Finite Group Databasebeta

← #30670900

Subgroups — #30670900

Order 16768.

760 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16862234C211yesmaximal, characteristic, complemented
2C2×#16862225C212yesmaximal, complemented
3C2×#16862220C212yesmaximal, complemented
4#22553247C212yesmaximal, complemented
5#22553220C212yesmaximal, complemented
6#22553212C214yesmaximal, complemented
7C4×C4⋊Dic131C211yesmaximal, characteristic, complemented
8C2×C4×D4×C131C211yesmaximal, characteristic, complemented
9C22×Dic262C2211yescharacteristic
10C22×D524C2211yescharacteristic
11C22×C22⋊D131C2214yes—
12#16862234—24no—
13#16862225—28no—
14#16862220—28no—
15C2×C4⋊Dic131C2212yes—
16C2×Dic131⋊C4C2214yes—
17C2×C22⋊Dic131C2214yes—
18C2×(C2×D131)⋊C4C2212yes—
19C2×(C2×D131)⋊C4C2214yes—
20C2×(C2×D131)⋊C4C2212yes—
21C2×C4×Dic131C2212yes—
22C131⋊#95C2212yes—
23C22×D4×C131C2211yescharacteristic
24C23×C4×C131C2212yes—
25C4×D4×C131—24no—
26C2×C4⋊C4×C131C2211yescharacteristic
27C2×C22⋊C4×C131C2212yes—
28C2×C42×C131C2211yescharacteristic
29C23×D131C2312yes—
30C2×Dic262—24no—
31C2×Dic262—22no—
32C2×D524—24no—
33C2×D524—22no—
34C2×C22⋊D131—28no—
35C2×C22⋊D131—28no—
36C2×C22⋊D131—44no—
37C22×Dic131C2312yes—
38C22×Dic131C2314yes—
39C4⋊Dic131D418yescomplemented
40Dic131⋊C4—28no—
41C22⋊Dic131—28no—
42(C2×D131)⋊C4—28no—
43(C2×D131)⋊C4—24no—
44(C2×D131)⋊C4—24no—
45C4×Dic131—42no—
46C24×C131C2312yes—
47C2×D4×C131—24no—
48C2×D4×C131—22no—
49C22×C4×C131—24no—
50C22×C4×C131—24no—
51C22×C4×C131—24no—
52C22×C4×C131C2312yes—
53C22×C4×C131C2311yescharacteristic
54C22×C4×C131C2311yescharacteristic
55C22×C4×C131C2311yescharacteristic
56C4⋊C4×C131—22no—
57C22⋊C4×C131—24no—
58C42×C131D414yescomplemented
59C22×D131—22no—
60C22×D131—24no—
61C22×D131—24no—
62C22×D131—22no—
63C22×D131—22no—
64Dic262—44no—
65D524—44no—
66C22⋊D131—416no—
67C2×Dic131—24no—
68C2×Dic131—24no—
69C2×Dic131—22no—
70C2×Dic131—24no—
71C2×Dic131C2×D418yes—
72C23×C131—22no—
73C23×C131—22no—
74C23×C131—22no—
75C23×C131—22no—
76C23×C131—22no—
77C23×C131—22no—
78C23×C131#4114yes—
79C23×C131C2411yescharacteristic
80D4×C131—28no—
81C2×C4×C131—22no—
82C2×C4×C131—21no—
83C2×C4×C131—21no—
84C2×C4×C131—22no—
85C2×C4×C131—22no—
86C2×C4×C131—22no—
87C2×C4×C131—48no—
88C2×C4×C131C2×D414yes—
89C2×C4×C131C2×D412yes—
90C2×C4×C131C2×D412yes—
91C2×C4×C131C2×D412yes—
92C22×C131—24no—
93C22×C131—24no—
94C22×C131—24no—
95C22×C131—42no—
96C22×C131—42no—
97C22×C131—42no—
98C22×C131—42no—
99C22×C131C22×D412yes—
100C22×C131C22×D411yescharacteristic
101C22×C131C2×#4112yes—
102C22×C131H(5,𝔽2)11yescharacteristic
103C22×C1312−1+411yescharacteristic
104C2×D131—42no—
105C2×D131—42no—
106C2×D131—42no—
107C2×D131—44no—
108C2×D131—44no—
109C4×C131—22no—
110C4×C131—22no—
111C4×C131—42no—
112C4×C131#12714yes—
113Dic131—28no—
114Dic131—42no—
115C2×C131—44no—
116C2×C131#52511yescharacteristic
117C2×C131C2×#12711yescharacteristic
118C2×C131#53711yescharacteristic
119C2×C131#54612yes—
120C2×C131#54812yes—
121D131—82no—
122C131#254011yescharacteristic, complemented, Sylow
123#2540—1311nomaximal, Sylow
124C2×#124—1311no—
125C2×C22⋊Q8—1312no—
126C2×#121—1312no—
127C2×C4×D4D13111yescharacteristic, complemented
128#399—1312no—
129#392—1312no—
130#390—1314no—
131C4×C4⋊C4—1311no—
132C22×Q8—1311no—
133C22×D4—1314no—
134C22×D4—1311no—
135C22×D4C2×D13111yescharacteristic
136C23×C4C2×D13112yes—
137#124—2624no—
138C22⋊Q8—2628no—
139#121—2628no—
140C4×D4—24no—
141C2×C4⋊C4—1314no—
142C2×C4⋊C4—1312no—
143C2×C4⋊C4C2×D13111yescharacteristic
144C2×C22⋊C4—1314no—
145C2×C22⋊C4—1314no—
146C2×C22⋊C4—1312no—
147C2×C22⋊C4—1312no—
148C2×C22⋊C4C2×D13112yes—
149C2×C42—1312no—
150C2×C42C2×D13111yescharacteristic
151#95—1312no—
152C24—1312no—
153C24C22×D13112yes—
154C2×Q8—2624no—
155C2×Q8—2622no—
156C2×D4—24no—
157C2×D4—22no—
158C2×D4—2628no—
159C2×D4—2628no—
160C2×D4—2624no—
161C2×D4—2622no—
162C2×D4—5244no—
163C22×C4—24no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—1312no—
167C22×C4—1314no—
168C22×C4C22×D13112yes—
169C22×C4C22×D13111yescharacteristic
170C22×C4C22×D13111yescharacteristic
171C22×C4C22×D13111yescharacteristic
172C4⋊C4—22no—
173C4⋊C4—2628no—
174C4⋊C4—1318no—
175C22⋊C4—24no—
176C22⋊C4—2628no—
177C22⋊C4—2628no—
178C22⋊C4—2624no—
179C22⋊C4—2624no—
180C42—5242no—
181C42C22⋊D13114yescomplemented
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—22no—
187C23—22no—
188C23—2624no—
189C23—2622no—
190C23—2624no—
191C23—2622no—
192C23—2622no—
193C23#1452351314yes—
194C23C23×D13111yescharacteristic
195Q8—5244no—
196D4—28no—
197D4—52416no—
198D4—5244no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—48no—
206C2×C4—2624no—
207C2×C4—2624no—
208C2×C4—2624no—
209C2×C4—2622no—
210C2×C4—1318no—
211C2×C4D4×D13114yes—
212C2×C4C2×C22⋊D13112yes—
213C2×C4C2×C22⋊D13112yes—
214C2×C4C2×C22⋊D13112yes—
215C22—24no—
216C22—24no—
217C22—24no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—5242no—
223C22—5244no—
224C22—5244no—
225C22—5242no—
226C22—5242no—
227C22C2×D4×D13112yes—
228C22C22×C22⋊D13111yescharacteristic
229C22C2×#1452351312yes—
230C22#1686229211yescharacteristic
231C22#1686229311yescharacteristic
232C4—22no—
233C4—22no—
234C4—42no—
235C4—2628no—
236C4—5242no—
237C4#1686224214yes—
238C2—44no—
239C2—10482no—
240C2#2255382311yescharacteristic
241C2C2×#1686224211yescharacteristic
242C2#2255389011yescharacteristic
243C2#2255395812yes—
244C2#2255397112yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC2×C4×D4×C131C2
hypercenterC2×C4×D4D131
hypocenterC131#2540
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16862234C2semidirect
2. C2×#16862225C2semidirect
3. C2×#16862220C2semidirect
4. #22553247C2semidirect
5. #22553220C2semidirect
6. #22553212C2semidirect
7. C4×C4⋊Dic131C2semidirect
8. C2×C4×D4×C131C2semidirect
39. C4⋊Dic131D4semidirect
58. C42×C131D4semidirect
122. C131#2540semidirect
127. C2×C4×D4D131semidirect
181. C42C22⋊D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622342
C2×#168622252
C2×#168622202
#225532472
#225532202
#225532122
C4×C4⋊Dic1312
C2×C4×D4×C1312
#2540131