Finite Group Databasebeta

← #13121207

Subgroups — #13121207

Order 1664.

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

normal subgroups only

This group has 592 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×#88151C212yesmaximal, complemented
2C2×#88150C212yesmaximal, complemented
3C13×#389C211yesmaximal, characteristic, complemented
4#11718655C212yesmaximal, complemented
5#11718655C212yesmaximal, complemented
6#11718600C211yesmaximal, characteristic
7#11718589C211yesmaximal, characteristic, complemented
8#11718326C212yesmaximal
9#11718317C212yesmaximal, complemented
10C22×D52C2211yescharacteristic
11C22×C4×D13C2212yes—
12C22×Dic26C2212yes—
13C2×C13×C4⋊C4C2212yes—
14C2×C13×C4⋊C4C2211yescharacteristic
15C2×C13×C4⋊C4C2211yescharacteristic
16C2×C42×C13C2211yescharacteristic
17C2×#6103C2212yes—
18C2×#6103C2212yes—
19C2×#6103C2212yes—
20C2×#6103C2212yes—
21C2×#6103C2212yes—
22C2×C4⋊Dic13C2212yes—
23C2×C4⋊Dic13C2211yescharacteristic
24C2×C4⋊Dic13C2211yescharacteristic
25C2×C4⋊Dic13C2211yescharacteristic
26C2×#6101C2212yes—
27C2×#6101C2212yes—
28#88151—28no—
29#88150—28no—
30C13×#95C2212yes—
31#88071C2212yes—
32#88071C2212yes—
33C23×D13C2312yes—
34C22×C4×C13C2312yes—
35C22×C4×C13C2312yes—
36C22×C4×C13C2311yescharacteristic
37C22×C4×C13C2311yescharacteristic
38C22×C4×C13C2311yescharacteristic
39C22×Dic13C2312yes—
40C22×Dic13C2312yes—
41C22×Dic13C2312yes—
42C2×D52—41no—
43C2×D52—41no—
44C2×D52Q814yescomplemented
45C2×C4×D13—24no—
46C2×C4×D13—24no—
47C2×C4×D13—42no—
48C2×Dic26—24no—
49C2×Dic26—24no—
50C2×Dic26—42no—
51C13×C4⋊C4—24no—
52C13×C4⋊C4—41no—
53C13×C4⋊C4D414yescomplemented
54C42×C13—41no—
55#6103—24no—
56#6103—24no—
57#6103—24no—
58#6103—24no—
59#6103—42no—
60C4⋊Dic13—24no—
61C4⋊Dic13—41no—
62C4⋊Dic13—41no—
63C4⋊Dic13D414yescomplemented
64#6101—24no—
65#6101—24no—
66#2683—131nomaximal, Sylow
67C23×C13C2411yescharacteristic
68C22×D13—22no—
69C22×D13—22no—
70C22×D13—22no—
71C22×D13—22no—
72C22×D13—22no—
73C22×D13—22no—
74C22×D13C2×Q814yes—
75C2×C4×C13—22no—
76C2×C4×C13—21no—
77C2×C4×C13—21no—
78C2×C4×C13—21no—
79C2×C4×C13—21no—
80C2×C4×C13—22no—
81C2×C4×C13—21no—
82C2×C4×C13—21no—
83C2×C4×C13—22no—
84C2×C4×C13—22no—
85C2×C4×C13—22no—
86C2×C4×C13C2×D412yes—
87C2×C4×C13C2×D414yes—
88C2×C4×C13C2×D412yes—
89C2×C4×C13C2×Q812yes—
90C2×Dic13—22no—
91C2×Dic13—22no—
92C2×Dic13—22no—
93C2×Dic13—22no—
94C2×Dic13—22no—
95C2×Dic13—22no—
96C2×Dic13—22no—
97C2×Dic13—22no—
98C2×Dic13C2×D414yes—
99D52—44no—
100C4×D13—48no—
101Dic26—48no—
102C2×C22⋊Q8—132no—
103C2×C22⋊Q8—132no—
104#396—132no—
105#396—132no—
106#395—132no—
107#390—131no—
108#389—131no—
109#389D1311yescharacteristic, complemented
110#386—132no—
111C22×C13C22×D411yescharacteristic
112C22×C13C22×D411yescharacteristic
113C22×C13C22×Q811yescharacteristic
114C22×C13H(5,𝔽2)11yescharacteristic
115C22×C13H(5,𝔽2)11yescharacteristic
116C22×C132−1+411yescharacteristic
117C22×C132−1+411yescharacteristic
118C2×D13—24no—
119C2×D13—24no—
120C2×D13—24no—
121C2×D13—42no—
122C2×D13—42no—
123C2×D13—42no—
124C2×D13—42no—
125C4×C13—22no—
126C4×C13—22no—
127C4×C13—22no—
128C4×C13—24no—
129C4×C13—42no—
130Dic13—24no—
131Dic13—42no—
132Dic13—42no—
133C22×Q8—132no—
134C22×D4—131no—
135C23×C4—132no—
136C22⋊Q8—268no—
137C22⋊Q8—268no—
138C2×C4⋊C4—131no—
139C2×C4⋊C4—132no—
140C2×C4⋊C4—132no—
141C2×C4⋊C4—132no—
142C2×C4⋊C4—131no—
143C2×C4⋊C4—131no—
144C2×C4⋊C4C2×D1312yes—
145C2×C4⋊C4C2×D1311yescharacteristic
146C2×C4⋊C4C2×D1311yescharacteristic
147C2×C22⋊C4—132no—
148C2×C22⋊C4—132no—
149C2×C22⋊C4—132no—
150C2×C22⋊C4—132no—
151C2×C22⋊C4—132no—
152C2×C42C2×D1311yescharacteristic
153#95—132no—
154#95—132no—
155#95C2×D1312yes—
156C2×C13#53711yescharacteristic
157C2×C13#53711yescharacteristic
158C2×C13#54311yescharacteristic
159C2×C13D4211yescharacteristic
160C2×C13D4×Q811yescharacteristic
161C2×C13D4×Q811yescharacteristic
162C2×C13#56311yescharacteristic
163D13—44no—
164C24—132no—
165C2×Q8—264no—
166C2×Q8—264no—
167C2×Q8—522no—
168C2×D4—134no—
169C2×D4—521no—
170C2×D4—521no—
171C22×C4—132no—
172C22×C4—132no—
173C22×C4—132no—
174C22×C4—264no—
175C22×C4—264no—
176C22×C4—522no—
177C22×C4C22×D1312yes—
178C22×C4C22×D1312yes—
179C22×C4C22×D1311yescharacteristic
180C22×C4C22×D1311yescharacteristic
181C22×C4C22×D1311yescharacteristic
182C4⋊C4—24no—
183C4⋊C4—41no—
184C4⋊C4—134no—
185C4⋊C4—264no—
186C4⋊C4—264no—
187C4⋊C4—264no—
188C4⋊C4—521no—
189C4⋊C4—521no—
190C4⋊C4D5214yescomplemented
191C22⋊C4—264no—
192C22⋊C4—264no—
193C22⋊C4—264no—
194C22⋊C4—264no—
195C22⋊C4—522no—
196C42—41no—
197C13#268311yescharacteristic, complemented, Sylow
198C23—134no—
199C23—262no—
200C23—262no—
201C23—262no—
202C23—262no—
203C23—262no—
204C23—262no—
205C23C23×D1311yescharacteristic
206Q8—528no—
207D4—524no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—134no—
220C2×C4—262no—
221C2×C4—262no—
222C2×C4—262no—
223C2×C4—262no—
224C2×C4—262no—
225C2×C4—262no—
226C2×C4—262no—
227C2×C4—262no—
228C2×C4—528no—
229C2×C4C2×D5212yes—
230C2×C4C2×D5214yes—
231C2×C4D4×D1312yes—
232C2×C4Q8×D1312yes—
233C22—264no—
234C22—264no—
235C22—264no—
236C22—522no—
237C22—522no—
238C22—522no—
239C22—522no—
240C22C22×D5211yescharacteristic
241C22C2×D4×D1311yescharacteristic
242C22#8825111yescharacteristic
243C22C2×Q8×D1311yescharacteristic
244C22#8825411yescharacteristic
245C22#8825611yescharacteristic
246C22#8825711yescharacteristic
247C4—22no—
248C4—24no—
249C4—22no—
250C4—22no—
251C4—42no—
252C4—264no—
253C4—522no—
254C4—522no—
255C2—524no—
256C2#1171921511yescharacteristic
257C2D4×D5211yescharacteristic
258C2Q8×D5211yescharacteristic
259C2#1171930211yescharacteristic
260C2#1171933911yescharacteristic
261C2#1171940411yescharacteristic
262C2#1171943211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#389C2
prime radicalC22×D52C22
hypercenter#389D13
hypocenterC13#2683
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#88151C2semidirect
2. C2×#88150C2semidirect
3. C13×#389C2semidirect
4. #11718655C2semidirect
5. #11718655C2semidirect
7. #11718589C2semidirect
9. #11718317C2semidirect
44. C2×D52Q8semidirect
53. C13×C4⋊C4D4semidirect
63. C4⋊Dic13D4semidirect
109. #389D13semidirect
190. C4⋊C4D52semidirect
197. C13#2683semidirect

Maximal subgroups

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

subgroupindex
C2×#881512
C2×#881502
C13×#3892
#117186552
#117186002
#117185892
#117183262
#117183172
#268313