Finite Group Databasebeta

← #16828870

Subgroups — #16828870

Order 3712.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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#14271999C211yesmaximal, characteristic
2C2×#11745304C212yesmaximal, complemented
3#14271345C214yesmaximal, complemented
4#14271340C214yesmaximal, complemented
5#14271181C212yesmaximal, complemented
6#14271147C211yesmaximal, characteristic, complemented
7C29×#536C211yesmaximal, characteristic, complemented
8C2×#41×C29C2211yescharacteristic
9C22×Q8×C29C2211yescharacteristic, complemented
10#11745365—22no—
11#11745364—22no—
12#41×D29—24no—
13#11745362C2212yes—
14#11745359—22no—
15#11745359C2212yes—
16C2×#90805—22no—
17C2×#90805C2212yes—
18C29×C4⋊Q8C2212yescomplemented
19C29×#124C2212yescomplemented
20C29×#123—22no—
21C29×C22⋊Q8—22no—
22C29×#117C2211yescharacteristic, complemented
23C2×#90785—22no—
24#11745304—24no—
25#11745304C2218yescomplemented
26C2×#90784—22no—
27C2×#90777C2212yescomplemented
28#11745184C2212yescomplemented
29#11745181C2212yescomplemented
30#11745153C2218yescomplemented
31#41×C29—22no—
32#41×C29—22no—
33C2×Q8×C29—21no—
34C2×Q8×C29—22no—
35C2×Q8×C29—22no—
36C2×Q8×C29C2312yes—
37C2×Q8×C29C2311yescharacteristic
38C2×D4×C29D412yes—
39C2×D4×C29C2311yescharacteristic
40C22×C4×C29—21no—
41C22×C4×C29D412yes—
42C22×C4×C29C2311yescharacteristic
43C2×C22⋊D29—24no—
44C2×C22⋊D29—22no—
45#90809—24no—
46Q8×D29—24no—
47#90807—28no—
48#90807—24no—
49D4×D29—28no—
50D4×D29—24no—
51#90805—24no—
52#90805—24no—
53#90805—22no—
54#90805—24no—
55#90805—22no—
56#90805D414yes—
57#90805C2314yes—
58C2×D116—22no—
59C2×D116D412yes—
60C2×C4×D29—24no—
61C2×C4×D29—22no—
62C2×Dic58—22no—
63C2×Dic58D412yes—
64C4⋊C4×C29—21no—
65C4⋊C4×C29—24no—
66C22⋊C4×C29—21no—
67C22⋊C4×C29—24no—
68C42×C29C2312yes—
69#90785—28no—
70#90784—28no—
71#90777—22no—
72#90777C2314yes—
73C2×#6597—22no—
74C23×C29C2×D412yes—
75C23×C29C2×D411yescharacteristic
76C22×D29—24no—
77C22×D29—22no—
78Q8×C29—22no—
79Q8×C29—24no—
80Q8×C29—41no—
81Q8×C29—41no—
82D4×C29—24no—
83D4×C29—22no—
84C2×C4×C29—22no—
85C2×C4×C29—22no—
86C2×C4×C29—22no—
87C2×C4×C29—22no—
88C2×C4×C29—42no—
89C2×C4×C29C2×D412yes—
90C2×C4×C29C2×D412yes—
91C2×C4×C29C2×D412yes—
92C2×C4×C29C2×D411yescharacteristic
93C2×C4×C29C2411yescharacteristic
94C22⋊D29—28no—
95C22⋊D29—24no—
96C22⋊D29—24no—
97C2×Dic29—24no—
98C2×Dic29—22no—
99D116—24no—
100D116—22no—
101D116C2×D414yes—
102C4×D29—28no—
103C4×D29—24no—
104C4×D29—24no—
105Dic58—24no—
106Dic58—22no—
107Dic58C2×D414yes—
108#6597—24no—
109#3022—291nomaximal, Sylow
110C22×C29—22no—
111C22×C29—41no—
112C22×C29—42no—
113C22×C29#12012yes—
114C22×C29C22×D411yescharacteristic
115C2×D29—24no—
116C2×D29—82no—
117C2×D29—84no—
118C4×C29—22no—
119C4×C29—42no—
120C4×C29—42no—
121C4×C29#12011yescharacteristic
122C4×C29#12011yescharacteristic
123C4×C29C22×D412yes—
124Dic29—24no—
125#585—291no—
126C2×#137—292no—
127#536D2911yescharacteristic, complemented
128#456—294no—
129#455—294no—
130#421—292no—
131#413—291no—
132C2×C29—21no—
133C2×C29—22no—
134C2×C29—42no—
135C2×C29C2×#12011yescharacteristic
136D29—44no—
1372−1+4—292no—
1382−1+4—582no—
139H(5,𝔽2)—292no—
140H(5,𝔽2)—582no—
141H(5,𝔽2)—582no—
142C2×#41—292no—
143C2×#41—584no—
144C2×#41—582no—
145C2×#41C2×D2911yescharacteristic
146C22×Q8C2×D2911yescharacteristic, complemented
147#137—298no—
148#137—584no—
149C2×Q16—582no—
150C2×SD16—582no—
151C2×M16—292no—
152C4⋊Q8C2×D2912yescomplemented
153#124C2×D2912yescomplemented
154#123—22no—
155C22⋊Q8—22no—
156#117C2×D2911yescharacteristic, complemented
157#104—298no—
158#101—292no—
159#100—292no—
160C29#302211yescharacteristic, complemented, Sylow
161#41—22no—
162#41—22no—
163#41—294no—
164#41—294no—
165#41—584no—
166#41—584no—
167#41—584no—
168#41—588no—
169#41—584no—
170#41—582no—
171#41—582no—
172#41—584no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—292no—
177C2×Q8—582no—
178C2×Q8—584no—
179C2×Q8C22×D2912yes—
180C2×Q8C22×D2911yescharacteristic
181C2×D4—292no—
182C2×D4—582no—
183C2×D4—584no—
184C2×D4—588no—
185C2×D4—582no—
186C2×D4—584no—
187C2×D4C22⋊D2912yes—
188C2×D4C22×D2911yescharacteristic
189C22×C4—21no—
190C22×C4—584no—
191C22×C4—582no—
192C22×C4C22⋊D2912yes—
193C22×C4C22×D2911yescharacteristic
194Q16—588no—
195SD16—588no—
196M16—294no—
197M16—582no—
198C2×C8—582no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D2912yes—
204C23—584no—
205C23—582no—
206C23C2×C22⋊D2912yes—
207C23C2×C22⋊D2911yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—294no—
213Q8—584no—
214Q8—582no—
215D4—24no—
216D4—22no—
217D4—294no—
218D4—584no—
219D4—588no—
220D4—584no—
221D4—584no—
222D4—582no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—588no—
229C2×C4—584no—
230C2×C4—582no—
231C2×C4—584no—
232C2×C4—584no—
233C2×C4D4×D2912yes—
234C2×C4D4×D2912yes—
235C2×C4C2×C22⋊D2912yes—
236C2×C4C2×C22⋊D2911yescharacteristic
237C2×C4C23×D2911yescharacteristic
238C8—584no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—584no—
243C22—2322no—
244C22—2324no—
245C22#1174529912yes—
246C22C22×C22⋊D2911yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—584no—
251C4#1174529911yescharacteristic
252C4#1174529911yescharacteristic
253C4C2×D4×D2912yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—1164no—
258C2C2×#1174529911yescharacteristic
subgroupG/H
centerC2C2×#11745299
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
FittingC29×#536C2
socleC2×C29C2×#120
prime radical#14271999C2
hypercenter#536D29
hypocenterC29#3022
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#11745304C2semidirect
3. #14271345C2semidirect
4. #14271340C2semidirect
5. #14271181C2semidirect
6. #14271147C2semidirect
7. C29×#536C2semidirect
9. C22×Q8×C29C22semidirect
18. C29×C4⋊Q8C22semidirect
19. C29×#124C22semidirect
22. C29×#117C22semidirect
25. #11745304C22semidirect
27. C2×#90777C22semidirect
28. #11745184C22semidirect
29. #11745181C22semidirect
30. #11745153C22semidirect
127. #536D29semidirect
146. C22×Q8C2×D29semidirect
152. C4⋊Q8C2×D29semidirect
153. #124C2×D29semidirect
156. #117C2×D29semidirect
160. C29#3022semidirect

Maximal subgroups

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

subgroupindex
#142719992
C2×#117453042
#142713452
#142713402
#142711812
#142711472
C29×#5362
#302229