Finite Group Databasebeta

← #30670235

Subgroups — #30670235

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×#16862205C211yesmaximal, characteristic, complemented
2#22553248C212yesmaximal, complemented
3#22553237C212yesmaximal, complemented
4#22553200C214yesmaximal, complemented
5#22553185C212yesmaximal, complemented
6#22553175C211yesmaximal, characteristic, complemented
7#22553168C212yesmaximal
8C22×C22⋊C4×C131C211yesmaximal, characteristic
9#16862205—24no—
10C2×C4⋊Dic131C2211yescharacteristic, complemented
11C2×C4⋊Dic131C2211yescharacteristic, complemented
12C2×Dic131⋊C4C2214yescomplemented
13C2×C22⋊Dic131C2212yes—
14C2×C22⋊Dic131C2212yes—
15C2×C22⋊Dic131C2214yes—
16C2×C22⋊Dic131C2214yes—
17C2×C4×Dic131C2212yescomplemented
18C131⋊#95C2212yescomplemented
19C131⋊#95C2212yescomplemented
20C131⋊#95C2214yescomplemented
21C25×C131C2211yescharacteristic
22C23×C4×C131C2211yescharacteristic
23C23×C4×C131C2211yescharacteristic
24C2×C22⋊C4×C131—24no—
25C2×C22⋊C4×C131C2212yes—
26C2×C22⋊C4×C131C2212yes—
27C22×Dic131C2314yes—
28C22×Dic131C2312yes—
29C22×Dic131C2312yes—
30C4⋊Dic131—22no—
31C4⋊Dic131—22no—
32Dic131⋊C4—44no—
33C22⋊Dic131—24no—
34C22⋊Dic131—24no—
35C22⋊Dic131—28no—
36C22⋊Dic131—44no—
37C4×Dic131—24no—
38C24×C131—22no—
39C24×C131—24no—
40C24×C131—24no—
41C24×C131—42no—
42C24×C131C2312yes—
43C24×C131C2311yescharacteristic
44C22×C4×C131—24no—
45C22×C4×C131—22no—
46C22×C4×C131—42no—
47C22×C4×C131D414yes—
48C22×C4×C131C2312yes—
49C22×C4×C131C2311yescharacteristic
50C22×C4×C131C2311yescharacteristic
51C22⋊C4×C131—24no—
52C22⋊C4×C131—24no—
53C2×Dic131—22no—
54C2×Dic131—22no—
55C2×Dic131—24no—
56C2×Dic131—24no—
57C2×Dic131—24no—
58C2×Dic131—24no—
59C2×Dic131—24no—
60C23×C131—22no—
61C23×C131—22no—
62C23×C131—28no—
63C23×C131—22no—
64C23×C131—24no—
65C23×C131—22no—
66C23×C131—24no—
67C23×C131—416no—
68C23×C131—44no—
69C23×C131—44no—
70C23×C131C2×D412yes—
71C23×C131#4114yes—
72C23×C131#4114yes—
73C23×C131C2411yescharacteristic
74C2×C4×C131—28no—
75C2×C4×C131—22no—
76C2×C4×C131—22no—
77C2×C4×C131—22no—
78C2×C4×C131—24no—
79C2×C4×C131—44no—
80C2×C4×C131C2×D412yes—
81C2×C4×C131C2×D412yes—
82C22×C131—22no—
83C22×C131—22no—
84C22×C131—28no—
85C22×C131—24no—
86C22×C131—28no—
87C22×C131—416no—
88C22×C131—42no—
89C22×C131—42no—
90C22×C131—44no—
91C22×C131#12314yes—
92C22×C131C22×D411yescharacteristic
93C22×C131C2×#4112yes—
94C22×C131C2×#4112yes—
95C22×C131H(5,𝔽2)12yes—
96C4×C131—22no—
97C4×C131—22no—
98C4×C131—42no—
99Dic131—42no—
100Dic131—42no—
101Dic131—44no—
102C2×C131—24no—
103C2×C131—44no—
104C2×C131C2×#12311yescharacteristic
105C2×C131#52511yescharacteristic
106C2×C131#53511yescharacteristic
107C2×C131#55114yes—
108C131#248211yescharacteristic, complemented, Sylow
109#2482—1311nomaximal, Sylow
110C2×#117—1311no—
111C22×C22⋊C4D13111yescharacteristic
112#399—1312no—
113#397—1312no—
114#388—1314no—
115#385—1312no—
116#381—1311no—
117#379—1312no—
118C25C2×D13111yescharacteristic
119C23×C4C2×D13111yescharacteristic
120C23×C4C2×D13111yescharacteristic
121#117—2624no—
122C2×C4⋊C4—1311no—
123C2×C4⋊C4—1311no—
124C2×C4⋊C4—1314no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—1312no—
127C2×C22⋊C4—1312no—
128C2×C22⋊C4—1314no—
129C2×C22⋊C4—1314no—
130C2×C22⋊C4C2×D13112yes—
131C2×C22⋊C4C2×D13112yes—
132C2×C42—1312no—
133#95—1312no—
134#95—1312no—
135#95—1314no—
136C24—22no—
137C24—24no—
138C24—24no—
139C24—42no—
140C24C22×D13112yes—
141C24C22×D13111yescharacteristic
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—1314no—
146C22×C4—1312no—
147C22×C4—1312no—
148C22×C4C22⋊D13114yes—
149C22×C4C22×D13112yes—
150C22×C4C22×D13111yescharacteristic
151C22×C4C22×D13111yescharacteristic
152C4⋊C4—2622no—
153C4⋊C4—2622no—
154C4⋊C4—5244no—
155C22⋊C4—24no—
156C22⋊C4—24no—
157C22⋊C4—2624no—
158C22⋊C4—2624no—
159C22⋊C4—2628no—
160C22⋊C4—5244no—
161C42—2624no—
162C23—22no—
163C23—22no—
164C23—28no—
165C23—22no—
166C23—24no—
167C23—22no—
168C23—24no—
169C23—416no—
170C23—44no—
171C23—44no—
172C23C2×C22⋊D13112yes—
173C23#1452351214yes—
174C23#1452351314yes—
175C23C23×D13111yescharacteristic
176C2×C4—28no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—44no—
182C2×C4—2622no—
183C2×C4—2622no—
184C2×C4—2624no—
185C2×C4—2624no—
186C2×C4—2624no—
187C2×C4—2624no—
188C2×C4—2624no—
189C2×C4C2×C22⋊D13112yes—
190C2×C4C2×C22⋊D13112yes—
191C22—22no—
192C22—28no—
193C22—22no—
194C22—28no—
195C22—24no—
196C22—416no—
197C22—42no—
198C22—44no—
199C22—42no—
200C22#1686222714yes—
201C22C22×C22⋊D13111yescharacteristic
202C22C2×#1452351212yes—
203C22C2×#1452351312yes—
204C22#1686229112yes—
205C4—22no—
206C4—22no—
207C4—42no—
208C4—5242no—
209C4—5244no—
210C4—5242no—
211C2—24no—
212C2—44no—
213C2C2×#1686222711yescharacteristic
214C2#2255382311yescharacteristic
215C2#2255387911yescharacteristic
216C2#2255398814yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC22×C22⋊C4×C131C2
prime radicalC25×C131C22
hypercenterC22×C22⋊C4D131
hypocenterC131#2482
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×#16862205C2semidirect
2. #22553248C2semidirect
3. #22553237C2semidirect
4. #22553200C2semidirect
5. #22553185C2semidirect
6. #22553175C2semidirect
10. C2×C4⋊Dic131C22semidirect
11. C2×C4⋊Dic131C22semidirect
12. C2×Dic131⋊C4C22semidirect
17. C2×C4×Dic131C22semidirect
18. C131⋊#95C22semidirect
19. C131⋊#95C22semidirect
20. C131⋊#95C22semidirect
108. C131#2482semidirect

Maximal subgroups

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

subgroupindex
C2×#168622052
#225532482
#225532372
#225532002
#225531852
#225531752
#225531682
C22×C22⋊C4×C1312
#2482131