Finite Group Databasebeta

← #13127707

Subgroups — #13127707

Order 1664.

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

normal subgroups only

This group has 338 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
1C13×#557C211yesmaximal, characteristic, complemented
2#11719284C211yesmaximal, characteristic
3#11718748C211yesmaximal, characteristic
4#11718740C211yesmaximal, characteristic, complemented
5#11718727C212yesmaximal
6#11718725C212yesmaximal, complemented
7C4×#6106C212yesmaximal
8C4×#6106C211yesmaximal, characteristic
9Q8×#1055C211yesmaximal, characteristic, complemented
10#11718683C212yesmaximal, complemented
11#11718682C211yesmaximal, characteristic, complemented
12C13×C4⋊Q8—21no—
13C13×C4⋊Q8C2211yescharacteristic
14C13×#125—22no—
15C13×#125C2212yes—
16C4×Q8×C13—21no—
17C4×Q8×C13C2211yescharacteristic
18C4×Q8×C13C2212yes—
19C4×Q8×C13C2211yescharacteristic
20#88200—21no—
21C2×#6106C2211yescharacteristic
22#88150—22no—
23C4⋊C4×D13—21no—
24C4×D52C2212yes—
25C4×D52C2211yescharacteristic
26#88124C2212yes—
27#88123C2211yescharacteristic
28#88075C2212yes—
29#88075C2211yescharacteristic
30#88049C2211yescharacteristic
31#88049C2212yes—
32#88048C2212yes—
33#88048C2211yescharacteristic
34#88048C2212yes—
35#88047C2212yes—
36#88047C2212yes—
37C4⋊#1055C2211yescharacteristic
38C4⋊#1055C2212yes—
39C4×#1055C2212yes—
40C4×#1055C2211yescharacteristic
41C2×Q8×C13—21no—
42C2×Q8×C13D411yescharacteristic
43C2×Q8×C13C2311yescharacteristic
44C2×D52C2311yescharacteristic
45C2×C4×D13—21no—
46C2×C4×D13—22no—
47C2×Dic26—21no—
48C13×C4⋊C4—22no—
49C13×C4⋊C4—21no—
50C13×C4⋊C4—22no—
51C13×C4⋊C4—21no—
52C13×C4⋊C4—22no—
53C13×C4⋊C4D411yescharacteristic
54C13×C4⋊C4D412yes—
55C13×C4⋊C4C2311yescharacteristic
56C13×C4⋊C4C2312yes—
57C42×C13—22no—
58C42×C13—21no—
59C42×C13C2311yescharacteristic
60C42×C13C2312yes—
61#6106Q814yescomplemented
62#6103—21no—
63#6103—22no—
64C4⋊Dic13—21no—
65C4⋊Dic13C2312yes—
66C4⋊Dic13C2311yescharacteristic
67#6101—22no—
68#6101—22no—
69C2×#1055C2312yes—
70C2×#1055C2311yescharacteristic
71C2×#1055C2311yescharacteristic
72#3381—131nomaximal, Sylow
73C22×D13—21no—
74Q8×C13—22no—
75Q8×C13—21no—
76Q8×C13—42no—
77Q8×C13C2×Q812yes—
78C2×C4×C13—21no—
79C2×C4×C13—21no—
80C2×C4×C13—22no—
81C2×C4×C13C2×D412yes—
82C2×C4×C13C2×D412yes—
83C2×C4×C13C2×D411yescharacteristic
84C2×C4×C13C2×D411yescharacteristic
85C2×C4×C13C2411yescharacteristic
86C2×Dic13—21no—
87C2×Dic13—22no—
88D52—21no—
89D52C2×Q812yes—
90C4×D13—42no—
91C4×D13—82no—
92Dic26—81no—
93#1055—22no—
94#1055—21no—
95#1055C2×Q812yes—
96#557D1311yescharacteristic, complemented
97#554—131no—
98#499—132no—
99#498—131no—
100#479—132no—
101#478—132no—
102#477—131no—
103#474—131no—
104C8×Q8—131no—
105C4×SD16—131no—
106C4×SD16—132no—
107C22×C13C22×D411yescharacteristic
108C2×D13—22no—
109C2×D13—41no—
110C2×D13—41no—
111C4×C13—22no—
112C4×C13—22no—
113C4×C13—21no—
114C4×C13—22no—
115C4×C13—42no—
116C4×C13—41no—
117C4×C13#13512yes—
118C4×C13C22×Q811yescharacteristic
119C4×C132−1+411yescharacteristic
120Dic13—41no—
121Dic13—42no—
122C2×SD16—131no—
123C4⋊Q8—21no—
124C4⋊Q8—131no—
125C4⋊Q8C2×D1311yescharacteristic
126#125—22no—
127#125—132no—
128#125C2×D1312yes—
129C22⋊Q8—261no—
130C22⋊Q8—262no—
131C4×Q8—21no—
132C4×Q8C2×D1311yescharacteristic
133C4×Q8C2×D1312yes—
134C4×Q8C2×D1311yescharacteristic
135C4×D4—131no—
136C4×D4—132no—
137C2×C4⋊C4—261no—
138#107—132no—
139#107—132no—
140#106—132no—
141#106—131no—
142#106—132no—
143C4⋊C8—131no—
144C4⋊C8—132no—
145#103—132no—
146#103—131no—
147#102—132no—
148#102—131no—
149C4×C8—131no—
150C4×C8—132no—
151C2×C13D4×Q811yescharacteristic
152C2×C13C2×#13511yescharacteristic
153C2×C13#57711yescharacteristic
154D13—42no—
155C2×Q8—21no—
156C2×Q8—261no—
157C2×Q8C22⋊D1311yescharacteristic
158C2×Q8C22×D1311yescharacteristic
159C2×D4—131no—
160C22×C4—261no—
161C22×C4—262no—
162SD16—134no—
163C2×C8—131no—
164C2×C8—131no—
165C2×C8—132no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—132no—
172C4⋊C4—131no—
173C4⋊C4—262no—
174C4⋊C4—261no—
175C4⋊C4—262no—
176C4⋊C4C22⋊D1311yescharacteristic
177C4⋊C4C22⋊D1312yes—
178C4⋊C4C22×D1312yes—
179C4⋊C4C22×D1311yescharacteristic
180C22⋊C4—261no—
181C22⋊C4—262no—
182C42—22no—
183C42—21no—
184C42C22×D1312yes—
185C42C22×D1311yescharacteristic
186C13#338111yescharacteristic, complemented, Sylow
187C23—261no—
188Q8—22no—
189Q8—21no—
190Q8—42no—
191Q8—1041no—
192Q8Q8×D1312yes—
193D4—132no—
194D4—261no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—261no—
199C2×C4—262no—
200C2×C4—522no—
201C2×C4—1042no—
202C2×C4C2×C22⋊D1312yes—
203C2×C4C2×C22⋊D1312yes—
204C2×C4C2×C22⋊D1311yescharacteristic
205C2×C4C2×C22⋊D1311yescharacteristic
206C2×C4C23×D1311yescharacteristic
207C8—132no—
208C8—261no—
209C8—262no—
210C22—262no—
211C22—521no—
212C22—521no—
213C22C22×C22⋊D1311yescharacteristic
214C4—22no—
215C4—22no—
216C4—22no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—521no—
221C4—522no—
222C4#8820412yes—
223C4C2×Q8×D1311yescharacteristic
224C4#8825411yescharacteristic
225C2—522no—
226C2Q8×C22⋊D1311yescharacteristic
227C2C2×#8820411yescharacteristic
228C2#1171954211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#557C2
socleC22×C13C22×D4
prime radicalC2×D52C23
hypercenter#557D13
hypocenterC13#3381
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #557

Complements

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

NH ≅ G/Ntype
1. C13×#557C2semidirect
4. #11718740C2semidirect
6. #11718725C2semidirect
9. Q8×#1055C2semidirect
10. #11718683C2semidirect
11. #11718682C2semidirect
61. #6106Q8semidirect
96. #557D13semidirect
186. C13#3381semidirect

Maximal subgroups

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

subgroupindex
C13×#5572
#117192842
#117187482
#117187402
#117187272
#117187252
C4×#61062
Q8×#10552
#117186832
#117186822
#338113