Finite Group Databasebeta

← #13122869

Subgroups — #13122869

Order 1664.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#88150C211yesmaximal, characteristic, complemented
2C2×#88149C211yesmaximal, characteristic
3C13×#398C211yesmaximal, characteristic, complemented
4#11718653C212yesmaximal, complemented
5#11718652C211yesmaximal, characteristic, complemented
6#11718652C212yesmaximal, complemented
7#11718641C211yesmaximal, characteristic, complemented
8#11718334C212yesmaximal, complemented
9#11718333C211yesmaximal, characteristic, complemented
10#11718333C212yesmaximal, complemented
11#11718325C211yesmaximal, characteristic, complemented
12C22×D52C2212yes—
13C22×D52C2211yescharacteristic
14C22×C4×D13C2211yescharacteristic
15C22×Dic26C2211yescharacteristic
16C2×C13×C4⋊C4C2211yescharacteristic
17C2×C13×C4⋊C4C2211yescharacteristic
18C2×C13×C4⋊C4C2212yes—
19C2×#6103C2212yes—
20C2×#6103C2211yescharacteristic
21C2×#6103C2211yescharacteristic
22C2×#6103C2212yes—
23C2×#6103C2212yes—
24C2×#6103C2212yes—
25C2×#6103C2212yes—
26C2×#6103C2212yes—
27C2×#6101C2211yescharacteristic
28C2×#6101C2211yescharacteristic
29C2×#6101C2212yes—
30#88150—24no—
31#88149—24no—
32C13×#95C2211yescharacteristic
33C13×#95C2212yes—
34#88071C2212yes—
35#88071C2211yescharacteristic
36#88071C2212yes—
37C23×D13C2312yes—
38C23×D13C2311yescharacteristic
39C22×C4×C13C2312yes—
40C22×C4×C13C2312yes—
41C22×C4×C13C2311yescharacteristic
42C22×C4×C13C2311yescharacteristic
43C22×C4×C13C2311yescharacteristic
44C22×Dic13C2312yes—
45C22×Dic13C2311yescharacteristic
46C22×Dic13C2311yescharacteristic
47C22×Dic13C2311yescharacteristic
48C2×D52—24no—
49C2×D52—22no—
50C2×D52—42no—
51C2×D52—42no—
52C2×D52—42no—
53C2×C4×D13—42no—
54C2×C4×D13D414yes—
55C2×Dic26—22no—
56C2×Dic26—42no—
57C13×C4⋊C4—22no—
58C13×C4⋊C4—22no—
59C13×C4⋊C4—42no—
60#6103—22no—
61#6103—22no—
62#6103—24no—
63#6103—42no—
64#6103—42no—
65#6103—42no—
66#6103—42no—
67#6103—42no—
68#6101—22no—
69#6101—22no—
70#6101—42no—
71#2816—131nomaximal, Sylow
72C23×C13C2411yescharacteristic
73C22×D13—22no—
74C22×D13—21no—
75C22×D13—21no—
76C22×D13—22no—
77C22×D13—22no—
78C22×D13—22no—
79C22×D13—22no—
80C22×D13—22no—
81C22×D13—22no—
82C22×D13—22no—
83C22×D13—22no—
84C22×D13C2×D412yes—
85C2×C4×C13—22no—
86C2×C4×C13—21no—
87C2×C4×C13—21no—
88C2×C4×C13—22no—
89C2×C4×C13—22no—
90C2×C4×C13—21no—
91C2×C4×C13—22no—
92C2×C4×C13—22no—
93C2×C4×C13—22no—
94C2×C4×C13—21no—
95C2×C4×C13—22no—
96C2×C4×C13—22no—
97C2×C4×C13C2×D412yes—
98C2×Dic13—22no—
99C2×Dic13—21no—
100C2×Dic13—21no—
101C2×Dic13—21no—
102C2×Dic13—22no—
103C2×Dic13—21no—
104C2×Dic13—22no—
105C2×Dic13—22no—
106C2×Dic13—22no—
107C2×Dic13C2×D412yes—
108D52—44no—
109D52—48no—
110C4×D13—44no—
111Dic26—44no—
112C2×C22⋊Q8—131no—
113C2×#121—131no—
114#399—132no—
115#398—131no—
116#398D1311yescharacteristic, complemented
117#394—131no—
118#394—132no—
119#394—131no—
120#394—132no—
121#393—132no—
122#381—131no—
123C22×C13C22×D411yescharacteristic
124C22×C13H(5,𝔽2)11yescharacteristic
125C22×C13H(5,𝔽2)12yes—
126C22×C13H(5,𝔽2)11yescharacteristic
127C22×C132−1+412yes—
128C2×D13—22no—
129C2×D13—22no—
130C2×D13—22no—
131C2×D13—42no—
132C2×D13—42no—
133C2×D13—42no—
134C2×D13—42no—
135C2×D13—42no—
136C2×D13—42no—
137C2×D13—42no—
138C2×D13—42no—
139C2×D13—42no—
140C4×C13—22no—
141C4×C13—41no—
142C4×C13—42no—
143C4×C13—41no—
144C4×C13—42no—
145Dic13—22no—
146Dic13—41no—
147Dic13—41no—
148Dic13—42no—
149C22×Q8—131no—
150C22×D4—132no—
151C22×D4—131no—
152C23×C4—131no—
153C22⋊Q8—264no—
154#121—264no—
155C2×C4⋊C4—131no—
156C2×C4⋊C4—131no—
157C2×C4⋊C4—132no—
158C2×C4⋊C4C2×D1311yescharacteristic
159C2×C4⋊C4C2×D1311yescharacteristic
160C2×C4⋊C4C2×D1312yes—
161C2×C22⋊C4—132no—
162C2×C22⋊C4—131no—
163C2×C22⋊C4—131no—
164C2×C22⋊C4—132no—
165C2×C22⋊C4—132no—
166C2×C22⋊C4—132no—
167C2×C22⋊C4—132no—
168C2×C22⋊C4—132no—
169#95—132no—
170#95—131no—
171#95—132no—
172#95C2×D1311yescharacteristic
173#95C2×D1312yes—
174C2×C13#53411yescharacteristic
175C2×C13#53511yescharacteristic
176C2×C13#53611yescharacteristic
177C2×C13#56212yes—
178C2×C13#56212yes—
179D13—42no—
180D13—82no—
181C24—131no—
182C24—132no—
183C2×Q8—262no—
184C2×Q8—522no—
185C2×D4—264no—
186C2×D4—262no—
187C2×D4—522no—
188C2×D4—522no—
189C2×D4—522no—
190C22×C4—131no—
191C22×C4—134no—
192C22×C4—132no—
193C22×C4—131no—
194C22×C4—131no—
195C22×C4—522no—
196C22×C4C22×D1312yes—
197C22×C4C22×D1312yes—
198C22×C4C22×D1311yescharacteristic
199C22×C4C22×D1311yescharacteristic
200C22×C4C22×D1311yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—42no—
204C4⋊C4—262no—
205C4⋊C4—262no—
206C4⋊C4—522no—
207C22⋊C4—264no—
208C22⋊C4—262no—
209C22⋊C4—262no—
210C22⋊C4—522no—
211C22⋊C4—522no—
212C22⋊C4—522no—
213C22⋊C4—522no—
214C22⋊C4—522no—
215C13#281611yescharacteristic, complemented, Sylow
216C23—132no—
217C23—261no—
218C23—262no—
219C23—262no—
220C23—261no—
221C23—262no—
222C23—262no—
223C23—262no—
224C23—262no—
225C23—262no—
226C23—262no—
227C23—262no—
228C23C23×D1311yescharacteristic
229Q8—524no—
230D4—524no—
231D4—528no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—132no—
245C2×C4—262no—
246C2×C4—261no—
247C2×C4—261no—
248C2×C4—261no—
249C2×C4—262no—
250C2×C4—262no—
251C2×C4—262no—
252C2×C4—262no—
253C2×C4—261no—
254C2×C4—524no—
255C2×C4D4×D1312yes—
256C22—262no—
257C22—262no—
258C22—262no—
259C22—522no—
260C22—522no—
261C22—522no—
262C22—522no—
263C22—522no—
264C22—522no—
265C22—522no—
266C22—522no—
267C22—522no—
268C22C2×D4×D1311yescharacteristic
269C22#8825111yescharacteristic
270C22#8825412yes—
271C22#8825611yescharacteristic
272C22#8825612yes—
273C4—22no—
274C4—41no—
275C4—42no—
276C4—42no—
277C4—41no—
278C4—262no—
279C4—521no—
280C4—521no—
281C4—522no—
282C2—522no—
283C2—1042no—
284C2#1171931211yescharacteristic
285C2#1171935212yes—
286C2#1171935811yescharacteristic
287C2#1171940512yes—
288C2#1171942911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#398C2
prime radicalC2×#88149C2
hypercenter#398D13
hypocenterC13#2816
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#88150C2semidirect
3. C13×#398C2semidirect
4. #11718653C2semidirect
5. #11718652C2semidirect
6. #11718652C2semidirect
7. #11718641C2semidirect
8. #11718334C2semidirect
9. #11718333C2semidirect
10. #11718333C2semidirect
11. #11718325C2semidirect
116. #398D13semidirect
215. C13#2816semidirect

Maximal subgroups

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

subgroupindex
C2×#881502
C2×#881492
C13×#3982
#117186532
#117186522
#117186412
#117183342
#117183332
#117183252
#281613