Finite Group Databasebeta

← #13120008

Subgroups — #13120008

Order 1664.

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

normal subgroups only

This group has 564 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×#88147C211yesmaximal, characteristic
2C2×#88142C211yesmaximal, characteristic, complemented
3C13×#382C211yesmaximal, characteristic, complemented
4#11718649C211yesmaximal, characteristic
5#11718638C211yesmaximal, characteristic, complemented
6#11718600C211yesmaximal, characteristic
7C4×#6101C211yesmaximal, characteristic, complemented
8#11718334C214yesmaximal, complemented
9#11718328C214yesmaximal, complemented
10C22×D52C2211yescharacteristic
11C22×C4×D13C2212yes—
12C22×Dic26C2212yes—
13C2×C13×C4⋊C4C2211yescharacteristic
14C2×C13×C4⋊C4C2211yescharacteristic
15C2×C42×C13C2211yescharacteristic
16C2×#6103C2212yes—
17C2×#6103C2214yes—
18C2×#6103C2214yes—
19C2×C4⋊Dic13C2211yescharacteristic
20C2×#6101C2212yes—
21C2×#6101C2212yes—
22C2×C4×Dic13C2211yescharacteristic
23C2×C4×Dic13C2211yescharacteristic
24#88147—24no—
25#88142—24no—
26C13×#95C2214yes—
27#88071C2214yes—
28#88071C2212yes—
29C23×D13C2312yes—
30C22×C4×C13C2312yes—
31C22×C4×C13C2312yes—
32C22×C4×C13C2312yes—
33C22×C4×C13C2311yescharacteristic
34C22×Dic13C2312yes—
35C22×Dic13C2312yes—
36C22×Dic13C2311yescharacteristic
37C22×Dic13C2311yescharacteristic
38C2×D52—22no—
39C2×D52—22no—
40C2×D52Q814yescomplemented
41C2×C4×D13—24no—
42C2×C4×D13—28no—
43C2×Dic26—24no—
44C2×Dic26—44no—
45C13×C4⋊C4—22no—
46C13×C4⋊C4—22no—
47C42×C13—22no—
48#6103—24no—
49#6103—44no—
50#6103—44no—
51C4⋊Dic13—22no—
52#6101—24no—
53#6101—24no—
54C4×Dic13—22no—
55C4×Dic13D414yes—
56#2563—131nomaximal, Sylow
57C23×C13C2411yescharacteristic
58C22×D13—22no—
59C22×D13—24no—
60C22×D13—22no—
61C22×D13—24no—
62C22×D13C2×Q814yes—
63C2×C4×C13—22no—
64C2×C4×C13—22no—
65C2×C4×C13—22no—
66C2×C4×C13—22no—
67C2×C4×C13—24no—
68C2×C4×C13—22no—
69C2×C4×C13—24no—
70C2×C4×C13C2×D412yes—
71C2×C4×C13C2×Q812yes—
72C2×C4×C13#4112yes—
73C2×Dic13—22no—
74C2×Dic13—22no—
75C2×Dic13—22no—
76C2×Dic13—22no—
77C2×Dic13—22no—
78C2×Dic13—24no—
79C2×Dic13C2×D412yes—
80C2×Dic13C2×D412yes—
81C2×Dic13#4114yes—
82D52—28no—
83C4×D13—48no—
84Dic26—48no—
85C2×C4⋊Q8—131no—
86C2×C4×D4—131no—
87#393—134no—
88#390—131no—
89#390—131no—
90#382—131no—
91#382D1311yescharacteristic, complemented
92#380—134no—
93C4×C4⋊C4—131no—
94C22×C13C22×D411yescharacteristic
95C22×C13C22×Q811yescharacteristic
96C22×C13C2×#4111yescharacteristic
97C22×C13C2×#4112yes—
98C22×C13H(5,𝔽2)12yes—
99C2×D13—24no—
100C2×D13—28no—
101C2×D13—42no—
102C2×D13—42no—
103C2×D13—44no—
104C4×C13—42no—
105C4×C13—42no—
106C4×C13—42no—
107C4×C13C22⋊Q814yes—
108Dic13—22no—
109Dic13—42no—
110Dic13—42no—
111Dic13#12414yes—
112C22×Q8—132no—
113C22×D4—131no—
114C23×C4—132no—
115C4⋊Q8—264no—
116C4×D4—264no—
117C2×C4⋊C4—132no—
118C2×C4⋊C4—131no—
119C2×C4⋊C4—132no—
120C2×C4⋊C4C2×D1311yescharacteristic
121C2×C4⋊C4C2×D1311yescharacteristic
122C2×C22⋊C4—132no—
123C2×C22⋊C4—134no—
124C2×C22⋊C4—134no—
125C2×C42—131no—
126C2×C42—131no—
127C2×C42C2×D1311yescharacteristic
128#95—132no—
129#95—134no—
130#95C2×D1314yes—
131C2×C13C2×C22⋊Q811yescharacteristic
132C2×C13C2×#12411yescharacteristic
133C2×C13#53511yescharacteristic
134C2×C13#55412yes—
135C2×C13#55512yes—
136D13—44no—
137C24—132no—
138C2×Q8—264no—
139C2×Q8—524no—
140C2×D4—134no—
141C2×D4—262no—
142C2×D4—262no—
143C22×C4—131no—
144C22×C4—131no—
145C22×C4—132no—
146C22×C4—132no—
147C22×C4—264no—
148C22×C4—268no—
149C22×C4C22×D1312yes—
150C22×C4C22×D1312yes—
151C22×C4C22×D1312yes—
152C22×C4C22×D1311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—264no—
156C4⋊C4—264no—
157C4⋊C4—262no—
158C22⋊C4—264no—
159C22⋊C4—524no—
160C22⋊C4—524no—
161C42—22no—
162C42—134no—
163C42—262no—
164C13#256311yescharacteristic, complemented, Sylow
165C23—134no—
166C23—262no—
167C23—262no—
168C23—264no—
169C23—264no—
170C23C23×D1311yescharacteristic
171Q8—528no—
172D4—268no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—24no—
180C2×C4—132no—
181C2×C4—134no—
182C2×C4—132no—
183C2×C4—262no—
184C2×C4—262no—
185C2×C4—262no—
186C2×C4—262no—
187C2×C4—262no—
188C2×C4—264no—
189C2×C4—528no—
190C2×C4#612712yes—
191C2×C4D4×D1312yes—
192C2×C4Q8×D1312yes—
193C22—264no—
194C22—268no—
195C22—524no—
196C22—522no—
197C22—522no—
198C22C2×#612711yescharacteristic
199C22C2×D4×D1311yescharacteristic
200C22C2×Q8×D1311yescharacteristic
201C22#41×D1312yes—
202C22#8825612yes—
203C4—42no—
204C4—42no—
205C4—42no—
206C4—134no—
207C4—262no—
208C4—522no—
209C4—522no—
210C4#8815014yes—
211C2—524no—
212C2C2×#8815011yescharacteristic
213C2#1171923712yes—
214C2D13×#12411yescharacteristic
215C2#1171937911yescharacteristic
216C2#1171939512yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#382C2
prime radicalC22×D52C22
hypercenter#382D13
hypocenterC13#2563
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C2×#88142C2semidirect
3. C13×#382C2semidirect
5. #11718638C2semidirect
7. C4×#6101C2semidirect
8. #11718334C2semidirect
9. #11718328C2semidirect
40. C2×D52Q8semidirect
91. #382D13semidirect
164. C13#2563semidirect

Maximal subgroups

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

subgroupindex
C2×#881472
C2×#881422
C13×#3822
#117186492
#117186382
#117186002
C4×#61012
#117183342
#117183282
#256313