Finite Group Databasebeta

← #13122863

Subgroups — #13122863

Order 1664.

528 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 133 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×#88184C211yesmaximal, characteristic
3#11718892C212yesmaximal, complemented
4#11718872C211yesmaximal, characteristic, complemented
5#11718653C212yesmaximal, complemented
6#11718652C212yesmaximal, complemented
7#11718644C212yesmaximal, complemented
8#11718620C214yesmaximal, complemented
9C22×Q8×C13C2211yescharacteristic
10C23×C4×C13C2211yescharacteristic
11C22×C22⋊D13C2212yes—
12C22×D52C2211yescharacteristic
13C13×C22⋊Q8—24no—
14C2×C13×C4⋊C4C2212yes—
15C2×C13×C4⋊C4C2211yescharacteristic
16C2×C13×C22⋊C4C2212yes—
17C2×C22⋊Dic13C2212yes—
18#88184—24no—
19C2×#6103C2212yes—
20C2×#6103C2214yes—
21C2×#6103C2214yes—
22C2×C4⋊Dic13C2211yescharacteristic
23C2×#6101C2214yes—
24#88071C2212yes—
25#88071C2212yes—
26#88071C2214yes—
27C24×C13C2311yescharacteristic
28C23×D13C2312yes—
29C2×Q8×C13—22no—
30C2×Q8×C13—42no—
31C22×C4×C13—42no—
32C22×C4×C13D414yes—
33C22×C4×C13C2312yes—
34C22×C4×C13C2312yes—
35C22×C4×C13C2311yescharacteristic
36C22×C4×C13C2311yescharacteristic
37C2×C22⋊D13—24no—
38C2×C22⋊D13—44no—
39C22×Dic13C2314yes—
40C22×Dic13C2312yes—
41C2×D52—22no—
42C2×D52—42no—
43C13×C4⋊C4—22no—
44C13×C4⋊C4—24no—
45C13×C22⋊C4—24no—
46C22⋊Dic13—42no—
47#6103—24no—
48#6103—44no—
49#6103—44no—
50C4⋊Dic13—22no—
51#6101—44no—
52#2816—131nomaximal, Sylow
53C23×C13—22no—
54C23×C13—22no—
55C23×C13—22no—
56C23×C13C2×D412yes—
57C23×C13C2411yescharacteristic
58C22×D13—22no—
59C22×D13—24no—
60C22×D13—24no—
61C22×D13—24no—
62Q8×C13—44no—
63C2×C4×C13—22no—
64C2×C4×C13—22no—
65C2×C4×C13—22no—
66C2×C4×C13—22no—
67C2×C4×C13—22no—
68C2×C4×C13—22no—
69C2×C4×C13—24no—
70C2×C4×C13—44no—
71C2×C4×C13C2×D412yes—
72C2×C4×C13C2×D412yes—
73C22⋊D13—48no—
74C2×Dic13—22no—
75C2×Dic13—24no—
76C2×Dic13—24no—
77C2×Dic13—24no—
78C2×Dic13—24no—
79D52—44no—
80C2×C22⋊Q8D1311yescharacteristic, complemented
81C2×#121—131no—
82#399—132no—
83#398—132no—
84#394—132no—
85#394—134no—
86#393—132no—
87#381—131no—
88C22×C13—22no—
89C22×C13—22no—
90C22×C13—22no—
91C22×C13—44no—
92C22×C13C22×D411yescharacteristic
93C22×C13H(5,𝔽2)12yes—
94C22×C13H(5,𝔽2)12yes—
95C22×C132−1+412yes—
96C2×D13—42no—
97C2×D13—42no—
98C2×D13—42no—
99C2×D13—48no—
100C4×C13—22no—
101C4×C13—22no—
102C4×C13—42no—
103C4×C13—42no—
104Dic13—42no—
105Dic13—44no—
106C22×Q8C2×D1311yescharacteristic
107C22×D4—132no—
108C22×D4—131no—
109C23×C4C2×D1311yescharacteristic
110C22⋊Q8—24no—
111#121—264no—
112C2×C4⋊C4—131no—
113C2×C4⋊C4—134no—
114C2×C4⋊C4C2×D1312yes—
115C2×C4⋊C4C2×D1311yescharacteristic
116C2×C22⋊C4—132no—
117C2×C22⋊C4—132no—
118C2×C22⋊C4—134no—
119C2×C22⋊C4—134no—
120C2×C22⋊C4C2×D1312yes—
121#95—132no—
122#95—132no—
123#95—134no—
124C2×C13—42no—
125C2×C13#53411yescharacteristic
126C2×C13#53511yescharacteristic
127C2×C13#53611yescharacteristic
128C2×C13#56214yes—
129D13—82no—
130C24—132no—
131C24C22×D1311yescharacteristic
132C2×Q8—22no—
133C2×Q8—42no—
134C2×D4—264no—
135C2×D4—262no—
136C2×D4—522no—
137C2×D4—524no—
138C22×C4—42no—
139C22×C4—134no—
140C22×C4—132no—
141C22×C4C22⋊D1314yes—
142C22×C4C22×D1312yes—
143C22×C4C22×D1312yes—
144C22×C4C22×D1311yescharacteristic
145C22×C4C22×D1311yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—24no—
148C4⋊C4—262no—
149C4⋊C4—524no—
150C22⋊C4—24no—
151C22⋊C4—264no—
152C22⋊C4—524no—
153C22⋊C4—524no—
154C22⋊C4—522no—
155C13#281611yescharacteristic, complemented, Sylow
156C23—22no—
157C23—22no—
158C23—22no—
159C23—264no—
160C23—264no—
161C23—264no—
162C23—262no—
163C23C2×C22⋊D1312yes—
164C23C23×D1311yescharacteristic
165Q8—44no—
166D4—524no—
167D4—528no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—44no—
176C2×C4—264no—
177C2×C4—264no—
178C2×C4—264no—
179C2×C4—264no—
180C2×C4—262no—
181C2×C4C2×C22⋊D1312yes—
182C2×C4C2×C22⋊D1312yes—
183C22—22no—
184C22—22no—
185C22—22no—
186C22—44no—
187C22—522no—
188C22—522no—
189C22—522no—
190C22—528no—
191C22#8825112yes—
192C22#8825412yes—
193C22#8825612yes—
194C22C22×C22⋊D1311yescharacteristic
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—522no—
200C4—524no—
201C2—42no—
202C2—1042no—
203C2#1171935214yes—
204C2#1171950911yescharacteristic
205C2#1171952111yescharacteristic
206C2#1171953511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC2×C13×C22⋊Q8C2
prime radicalC2×#88184C2
hypercenterC2×C22⋊Q8D13
hypocenterC13#2816
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
3. #11718892C2semidirect
4. #11718872C2semidirect
5. #11718653C2semidirect
6. #11718652C2semidirect
7. #11718644C2semidirect
8. #11718620C2semidirect
80. C2×C22⋊Q8D13semidirect
155. C13#2816semidirect

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×#881842
#117188922
#117188722
#117186532
#117186522
#117186442
#117186202
#281613