Finite Group Databasebeta

← #13125919

Subgroups — #13125919

Order 1664.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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×#536C211yesmaximal, characteristic
2#11719223C211yesmaximal, characteristic, complemented
3#11718900C212yesmaximal
4#11718889C212yesmaximal, complemented
5#11718753C211yesmaximal, characteristic, complemented
6#11718751C212yesmaximal, complemented
7#11718743C211yesmaximal, characteristic, complemented
8#11718721C211yesmaximal, characteristic, complemented
9#11718717C211yesmaximal, characteristic, complemented
10#11718715C212yesmaximal, complemented
11#11718661C211yesmaximal, characteristic, complemented
12C2×C13×#41C2211yescharacteristic
13C22×Q8×C13C2211yescharacteristic
14C13×C4⋊Q8C2211yescharacteristic
15C13×C4⋊Q8C2211yescharacteristic
16C13×#124C2211yescharacteristic
17C13×#124C2211yescharacteristic
18C13×#123—22no—
19C13×C22⋊Q8—22no—
20C13×#117C2211yescharacteristic
21C2×#6107C2212yes—
22C2×#6107C2212yes—
23C2×#6107C2212yes—
24C2×#6105C2212yes—
25#88178C2212yes—
26#88143—22no—
27#88133—22no—
28#88124C2211yescharacteristic
29#88123C2211yescharacteristic
30C4×Dic26C2212yes—
31#88075C2212yes—
32#88075C2212yes—
33#88075C2212yes—
34#88072C2212yes—
35C22⋊#1055C2212yes—
36C4⋊#1055C2212yes—
37#88043C2211yescharacteristic
38C4×#1055C2211yescharacteristic
39C13×#41—42no—
40C2×Q8×C13—21no—
41C2×Q8×C13—42no—
42C2×Q8×C13C2311yescharacteristic
43C2×Q8×C13C2311yescharacteristic
44C2×Q8×C13C2311yescharacteristic
45C2×D4×C13—21no—
46C2×D4×C13C2311yescharacteristic
47C22×C4×C13—21no—
48C22×C4×C13—21no—
49C22×C4×C13C2311yescharacteristic
50C2×Dic26C2312yes—
51C13×C4⋊C4—22no—
52C13×C4⋊C4—22no—
53C13×C4⋊C4D412yes—
54C13×C22⋊C4—22no—
55C13×C22⋊C4—22no—
56C13×C22⋊C4D412yes—
57C42×C13C2311yescharacteristic
58C42×C13C2311yescharacteristic
59C22⋊Dic13—22no—
60#6107—42no—
61#6107—42no—
62#6107—42no—
63#6105—42no—
64C4⋊Dic13—22no—
65C4⋊Dic13C2312yes—
66#6101—22no—
67#6101—22no—
68#6101—22no—
69C4×Dic13—22no—
70C2×#1055C2312yes—
71C2×#1055C2312yes—
72#3223—131nomaximal, Sylow
73C23×C13—21no—
74C23×C13C2×D411yescharacteristic
75Q8×C13—22no—
76Q8×C13—22no—
77Q8×C13—22no—
78Q8×C13—42no—
79D4×C13—22no—
80D4×C13—42no—
81C2×C4×C13—22no—
82C2×C4×C13—21no—
83C2×C4×C13—21no—
84C2×C4×C13—21no—
85C2×C4×C13—42no—
86C2×C4×C13—42no—
87C2×C4×C13C2×D412yes—
88C2×C4×C13C2×D411yescharacteristic
89C2×C4×C13C2×D412yes—
90C2×C4×C13C2411yescharacteristic
91C2×Dic13—22no—
92C2×Dic13—22no—
93Dic26—22no—
94Dic26—22no—
95#1055—22no—
96#1055—22no—
97#541—131no—
98#536D1311yescharacteristic
99#497—131no—
100#494—131no—
101#491—131no—
102#488—131no—
103#464—132no—
104#462—132no—
105#452—132no—
106#451—132no—
107#433—131no—
108C22×C13—21no—
109C22×C13—22no—
110C22×C13—41no—
111C22×C13—42no—
112C22×C13C22×D411yescharacteristic
113C4×C13—21no—
114C4×C13—22no—
115C4×C13—22no—
116C4×C13—41no—
117C4×C13—41no—
118C4×C13—41no—
119C4×C13H(5,𝔽2)12yes—
120Dic13—42no—
121Dic13—42no—
122C2×#41C2×D1311yescharacteristic
123C22×Q8C2×D1311yescharacteristic
124C2×Q16—132no—
125C2×Q16—132no—
126C2×Q16—132no—
127C2×SD16—132no—
128C4⋊Q8—131no—
129C4⋊Q8C2×D1311yescharacteristic
130C4⋊Q8C2×D1311yescharacteristic
131#126—262no—
132#125—131no—
133#125—262no—
134#124C2×D1311yescharacteristic
135#124C2×D1311yescharacteristic
136#123—22no—
137C22⋊Q8—22no—
138C22⋊Q8—132no—
139C4×Q8—132no—
140#117C2×D1311yescharacteristic
141C4⋊C8—132no—
142#103—132no—
143#103—132no—
144#103—132no—
145#102—132no—
146C22⋊C8—132no—
147#97—131no—
148C4×C8—131no—
149C2×C13—41no—
150C2×C13—81no—
151C2×C13#53511yescharacteristic
152C2×C13#57812yes—
153#41—42no—
154C2×Q8—21no—
155C2×Q8—42no—
156C2×Q8—132no—
157C2×Q8C22×D1311yescharacteristic
158C2×Q8C22×D1311yescharacteristic
159C2×Q8C22×D1311yescharacteristic
160C2×D4—21no—
161C2×D4C22×D1311yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4C22×D1311yescharacteristic
165Q16—522no—
166Q16—522no—
167Q16—522no—
168SD16—522no—
169C2×C8—132no—
170C2×C8—132no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—132no—
174C4⋊C4—262no—
175C4⋊C4—262no—
176C4⋊C4—262no—
177C4⋊C4—262no—
178C4⋊C4C22⋊D1312yes—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—262no—
182C22⋊C4C22⋊D1312yes—
183C42—262no—
184C42C22×D1311yescharacteristic
185C42C22×D1311yescharacteristic
186C13#322311yescharacteristic, complemented, Sylow
187C23—21no—
188C23C2×C22⋊D1311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—42no—
193Q8—262no—
194Q8—262no—
195D4—22no—
196D4—42no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—262no—
204C2×C4—262no—
205C2×C4C2×C22⋊D1312yes—
206C2×C4C2×C22⋊D1312yes—
207C2×C4C2×C22⋊D1311yescharacteristic
208C2×C4C23×D1311yescharacteristic
209C8—262no—
210C8—262no—
211C22—21no—
212C22—22no—
213C22—41no—
214C22—42no—
215C22C22×C22⋊D1311yescharacteristic
216C4—21no—
217C4—22no—
218C4—22no—
219C4—41no—
220C4—41no—
221C4—41no—
222C4—522no—
223C4—522no—
224C4#8825112yes—
225C2—41no—
226C2—81no—
227C2#1171951011yescharacteristic
228C2#1171954312yes—
subgroupG/H
centerC22C22×C22⋊D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#536C2
socleC22×C13C22×D4
prime radicalC2×C13×#41C22
hypercenter#536D13
hypocenterC13#3223
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #11719223C2semidirect
4. #11718889C2semidirect
5. #11718753C2semidirect
6. #11718751C2semidirect
7. #11718743C2semidirect
8. #11718721C2semidirect
9. #11718717C2semidirect
10. #11718715C2semidirect
11. #11718661C2semidirect
186. C13#3223semidirect

Maximal subgroups

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

subgroupindex
C13×#5362
#117192232
#117189002
#117188892
#117187532
#117187512
#117187432
#117187212
#117187172
#117187152
#117186612
#322313