Finite Group Databasebeta

← C2×#21453859 #29886570

Subgroups — C2×#21453859 #29886570

Order 15616.

3666 conjugacy classes of subgroups; only the 667 normal subgroups are shown.

normal subgroups only

This group has 3666 conjugacy classes of subgroups — too many to draw. Only its 667 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×#14519774C211yesmaximal, characteristic, complemented
2#21453859C2116yesmaximal, direct product factor
3C2×#16839679C216yesmaximal, complemented
4C2×#16839671C216yesmaximal, complemented
5C2×#16839630C211yesmaximal, characteristic, complemented
6C22×#126×C61C211yesmaximal, characteristic, complemented
7C23×D244C2216yes—
8C23×C22⋊D61C2214yes—
9C2×#14519774—28no—
10C2×#14519774C2218yescomplemented
11C22×#11758198C22112yes—
12#16839679C22148yescomplemented
13#16839671C22148yescomplemented
14#16839630C2218yescomplemented
15C2×#14519684C2216yes—
16C2×#126×C61—28no—
17C2×#126×C61C2218yescomplemented
18C22×C4⋊C4×C61C2213yes—
19C22×C22⋊C4×C61C2213yes—
20C22×C42×C61C2211yescharacteristic
21C24×D61C2314yes—
22C22×D244—224no—
23C22×D244—424no—
24C22×D244C23124yes—
25C22×C22⋊D61—424no—
26C22×C22⋊D61C23116yes—
27C23×Dic61C2314yes—
28#14519774—232no—
29C2×#11758198—424no—
30C2×#11758198C23148yes—
31#14519684C23124yes—
32C25×C61C2311yescharacteristic
33C23×C4×C61C2313yes—
34C23×C4×C61C2313yes—
35#126×C61—232no—
36C2×C4⋊C4×C61—212no—
37C2×C4⋊C4×C61C23112yes—
38C2×C22⋊C4×C61—212no—
39C2×C22⋊C4×C61C23112yes—
40C2×C42×C61D418yescomplemented
41C2×C42×C61C2314yes—
42C24×C61—22no—
43C24×C61—212no—
44C24×C61C2412yes—
45C24×C61C2411yescharacteristic
46C23×D61—28no—
47C23×D61—224no—
48C23×D61—224no—
49C23×D61C2418yes—
50C22×C4×C61—212no—
51C22×C4×C61—26no—
52C22×C4×C61—212no—
53C22×C4×C61C2×D4112yes—
54C22×C4×C61C2416yes—
55C22×C4×C61C2416yes—
56C2×C22⋊D61—448no—
57C2×C22⋊D61—464no—
58C22×Dic61—224no—
59C22×Dic61C2418yes—
60C2×D244—248no—
61C2×D244—448no—
62C2×D244—496no—
63C4⋊C4×C61—224no—
64C22⋊C4×C61—224no—
65C42×C61C2×D4116yescomplemented
66#11758198—448no—
67C23×C61—22no—
68C23×C61—212no—
69C23×C61—412no—
70C23×C61—416no—
71C23×C61C22×D412yes—
72C23×C61H(5,𝔽2)112yes—
73C23×C61C2511yescharacteristic
74C22×D61—28no—
75C22×D61—224no—
76C22×D61—224no—
77C22×D61—448no—
78C22×D61—416no—
79C22×D61—448no—
80C2×C4×C61—224no—
81C2×C4×C61—212no—
82C2×C4×C61—26no—
83C2×C4×C61—212no—
84C2×C4×C61—412no—
85C2×C4×C61C22×D4112yes—
86C22⋊D61—464no—
87C2×Dic61—224no—
88C2×Dic61—416no—
89D244—496no—
90C2×#2818—611nomaximal, Sylow
91C22×C61—46no—
92C22×C61—48no—
93C22×C61—88no—
94C22×C61#535112yes—
95C22×C61#560116yes—
96C22×C61C23×D411yescharacteristic
97C22×C61C2×H(5,𝔽2)16yes—
98C2×D61—424no—
99C2×D61—48no—
100C2×D61—424no—
101C2×D61—832no—
102C4×C61—212no—
103C4×C61—46no—
104Dic61—48no—
105C22×#127—611no—
106C22×#126D6111yescharacteristic, complemented
107#2818—6116no—
108C2×#394—616no—
109C2×#392—616no—
110C2×#383—611no—
111C2×C61—82no—
112C2×C61#281818yescomplemented
113C2×C61C2×#53513yes—
114C2×C61C2×#56014yes—
115D61—88no—
116C23×D4—614no—
117C23×D4—616no—
118C2×#127—618no—
119C2×#127—1228no—
120C2×#126—28no—
121C2×#126C2×D6118yescomplemented
122C22×C4⋊C4C2×D6113yes—
123C22×C22⋊C4—6112no—
124C22×C22⋊C4C2×D6113yes—
125C22×C42C2×D6111yescharacteristic
126#394—6148no—
127#392—6148no—
128#383—618no—
129C2×#95—616no—
130C61C2×#281811yescharacteristic, complemented, Sylow
131C25—614no—
132C25C22×D6111yescharacteristic
133C22×D4—6116no—
134C22×D4—6124no—
135C22×D4—12224no—
136C22×D4—24424no—
137C22×D4—24424no—
138C23×C4—614no—
139C23×C4C22×D6113yes—
140C23×C4C22×D6113yes—
141#127—12232no—
142#126—232no—
143C2×C4⋊C4—212no—
144C2×C4⋊C4C22×D61112yes—
145C2×C22⋊C4—212no—
146C2×C22⋊C4—6148no—
147C2×C22⋊C4—24424no—
148C2×C22⋊C4C22×D61112yes—
149C2×C42C22⋊D6118yescomplemented
150C2×C42C22×D6114yes—
151#95—6124no—
152C24—22no—
153C24—212no—
154C24—618no—
155C24—1228no—
156C24—12224no—
157C24—12224no—
158C24C23×D6112yes—
159C24C23×D6111yescharacteristic
160C2×D4—12248no—
161C2×D4—24448no—
162C2×D4—24496no—
163C2×D4—24448no—
164C2×D4—24464no—
165C22×C4—212no—
166C22×C4—26no—
167C22×C4—212no—
168C22×C4—618no—
169C22×C4—12224no—
170C22×C4C2×C22⋊D61112yes—
171C22×C4C23×D6116yes—
172C22×C4C23×D6116yes—
173C4⋊C4—224no—
174C22⋊C4—224no—
175C22⋊C4—24448no—
176C42C2×C22⋊D61116yescomplemented
177C23—22no—
178C23—212no—
179C23—416no—
180C23—412no—
181C23—1228no—
182C23—12224no—
183C23—12224no—
184C23—24416no—
185C23—24448no—
186C23—24448no—
187C23C22×C22⋊D6112yes—
188C23#14519823112yes—
189C23C24×D6111yescharacteristic
190D4—24496no—
191D4—24464no—
192C2×C4—224no—
193C2×C4—212no—
194C2×C4—26no—
195C2×C4—212no—
196C2×C4—412no—
197C2×C4—12224no—
198C2×C4—24416no—
199C2×C4C22×C22⋊D61112yes—
200C22—46no—
201C22—48no—
202C22—88no—
203C22—48832no—
204C22—24424no—
205C22—2448no—
206C22—24424no—
207C22#16840326112yes—
208C22#16840503116yes—
209C22C23×C22⋊D6111yescharacteristic
210C22C2×#1451982316yes—
211C4—212no—
212C4—46no—
213C4—2448no—
214C2—82no—
215C2—4888no—
216C2#2145385918yesdirect product factor
217C2C2×#1684032613yes—
218C2C2×#1684050314yes—
subgroupG/H
centerC24C23×D61
commutator, cosocleC23×C61C25
FrattiniC23C24×D61
FittingC22×#126×C61C2
socleC24×C61C24
hypercenterC22×#126D61
hypocenterC61C2×#2818
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C22×#14519774C2semidirect
2. #21453859C2direct (N × H)
3. C2×#16839679C2semidirect
4. C2×#16839671C2semidirect
5. C2×#16839630C2semidirect
6. C22×#126×C61C2semidirect
10. C2×#14519774C22semidirect
12. #16839679C22semidirect
13. #16839671C22semidirect
14. #16839630C22semidirect
17. C2×#126×C61C22semidirect
40. C2×C42×C61D4semidirect
65. C42×C61C2×D4semidirect
106. C22×#126D61semidirect
112. C2×C61#2818semidirect
121. C2×#126C2×D61semidirect
130. C61C2×#2818semidirect
149. C2×C42C22⋊D61semidirect
176. C42C2×C22⋊D61semidirect
216. C2#21453859direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145197742
#214538592
C2×#168396792
C2×#168396712
C2×#168396302
C22×#126×C612
C2×#281861