Finite Group Databasebeta

← #13120598

Subgroups — #13120598

Order 1664.

612 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 157 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×C13×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#88201C211yesmaximal, characteristic, complemented
3C2×#88184C211yesmaximal, characteristic
4C2×#88180C211yesmaximal, characteristic, complemented
5#11718891C211yesmaximal, characteristic, complemented
6#11718655C214yesmaximal, complemented
7#11718649C211yesmaximal, characteristic, complemented
8#11718637C211yesmaximal, characteristic, complemented
9#11718620C214yesmaximal, complemented
10C22×Q8×C13C2211yescharacteristic
11C23×C4×C13C2211yescharacteristic
12C22×C22⋊D13C2212yes—
13C22×D52C2211yescharacteristic
14C13×C22⋊Q8—24no—
15C2×C13×C4⋊C4C2211yescharacteristic
16C2×C13×C4⋊C4C2212yes—
17C2×C13×C22⋊C4C2212yes—
18C2×C22⋊Dic13C2212yes—
19#88201—24no—
20#88184—24no—
21C2×#6103C2212yes—
22C2×#6103C2214yes—
23C2×#6103C2214yes—
24#88180—24no—
25C2×C4⋊Dic13C2212yes—
26C2×C4⋊Dic13C2211yescharacteristic
27C2×#6101C2214yes—
28C2×C4×Dic13C2211yescharacteristic
29C2×C4×Dic13C2211yescharacteristic
30#88071C2214yes—
31C24×C13C2311yescharacteristic
32C23×D13C2312yes—
33C2×Q8×C13—22no—
34C2×Q8×C13—22no—
35C2×Q8×C13—22no—
36C22×C4×C13—22no—
37C22×C4×C13—22no—
38C22×C4×C13D414yes—
39C22×C4×C13C2312yes—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C2×C22⋊D13—24no—
44C2×C22⋊D13—44no—
45C22×Dic13C2312yes—
46C22×Dic13C2312yes—
47C22×Dic13C2312yes—
48C2×D52—22no—
49C2×D52—22no—
50C2×D52—22no—
51C13×C4⋊C4—22no—
52C13×C4⋊C4—24no—
53C13×C22⋊C4—24no—
54C22⋊Dic13—24no—
55#6103—24no—
56#6103—28no—
57#6103—44no—
58C4⋊Dic13—22no—
59C4⋊Dic13—24no—
60#6101—44no—
61C4×Dic13—22no—
62C4×Dic13D414yescomplemented
63#2620—131nomaximal, Sylow
64C23×C13—21no—
65C23×C13—21no—
66C23×C13—22no—
67C23×C13—22no—
68C23×C13C2×D412yes—
69C23×C13C2411yescharacteristic
70C22×D13—22no—
71C22×D13—22no—
72C22×D13—22no—
73C22×D13—24no—
74C22×D13—24no—
75Q8×C13—44no—
76C2×C4×C13—21no—
77C2×C4×C13—21no—
78C2×C4×C13—28no—
79C2×C4×C13—22no—
80C2×C4×C13—22no—
81C2×C4×C13—22no—
82C2×C4×C13—22no—
83C2×C4×C13—24no—
84C2×C4×C13C2×D412yes—
85C2×C4×C13C2×D412yes—
86C2×C4×C13C2×D412yes—
87C2×C4×C13#4112yes—
88C22⋊D13—48no—
89C2×Dic13—22no—
90C2×Dic13—22no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—24no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13C2×D414yes—
97D52—44no—
98C2×#124—131no—
99C2×C22⋊Q8D1311yescharacteristic, complemented
100C2×#121—131no—
101C2×#117—131no—
102#396—134no—
103#394—134no—
104#391—131no—
105#390—131no—
106#384—131no—
107C22×C13—22no—
108C22×C13—22no—
109C22×C13—22no—
110C22×C13—42no—
111C22×C13—42no—
112C22×C13C22×D411yescharacteristic
113C22×C13C22×D411yescharacteristic
114C22×C13C2×#4111yescharacteristic
115C22×C13H(5,𝔽2)12yes—
116C22×C132−1+412yes—
117C2×D13—42no—
118C2×D13—42no—
119C2×D13—42no—
120C2×D13—44no—
121C2×D13—44no—
122C4×C13—22no—
123C4×C13—42no—
124C4×C13—42no—
125C4×C13#12114yes—
126Dic13—24no—
127Dic13—42no—
128Dic13—42no—
129C22×Q8C2×D1311yescharacteristic
130C22×D4—132no—
131C22×D4—131no—
132C23×C4C2×D1311yescharacteristic
133#124—264no—
134C22⋊Q8—24no—
135#121—264no—
136#117—264no—
137C2×C4⋊C4—132no—
138C2×C4⋊C4—131no—
139C2×C4⋊C4—134no—
140C2×C4⋊C4C2×D1311yescharacteristic
141C2×C4⋊C4C2×D1312yes—
142C2×C22⋊C4—132no—
143C2×C22⋊C4—132no—
144C2×C22⋊C4—134no—
145C2×C22⋊C4—134no—
146C2×C22⋊C4C2×D1312yes—
147C2×C42—131no—
148C2×C42—131no—
149#95—134no—
150C2×C13—42no—
151C2×C13C2×#12111yescharacteristic
152C2×C13#53511yescharacteristic
153C2×C13#53611yescharacteristic
154C2×C13#53712yes—
155C2×C13#54212yes—
156D13—82no—
157C24—132no—
158C24C22×D1311yescharacteristic
159C2×Q8—22no—
160C2×Q8—22no—
161C2×Q8—22no—
162C2×D4—264no—
163C2×D4—262no—
164C2×D4—262no—
165C2×D4—262no—
166C2×D4—524no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—132no—
170C22×C4—132no—
171C22×C4—132no—
172C22×C4C22⋊D1314yes—
173C22×C4C22×D1312yes—
174C22×C4C22×D1312yes—
175C22×C4C22×D1311yescharacteristic
176C22×C4C22×D1311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—24no—
179C4⋊C4—264no—
180C4⋊C4—262no—
181C4⋊C4—524no—
182C22⋊C4—24no—
183C22⋊C4—268no—
184C22⋊C4—264no—
185C22⋊C4—264no—
186C22⋊C4—524no—
187C42—134no—
188C42—262no—
189C13#262011yescharacteristic, complemented, Sylow
190C23—21no—
191C23—21no—
192C23—22no—
193C23—22no—
194C23—262no—
195C23—262no—
196C23—264no—
197C23—264no—
198C23—262no—
199C23C2×C22⋊D1312yes—
200C23C23×D1311yescharacteristic
201Q8—44no—
202D4—528no—
203D4—524no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—28no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—134no—
213C2×C4—262no—
214C2×C4—262no—
215C2×C4—262no—
216C2×C4—262no—
217C2×C4—262no—
218C2×C4—264no—
219C2×C4—262no—
220C2×C4D4×D1312yes—
221C2×C4#612912yes—
222C2×C4C2×C22⋊D1312yes—
223C2×C4C2×C22⋊D1312yes—
224C22—22no—
225C22—22no—
226C22—22no—
227C22—42no—
228C22—42no—
229C22—522no—
230C22—522no—
231C22—522no—
232C22—524no—
233C22—524no—
234C22C2×D4×D1311yescharacteristic
235C22C2×#612911yescharacteristic
236C22#8825412yes—
237C22#8825612yes—
238C22C22×C22⋊D1311yescharacteristic
239C4—22no—
240C4—42no—
241C4—42no—
242C4—264no—
243C4—522no—
244C4—522no—
245C4#8819314yes—
246C2—42no—
247C2—1042no—
248C2#1171933412yes—
249C2#1171934812yes—
250C2C2×#8819311yescharacteristic
251C2#1171952111yescharacteristic
252C2#1171953511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC2×C13×C22⋊Q8C2
prime radicalC2×#88184C2
hypercenterC2×C22⋊Q8D13
hypocenterC13#2620
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C13×C22⋊Q8C2semidirect
2. C2×#88201C2semidirect
4. C2×#88180C2semidirect
5. #11718891C2semidirect
6. #11718655C2semidirect
7. #11718649C2semidirect
8. #11718637C2semidirect
9. #11718620C2semidirect
62. C4×Dic13D4semidirect
99. C2×C22⋊Q8D13semidirect
189. C13#2620semidirect

Maximal subgroups

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

subgroupindex
C2×C13×C22⋊Q82
C2×#882012
C2×#881842
C2×#881802
#117188912
#117186552
#117186492
#117186372
#117186202
#262013