Finite Group Databasebeta

← #16822827

Subgroups — #16822827

Order 3712.

864 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#90805C211yesmaximal, characteristic
2C2×#11745291C212yesmaximal, complemented
3#14271083C212yesmaximal, complemented
4#14271080C212yesmaximal, complemented
5#14271042C212yesmaximal, complemented
6#14271034C214yesmaximal, complemented
7#14271026C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C29C211yesmaximal, characteristic, complemented
9C23×C4×C29C2211yescharacteristic
10C23×C4×C29C2211yescharacteristic
11C23×C4×C29C2211yescharacteristic
12C22×C22⋊D29C2212yes—
13C2×#90805—24no—
14C2×#90805—28no—
15C22×D116C2211yescharacteristic
16C22×C4×D29C2212yes—
17C22×Dic58C2211yescharacteristic
18C2×C4⋊C4×C29—24no—
19C2×C4⋊C4×C29C2212yes—
20C2×C4⋊C4×C29C2212yes—
21C2×C22⋊Dic29C2212yes—
22#11745291—28no—
23C2×#90781C2214yes—
24C2×#90781C2214yes—
25C2×#90779C2214yes—
26#11745179C2212yes—
27#11745179C2212yes—
28#11745179C2214yes—
29C24×C29C2311yescharacteristic
30C23×D29C2312yes—
31C22×C4×C29—22no—
32C22×C4×C29—24no—
33C22×C4×C29—24no—
34C22×C4×C29—24no—
35C22×C4×C29—22no—
36C22×C4×C29D414yes—
37C22×C4×C29C2312yes—
38C22×C4×C29C2311yescharacteristic
39C22×C4×C29C2311yescharacteristic
40C22×C4×C29C2312yes—
41C2×C22⋊D29—24no—
42C2×C22⋊D29—24no—
43C2×C22⋊D29D418yes—
44C22×Dic29C2314yes—
45C22×Dic29C2312yes—
46#90805—232no—
47C2×D116—24no—
48C2×D116—22no—
49C2×C4×D29—24no—
50C2×C4×D29—24no—
51C2×C4×D29—24no—
52C2×Dic58—22no—
53C2×Dic58—24no—
54C4⋊C4×C29—24no—
55C4⋊C4×C29—24no—
56C22⋊Dic29—24no—
57#90781—28no—
58#90781—28no—
59#90779—28no—
60C23×C29—22no—
61C23×C29—22no—
62C23×C29C2×D414yes—
63C23×C29C2×D412yes—
64C23×C29C2411yescharacteristic
65C22×D29—22no—
66C22×D29—22no—
67C22×D29—24no—
68C22×D29—24no—
69C22×D29C2×D414yes—
70C2×C4×C29—28no—
71C2×C4×C29—28no—
72C2×C4×C29—22no—
73C2×C4×C29—22no—
74C2×C4×C29—22no—
75C2×C4×C29—22no—
76C2×C4×C29—24no—
77C2×C4×C29—22no—
78C2×C4×C29—44no—
79C2×C4×C29C2×D412yes—
80C2×C4×C29C2×D412yes—
81C2×C4×C29#4114yescomplemented
82C22⋊D29—216no—
83C2×Dic29—22no—
84C2×Dic29—22no—
85C2×Dic29—24no—
86C2×Dic29—24no—
87C2×Dic29—24no—
88C2×Dic29C2×D414yes—
89D116—28no—
90C4×D29—216no—
91Dic58—28no—
92#2406—291nomaximal, Sylow
93C22×C29—22no—
94C22×C29—22no—
95C22×C29—28no—
96C22×C29#12014yes—
97C22×C29C22×D412yes—
98C22×C29C22×D411yescharacteristic
99C22×C29C2×#4112yes—
100C22×C292−1+412yes—
101C2×D29—24no—
102C2×D29—28no—
103C2×D29—42no—
104C2×D29—42no—
105C2×D29—44no—
106C4×C29—22no—
107C4×C29—22no—
108C4×C29—24no—
109C4×C29—42no—
110Dic29—24no—
111Dic29—44no—
112C22×#41—291no—
113C2×#123—292no—
114C22×C4⋊C4D2911yescharacteristic, complemented
115#395—292no—
116#394—292no—
117#386—292no—
118#385—294no—
119#381—291no—
120C2×C29—24no—
121C2×C29C2×#12011yescharacteristic
122C2×C29#52511yescharacteristic
123C2×C29#53611yescharacteristic
124C2×C29#54714yes—
125D29—44no—
126C2×#41—588no—
127C2×#41—584no—
128C22×Q8—291no—
129C22×D4—292no—
130C22×D4—291no—
131C23×C4—292no—
132C23×C4C2×D2911yescharacteristic
133C23×C4C2×D2911yescharacteristic
134C23×C4C2×D2911yescharacteristic
135#123—588no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—294no—
138C2×C4⋊C4C2×D2912yes—
139C2×C4⋊C4C2×D2912yes—
140C2×C22⋊C4—292no—
141C2×C22⋊C4—294no—
142C2×C22⋊C4—294no—
143#95—292no—
144#95—292no—
145#95—294no—
146C29#240611yescharacteristic, complemented, Sylow
147C24—292no—
148C24C22×D2911yescharacteristic
149#41—5832no—
150C2×Q8—584no—
151C2×Q8—582no—
152C2×D4—298no—
153C2×D4—584no—
154C2×D4—584no—
155C2×D4—584no—
156C2×D4—582no—
157C22×C4—24no—
158C22×C4—22no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—22no—
162C22×C4—292no—
163C22×C4—294no—
164C22×C4—584no—
165C22×C4—584no—
166C22×C4—584no—
167C22×C4C22⋊D2914yes—
168C22×C4C22×D2912yes—
169C22×C4C22×D2912yes—
170C22×C4C22×D2911yescharacteristic
171C22×C4C22×D2911yescharacteristic
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—588no—
175C22⋊C4—588no—
176C22⋊C4—584no—
177C22⋊C4—588no—
178C23—22no—
179C23—22no—
180C23—294no—
181C23—584no—
182C23—582no—
183C23—584no—
184C23—582no—
185C23D4×D2914yes—
186C23C2×C22⋊D2912yes—
187C23C23×D2911yescharacteristic
188Q8—588no—
189D4—588no—
190D4—5816no—
191C2×C4—28no—
192C2×C4—28no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—22no—
199C2×C4—44no—
200C2×C4—294no—
201C2×C4—5816no—
202C2×C4—582no—
203C2×C4—582no—
204C2×C4—584no—
205C2×C4—584no—
206C2×C4—584no—
207C2×C4#9080514yescomplemented
208C2×C4C2×C22⋊D2912yes—
209C2×C4C2×C22⋊D2912yes—
210C22—28no—
211C22—22no—
212C22—22no—
213C22—588no—
214C22—584no—
215C22—1164no—
216C22—1162no—
217C22—1162no—
218C22#1174529914yes—
219C22C2×#9080512yes—
220C22C2×D4×D2912yes—
221C22#1174536212yes—
222C22C22×C22⋊D2911yescharacteristic
223C4—22no—
224C4—22no—
225C4—24no—
226C4—42no—
227C4—584no—
228C4—1164no—
229C2—24no—
230C2—1164no—
231C2C2×#1174529911yescharacteristic
232C2#1427167411yescharacteristic
233C2#1427173711yescharacteristic
234C2#1427181414yes—
subgroupG/H
center, FrattiniC23C23×D29
commutator, socle, cosocleC23×C29C24
FittingC22×C4⋊C4×C29C2
prime radicalC22×#90805C2
hypercenterC22×C4⋊C4D29
hypocenterC29#2406
LCSG ▷ C23×C29 ▷ C29
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#11745291C2semidirect
3. #14271083C2semidirect
4. #14271080C2semidirect
5. #14271042C2semidirect
6. #14271034C2semidirect
7. #14271026C2semidirect
8. C22×C4⋊C4×C29C2semidirect
81. C2×C4×C29#41semidirect
114. C22×C4⋊C4D29semidirect
146. C29#2406semidirect
207. C2×C4#90805semidirect

Maximal subgroups

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

subgroupindex
C22×#908052
C2×#117452912
#142710832
#142710802
#142710422
#142710342
#142710262
C22×C4⋊C4×C292
#240629