Finite Group Databasebeta

← #13124656

Subgroups — #13124656

Order 1664.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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
1#41×Dic13C211yesmaximal, characteristic
2C2×#88144C211yesmaximal, characteristic, complemented
3C13×#417C211yesmaximal, characteristic
4#11718893C211yesmaximal, characteristic
5#11718782C211yesmaximal, characteristic, complemented
6#11718779C211yesmaximal, characteristic, complemented
7#11718625C211yesmaximal, characteristic, complemented
8#11718460C212yesmaximal, complemented
9#11718457C212yesmaximal, complemented
10#11718427C212yesmaximal, complemented
11#11718424C212yesmaximal, complemented
12C2×C13×#41C2211yescharacteristic
13C2×C13×M16C2211yescharacteristic
14C2×C13×C4⋊C4C2211yescharacteristic
15Q8×Dic13C2211yescharacteristic
16Q8×Dic13C2211yescharacteristic
17D4×Dic13—21no—
18D4×Dic13—21no—
19D4×Dic13C2211yescharacteristic
20D4×Dic13C2211yescharacteristic
21#88180—21no—
22#88180C2211yescharacteristic
23C2×C4⋊Dic13—21no—
24C2×C4⋊Dic13C2211yescharacteristic
25C2×#6101—21no—
26C2×C4×Dic13—21no—
27C2×C4×Dic13C2211yescharacteristic
28C2×#6099C2211yescharacteristic
29#88144—21no—
30#88144—21no—
31#88144C2212yes—
32#88144C2212yes—
33C13×#103C2212yes—
34C13×#102C2212yes—
35#88075C2212yes—
36#88072C2212yes—
37#88058C2212yes—
38#88057C2212yes—
39#88054C2212yes—
40#88054C2212yes—
41#88048C2212yes—
42#88047C2212yes—
43C13×#41—22no—
44C13×#41Q814yes—
45C2×Q8×C13C2311yescharacteristic
46C2×D4×C13—21no—
47C2×D4×C13C2311yescharacteristic
48C22×C4×C13—21no—
49C22×C4×C13—21no—
50C22×C4×C13C2311yescharacteristic
51C22×Dic13—21no—
52C22×Dic13—21no—
53C22×Dic13—21no—
54C22×Dic13D411yescharacteristic
55C22×Dic13D411yescharacteristic
56C13×M16—22no—
57C2×C8×C13C2312yes—
58C13×C4⋊C4—21no—
59C13×C4⋊C4C2312yes—
60C22⋊Dic13—21no—
61C22⋊Dic13—21no—
62C22⋊Dic13—21no—
63C4⋊Dic13—21no—
64C4⋊Dic13—22no—
65C4⋊Dic13—22no—
66C4⋊Dic13—21no—
67C4⋊Dic13—21no—
68C4⋊Dic13C2312yes—
69C4⋊Dic13C2311yescharacteristic
70C4⋊Dic13C2311yescharacteristic
71#6101—22no—
72#6101—22no—
73C4×Dic13—21no—
74C4×Dic13—21no—
75C4×Dic13—21no—
76C4×Dic13D411yescharacteristic
77C4×Dic13D411yescharacteristic
78C4×Dic13C2311yescharacteristic
79C4×Dic13C2311yescharacteristic
80#6099—22no—
81C2×#1055C2312yes—
82#3080—131nomaximal, Sylow
83C23×C13—21no—
84C23×C13C2×D411yescharacteristic
85Q8×C13—21no—
86Q8×C13C2×Q812yes—
87D4×C13—24no—
88D4×C13—21no—
89D4×C13C2×Q812yes—
90C2×C4×C13—21no—
91C2×C4×C13—24no—
92C2×C4×C13—22no—
93C2×C4×C13—42no—
94C2×C4×C13C2×D411yescharacteristic
95C2×C4×C13C2×Q812yes—
96C2×C4×C13#4112yes—
97C2×C4×C13C2411yescharacteristic
98C2×Dic13—21no—
99C2×Dic13—21no—
100C2×Dic13—22no—
101C2×Dic13—41no—
102C2×Dic13—41no—
103C2×Dic13—42no—
104C2×Dic13—42no—
105C2×Dic13—42no—
106C2×Dic13C2×D411yescharacteristic
107C2×Dic13C2×D411yescharacteristic
108C2×Dic13C2×D411yescharacteristic
109C2×Dic13C2×D411yescharacteristic
110C8×C13—22no—
111#1055—22no—
112C2×#125—131no—
113C4×#41—131no—
114#479—132no—
115#479—132no—
116#478—132no—
117#478—132no—
118#428—131no—
119#428—131no—
120#423—131no—
121#417—131no—
122#417D1311yescharacteristic
123C22×C13—21no—
124C22×C13—21no—
125C22×C13—22no—
126C22×C13—41no—
127C22×C13—41no—
128C22×C13C22⋊Q812yes—
129C22×C13C22×D411yescharacteristic
130C4×C13—22no—
131C4×C13—42no—
132C4×C13C22⋊Q812yes—
133C4×C13C22×Q811yescharacteristic
134C4×C13C2×#4111yescharacteristic
135Dic13—21no—
136Dic13—21no—
137Dic13—21no—
138Dic13—21no—
139Dic13—41no—
140Dic13—41no—
141Dic13—42no—
142C2×#41C2×D1311yescharacteristic
143C2×M16—131no—
144C2×M16C2×D1311yescharacteristic
145#125—132no—
146#125—132no—
147#125—261no—
148#125—261no—
149C4×Q8—131no—
150C4×Q8—131no—
151C4×D4—131no—
152C4×D4—131no—
153C4×D4—261no—
154C4×D4—261no—
155#117—131no—
156#117—261no—
157C2×C4⋊C4—131no—
158C2×C4⋊C4—261no—
159C2×C4⋊C4—261no—
160C2×C4⋊C4C2×D1311yescharacteristic
161C2×C42—131no—
162C2×C42—261no—
163#107—132no—
164#107—132no—
165#106—132no—
166#106—132no—
167C4⋊C8—132no—
168C4⋊C8—132no—
169#103—132no—
170#103C2×D1312yes—
171#102—132no—
172#102C2×D1312yes—
173C2×C13—22no—
174C2×C13—42no—
175C2×C13C2×C22⋊Q811yescharacteristic
176C2×C13#57511yescharacteristic
177C2×C13#57511yescharacteristic
178#41—22no—
179#41Dic2614yes—
180C2×Q8C22×D1311yescharacteristic
181C2×D4—21no—
182C2×D4C22×D1311yescharacteristic
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—131no—
186C22×C4—131no—
187C22×C4—261no—
188C22×C4—261no—
189C22×C4—261no—
190C22×C4C22×D1311yescharacteristic
191M16—22no—
192M16—262no—
193C2×C8—132no—
194C2×C8C22×D1312yes—
195C4⋊C4—21no—
196C4⋊C4—132no—
197C4⋊C4—131no—
198C4⋊C4—131no—
199C4⋊C4—262no—
200C4⋊C4—262no—
201C4⋊C4—262no—
202C4⋊C4—262no—
203C4⋊C4—261no—
204C4⋊C4—261no—
205C4⋊C4—261no—
206C4⋊C4C22×D1312yes—
207C22⋊C4—261no—
208C22⋊C4—261no—
209C22⋊C4—261no—
210C42—131no—
211C42—131no—
212C42—131no—
213C42—131no—
214C42—261no—
215C42—261no—
216C42—261no—
217C13#308011yescharacteristic, complemented, Sylow
218C23—21no—
219C23D4×D1311yescharacteristic
220Q8—21no—
221Q8C2×Dic2612yes—
222D4—24no—
223D4—21no—
224D4C2×Dic2612yes—
225C2×C4—21no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—42no—
229C2×C4—131no—
230C2×C4—131no—
231C2×C4—131no—
232C2×C4—131no—
233C2×C4—261no—
234C2×C4—261no—
235C2×C4—262no—
236C2×C4—521no—
237C2×C4—521no—
238C2×C4—522no—
239C2×C4—522no—
240C2×C4—522no—
241C2×C4C2×Dic2612yes—
242C2×C4D4×D1311yescharacteristic
243C2×C4#612912yes—
244C2×C4C23×D1311yescharacteristic
245C8—22no—
246C8—262no—
247C22—21no—
248C22—21no—
249C22—22no—
250C22—41no—
251C22—41no—
252C22#8813212yes—
253C22C2×D4×D1311yescharacteristic
254C4—22no—
255C4—42no—
256C4—261no—
257C4—261no—
258C4—261no—
259C4—261no—
260C4—521no—
261C4—521no—
262C4—522no—
263C4#8813212yes—
264C4C22×Dic2611yescharacteristic
265C4C2×#612911yescharacteristic
266C2—22no—
267C2—42no—
268C2C2×#8813211yescharacteristic
269C2#1171947811yescharacteristic
270C2#1171948211yescharacteristic
subgroupG/H
centerC22C2×D4×D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#417C2
socleC22×C13C22×D4
prime radicalC2×C13×#41C22
hypercenter#417D13
hypocenterC13#3080
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#88144C2semidirect
5. #11718782C2semidirect
6. #11718779C2semidirect
7. #11718625C2semidirect
8. #11718460C2semidirect
9. #11718457C2semidirect
10. #11718427C2semidirect
11. #11718424C2semidirect
217. C13#3080semidirect

Maximal subgroups

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

subgroupindex
#41×Dic132
C2×#881442
C13×#4172
#117188932
#117187822
#117187792
#117186252
#117184602
#117184572
#117184272
#117184242
#308013