Finite Group Databasebeta

← #13120566

Subgroups — #13120566

Order 1664.

816 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×#6127C211yesmaximal, characteristic
2C2×C13×C22⋊Q8C211yesmaximal, characteristic, complemented
3C2×#88200C212yesmaximal, complemented
4#11718873C211yesmaximal, characteristic, complemented
5#11718652C212yesmaximal, complemented
6#11718648C212yesmaximal, complemented
7#11718640C212yesmaximal, complemented
8#11718620C214yesmaximal, complemented
9C22×Q8×C13C2211yescharacteristic
10C23×C4×C13C2211yescharacteristic
11C22×C22⋊D13C2212yes—
12C2×#6127—24no—
13C2×#6127—28no—
14C22×D52C2211yescharacteristic
15C22×C4×D13C2212yes—
16C22×Dic26C2211yescharacteristic
17C13×C22⋊Q8—24no—
18C2×C13×C4⋊C4C2212yes—
19C2×C13×C4⋊C4C2211yescharacteristic
20C2×C13×C22⋊C4C2212yes—
21C2×C22⋊Dic13C2212yes—
22#88200—28no—
23C2×#6103C2214yes—
24C2×#6103C2214yes—
25C2×C4⋊Dic13C2212yes—
26C2×#6101C2214yes—
27#88071C2212yes—
28#88071C2214yes—
29C24×C13C2311yescharacteristic
30C23×D13C2312yes—
31C2×Q8×C13—24no—
32C2×Q8×C13—22no—
33C22×C4×C13—24no—
34C22×C4×C13D414yes—
35C22×C4×C13C2312yes—
36C22×C4×C13C2312yes—
37C22×C4×C13C2311yescharacteristic
38C22×C4×C13C2311yescharacteristic
39C2×C22⋊D13—24no—
40C2×C22⋊D13—24no—
41C2×C22⋊D13—24no—
42C22×Dic13C2314yes—
43C22×Dic13C2312yes—
44#6127—232no—
45C2×D52—24no—
46C2×D52—22no—
47C2×C4×D13—24no—
48C2×C4×D13—24no—
49C2×C4×D13D418yes—
50C2×Dic26—22no—
51C2×Dic26—24no—
52C13×C4⋊C4—22no—
53C13×C4⋊C4—24no—
54C13×C22⋊C4—24no—
55C22⋊Dic13—42no—
56#6103—28no—
57#6103—44no—
58C4⋊Dic13—24no—
59#6101—28no—
60#2616—131nomaximal, Sylow
61C23×C13—22no—
62C23×C13—21no—
63C23×C13—22no—
64C23×C13—21no—
65C23×C13C2×D412yes—
66C23×C13C2411yescharacteristic
67C22×D13—22no—
68C22×D13—22no—
69C22×D13—22no—
70C22×D13—22no—
71C22×D13—22no—
72C22×D13—22no—
73C22×D13C2×D414yes—
74Q8×C13—44no—
75C2×C4×C13—22no—
76C2×C4×C13—28no—
77C2×C4×C13—24no—
78C2×C4×C13—22no—
79C2×C4×C13—22no—
80C2×C4×C13—22no—
81C2×C4×C13—22no—
82C2×C4×C13C2×D412yes—
83C2×C4×C13C2×D414yes—
84C2×C4×C13C2×D412yes—
85C22⋊D13—216no—
86C2×Dic13—22no—
87C2×Dic13—22no—
88C2×Dic13—24no—
89C2×Dic13—24no—
90C2×Dic13—24no—
91C2×Dic13C2×D414yes—
92D52—28no—
93C4×D13—216no—
94Dic26—28no—
95C22×#41—131no—
96C2×C22⋊Q8—132no—
97C2×C22⋊Q8D1311yescharacteristic, complemented
98#395—132no—
99#394—132no—
100#394—134no—
101#380—131no—
102#380—132no—
103C22×C13—22no—
104C22×C13—22no—
105C22×C13—22no—
106C22×C13—42no—
107C22×C13—42no—
108C22×C13C22×D412yes—
109C22×C13C22×D411yescharacteristic
110C22×C13H(5,𝔽2)11yescharacteristic
111C22×C132−1+412yes—
112C22×C132−1+411yescharacteristic
113C2×D13—24no—
114C2×D13—24no—
115C2×D13—24no—
116C2×D13—42no—
117C2×D13—42no—
118C2×D13—42no—
119C2×D13—42no—
120C4×C13—22no—
121C4×C13—42no—
122C4×C13—42no—
123C4×C13#12014yes—
124Dic13—24no—
125Dic13—44no—
126C2×#41—268no—
127C2×#41—264no—
128C22×Q8—131no—
129C22×Q8C2×D1311yescharacteristic
130C22×D4—132no—
131C22×D4—131no—
132C23×C4—132no—
133C23×C4C2×D1311yescharacteristic
134C22⋊Q8—24no—
135C22⋊Q8—268no—
136C2×C4⋊C4—134no—
137C2×C4⋊C4—132no—
138C2×C4⋊C4C2×D1311yescharacteristic
139C2×C4⋊C4C2×D1312yes—
140C2×C22⋊C4—134no—
141C2×C22⋊C4—132no—
142C2×C22⋊C4—134no—
143C2×C22⋊C4C2×D1312yes—
144#95—134no—
145#95—132no—
146C2×C13—42no—
147C2×C13C2×#12011yescharacteristic
148C2×C13#53611yescharacteristic
149C2×C13#53612yes—
150C2×C13#53711yescharacteristic
151C2×C13#53712yes—
152D13—44no—
153C24—132no—
154C24C22×D1311yescharacteristic
155#41—2632no—
156C2×Q8—24no—
157C2×Q8—22no—
158C2×Q8—264no—
159C2×Q8—262no—
160C2×D4—264no—
161C2×D4—264no—
162C2×D4—264no—
163C2×D4—264no—
164C2×D4—262no—
165C22×C4—24no—
166C22×C4—132no—
167C22×C4—138no—
168C22×C4—134no—
169C22×C4—264no—
170C22×C4—264no—
171C22×C4C22⋊D1314yes—
172C22×C4C22×D1312yes—
173C22×C4C22×D1312yes—
174C22×C4C22×D1311yescharacteristic
175C22×C4C22×D1311yescharacteristic
176C4⋊C4—22no—
177C4⋊C4—24no—
178C4⋊C4—268no—
179C4⋊C4—264no—
180C22⋊C4—24no—
181C22⋊C4—268no—
182C22⋊C4—524no—
183C22⋊C4—522no—
184C13#261611yescharacteristic, complemented, Sylow
185C23—22no—
186C23—22no—
187C23—21no—
188C23—21no—
189C23—134no—
190C23—262no—
191C23—262no—
192C23—262no—
193C23—262no—
194C23—262no—
195C23—262no—
196C23C2×C22⋊D1312yes—
197C23C23×D1311yescharacteristic
198Q8—44no—
199Q8—268no—
200D4—268no—
201D4—2616no—
202C2×C4—22no—
203C2×C4—28no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—134no—
210C2×C4—262no—
211C2×C4—262no—
212C2×C4—2616no—
213C2×C4—264no—
214C2×C4—264no—
215C2×C4—264no—
216C2×C4D4×D1314yes—
217C2×C4C2×C22⋊D1312yes—
218C2×C4C2×C22⋊D1312yes—
219C22—22no—
220C22—22no—
221C22—22no—
222C22—42no—
223C22—42no—
224C22—264no—
225C22—264no—
226C22—264no—
227C22—522no—
228C22—522no—
229C22—522no—
230C22—522no—
231C22C2×D4×D1312yes—
232C22#8825412yes—
233C22#8825611yescharacteristic
234C22#8825711yescharacteristic
235C22C22×C22⋊D1311yescharacteristic
236C4—22no—
237C4—42no—
238C4—42no—
239C4—264no—
240C4—524no—
241C4#8819114yes—
242C2—42no—
243C2—524no—
244C2#1171933312yes—
245C2#1171933412yes—
246C2C2×#8819111yescharacteristic
247C2#1171952111yescharacteristic
248C2#1171953811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC2×C13×C22⋊Q8C2
prime radicalC22×#6127C2
hypercenterC2×C22⋊Q8D13
hypocenterC13#2616
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×C13×C22⋊Q8C2semidirect
3. C2×#88200C2semidirect
4. #11718873C2semidirect
5. #11718652C2semidirect
6. #11718648C2semidirect
7. #11718640C2semidirect
8. #11718620C2semidirect
97. C2×C22⋊Q8D13semidirect
184. C13#2616semidirect

Maximal subgroups

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

subgroupindex
C22×#61272
C2×C13×C22⋊Q82
C2×#882002
#117188732
#117186522
#117186482
#117186402
#117186202
#261613