Finite Group Databasebeta

← #16828871

Subgroups — #16828871

Order 3712.

704 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 161 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×#11745313C212yesmaximal, complemented
2#14271734C211yesmaximal, characteristic, complemented
3#14271346C214yesmaximal, complemented
4#14271341C214yesmaximal, complemented
5#14271180C212yesmaximal, complemented
6#14271147C211yesmaximal, characteristic, complemented
7C29×#585C211yesmaximal, characteristic
8C29×2−1+4—22no—
9C29×2−1+4C2212yes—
10C29×H(5,𝔽2)—22no—
11C29×H(5,𝔽2)—22no—
12C29×H(5,𝔽2)C2212yes—
13C2×#41×C29—24no—
14C2×#41×C29—22no—
15C2×#41×C29C2212yes—
16C2×#41×C29C2211yescharacteristic
17C22×Dic58C2211yescharacteristic, complemented
18#11745313—24no—
19#11745313C2218yescomplemented
20#11745306C2212yescomplemented
21C2×#90785—22no—
22#11745298C2212yescomplemented
23#11745297—22no—
24C2×#90783—22no—
25#11745288C2211yescharacteristic, complemented
26#11745286—22no—
27C2×#90777C2212yescomplemented
28#11745185C2218yescomplemented
29#11745184C2212yescomplemented
30#11745181C2212yescomplemented
31#41×C29—24no—
32#41×C29—28no—
33#41×C29—24no—
34#41×C29—24no—
35#41×C29—24no—
36#41×C29—22no—
37#41×C29—22no—
38#41×C29—24no—
39#41×C29—22no—
40#41×C29—22no—
41#41×C29D414yes—
42#41×C29C2314yes—
43C2×Q8×C29—22no—
44C2×Q8×C29—24no—
45C2×Q8×C29D412yes—
46C2×Q8×C29C2311yescharacteristic
47C2×D4×C29—28no—
48C2×D4×C29—22no—
49C2×D4×C29—24no—
50C2×D4×C29—22no—
51C2×D4×C29—24no—
52C2×D4×C29D412yes—
53C2×D4×C29D412yes—
54C2×D4×C29C2311yescharacteristic
55C22×C4×C29—24no—
56C22×C4×C29—22no—
57C22×C4×C29D412yes—
58C22×C4×C29C2311yescharacteristic
59C22×Dic29—21no—
60C2×Dic58—22no—
61C2×Dic58—22no—
62C2×Dic58—21no—
63C2×Dic58C2312yes—
64C22⋊Dic29—24no—
65C22⋊Dic29—21no—
66#90785—28no—
67#90783—28no—
68C4⋊Dic29—21no—
69#90779—24no—
70C4×Dic29C2312yes—
71#90777—22no—
72#90777C2314yes—
73C2×#6597—22no—
74C23×C29—24no—
75C23×C29—22no—
76C23×C29C2×D412yes—
77C23×C29C2×D411yescharacteristic
78Q8×C29—24no—
79Q8×C29—22no—
80Q8×C29—22no—
81Q8×C29C2×D414yes—
82D4×C29—28no—
83D4×C29—24no—
84D4×C29—24no—
85D4×C29—24no—
86D4×C29—24no—
87D4×C29—22no—
88D4×C29—22no—
89D4×C29C2×D414yes—
90C2×C4×C29—24no—
91C2×C4×C29—28no—
92C2×C4×C29—22no—
93C2×C4×C29—24no—
94C2×C4×C29—22no—
95C2×C4×C29—22no—
96C2×C4×C29—24no—
97C2×C4×C29C2×D412yes—
98C2×C4×C29C2×D411yescharacteristic
99C2×C4×C29C2×D412yes—
100C2×C4×C29C2×D412yes—
101C2×C4×C29C2411yescharacteristic
102C2×Dic29—22no—
103C2×Dic29—22no—
104C2×Dic29—42no—
105Dic58—24no—
106Dic58—41no—
107Dic58—41no—
108#6597—24no—
109#3022—291nomaximal, Sylow
110C22×C29—22no—
111C22×C29—24no—
112C22×C29—41no—
113C22×C29—42no—
114C22×C29—82no—
115C22×C29—84no—
116C22×C29#12012yes—
117C22×C29C22×D411yescharacteristic
118C4×C29—24no—
119C4×C29—22no—
120C4×C29#12011yescharacteristic
121C4×C29#12011yescharacteristic
122C4×C29C22×D412yes—
123Dic29—42no—
124Dic29—42no—
125#585D2911yescharacteristic
126C2×#137—292no—
127#536—291no—
128#456—294no—
129#455—294no—
130#421—292no—
131#413—291no—
132C2×C29—21no—
133C2×C29—22no—
134C2×C29—42no—
135C2×C29—44no—
136C2×C29C2×#12011yescharacteristic
1372−1+4—22no—
1382−1+4C2×D2912yes—
139H(5,𝔽2)—22no—
140H(5,𝔽2)—22no—
141H(5,𝔽2)C2×D2912yes—
142C2×#41—24no—
143C2×#41—22no—
144C2×#41C2×D2912yes—
145C2×#41C2×D2911yescharacteristic
146C22×Q8—291no—
147#137—298no—
148#137—584no—
149C2×Q16—582no—
150C2×SD16—582no—
151C2×M16—292no—
152C4⋊Q8—292no—
153#124—292no—
154#123—582no—
155C22⋊Q8—582no—
156#117—291no—
157#104—298no—
158#101—292no—
159#100—292no—
160C29#302211yescharacteristic, complemented, Sylow
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41C22⋊D2914yes—
172#41C22×D2914yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—292no—
176C2×Q8—581no—
177C2×Q8—582no—
178C2×Q8—582no—
179C2×Q8C22⋊D2912yes—
180C2×Q8C22×D2911yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4C22⋊D2912yes—
187C2×D4C22⋊D2912yes—
188C2×D4C22×D2911yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—581no—
192C22×C4C22⋊D2912yes—
193C22×C4C22×D2911yescharacteristic
194Q16—588no—
195SD16—588no—
196M16—294no—
197M16—582no—
198C2×C8—582no—
199C4⋊C4—581no—
200C4⋊C4—584no—
201C22⋊C4—581no—
202C22⋊C4—584no—
203C42—292no—
204C23—24no—
205C23—22no—
206C23C2×C22⋊D2912yes—
207C23C2×C22⋊D2911yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—584no—
212Q8—1161no—
213Q8—1161no—
214Q8C2×C22⋊D2914yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×C22⋊D2914yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—582no—
231C2×C4—582no—
232C2×C4—1162no—
233C2×C4C2×C22⋊D2912yes—
234C2×C4C2×C22⋊D2911yescharacteristic
235C2×C4C2×C22⋊D2912yes—
236C2×C4C2×C22⋊D2912yes—
237C2×C4C23×D2911yescharacteristic
238C8—584no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#1174531512yes—
246C22C22×C22⋊D2911yescharacteristic
247C4—24no—
248C4—22no—
249C4—1162no—
250C4—1162no—
251C4#1174531511yescharacteristic
252C4#1174531511yescharacteristic
253C4C22×C22⋊D2912yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#1174531511yescharacteristic
subgroupG/H
centerC2C2×#11745315
commutator, cosocleC2×C4×C29C24
FrattiniC2×C4C23×D29
Fitting, prime radicalC29×#585C2
socleC2×C29C2×#120
hypercenter#585D29
hypocenterC29#3022
LCSG ▷ C2×C4×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#11745313C2semidirect
2. #14271734C2semidirect
3. #14271346C2semidirect
4. #14271341C2semidirect
5. #14271180C2semidirect
6. #14271147C2semidirect
17. C22×Dic58C22semidirect
19. #11745313C22semidirect
20. #11745306C22semidirect
22. #11745298C22semidirect
25. #11745288C22semidirect
27. C2×#90777C22semidirect
28. #11745185C22semidirect
29. #11745184C22semidirect
30. #11745181C22semidirect
160. C29#3022semidirect

Maximal subgroups

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

subgroupindex
C2×#117453132
#142717342
#142713462
#142713412
#142711802
#142711472
C29×#5852
#302229