Finite Group Databasebeta

← #30671119

Subgroups — #30671119

Order 16768.

672 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#16862225C211yesmaximal, characteristic, complemented
2#22553237C212yesmaximal, complemented
3#22553224C212yesmaximal, complemented
4#22553206C214yesmaximal, complemented
5#22553175C212yesmaximal, complemented
6#22553168C212yesmaximal
7C4×C22⋊Dic131C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C131C211yesmaximal, characteristic
9C22×Dic262C2211yescharacteristic, complemented
10#16862225—24no—
11C2×C4⋊Dic131C2211yescharacteristic, complemented
12C2×Dic131⋊C4C2212yescomplemented
13C2×C22⋊Dic131C2212yes—
14C2×C22⋊Dic131C2212yes—
15C2×C22⋊Dic131C2214yes—
16C2×C22⋊Dic131C2214yes—
17C2×C4×Dic131C2212yescomplemented
18C131⋊#95C2212yescomplemented
19C131⋊#95C2214yescomplemented
20C131⋊#95C2214yescomplemented
21C25×C131C2211yescharacteristic
22C23×C4×C131C2211yescharacteristic
23C23×C4×C131C2211yescharacteristic
24C2×C22⋊C4×C131—24no—
25C2×C22⋊C4×C131C2212yes—
26C2×C22⋊C4×C131C2212yes—
27C2×Dic262—22no—
28C2×Dic262—42no—
29C22×Dic131C2312yes—
30C22×Dic131C2314yes—
31C22×Dic131C2312yes—
32C4⋊Dic131—22no—
33Dic131⋊C4—24no—
34C22⋊Dic131—24no—
35C22⋊Dic131—24no—
36C22⋊Dic131—28no—
37C22⋊Dic131—44no—
38C4×Dic131—42no—
39C24×C131—22no—
40C24×C131—24no—
41C24×C131—24no—
42C24×C131—42no—
43C24×C131C2312yes—
44C24×C131C2311yescharacteristic
45C22×C4×C131—22no—
46C22×C4×C131—24no—
47C22×C4×C131—42no—
48C22×C4×C131D414yes—
49C22×C4×C131C2312yes—
50C22×C4×C131C2311yescharacteristic
51C22×C4×C131C2311yescharacteristic
52C22⋊C4×C131—24no—
53C22⋊C4×C131—24no—
54Dic262—44no—
55C2×Dic131—22no—
56C2×Dic131—22no—
57C2×Dic131—24no—
58C2×Dic131—24no—
59C2×Dic131—24no—
60C2×Dic131—24no—
61C2×Dic131—24no—
62C23×C131—22no—
63C23×C131—22no—
64C23×C131—28no—
65C23×C131—22no—
66C23×C131—24no—
67C23×C131—22no—
68C23×C131—24no—
69C23×C131—416no—
70C23×C131—44no—
71C23×C131—44no—
72C23×C131C2×D412yes—
73C23×C131#4114yes—
74C23×C131#4114yes—
75C23×C131C2411yescharacteristic
76C2×C4×C131—22no—
77C2×C4×C131—22no—
78C2×C4×C131—22no—
79C2×C4×C131—24no—
80C2×C4×C131—44no—
81C2×C4×C131—44no—
82C2×C4×C131C2×D412yes—
83C2×C4×C131C2×D412yes—
84C22×C131—22no—
85C22×C131—22no—
86C22×C131—28no—
87C22×C131—24no—
88C22×C131—28no—
89C22×C131—416no—
90C22×C131—42no—
91C22×C131—42no—
92C22×C131—44no—
93C22×C131#12414yes—
94C22×C131C22×D411yescharacteristic
95C22×C131C2×#4112yes—
96C22×C131C2×#4112yes—
97C22×C131H(5,𝔽2)12yes—
98C4×C131—22no—
99C4×C131—22no—
100C4×C131—42no—
101Dic131—42no—
102Dic131—42no—
103Dic131—44no—
104C2×C131—24no—
105C2×C131—44no—
106C2×C131#52511yescharacteristic
107C2×C131C2×#12411yescharacteristic
108C2×C131#53411yescharacteristic
109C2×C131#55114yes—
110C131#256111yescharacteristic, complemented, Sylow
111#2561—1311nomaximal, Sylow
112C2×C22⋊Q8—1311no—
113C22×C22⋊C4D13111yescharacteristic
114#397—1312no—
115#393—1312no—
116#388—1314no—
117#381—1312no—
118#379—1312no—
119C4×C22⋊C4—1311no—
120C25C2×D13111yescharacteristic
121C22×Q8—1311no—
122C23×C4C2×D13111yescharacteristic
123C23×C4C2×D13111yescharacteristic
124C22⋊Q8—2624no—
125C2×C4⋊C4—1312no—
126C2×C4⋊C4—1311no—
127C2×C22⋊C4—24no—
128C2×C22⋊C4—1312no—
129C2×C22⋊C4—1312no—
130C2×C22⋊C4—1314no—
131C2×C22⋊C4—1314no—
132C2×C22⋊C4C2×D13112yes—
133C2×C22⋊C4C2×D13112yes—
134C2×C42—1312no—
135#95—1312no—
136#95—1314no—
137#95—1314no—
138C24—22no—
139C24—24no—
140C24—24no—
141C24—42no—
142C24C22×D13112yes—
143C24C22×D13111yescharacteristic
144C2×Q8—2622no—
145C2×Q8—5242no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—42no—
149C22×C4—1314no—
150C22×C4—1312no—
151C22×C4—1312no—
152C22×C4C22⋊D13114yes—
153C22×C4C22×D13112yes—
154C22×C4C22×D13111yescharacteristic
155C22×C4C22×D13111yescharacteristic
156C4⋊C4—2624no—
157C4⋊C4—2622no—
158C22⋊C4—24no—
159C22⋊C4—24no—
160C22⋊C4—2624no—
161C22⋊C4—2624no—
162C22⋊C4—2628no—
163C22⋊C4—5244no—
164C42—5242no—
165C23—22no—
166C23—22no—
167C23—28no—
168C23—22no—
169C23—24no—
170C23—22no—
171C23—24no—
172C23—416no—
173C23—44no—
174C23—44no—
175C23C2×C22⋊D13112yes—
176C23#1452351214yes—
177C23#1452351314yes—
178C23C23×D13111yescharacteristic
179Q8—5244no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—44no—
185C2×C4—44no—
186C2×C4—2622no—
187C2×C4—2622no—
188C2×C4—2624no—
189C2×C4—2624no—
190C2×C4—2624no—
191C2×C4—2624no—
192C2×C4—2624no—
193C2×C4C2×C22⋊D13112yes—
194C2×C4C2×C22⋊D13112yes—
195C22—22no—
196C22—28no—
197C22—22no—
198C22—28no—
199C22—24no—
200C22—416no—
201C22—42no—
202C22—44no—
203C22—42no—
204C22#1686223314yes—
205C22C22×C22⋊D13111yescharacteristic
206C22C2×#1452351212yes—
207C22C2×#1452351312yes—
208C22#1686229112yes—
209C4—22no—
210C4—22no—
211C4—42no—
212C4—5242no—
213C4—5244no—
214C4—5242no—
215C2—24no—
216C2—44no—
217C2#2255382311yescharacteristic
218C2C2×#1686223311yescharacteristic
219C2#2255387111yescharacteristic
220C2#2255398814yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC22×C22⋊C4×C131C2
prime radicalC25×C131C22
hypercenterC22×C22⋊C4D131
hypocenterC131#2561
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16862225C2semidirect
2. #22553237C2semidirect
3. #22553224C2semidirect
4. #22553206C2semidirect
5. #22553175C2semidirect
7. C4×C22⋊Dic131C2semidirect
9. C22×Dic262C22semidirect
11. C2×C4⋊Dic131C22semidirect
12. C2×Dic131⋊C4C22semidirect
17. C2×C4×Dic131C22semidirect
18. C131⋊#95C22semidirect
19. C131⋊#95C22semidirect
20. C131⋊#95C22semidirect
110. C131#2561semidirect

Maximal subgroups

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

subgroupindex
C2×#168622252
#225532372
#225532242
#225532062
#225531752
#225531682
C4×C22⋊Dic1312
C22×C22⋊C4×C1312
#2561131