Finite Group Databasebeta

← #13123007

Subgroups — #13123007

Order 1664.

416 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C13×#394C211yesmaximal, characteristic
2#11718883C212yesmaximal
3#11718640C211yesmaximal, characteristic, complemented
4#11718635C211yesmaximal, characteristic, complemented
5#11718614C212yesmaximal, complemented
6#11718612C212yesmaximal, complemented
7#11718608C212yesmaximal, complemented
8#11718608C212yesmaximal, complemented
9#11718321C212yesmaximal, complemented
10C22×D4×C13C2211yescharacteristic
11C22×Dic26C2211yescharacteristic
12C22×Dic26C2211yescharacteristic
13C2×C13×C4⋊C4C2211yescharacteristic
14C2×C13×C22⋊C4C2212yes—
15C2×C13×C22⋊C4C2212yes—
16C2×C22⋊Dic13C2212yes—
17C2×C22⋊Dic13C2212yes—
18C2×C22⋊Dic13C2212yes—
19C2×C22⋊Dic13C2212yes—
20C2×C4⋊Dic13C2212yes—
21C2×#6101C2212yes—
22C2×C4×Dic13C2211yescharacteristic
23C13×#95C2211yescharacteristic
24#88071C2212yes—
25#88071C2212yes—
26#88071C2212yes—
27#88071C2212yes—
28#88071C2212yes—
29#88071C2212yes—
30#88071C2211yescharacteristic
31C24×C13C2312yes—
32C2×D4×C13—42no—
33C2×D4×C13—41no—
34C22×C4×C13C2312yes—
35C22×C4×C13C2311yescharacteristic
36C22×C4×C13C2311yescharacteristic
37C22×C4×C13C2311yescharacteristic
38C22×Dic13C2312yes—
39C22×Dic13C2312yes—
40C22×Dic13C2312yes—
41C22×Dic13C2312yes—
42C2×Dic26—22no—
43C2×Dic26—41no—
44C2×Dic26—41no—
45C2×Dic26—42no—
46C2×Dic26—41no—
47C13×C4⋊C4—41no—
48C13×C22⋊C4—24no—
49C13×C22⋊C4—42no—
50C22⋊Dic13—42no—
51C22⋊Dic13—42no—
52C22⋊Dic13—42no—
53C22⋊Dic13—42no—
54C4⋊Dic13—42no—
55#6101—42no—
56C4×Dic13—41no—
57#2832—131nomaximal, Sylow
58C23×C13—22no—
59C23×C13—22no—
60C23×C13—22no—
61C23×C13—22no—
62C23×C13—22no—
63C23×C13—22no—
64C23×C13—22no—
65C23×C13C2411yescharacteristic
66D4×C13—44no—
67C2×C4×C13—21no—
68C2×C4×C13—21no—
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#4112yes—
77C2×Dic13—22no—
78C2×Dic13—22no—
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—
89Dic26—44no—
90Dic26—44no—
91#400—132no—
92#397—132no—
93#395—131no—
94#394—132no—
95#394D1311yescharacteristic
96#393—132no—
97#393—132no—
98#391—131no—
99#388—132no—
100C22×C13—42no—
101C22×C13—42no—
102C22×C13—42no—
103C22×C13—42no—
104C22×C13—42no—
105C22×C13—42no—
106C22×C13—42no—
107C22×C13C2×#4111yescharacteristic
108C22×C13H(5,𝔽2)11yescharacteristic
109C22×C13H(5,𝔽2)12yes—
110C22×C13H(5,𝔽2)11yescharacteristic
111C22×C132−1+412yes—
112C4×C13—22no—
113C4×C13—42no—
114C4×C13—41no—
115C4×C13—41no—
116Dic13—42no—
117Dic13—42no—
118Dic13—42no—
119Dic13—42no—
120C22×Q8—131no—
121C22×Q8—131no—
122C22×D4C2×D1311yescharacteristic
123C2×C4⋊C4—132no—
124C2×C4⋊C4—132no—
125C2×C4⋊C4C2×D1311yescharacteristic
126C2×C22⋊C4—132no—
127C2×C22⋊C4—132no—
128C2×C22⋊C4—132no—
129C2×C22⋊C4—132no—
130C2×C22⋊C4C2×D1312yes—
131C2×C22⋊C4C2×D1312yes—
132C2×C42—131no—
133#95—132no—
134#95—132no—
135#95—132no—
136#95—132no—
137#95—132no—
138#95—132no—
139#95—131no—
140#95C2×D1311yescharacteristic
141C2×C13—82no—
142C2×C13#53811yescharacteristic
143C2×C13#54212yes—
144C2×C13H(3,𝔽4)11yescharacteristic
145C2×C13#56212yes—
146C2×C13#56311yescharacteristic
147C24C22×D1312yes—
148C2×Q8—262no—
149C2×Q8—521no—
150C2×Q8—521no—
151C2×Q8—522no—
152C2×Q8—521no—
153C2×D4—41no—
154C2×D4—42no—
155C22×C4—132no—
156C22×C4—132no—
157C22×C4—132no—
158C22×C4—132no—
159C22×C4C22×D1312yes—
160C22×C4C22×D1311yescharacteristic
161C22×C4C22×D1311yescharacteristic
162C22×C4C22×D1311yescharacteristic
163C4⋊C4—41no—
164C4⋊C4—522no—
165C4⋊C4—522no—
166C22⋊C4—24no—
167C22⋊C4—42no—
168C22⋊C4—522no—
169C22⋊C4—522no—
170C22⋊C4—522no—
171C22⋊C4—522no—
172C42—521no—
173C13#283211yescharacteristic, complemented, Sylow
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23C23×D1311yescharacteristic
182Q8—524no—
183Q8—524no—
184D4—44no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—262no—
195C2×C4—262no—
196C2×C4—262no—
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#612912yes—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C2×#612911yescharacteristic
215C22#8825112yes—
216C22#8825111yescharacteristic
217C22#8825611yescharacteristic
218C22#8825712yes—
219C4—22no—
220C4—41no—
221C4—41no—
222C4—42no—
223C4—522no—
224C4—522no—
225C4—522no—
226C4—522no—
227C2—82no—
228C2#1171929911yescharacteristic
229C2#1171930812yes—
230C2#1171932211yescharacteristic
231C2#1171937212yes—
232C2#1171937711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#394C2
prime radicalC22×D4×C13C22
hypercenter#394D13
hypocenterC13#2832
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
3. #11718640C2semidirect
4. #11718635C2semidirect
5. #11718614C2semidirect
6. #11718612C2semidirect
7. #11718608C2semidirect
8. #11718608C2semidirect
9. #11718321C2semidirect
173. C13#2832semidirect

Maximal subgroups

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

subgroupindex
C13×#3942
#117188832
#117186402
#117186352
#117186142
#117186122
#117186082
#117183212
#283213