal, characteristic, complemented
5C29×#543C211yesmaximal, characteristic, complemented
6C22×C22⋊D29—21no—
7C22×D116C2211yescharacteristic
8C29×C22⋊Q8—22no—
9C29×C22⋊Q8—21no—
10C29×C22⋊Q8—22no—
11C29×C22⋊Q8C2212yes—
12C2×C22⋊C4×C29—21no—
13C2×C22⋊C4×C29C2211yescharacteristic
14C2×#90784—22no—
15C2×#90781—21no—
16#11745179C2211yescharacteristic
17C22⋊#6597C2212yes—
18#11745157—22no—
19C24×C29D411yescharacteristic
20C23×D29—21no—
21C2×Q8×C29—21no—
22C2×Q8×C29D412yescomplemented
23C22×C4×C29—21no—
24C22×C4×C29—21no—
25C22×C4×C29D411yescharacteristic
26C22×C4×C29C2311yescharacteristic
27C2×C22⋊D29—42no—
28C2×C22⋊D29—42no—
29C2×C22⋊D29—42no—
30C22×Dic29—21no—
31C2×D116—22no—
32C2×D116—42no—
33C4⋊C4×C29—21no—
34C4⋊C4×C29—22no—
35C4⋊C4×C29—22no—
36C4⋊C4×C29D412yescomplemented
37C22⋊C4×C29—22no—
38C22⋊C4×C29—22no—
39C22⋊C4×C29—22no—
40#90784—44no—
41#90781—42no—
42C2×#6597—22no—
43C23×C29—21no—
44C23×C29—21no—
45C23×C29—41no—
46C23×C29—41no—
47C23×C29SD1612yescomplemented
48C23×C29C2×D411yescharacteristic
49C22×D29—22no—
50C22×D29—41no—
51C22×D29—41no—
52C22×D29—41no—
53C22×D29—41no—
54C22×D29—41no—
55C22×D29—41no—
56Q8×C29—42no—
57Q8×C29—81no—
58C2×C4×C29—22no—
59C2×C4×C29—21no—
60C2×C4×C29—22no—
61C2×C4×C29—41no—
62C2×C4×C29—41no—
63C2×C4×C29—81no—
64C2×C4×C29—81no—
65C2×C4×C29C2×D412yes—
66C22⋊D29—48no—
67C2×Dic29—22no—
68C2×Dic29—41no—
69C2×Dic29—41no—
70D116—24no—
71D116—42no—
72#6597—42no—
73#1596—291nomaximal, Sylow
74C22×C29—21no—
75C22×C29—21no—
76C22×C29—22no—
77C22×C29—42no—
78C22×C29—41no—
79C22×C29—41no—
80C22×C29—81no—
81C22×C29#12011yescharacteristic
82C22×C29C2×SD1611yescharacteristic
83C22×C29#13711yescharacteristic
84C2×D29—42no—
85C2×D29—42no—
86C2×D29—42no—
87C2×D29—81no—
88C2×D29—81no—
89C2×D29—81no—
90C2×D29—81no—
91C4×C29—22no—
92C4×C29—41no—
93C4×C29—42no—
94C4×C29—42no—
95Dic29—42no—
96#543D2911yescharacteristic, complemented
97#450—292no—
98#392—291no—
99#328—292no—
100#323—291no—
101C2×C29—21no—
102C2×C29—21no—
103C2×C29—42no—
104C2×C29#44811yescharacteristic
105C2×C29#45311yescharacteristic
106C2×C29B(4,𝔽2)11yescharacteristic
107D29—82no—
108C22×D4—291no—
109C22×D4—581no—
110C2×SD16—582no—
111C22⋊Q8—22no—
112C22⋊Q8—21no—
113C22⋊Q8—22no—
114C22⋊Q8C2×D2912yes—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—581no—
117C2×C22⋊C4C2×D2911yescharacteristic
118#102—582no—
119C22⋊C8—292no—
120#95—291no—
121C29#159611yescharacteristic, complemented, Sylow
122C24—581no—
123C24C22⋊D2911yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D2912yescomplemented
126C2×D4—582no—
127C2×D4—1162no—
128C2×D4—1162no—
129C2×D4—1162no—
130C2×D4—1162no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—581no—
134C22×C4C22⋊D2911yescharacteristic
135C22×C4C22×D2911yescharacteristic
136SD16—1164no—
137C2×C8—582no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D2912yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—1162no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—582no—
151C23—1161no—
152C23—1161no—
153C23—1161no—
154C23—1161no—
155C23—1161no—
156C23—1161no—
157C23#9078412yescomplemented
158C23C2×C22⋊D2911yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—584no—
162D4—1162no—
163D4—1168no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—582no—
172C2×C4—1161no—
173C2×C4—1161no—
174C2×C4C2×C22⋊D2912yes—
175C8—1162no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—1162no—
184C22—1162no—
185C22—1162no—
186C22—2321no—
187C22—2321no—
188C22—2321no—
189C22—2321no—
190C22C2×#9078411yescharacteristic
191C22#1174530411yescharacteristic
192C22#1174531511yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—1162no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—2322no—
202C2#1427130211yescharacteristic
203C2#1427133111yescharacteristic
204C2#1427135011yescharacteristic
subgroupG/H
centerC22#11745315
commutator, cosocleC22×C4×C29C23
FrattiniC22×C4C22×D29
FittingC29×#543C2
socleC22×C29#120
prime radical#14271072C2
hypercenter#543D29
hypocenterC29#1596
LCSG ▷ C22×C4×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #14271313C2semidirect
3. #14270900C2semidirect
4. #14270887C2semidirect
5. C29×#543C2semidirect
22. C2×Q8×C29D4semidirect
36. C4⋊C4×C29D4semidirect
47. C23×C29SD16semidirect
96. #543D29semidirect
121. C29#1596semidirect
125. C2×Q8C22⋊D29semidirect
141. C4⋊C4C22⋊D29semidirect
157. C23#90784semidirect

Maximal subgroups

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

subgroupindex
#142713132
#142710722
#142709002
#142708872
C29×#5432
#159629