Finite Group Databasebeta

← #13120507

Subgroups — #13120507

Order 1664.

540 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 137 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×#88131C212yesmaximal, complemented
2C13×#394C211yesmaximal, characteristic
3#11718881C212yesmaximal
4#11718644C211yesmaximal, characteristic, complemented
5#11718638C211yesmaximal, characteristic, complemented
6#11718612C212yesmaximal, complemented
7#11718609C212yesmaximal, complemented
8#11718607C212yesmaximal, complemented
9#11718313C212yesmaximal, complemented
10C22×D4×C13C2211yescharacteristic
11C23×Dic13C2212yes—
12C2×C13×C4⋊C4C2211yescharacteristic
13C2×C13×C22⋊C4C2212yes—
14C2×C13×C22⋊C4C2212yes—
15C2×C22⋊Dic13C2212yes—
16C2×C22⋊Dic13C2212yes—
17C2×C22⋊Dic13C2212yes—
18C2×C4⋊Dic13C2211yescharacteristic
19C2×C4⋊Dic13C2211yescharacteristic
20C2×#6101C2212yes—
21C2×#6101C2212yes—
22C2×#6101C2212yes—
23C2×C4×Dic13C2212yes—
24C2×C4×Dic13C2211yescharacteristic
25#88131—28no—
26C13×#95C2211yescharacteristic
27#88071C2212yes—
28#88071C2212yes—
29#88071C2212yes—
30#88071C2211yescharacteristic
31#88071C2212yes—
32C24×C13C2312yes—
33C2×D4×C13—24no—
34C2×D4×C13—41no—
35C22×C4×C13C2311yescharacteristic
36C22×C4×C13C2311yescharacteristic
37C22×C4×C13C2312yes—
38C22×C4×C13C2311yescharacteristic
39C22×Dic13—24no—
40C22×Dic13—24no—
41C22×Dic13—24no—
42C22×Dic13C2312yes—
43C22×Dic13C2312yes—
44C22×Dic13C2312yes—
45C22×Dic13C2312yes—
46C13×C4⋊C4—41no—
47C13×C22⋊C4—24no—
48C13×C22⋊C4—24no—
49C22⋊Dic13—24no—
50C22⋊Dic13—24no—
51C22⋊Dic13—42no—
52C4⋊Dic13—41no—
53C4⋊Dic13—41no—
54#6101—24no—
55#6101—24no—
56#6101—24no—
57C4×Dic13—24no—
58C4×Dic13D414yescomplemented
59#2609—131nomaximal, Sylow
60C23×C13—22no—
61C23×C13—22no—
62C23×C13—22no—
63C23×C13—22no—
64C23×C13—22no—
65C23×C13—22no—
66C23×C13#4114yes—
67C23×C13C2411yescharacteristic
68D4×C13—44no—
69C2×C4×C13—21no—
70C2×C4×C13—21no—
71C2×C4×C13—22no—
72C2×C4×C13—22no—
73C2×C4×C13—22no—
74C2×C4×C13—22no—
75C2×C4×C13—22no—
76C2×C4×C13—21no—
77C2×C4×C13—21no—
78C2×C4×C13C2×D412yes—
79C2×Dic13—216no—
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×Dic13—22no—
90C2×Dic13C2×D414yes—
91C2×Dic13#4114yes—
92C2×#117—132no—
93#398—131no—
94#396—132no—
95#394D1311yescharacteristic
96#388—132no—
97#387—132no—
98#386—132no—
99#385—132no—
100#382—131no—
101C22×C13—24no—
102C22×C13—24no—
103C22×C13—24no—
104C22×C13—42no—
105C22×C13—42no—
106C22×C13—42no—
107C22×C13—42no—
108C22×C13C22×D411yescharacteristic
109C22×C13C2×#4112yes—
110C22×C13C2×#4112yes—
111C22×C13H(5,𝔽2)11yescharacteristic
112C22×C132−1+411yescharacteristic
113C4×C13—22no—
114C4×C13—42no—
115C4×C13—41no—
116C4×C13—41no—
117Dic13—24no—
118Dic13—24no—
119Dic13—42no—
120Dic13—42no—
121C22×D4C2×D1311yescharacteristic
122C23×C4—132no—
123#117—268no—
124C2×C4⋊C4—132no—
125C2×C4⋊C4—132no—
126C2×C4⋊C4—132no—
127C2×C4⋊C4—131no—
128C2×C4⋊C4—131no—
129C2×C4⋊C4C2×D1311yescharacteristic
130C2×C22⋊C4—132no—
131C2×C22⋊C4—132no—
132C2×C22⋊C4—132no—
133C2×C22⋊C4C2×D1312yes—
134C2×C22⋊C4C2×D1312yes—
135C2×C42—131no—
136C2×C42—132no—
137#95—132no—
138#95—132no—
139#95—131no—
140#95—132no—
141#95—132no—
142#95C2×D1311yescharacteristic
143C2×C13—44no—
144C2×C13#52512yes—
145C2×C13#53711yescharacteristic
146C2×C13#55111yescharacteristic
147C2×C13#55212yes—
148C2×C13#55511yescharacteristic
149C24C22×D1312yes—
150C2×D4—24no—
151C2×D4—41no—
152C22×C4—132no—
153C22×C4—132no—
154C22×C4—132no—
155C22×C4—132no—
156C22×C4—264no—
157C22×C4—264no—
158C22×C4—264no—
159C22×C4C22×D1311yescharacteristic
160C22×C4C22×D1311yescharacteristic
161C22×C4C22×D1312yes—
162C22×C4C22×D1311yescharacteristic
163C4⋊C4—41no—
164C4⋊C4—264no—
165C4⋊C4—264no—
166C4⋊C4—264no—
167C4⋊C4—521no—
168C4⋊C4—521no—
169C22⋊C4—24no—
170C22⋊C4—24no—
171C22⋊C4—264no—
172C22⋊C4—264no—
173C22⋊C4—522no—
174C42—134no—
175C42—264no—
176C13#260911yescharacteristic, complemented, Sylow
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23#612914yes—
184C23C23×D1311yescharacteristic
185D4—44no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—134no—
196C2×C4—134no—
197C2×C4—2616no—
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×C4—262no—
208C2×C4D4×D1312yes—
209C22—24no—
210C22—24no—
211C22—24no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—42no—
216C22C2×D4×D1311yescharacteristic
217C22C2×#612912yes—
218C22#8825111yescharacteristic
219C22#41×D1312yes—
220C22#8825711yescharacteristic
221C4—22no—
222C4—41no—
223C4—41no—
224C4—42no—
225C4—264no—
226C4—264no—
227C4—522no—
228C4—522no—
229C2—44no—
230C2#1171929211yescharacteristic
231C2#1171930211yescharacteristic
232C2#1171931112yes—
233C2#1171935512yes—
234C2#1171937511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#394C2
prime radicalC22×D4×C13C22
hypercenter#394D13
hypocenterC13#2609
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#88131C2semidirect
4. #11718644C2semidirect
5. #11718638C2semidirect
6. #11718612C2semidirect
7. #11718609C2semidirect
8. #11718607C2semidirect
9. #11718313C2semidirect
58. C4×Dic13D4semidirect
176. C13#2609semidirect

Maximal subgroups

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

subgroupindex
C2×#881312
C13×#3942
#117188812
#117186442
#117186382
#117186122
#117186092
#117186072
#117183132
#260913