Finite Group Databasebeta

← #13121151

Subgroups — #13121151

Order 1664.

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

normal subgroups only

This group has 544 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×#88178C212yesmaximal
2C2×#88142C212yesmaximal, complemented
3C13×#385C211yesmaximal, characteristic
4#11718645C212yesmaximal, complemented
5#11718612C212yesmaximal
6#11718610C211yesmaximal, characteristic
7#11718608C211yesmaximal, characteristic
8#11718317C212yesmaximal, complemented
9#11718316C212yesmaximal, complemented
10C23×C4×C13C2211yescharacteristic
11C22×Dic26C2211yescharacteristic
12C22×Dic26C2212yes—
13C2×C13×C4⋊C4C2212yes—
14C2×C13×C22⋊C4C2211yescharacteristic
15C2×C13×C22⋊C4C2211yescharacteristic
16C2×C22⋊Dic13C2212yes—
17C2×C22⋊Dic13C2212yes—
18C2×C4⋊Dic13C2212yes—
19C2×C4⋊Dic13C2212yes—
20C2×C4⋊Dic13C2211yescharacteristic
21#88178—28no—
22C2×#6101C2212yes—
23C2×#6101C2212yes—
24C2×#6101C2212yes—
25C2×C4×Dic13C2212yes—
26#88142—28no—
27C13×#95C2212yes—
28#88071C2211yescharacteristic
29#88071C2212yes—
30#88071C2212yes—
31#88071C2212yes—
32#88071C2211yescharacteristic
33C24×C13C2311yescharacteristic
34C22×C4×C13—24no—
35C22×C4×C13—41no—
36C22×C4×C13C2311yescharacteristic
37C22×C4×C13C2311yescharacteristic
38C22×C4×C13C2312yes—
39C22×C4×C13C2312yes—
40C22×Dic13C2312yes—
41C22×Dic13C2312yes—
42C22×Dic13C2312yes—
43C22×Dic13C2312yes—
44C2×Dic26—24no—
45C2×Dic26—24no—
46C2×Dic26—24no—
47C2×Dic26—41no—
48C2×Dic26—42no—
49C13×C4⋊C4—24no—
50C13×C22⋊C4—41no—
51C13×C22⋊C4Q814yescomplemented
52C22⋊Dic13—24no—
53C22⋊Dic13—24no—
54C4⋊Dic13—24no—
55C4⋊Dic13—24no—
56C4⋊Dic13—41no—
57#6101—24no—
58#6101—24no—
59#6101D418yes—
60C4×Dic13—24no—
61#2677—131nomaximal, Sylow
62C23×C13—22no—
63C23×C13—21no—
64C23×C13—21no—
65C23×C13—22no—
66C23×C13C2×Q812yes—
67C23×C13C2411yescharacteristic
68C2×C4×C13—22no—
69C2×C4×C13—22no—
70C2×C4×C13—22no—
71C2×C4×C13—22no—
72C2×C4×C13—22no—
73C2×C4×C13—21no—
74C2×C4×C13—21no—
75C2×C4×C13—22no—
76C2×C4×C13—44no—
77C2×C4×C13C2×D414yes—
78C2×C4×C13C2×Q814yes—
79C2×Dic13—22no—
80C2×Dic13—22no—
81C2×Dic13—22no—
82C2×Dic13—22no—
83C2×Dic13—22no—
84C2×Dic13—22no—
85C2×Dic13—22no—
86C2×Dic13—22no—
87C2×Dic13—22no—
88C2×Dic13—22no—
89C2×Dic13C2×D414yes—
90C2×Dic13C2×D414yes—
91Dic26—44no—
92Dic26—48no—
93C2×C4⋊Q8—132no—
94C2×C22⋊Q8—132no—
95#399—131no—
96#398—132no—
97#395—132no—
98#393—131no—
99#388—132no—
100#387—132no—
101#385D1311yescharacteristic
102C22×C13—22no—
103C22×C13—24no—
104C22×C13—42no—
105C22×C13—41no—
106C22×C13—41no—
107C22×C13C22×D412yes—
108C22×C13C22×Q811yescharacteristic
109C22×C13H(5,𝔽2)11yescharacteristic
110C22×C132−1+411yescharacteristic
111C22×C132−1+412yes—
112C4×C13—24no—
113C4×C13—24no—
114C4×C13—41no—
115C4×C13—41no—
116Dic13—24no—
117Dic13—24no—
118Dic13—42no—
119Dic13—42no—
120C22×Q8—132no—
121C22×Q8—131no—
122C23×C4C2×D1311yescharacteristic
123C4⋊Q8—268no—
124C22⋊Q8—268no—
125C2×C4⋊C4—132no—
126C2×C4⋊C4—132no—
127C2×C4⋊C4—132no—
128C2×C4⋊C4—132no—
129C2×C4⋊C4—132no—
130C2×C4⋊C4—131no—
131C2×C4⋊C4C2×D1312yes—
132C2×C22⋊C4—132no—
133C2×C22⋊C4—132no—
134C2×C22⋊C4C2×D1311yescharacteristic
135C2×C22⋊C4C2×D1311yescharacteristic
136C2×C42—132no—
137#95—131no—
138#95—132no—
139#95—132no—
140#95—132no—
141#95—131no—
142#95C2×D1312yes—
143C2×C13—42no—
144C2×C13#53612yes—
145C2×C13#54411yescharacteristic
146C2×C13D4211yescharacteristic
147C2×C13D4×Q812yes—
148C2×C13#56311yescharacteristic
149C24C22×D1311yescharacteristic
150C2×Q8—264no—
151C2×Q8—264no—
152C2×Q8—264no—
153C2×Q8—522no—
154C2×Q8—521no—
155C22×C4—24no—
156C22×C4—41no—
157C22×C4—132no—
158C22×C4—132no—
159C22×C4—132no—
160C22×C4—132no—
161C22×C4C22×D1311yescharacteristic
162C22×C4C22×D1311yescharacteristic
163C22×C4C22×D1312yes—
164C22×C4C22×D1312yes—
165C4⋊C4—24no—
166C4⋊C4—138no—
167C4⋊C4—264no—
168C4⋊C4—264no—
169C4⋊C4—264no—
170C4⋊C4—264no—
171C4⋊C4—521no—
172C22⋊C4—41no—
173C22⋊C4—264no—
174C22⋊C4—264no—
175C22⋊C4Dic2614yescomplemented
176C42—264no—
177C13#267711yescharacteristic, complemented, Sylow
178C23—22no—
179C23—21no—
180C23—21no—
181C23—22no—
182C23C2×Dic2612yes—
183C23C23×D1311yescharacteristic
184Q8—528no—
185Q8—524no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—134no—
196C2×C4—134no—
197C2×C4—262no—
198C2×C4—262no—
199C2×C4—262no—
200C2×C4—262no—
201C2×C4—262no—
202C2×C4—262no—
203C2×C4—262no—
204C2×C4—262no—
205C2×C4—262no—
206C2×C4—262no—
207C2×C4C2×Dic2614yes—
208C2×C4D4×D1314yes—
209C22—22no—
210C22—24no—
211C22—41no—
212C22—42no—
213C22—41no—
214C22C22×Dic2611yescharacteristic
215C22C2×D4×D1312yes—
216C22#8825111yescharacteristic
217C22#8825411yescharacteristic
218C22#8825712yes—
219C4—24no—
220C4—24no—
221C4—41no—
222C4—41no—
223C4—264no—
224C4—264no—
225C4—522no—
226C4—522no—
227C2—42no—
228C2#1171920511yescharacteristic
229C2D4×Dic2612yes—
230C2#1171929511yescharacteristic
231C2#1171933312yes—
232C2#1171935611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#385C2
prime radicalC24×C13C23
hypercenter#385D13
hypocenterC13#2677
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#88142C2semidirect
4. #11718645C2semidirect
8. #11718317C2semidirect
9. #11718316C2semidirect
51. C13×C22⋊C4Q8semidirect
175. C22⋊C4Dic26semidirect
177. C13#2677semidirect

Maximal subgroups

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

subgroupindex
C2×#881782
C2×#881422
C13×#3852
#117186452
#117186122
#117186102
#117186082
#117183172
#117183162
#267713