Finite Group Databasebeta

← #16819902

Subgroups — #16819902

Order 3712.

386 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 83 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×#11745291C211yesmaximal, characteristic
2#14271636C211yesmaximal, characteristic
3C2×#11745154C211yesmaximal, characteristic
4#14271026C211yesmaximal, characteristic, complemented
5#14270930C211yesmaximal, characteristic, complemented
6#14270929C211yesmaximal, characteristic, complemented
7C29×#381C211yesmaximal, characteristic, complemented
8C23×C4×C29C2211yescharacteristic
9C22×C22⋊D29C2211yescharacteristic
10C2×C22⋊C4×C29C2211yescharacteristic
11C2×C22⋊C4×C29C2211yescharacteristic
12C2×C22⋊Dic29C2211yescharacteristic
13C2×C22⋊Dic29C2211yescharacteristic
14C2×C22⋊Dic29C2211yescharacteristic
15#11745291—22no—
16#11745291C414yescomplemented
17C4×C22⋊D29—22no—
18C2×#90781—21no—
19C2×#90779—21no—
20#11745243—22no—
21C22⋊C4×D29—21no—
22#11745239—21no—
23C29×#95—22no—
24#11745179—22no—
25#11745154—22no—
26C24×C29C2311yescharacteristic
27C23×D29D411yescharacteristic
28C22×C4×C29—21no—
29C22×C4×C29—21no—
30C22×C4×C29—21no—
31C22×C4×C29—21no—
32C22×C4×C29—21no—
33C22×C4×C29—21no—
34C22×C4×C29—21no—
35C22×C4×C29C2×C412yescomplemented
36C22×C4×C29D412yes—
37C2×C22⋊D29—21no—
38C2×C22⋊D29—21no—
39C2×C22⋊D29—21no—
40C2×C22⋊D29—21no—
41C2×C22⋊D29C2×C412yes—
42C2×C22⋊D29D412yes—
43C22×Dic29—21no—
44C22×Dic29—21no—
45C22×Dic29D411yescharacteristic
46C22×Dic29D412yes—
47C2×C4×D29—22no—
48C22⋊C4×C29—21no—
49C22⋊C4×C29—21no—
50C22⋊C4×C29—21no—
51C22⋊C4×C29—21no—
52C22⋊Dic29—21no—
53C22⋊Dic29—21no—
54C22⋊Dic29—21no—
55C22⋊Dic29—21no—
56C22⋊Dic29—21no—
57C22⋊Dic29C2×C412yescomplemented
58#90781—22no—
59#90781—24no—
60#90779—22no—
61#90779—24no—
62C4×Dic29—22no—
63C23×C29—42no—
64C23×C29C22×C411yescharacteristic
65C23×C29C2×D411yescharacteristic
66C23×C29C2×D411yescharacteristic
67C23×C29C2×D411yescharacteristic
68C23×C29C2×D411yescharacteristic
69C23×C29#4111yescharacteristic
70C23×C29#4111yescharacteristic
71C22×D29—21no—
72C22×D29—21no—
73C22×D29—42no—
74C22×D29C22⋊C412yes—
75C2×C4×C29—22no—
76C2×C4×C29—22no—
77C2×C4×C29—22no—
78C2×C4×C29—22no—
79C2×C4×C29—41no—
80C2×C4×C29—41no—
81C2×C4×C29—41no—
82C2×C4×C29—41no—
83C2×C4×C29—42no—
84C2×C4×C29—42no—
85C2×C4×C29—41no—
86C22⋊D29—22no—
87C22⋊D29—22no—
88C22⋊D29—22no—
89C22⋊D29—22no—
90C2×Dic29—21no—
91C2×Dic29—21no—
92C2×Dic29—22no—
93C2×Dic29—22no—
94C2×Dic29—42no—
95C2×Dic29—42no—
96C2×Dic29—42no—
97C2×Dic29—41no—
98C2×Dic29C22⋊C412yes—
99C4×D29—82no—
100#1890—291nomaximal, Sylow
101C22×C29—22no—
102C22×C29—22no—
103C22×C29—22no—
104C22×C29—41no—
105C22×C29—41no—
106C22×C29—41no—
107C22×C29—41no—
108C22×C29C2×C22⋊C411yescharacteristic
109C22×C29C4×D411yescharacteristic
110C22×C29C4×D411yescharacteristic
111C22×C29#12011yescharacteristic
112C22×C29#12111yescharacteristic
113C22×C29#12111yescharacteristic
114C22×C29#12311yescharacteristic
115C2×D29—22no—
116C2×D29—42no—
117C2×D29—42no—
118C2×D29—81no—
119C4×C29—42no—
120C4×C29—42no—
121C4×C29—81no—
122Dic29—22no—
123Dic29—42no—
124Dic29—42no—
125Dic29—81no—
126C2×#123—291no—
127#518—291no—
128C2×#99—291no—
129#381—291no—
130#381D2911yescharacteristic, complemented
131#342—291no—
132#342—291no—
133C2×C29—21no—
134C2×C29—21no—
135C2×C29—21no—
136C2×C29—21no—
137C2×C29—21no—
138C2×C29—21no—
139C2×C29#38611yescharacteristic
140C2×C29#45812yes—
141D29—42no—
142C22×D4—291no—
143C23×C4C2×D2911yescharacteristic
144#123—294no—
145#123—582no—
146C4×D4—582no—
147C4×D4—582no—
148#117—581no—
149C2×C4⋊C4—581no—
150C2×C22⋊C4—291no—
151C2×C22⋊C4—291no—
152C2×C22⋊C4—291no—
153C2×C22⋊C4—581no—
154C2×C22⋊C4—581no—
155C2×C22⋊C4C2×D2911yescharacteristic
156C2×C22⋊C4C2×D2911yescharacteristic
157#99—582no—
158#95—22no—
159#95—582no—
160C29#189011yescharacteristic, complemented, Sylow
161C24—291no—
162C24C22×D2911yescharacteristic
163C2×D4—292no—
164C2×D4—292no—
165C2×D4—581no—
166C2×D4—581no—
167C2×D4—581no—
168C2×D4—581no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—291no—
177C22×C4—292no—
178C22×C4—582no—
179C22×C4—581no—
180C22×C4—581no—
181C22×C4C4×D2912yescomplemented
182C22×C4C22⋊D2912yes—
183C4⋊C4—584no—
184C4⋊C4—582no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—292no—
190C22⋊C4—584no—
191C22⋊C4—582no—
192C22⋊C4—581no—
193C22⋊C4—581no—
194C22⋊C4—581no—
195C22⋊C4—581no—
196C22⋊C4—581no—
197C42—582no—
198C23—42no—
199C23—292no—
200C23—581no—
201C23—581no—
202C23—1162no—
203C23C2×C4×D2911yescharacteristic
204C23#9080511yescharacteristic
205C23D4×D2911yescharacteristic
206C23D4×D2911yescharacteristic
207C23D4×D2911yescharacteristic
208C23#9080711yescharacteristic
209C23C2×C22⋊D2911yescharacteristic
210D4—582no—
211D4—582no—
212D4—582no—
213D4—582no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—292no—
226C2×C4—581no—
227C2×C4—581no—
228C2×C4—582no—
229C2×C4—582no—
230C2×C4—1162no—
231C2×C4—1162no—
232C2×C4—1162no—
233C2×C4—1161no—
234C2×C4—2322no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—582no—
243C22—1162no—
244C22—1162no—
245C22—2321no—
246C22C22⋊C4×D2911yescharacteristic
247C22#1174524311yescharacteristic
248C22#1174524511yescharacteristic
249C22#1174524611yescharacteristic
250C22C4×C22⋊D2911yescharacteristic
251C22#1174529911yescharacteristic
252C22#1174530011yescharacteristic
253C4—42no—
254C4—42no—
255C4—81no—
256C4—582no—
257C4—1162no—
258C4—1162no—
259C4—2321no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—1162no—
267C2#1427103911yescharacteristic
268C2#1427135212yes—
subgroupG/H
centerC22#11745299
commutatorC23×C29C22×C4
FrattiniC24C22×D29
FittingC29×#381C2
socleC22×C29#120
prime radicalC22×C22⋊D29C22
cosocleC24×C29C23
hypercenter#381D29
hypocenterC29#1890
LCSG ▷ C23×C29 ▷ C2×C29 ▷ C29
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
4. #14271026C2semidirect
5. #14270930C2semidirect
6. #14270929C2semidirect
7. C29×#381C2semidirect
16. #11745291C4semidirect
35. C22×C4×C29C2×C4semidirect
57. C22⋊Dic29C2×C4semidirect
130. #381D29semidirect
160. C29#1890semidirect
181. C22×C4C4×D29semidirect

Maximal subgroups

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

subgroupindex
C2×#117452912
#142716362
C2×#117451542
#142710262
#142709302
#142709292
C29×#3812
#189029