Finite Group Databasebeta

← #13127521

Subgroups — #13127521

Order 1664.

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

normal subgroups only

This group has 336 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
1C13×#557C211yesmaximal, characteristic, complemented
2#11719289C211yesmaximal, characteristic
3#11718754C211yesmaximal, characteristic, complemented
4#11718748C211yesmaximal, characteristic
5#11718729C212yesmaximal
6#11718724C212yesmaximal, complemented
7#11718692C211yesmaximal, characteristic, complemented
8C4×#6107C212yesmaximal, complemented
9#11718689C212yesmaximal
10C4×#6106C211yesmaximal, characteristic
11Q8×#1055C211yesmaximal, 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×#6107C2212yes—
22C2×#6106C2211yescharacteristic
23#88152—22no—
24#88146—21no—
25#88129C2212yes—
26C4×D52C2211yescharacteristic
27#88123C2211yescharacteristic
28C4×Dic26C2212yes—
29#88075C2211yescharacteristic
30#88050C2212yes—
31#88050C2212yes—
32#88050C2212yes—
33#88049C2211yescharacteristic
34#88049C2212yes—
35#88048C2211yescharacteristic
36#88047C2212yes—
37C4⋊#1055C2212yes—
38C4⋊#1055C2211yescharacteristic
39C4×#1055C2211yescharacteristic
40C4×#1055C2212yes—
41C2×Q8×C13—21no—
42C2×Q8×C13D411yescharacteristic
43C2×Q8×C13C2311yescharacteristic
44C2×D52C2311yescharacteristic
45C2×C4×D13—21no—
46C2×Dic26C2312yes—
47C13×C4⋊C4—22no—
48C13×C4⋊C4—21no—
49C13×C4⋊C4—22no—
50C13×C4⋊C4—21no—
51C13×C4⋊C4—22no—
52C13×C4⋊C4D411yescharacteristic
53C13×C4⋊C4D412yes—
54C13×C4⋊C4C2311yescharacteristic
55C13×C4⋊C4C2312yes—
56C42×C13—22no—
57C42×C13—21no—
58C42×C13C2312yes—
59C42×C13C2311yescharacteristic
60#6107—24no—
61#6106—22no—
62#6103—21no—
63#6103—22no—
64#6103—22no—
65C4⋊Dic13—21no—
66C4⋊Dic13—21no—
67C4⋊Dic13C2311yescharacteristic
68#6101—22no—
69C4×Dic13—22no—
70C2×#1055C2311yescharacteristic
71C2×#1055C2311yescharacteristic
72C2×#1055C2312yes—
73#3364—131nomaximal, Sylow
74C22×D13—21no—
75Q8×C13—22no—
76Q8×C13—21no—
77Q8×C13—42no—
78Q8×C13#4112yes—
79C2×C4×C13—21no—
80C2×C4×C13—21no—
81C2×C4×C13—22no—
82C2×C4×C13C2×D412yes—
83C2×C4×C13C2×D412yes—
84C2×C4×C13C2×D411yescharacteristic
85C2×C4×C13C2×D411yescharacteristic
86C2×C4×C13C2411yescharacteristic
87C2×Dic13—21no—
88C2×Dic13—22no—
89D52—21no—
90D52—21no—
91C4×D13—42no—
92Dic26—22no—
93Dic26—22no—
94#1055—21no—
95#1055—22no—
96#1055—21no—
97#557D1311yescharacteristic, complemented
98#556—131no—
99#488—132no—
100#487—131no—
101#479—132no—
102#474—131no—
103#464—132no—
104#462—131no—
105C8×Q8—131no—
106C4×Q16—132no—
107C4×SD16—131no—
108C22×C13C22×D411yescharacteristic
109C2×D13—41no—
110C2×D13—41no—
111C2×D13—41no—
112C4×C13—22no—
113C4×C13—22no—
114C4×C13—22no—
115C4×C13—21no—
116C4×C13—42no—
117C4×C13—41no—
118C4×C13#13512yes—
119C4×C13C2×#4111yescharacteristic
120C4×C132−1+411yescharacteristic
121Dic13—42no—
122Dic13—41no—
123C2×Q16—132no—
124C2×SD16—131no—
125C4⋊Q8—21no—
126C4⋊Q8—131no—
127C4⋊Q8C2×D1311yescharacteristic
128#126—262no—
129#125—22no—
130#125C2×D1312yes—
131#124—132no—
132C22⋊Q8—261no—
133C4×Q8—21no—
134C4×Q8—132no—
135C4×Q8C2×D1311yescharacteristic
136C4×Q8C2×D1312yes—
137C4×Q8C2×D1311yescharacteristic
138C4×D4—131no—
139#117—261no—
140#107—132no—
141#106—131no—
142C4⋊C8—131no—
143C4⋊C8—132no—
144#103—131no—
145#103—132no—
146#103—132no—
147#103—132no—
148#102—131no—
149#102—132no—
150C4×C8—131no—
151C4×C8—132no—
152C2×C13#54811yescharacteristic
153C2×C13C2×#13511yescharacteristic
154C2×C13#57811yescharacteristic
155D13—81no—
156C2×Q8—21no—
157C2×Q8—132no—
158C2×Q8C22⋊D1311yescharacteristic
159C2×Q8C22×D1311yescharacteristic
160C2×D4—131no—
161C22×C4—261no—
162Q16—264no—
163SD16—262no—
164C2×C8—131no—
165C2×C8—132no—
166C2×C8—131no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—21no—
171C4⋊C4—22no—
172C4⋊C4—131no—
173C4⋊C4—261no—
174C4⋊C4—262no—
175C4⋊C4—261no—
176C4⋊C4C22⋊D1311yescharacteristic
177C4⋊C4C22⋊D1312yes—
178C4⋊C4C22×D1311yescharacteristic
179C4⋊C4C22×D1312yes—
180C22⋊C4—262no—
181C22⋊C4—261no—
182C22⋊C4—262no—
183C42—22no—
184C42—21no—
185C42—262no—
186C42C22×D1311yescharacteristic
187C42C22×D1312yes—
188C13#336411yescharacteristic, complemented, Sylow
189C23—261no—
190Q8—22no—
191Q8—21no—
192Q8—42no—
193Q8—262no—
194Q8—262no—
195Q8#613112yes—
196D4—261no—
197D4—261no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—261no—
202C2×C4—262no—
203C2×C4—522no—
204C2×C4C2×C22⋊D1312yes—
205C2×C4C2×C22⋊D1311yescharacteristic
206C2×C4C2×C22⋊D1312yes—
207C2×C4C2×C22⋊D1311yescharacteristic
208C2×C4C23×D1311yescharacteristic
209C8—261no—
210C8—262no—
211C8—261no—
212C22—521no—
213C22—521no—
214C22—521no—
215C22C22×C22⋊D1311yescharacteristic
216C4—22no—
217C4—22no—
218C4—22no—
219C4—21no—
220C4—42no—
221C4—41no—
222C4—522no—
223C4—521no—
224C4#8820412yes—
225C4C2×#613111yescharacteristic
226C4#8825411yescharacteristic
227C2—1041no—
228C2#1171952211yescharacteristic
229C2C2×#8820411yescharacteristic
230C2#1171954311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#557C2
socleC22×C13C22×D4
prime radicalC2×D52C23
hypercenter#557D13
hypocenterC13#3364
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
3. #11718754C2semidirect
6. #11718724C2semidirect
7. #11718692C2semidirect
8. C4×#6107C2semidirect
11. Q8×#1055C2semidirect
97. #557D13semidirect
188. C13#3364semidirect

Maximal subgroups

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

subgroupindex
C13×#5572
#117192892
#117187542
#117187482
#117187292
#117187242
#117186922
C4×#61072
#117186892
C4×#61062
Q8×#10552
#336413