Finite Group Databasebeta

← #16822771

Subgroups — #16822771

Order 3712.

840 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 177 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
1C2×C4×D116C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊D29C214yesmaximal, complemented
3#14271099C212yesmaximal, complemented
4#14271098C212yesmaximal, complemented
5#14271072C212yesmaximal, complemented
6#14271071C212yesmaximal, complemented
7#14271009C211yesmaximal, characteristic, complemented
8C2×C4×D4×C29C211yesmaximal, characteristic, complemented
9C22×D4×C29C2211yescharacteristic
10C23×C4×C29C2212yes—
11C22×C22⋊D29C2214yes—
12C22×D116C2211yescharacteristic
13C22×C4×D29C2212yes—
14C4×D4×C29—24no—
15C2×C4⋊C4×C29C2211yescharacteristic
16C2×C22⋊C4×C29C2212yes—
17C2×C42×C29C2211yescharacteristic
18C2×C22⋊Dic29C2214yes—
19C4×C22⋊D29—216no—
20C2×#90781C2212yes—
21C2×#90781C2214yes—
22C2×C4⋊Dic29C2211yescharacteristic
23C2×#90779C2214yes—
24C2×C4×Dic29C2212yes—
25C4×D116—24no—
26#11745179C2212yes—
27#11745179C2212yes—
28C24×C29C2312yes—
29C23×D29C2312yes—
30C2×D4×C29—24no—
31C2×D4×C29—22no—
32C22×C4×C29—28no—
33C22×C4×C29—24no—
34C22×C4×C29C2312yes—
35C22×C4×C29C2311yescharacteristic
36C22×C4×C29C2311yescharacteristic
37C22×C4×C29C2311yescharacteristic
38C2×C22⋊D29—28no—
39C2×C22⋊D29—28no—
40C2×C22⋊D29—28no—
41C22×Dic29C2312yes—
42C22×Dic29C2314yes—
43C2×D116—24no—
44C2×D116—22no—
45C2×C4×D29—28no—
46C2×C4×D29—24no—
47C4⋊C4×C29—22no—
48C22⋊C4×C29—24no—
49C42×C29D414yescomplemented
50C22⋊Dic29—28no—
51#90781—24no—
52#90781—28no—
53C4⋊Dic29—22no—
54#90779—28no—
55C4×Dic29D418yescomplemented
56C23×C29—24no—
57C23×C29—22no—
58C23×C29—24no—
59C23×C29—22no—
60C23×C29#4114yes—
61C23×C29C2411yescharacteristic
62C22×D29—22no—
63C22×D29—24no—
64C22×D29—22no—
65C22×D29—24no—
66C22×D29#4114yes—
67D4×C29—28no—
68C2×C4×C29—22no—
69C2×C4×C29—22no—
70C2×C4×C29—24no—
71C2×C4×C29—22no—
72C2×C4×C29—48no—
73C2×C4×C29C2×D414yes—
74C2×C4×C29C2×D412yes—
75C2×C4×C29C2×D412yes—
76C2×C4×C29C2×D412yes—
77C22⋊D29—232no—
78C2×Dic29—24no—
79C2×Dic29—24no—
80C2×Dic29—24no—
81C2×Dic29—22no—
82C2×Dic29C2×D418yes—
83D116—28no—
84C4×D29—48no—
85#2397—291nomaximal, Sylow
86C22×C29—28no—
87C22×C29—24no—
88C22×C29—42no—
89C22×C29—44no—
90C22×C29—42no—
91C22×C29C22×D412yes—
92C22×C29C22×D411yescharacteristic
93C22×C29C2×#4112yes—
94C22×C29C2×#4112yes—
95C2×D29—24no—
96C2×D29—28no—
97C2×D29—44no—
98C2×D29—42no—
99C2×D29—42no—
100C4×C29—22no—
101C4×C29—22no—
102C4×C29—42no—
103C4×C29#12714yes—
104Dic29—28no—
105Dic29—42no—
106C2×C4×D4—294no—
107C2×C4×D4—291no—
108C2×C4×D4D2911yescharacteristic, complemented
109#399—292no—
110#399—292no—
111#392—292no—
112#392—292no—
113#376—291no—
114C2×C29—44no—
115C2×C29#52511yescharacteristic
116C2×C29#52514yes—
117C2×C29#52511yescharacteristic
118C2×C29C2×#12711yescharacteristic
119D29—44no—
120C22×D4—294no—
121C22×D4—291no—
122C22×D4C2×D2911yescharacteristic
123C23×C4—292no—
124C23×C4C2×D2912yes—
125C4×D4—24no—
126C4×D4—5816no—
127C4×D4—584no—
128C2×C4⋊C4—294no—
129C2×C4⋊C4—291no—
130C2×C4⋊C4C2×D2911yescharacteristic
131C2×C22⋊C4—294no—
132C2×C22⋊C4—294no—
133C2×C22⋊C4—292no—
134C2×C22⋊C4C2×D2912yes—
135C2×C42—292no—
136C2×C42C2×D2911yescharacteristic
137#95—292no—
138#95—292no—
139C29#239711yescharacteristic, complemented, Sylow
140C24—292no—
141C24C22×D2912yes—
142C2×D4—24no—
143C2×D4—22no—
144C2×D4—584no—
145C2×D4—588no—
146C2×D4—588no—
147C2×D4—588no—
148C2×D4—582no—
149C22×C4—28no—
150C22×C4—24no—
151C22×C4—294no—
152C22×C4—292no—
153C22×C4—588no—
154C22×C4—584no—
155C22×C4C22×D2912yes—
156C22×C4C22×D2911yescharacteristic
157C22×C4C22×D2911yescharacteristic
158C22×C4C22×D2911yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—588no—
161C4⋊C4—582no—
162C22⋊C4—24no—
163C22⋊C4—588no—
164C22⋊C4—588no—
165C22⋊C4—584no—
166C42—298no—
167C42C22⋊D2914yescomplemented
168C23—22no—
169C23—24no—
170C23—24no—
171C23—22no—
172C23—294no—
173C23—584no—
174C23—584no—
175C23—582no—
176C23—582no—
177C23#9080514yes—
178C23C23×D2911yescharacteristic
179D4—28no—
180D4—588no—
181D4—5832no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—24no—
185C2×C4—22no—
186C2×C4—48no—
187C2×C4—298no—
188C2×C4—584no—
189C2×C4—584no—
190C2×C4—584no—
191C2×C4—582no—
192C2×C4—1168no—
193C2×C4D4×D2914yes—
194C2×C4C2×C22⋊D2912yes—
195C2×C4C2×C22⋊D2912yes—
196C2×C4C2×C22⋊D2912yes—
197C22—28no—
198C22—24no—
199C22—42no—
200C22—44no—
201C22—42no—
202C22—584no—
203C22—588no—
204C22—1164no—
205C22—1162no—
206C22—1162no—
207C22C2×#9080512yes—
208C22C2×D4×D2912yes—
209C22#41×D2912yes—
210C22C22×C22⋊D2911yescharacteristic
211C4—22no—
212C4—22no—
213C4—42no—
214C4—588no—
215C4—1162no—
216C4#1174530214yes—
217C2—44no—
218C2—1164no—
219C2#1427167011yescharacteristic
220C2#1427167214yes—
221C2#1427167411yescharacteristic
222C2C2×#1174530211yescharacteristic
subgroupG/H
centerC22×C4C22×D29
commutator, socle, cosocleC23×C29C24
FrattiniC23C23×D29
FittingC2×C4×D4×C29C2
hypercenterC2×C4×D4D29
hypocenterC29#2397
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C22×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4×D116C2semidirect
2. C2×C4×C22⋊D29C2semidirect
3. #14271099C2semidirect
4. #14271098C2semidirect
5. #14271072C2semidirect
6. #14271071C2semidirect
7. #14271009C2semidirect
8. C2×C4×D4×C29C2semidirect
49. C42×C29D4semidirect
55. C4×Dic29D4semidirect
108. C2×C4×D4D29semidirect
139. C29#2397semidirect
167. C42C22⋊D29semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D1162
C2×C4×C22⋊D292
#142710992
#142710982
#142710722
#142710712
#142710092
C2×C4×D4×C292
#239729