Finite Group Databasebeta

← #13125921

Subgroups — #13125921

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
1#11719231C211yesmaximal, characteristic
2#11719223C211yesmaximal, characteristic, complemented
3C13×#433C211yesmaximal, characteristic, complemented
4#11718510C212yesmaximal, complemented
5#11718506C212yesmaximal, complemented
6#11718400C212yesmaximal, complemented
7#11718394C212yesmaximal
8#11718376C211yesmaximal, characteristic, complemented
9#11718364C211yesmaximal, characteristic, complemented
10#11718360C211yesmaximal, characteristic, complemented
11#11718358C211yesmaximal, characteristic, complemented
12C2×#6127C2211yescharacteristic
13C22×Dic26C2211yescharacteristic
14C13×#117C2211yescharacteristic
15#88178C2212yes—
16C2×Dic52C2212yes—
17C2×Dic52C2212yes—
18C2×Dic52C2212yes—
19C2×#6095C2212yes—
20#88151—22no—
21#88143—22no—
22#88140—22no—
23#88133—22no—
24#88129C2211yescharacteristic
25#88129C2211yescharacteristic
26#88124C2211yescharacteristic
27#88123C2211yescharacteristic
28#88123C2211yescharacteristic
29#88123C2211yescharacteristic
30C4×Dic26C2212yes—
31C13×C4⋊C8C2212yes—
32C13×C22⋊C8C2212yes—
33C13×#97C2211yescharacteristic
34C4×C8×C13C2211yescharacteristic
35#88061C2212yes—
36#88056C2212yes—
37#88056C2212yes—
38#88056C2212yes—
39C22×C4×C13C2311yescharacteristic
40C2×C22⋊D13—21no—
41C22×Dic13—21no—
42#6127—42no—
43C2×D52C2311yescharacteristic
44C2×C4×D13—21no—
45C2×Dic26—21no—
46C2×Dic26—42no—
47C2×Dic26C2312yes—
48C2×Dic26C2311yescharacteristic
49C2×Dic26C2311yescharacteristic
50C2×Dic26C2311yescharacteristic
51C2×C8×C13C2312yes—
52C2×C8×C13C2312yes—
53C13×C4⋊C4D412yes—
54C13×C22⋊C4D412yes—
55C42×C13C2311yescharacteristic
56C42×C13C2311yescharacteristic
57C22⋊Dic13—22no—
58#6103—22no—
59#6103—22no—
60C4⋊Dic13—22no—
61C4⋊Dic13—22no—
62C4⋊Dic13—22no—
63C4⋊Dic13C2312yes—
64#6101—22no—
65#6101—22no—
66#6101—22no—
67C4×Dic13—22no—
68Dic52—42no—
69Dic52—42no—
70Dic52—42no—
71#6095—42no—
72#3223—131nomaximal, Sylow
73C23×C13C2×D411yescharacteristic
74C22×D13—21no—
75C2×C4×C13—22no—
76C2×C4×C13C2×D412yes—
77C2×C4×C13C2×D411yescharacteristic
78C2×C4×C13C2×D412yes—
79C2×C4×C13C2411yescharacteristic
80C22⋊D13—42no—
81C2×Dic13—22no—
82C2×Dic13—22no—
83C2×Dic13—21no—
84C2×Dic13—21no—
85C2×Dic13—21no—
86C2×Dic13—42no—
87D52—22no—
88C4×D13—42no—
89Dic26—22no—
90Dic26—22no—
91Dic26—22no—
92Dic26—22no—
93Dic26—22no—
94Dic26—42no—
95C8×C13—22no—
96C8×C13—22no—
97#541—131no—
98#536—131no—
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#433D1311yescharacteristic, complemented
108C22×C13—21no—
109C22×C13—22no—
110C22×C13C22×D411yescharacteristic
111C2×D13—41no—
112C2×D13—42no—
113C4×C13—21no—
114C4×C13—22no—
115C4×C13—22no—
116C4×C13H(5,𝔽2)12yes—
117Dic13—41no—
118Dic13—41no—
119Dic13—41no—
120Dic13—42no—
121Dic13—42no—
122C2×#41—131no—
123C22×Q8—131no—
124C2×Q16—132no—
125C2×Q16—132no—
126C2×Q16—132no—
127C2×SD16—132no—
128C4⋊Q8—131no—
129C4⋊Q8—131no—
130C4⋊Q8—131no—
131#126—262no—
132#125—131no—
133#125—262no—
134#124—131no—
135#124—131no—
136#123—262no—
137C22⋊Q8—132no—
138C22⋊Q8—262no—
139C4×Q8—132no—
140#117C2×D1311yescharacteristic
141C4⋊C8C2×D1312yes—
142#103—132no—
143#103—132no—
144#103—132no—
145#102—132no—
146C22⋊C8C2×D1312yes—
147#97C2×D1311yescharacteristic
148C4×C8C2×D1311yescharacteristic
149C2×C13—41no—
150C2×C13#53511yescharacteristic
151C2×C13#57812yes—
152D13—81no—
153#41—522no—
154C2×Q8—131no—
155C2×Q8—131no—
156C2×Q8—131no—
157C2×Q8—132no—
158C2×Q8—261no—
159C2×Q8—522no—
160C2×D4—131no—
161C2×D4—261no—
162C22×C4—261no—
163C22×C4—261no—
164C22×C4C22×D1311yescharacteristic
165Q16—522no—
166Q16—522no—
167Q16—522no—
168SD16—522no—
169C2×C8C22×D1312yes—
170C2×C8C22×D1312yes—
171C4⋊C4—132no—
172C4⋊C4—262no—
173C4⋊C4—262no—
174C4⋊C4—262no—
175C4⋊C4—262no—
176C4⋊C4—262no—
177C4⋊C4—262no—
178C4⋊C4D5212yes—
179C22⋊C4—262no—
180C22⋊C4—262no—
181C22⋊C4—262no—
182C22⋊C4D5212yes—
183C42—262no—
184C42C22×D1311yescharacteristic
185C42C22×D1311yescharacteristic
186C13#322311yescharacteristic, complemented, Sylow
187C23—261no—
188C23C2×D5211yescharacteristic
189Q8—262no—
190Q8—262no—
191Q8—262no—
192Q8—262no—
193Q8—262no—
194Q8—522no—
195D4—262no—
196D4—522no—
197C2×C4—22no—
198C2×C4—262no—
199C2×C4—262no—
200C2×C4—261no—
201C2×C4—261no—
202C2×C4—261no—
203C2×C4—522no—
204C2×C4—522no—
205C2×C4C2×D5212yes—
206C2×C4C2×D5212yes—
207C2×C4C2×D5211yescharacteristic
208C2×C4C23×D1311yescharacteristic
209C8—22no—
210C8—22no—
211C22—22no—
212C22—21no—
213C22—522no—
214C22—521no—
215C22C22×D5211yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—522no—
220C4—521no—
221C4—521no—
222C4—521no—
223C4—522no—
224C4#8825612yes—
225C2—41no—
226C2—1041no—
227C2#1171923011yescharacteristic
228C2#1171945812yes—
subgroupG/H
centerC22C22×D52
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#433C2
socleC22×C13C22×D4
prime radicalC2×#6127C22
hypercenter#433D13
hypocenterC13#3223
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #11719223C2semidirect
3. C13×#433C2semidirect
4. #11718510C2semidirect
5. #11718506C2semidirect
6. #11718400C2semidirect
8. #11718376C2semidirect
9. #11718364C2semidirect
10. #11718360C2semidirect
11. #11718358C2semidirect
107. #433D13semidirect
186. C13#3223semidirect

Maximal subgroups

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

subgroupindex
#117192312
#117192232
C13×#4332
#117185102
#117185062
#117184002
#117183942
#117183762
#117183642
#117183602
#117183582
#322313