Finite Group Databasebeta

← #13123185

Subgroups — #13123185

Order 1664.

644 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 189 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×#88200C211yesmaximal, characteristic
2C2×#88198C211yesmaximal, characteristic, complemented
3C2×#88151C211yesmaximal, characteristic
4C2×C42×D13C211yesmaximal, characteristic
5C13×#391C211yesmaximal, characteristic, complemented
6#11718637C211yesmaximal, characteristic, complemented
7#11718590C211yesmaximal, characteristic, complemented
8#11718328C214yesmaximal
9#11718308C214yesmaximal, complemented
10C22×Q8×C13C2211yescharacteristic
11C22×C4×D13C2211yescharacteristic
12C22×C4×D13C2212yes—
13C22×Dic26C2211yescharacteristic
14C2×C13×C4⋊C4C2211yescharacteristic
15C2×C42×C13C2211yescharacteristic
16#88200—24no—
17#88198—24no—
18C2×#6103C2212yes—
19C2×#6103C2212yes—
20C2×C4⋊Dic13C2212yes—
21C2×C4⋊Dic13C2211yescharacteristic
22C2×#6101C2214yes—
23C2×#6101C2212yes—
24C2×#6101C2214yes—
25C2×C4×Dic13C2212yes—
26C2×C4×Dic13C2211yescharacteristic
27#88151—24no—
28C42×D13—24no—
29C13×#95C2214yes—
30#88071C2214yes—
31C23×D13C2311yescharacteristic
32C2×Q8×C13—22no—
33C2×Q8×C13—22no—
34C2×Q8×C13—22no—
35C22×C4×C13C2312yes—
36C22×C4×C13C2312yes—
37C22×C4×C13C2312yes—
38C22×C4×C13C2311yescharacteristic
39C22×Dic13C2312yes—
40C22×Dic13C2312yes—
41C22×Dic13C2311yescharacteristic
42C22×Dic13C2312yes—
43C2×C4×D13—22no—
44C2×C4×D13—22no—
45C2×C4×D13—24no—
46C2×C4×D13—24no—
47C2×C4×D13D414yes—
48C2×C4×D13Q818yescomplemented
49C2×Dic26—22no—
50C2×Dic26—22no—
51C2×Dic26—22no—
52C13×C4⋊C4—22no—
53C42×C13—22no—
54#6103—24no—
55#6103—24no—
56C4⋊Dic13—22no—
57C4⋊Dic13—24no—
58#6101—28no—
59#6101—24no—
60#6101—28no—
61C4×Dic13—24no—
62C4×Dic13D414yes—
63#2857—131nomaximal, Sylow
64C23×C13C2411yescharacteristic
65C22×D13—21no—
66C22×D13—21no—
67C22×D13—22no—
68C22×D13C2×D412yes—
69C22×D13C2×Q814yes—
70Q8×C13—44no—
71C2×C4×C13—22no—
72C2×C4×C13—22no—
73C2×C4×C13—22no—
74C2×C4×C13—22no—
75C2×C4×C13—22no—
76C2×C4×C13—24no—
77C2×C4×C13—22no—
78C2×C4×C13C2×D412yes—
79C2×C4×C13C2×D412yes—
80C2×C4×C13C2×Q814yes—
81C2×C4×C13#4112yes—
82C2×Dic13—21no—
83C2×Dic13—21no—
84C2×Dic13—22no—
85C2×Dic13—22no—
86C2×Dic13—22no—
87C2×Dic13—24no—
88C2×Dic13—22no—
89C2×Dic13C2×D412yes—
90C2×Dic13C2×D414yes—
91C2×Dic13C2×Q814yes—
92C2×Dic13#4114yes—
93C4×D13—28no—
94C4×D13—216no—
95Dic26—44no—
96C2×C4⋊Q8—131no—
97C2×C22⋊Q8—131no—
98C2×C22⋊Q8—131no—
99C22×C42—131no—
100#391—131no—
101#391D1311yescharacteristic, complemented
102#389—134no—
103#384—131no—
104#380—134no—
105C22×C13C22×D411yescharacteristic
106C22×C13C22×D411yescharacteristic
107C22×C13C22×Q812yes—
108C22×C13C2×#4111yescharacteristic
109C22×C13C2×#4112yes—
110C2×D13—22no—
111C2×D13—24no—
112C2×D13—22no—
113C2×D13—24no—
114C2×D13C4⋊Q814yescomplemented
115C4×C13—24no—
116C4×C13—42no—
117C4×C13—42no—
118C4×C13#12114yes—
119Dic13—22no—
120Dic13—42no—
121Dic13—42no—
122Dic13C22⋊Q818yescomplemented
123C22×Q8—131no—
124C22×Q8C2×D1311yescharacteristic
125C23×C4—131no—
126C23×C4—132no—
127C4⋊Q8—264no—
128C22⋊Q8—264no—
129C22⋊Q8—264no—
130C2×C4⋊C4—132no—
131C2×C4⋊C4—131no—
132C2×C4⋊C4—132no—
133C2×C4⋊C4—134no—
134C2×C4⋊C4—134no—
135C2×C4⋊C4C2×D1311yescharacteristic
136C2×C22⋊C4—132no—
137C2×C22⋊C4—132no—
138C2×C42—131no—
139C2×C42—132no—
140C2×C42—264no—
141C2×C42C2×D1311yescharacteristic
142#95—134no—
143#95C2×D1314yes—
144C2×C13C2×#12111yescharacteristic
145C2×C13C2×C22⋊Q812yes—
146C2×C13C2×C4⋊Q811yescharacteristic
147C2×C13#53211yescharacteristic
148C2×C13#53312yes—
149D13—24no—
150C24—131no—
151C2×Q8—22no—
152C2×Q8—22no—
153C2×Q8—22no—
154C2×Q8—262no—
155C2×Q8—262no—
156C2×Q8—262no—
157C22×C4—132no—
158C22×C4—131no—
159C22×C4—134no—
160C22×C4—138no—
161C22×C4—132no—
162C22×C4—132no—
163C22×C4—264no—
164C22×C4—264no—
165C22×C4—262no—
166C22×C4—262no—
167C22×C4C22×D1312yes—
168C22×C4C22×D1312yes—
169C22×C4C22×D1312yes—
170C22×C4C22×D1311yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—264no—
173C4⋊C4—264no—
174C4⋊C4—268no—
175C4⋊C4—268no—
176C4⋊C4—262no—
177C22⋊C4—264no—
178C22⋊C4—264no—
179C42—22no—
180C42—134no—
181C42—264no—
182C13#285711yescharacteristic, complemented, Sylow
183C23—132no—
184C23—134no—
185C23—261no—
186C23—261no—
187C23—262no—
188C23C23×D1311yescharacteristic
189Q8—44no—
190Q8—524no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—24no—
198C2×C4—134no—
199C2×C4—134no—
200C2×C4—134no—
201C2×C4—132no—
202C2×C4—2616no—
203C2×C4—261no—
204C2×C4—261no—
205C2×C4—268no—
206C2×C4—262no—
207C2×C4—262no—
208C2×C4—262no—
209C2×C4—262no—
210C2×C4—264no—
211C2×C4#612712yes—
212C2×C4D4×D1312yes—
213C2×C4D4×D1312yes—
214C2×C4Q8×D1314yes—
215C22—134no—
216C22—262no—
217C22—264no—
218C22—264no—
219C22—262no—
220C22C2×#612711yescharacteristic
221C22C2×D4×D1311yescharacteristic
222C22C2×D4×D1311yescharacteristic
223C22C2×Q8×D1312yes—
224C22#41×D1312yes—
225C4—24no—
226C4—42no—
227C4—42no—
228C4—138no—
229C4—262no—
230C4—522no—
231C4—522no—
232C4#8813814yes—
233C2—264no—
234C2C2×#8813811yescharacteristic
235C2#1171928312yes—
236C2D13×C22⋊Q812yes—
237C2#1171937311yescharacteristic
238C2D13×C4⋊Q811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#391C2
prime radicalC23×D13C23
hypercenter#391D13
hypocenterC13#2857
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#88198C2semidirect
5. C13×#391C2semidirect
6. #11718637C2semidirect
7. #11718590C2semidirect
9. #11718308C2semidirect
48. C2×C4×D13Q8semidirect
101. #391D13semidirect
114. C2×D13C4⋊Q8semidirect
122. Dic13C22⋊Q8semidirect
182. C13#2857semidirect

Maximal subgroups

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

subgroupindex
C2×#882002
C2×#881982
C2×#881512
C2×C42×D132
C13×#3912
#117186372
#117185902
#117183282
#117183082
#285713