Finite Group Databasebeta

← C22×C61×#427 #29563536

Subgroups — C22×C61×#427 #29563536

Order 15616.

3006 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C24×C8×C61C211yesmaximal, characteristic
2C22×#117×C61C212yesmaximal
3C2×C61×#427C2124yesmaximal, direct product factor
4C22×#107×C61C212yesmaximal, complemented
5C22×#106×C61C212yesmaximal, complemented
6C24×C4×C61C2211yescharacteristic
7C23×C8×C61C4116yescomplemented
8C23×C8×C61C2211yescharacteristic
9C23×C8×C61C2211yescharacteristic
10C23×C8×C61C22112yes—
11C2×#117×C61—212no—
12C2×#117×C61C22124yes—
13C22×C4⋊C4×C61C2214yes—
14C22×C22⋊C4×C61—22no—
15C22×C42×C61—22no—
16C61×#427C22164yesdirect product factor
17C2×#107×C61C22124yescomplemented
18C2×#106×C61C22124yescomplemented
19C25×C61D411yescharacteristic
20C23×C4×C61—24no—
21C23×C4×C61C2×C418yes—
22C23×C4×C61D416yes—
23C23×C4×C61D411yescharacteristic
24C23×C4×C61Q818yes—
25C23×C4×C61C2311yescharacteristic
26C23×C4×C61C2316yes—
27C22×C8×C61C2×C418yes—
28C22×C8×C61C2×C418yes—
29C22×C8×C61C2×C4196yescomplemented
30C22×C8×C61C2316yes—
31C22×C8×C61C2316yes—
32C22×C8×C61C23116yes—
33#117×C61—248no—
34#117×C61C23132yes—
35C2×C4⋊C4×C61—212no—
36C2×C4⋊C4×C61C23124yes—
37C2×C22⋊C4×C61—224no—
38C2×C42×C61—224no—
39#107×C61C23132yescomplemented
40#106×C61C23132yescomplemented
41C24×C61—28no—
42C24×C61C4⋊C418yes—
43C24×C61C2×D416yes—
44C24×C61C2×D411yescharacteristic
45C22×C4×C61—224no—
46C22×C4×C61—416no—
47C22×C4×C61C4⋊C4148yes—
48C22×C4×C61C4⋊C418yes—
49C22×C4×C61C22×C414yes—
50C22×C4×C61C22×C4124yes—
51C22×C4×C61C2×D416yes—
52C22×C4×C61C2×D413yes—
53C22×C4×C61C2×D4112yes—
54C22×C4×C61C2×Q8124yes—
55C22×C4×C61C2×Q814yes—
56C22×C4×C61C2413yes—
57C22×C4×C61C2414yes—
58C2×C8×C61C22×C4124yes—
59C2×C8×C61C22×C4124yes—
60C2×C8×C61C22×C4164yescomplemented
61C2×C8×C61C2414yes—
62C2×C8×C61C2414yes—
63C4⋊C4×C61—224no—
64C4⋊C4×C61C24116yes—
65C22⋊C4×C61—232no—
66C42×C61—232no—
67C23×C61—224no—
68C23×C61—232no—
69C23×C61C2×C4⋊C4124yes—
70C23×C61C2×C4⋊C414yes—
71C23×C61#13518yes—
72C23×C61C22×D414yes—
73C23×C61C22×D413yes—
74C2×C4×C61—216no—
75C2×C4×C61—448no—
76C2×C4×C61C2×C4⋊C4148yes—
77C2×C4×C61C2×C4⋊C4112yes—
78C2×C4×C61C2×C4⋊C4124yes—
79C2×C4×C61C23×C416yes—
80C2×C4×C61C23×C418yes—
81C2×C4×C61C22×D414yes—
82C2×C4×C61C22×D413yes—
83C2×C4×C61C22×Q818yes—
84C2×C4×C61C22×Q816yes—
85C2×C4×C61C2511yescharacteristic
86C8×C61C23×C418yes—
87C8×C61C23×C418yes—
88C22×#427C6111yesmaximal, characteristic, direct product factor, Sylow
89C22×C61—248no—
90C22×C61—28no—
91C22×C61#427116yesdirect product factor
92C22×C61C22×C4⋊C418yes—
93C22×C61C22×C4⋊C416yes—
94C22×C61C2×#135112yes—
95C22×C61C23×D411yescharacteristic
96C4×C61—416no—
97C4×C61C22×C4⋊C416yes—
98C4×C61C22×C4⋊C418yes—
99C4×C61C24×C411yescharacteristic
100C4×C61C23×Q811yescharacteristic
101C24×C8C2×C6111yescharacteristic
102C22×#117C2×C6112yes—
103C2×#427C2×C61124yesdirect product factor
104C22×#107C2×C6112yescomplemented
105C22×#106C2×C6112yescomplemented
106C2×C61—28no—
107C2×C61C2×#427112yesdirect product factor
108C2×C61C23×C4⋊C411yescharacteristic
109C2×C61C22×#13512yes—
110C24×C4C22×C6111yescharacteristic
111C23×C8C4×C61116yescomplemented
112C23×C8C22×C6111yescharacteristic
113C23×C8C22×C6111yescharacteristic
114C23×C8C22×C61112yes—
115C2×#117—212no—
116C2×#117C22×C61124yes—
117C22×C4⋊C4C22×C6114yes—
118C22×C22⋊C4—22no—
119C22×C42—22no—
120#427C22×C61164yesdirect product factor
121C2×#107C22×C61124yescomplemented
122C2×#106C22×C61124yescomplemented
123C61C22×#42711yescharacteristic, direct product factor, Sylow
124C25D4×C6111yescharacteristic
125C23×C4—24no—
126C23×C4C2×C4×C6118yes—
127C23×C4D4×C6116yes—
128C23×C4D4×C6111yescharacteristic
129C23×C4Q8×C6118yes—
130C23×C4C23×C6111yescharacteristic
131C23×C4C23×C6116yes—
132C22×C8C2×C4×C6118yes—
133C22×C8C2×C4×C6118yes—
134C22×C8C2×C4×C61196yescomplemented
135C22×C8C23×C6116yes—
136C22×C8C23×C6116yes—
137C22×C8C23×C61116yes—
138#117—248no—
139#117C23×C61132yes—
140C2×C4⋊C4—212no—
141C2×C4⋊C4C23×C61124yes—
142C2×C22⋊C4—224no—
143C2×C42—224no—
144#107C23×C61132yescomplemented
145#106C23×C61132yescomplemented
146C24—28no—
147C24C4⋊C4×C6118yes—
148C24C2×D4×C6116yes—
149C24C2×D4×C6111yescharacteristic
150C22×C4—224no—
151C22×C4—416no—
152C22×C4C4⋊C4×C61148yes—
153C22×C4C4⋊C4×C6118yes—
154C22×C4C22×C4×C6114yes—
155C22×C4C22×C4×C61124yes—
156C22×C4C2×D4×C6116yes—
157C22×C4C2×D4×C6113yes—
158C22×C4C2×D4×C61112yes—
159C22×C4C2×Q8×C61124yes—
160C22×C4C2×Q8×C6114yes—
161C22×C4C24×C6113yes—
162C22×C4C24×C6114yes—
163C2×C8C22×C4×C61124yes—
164C2×C8C22×C4×C61124yes—
165C2×C8C22×C4×C61164yescomplemented
166C2×C8C24×C6114yes—
167C2×C8C24×C6114yes—
168C4⋊C4—224no—
169C4⋊C4C24×C61116yes—
170C22⋊C4—232no—
171C42—232no—
172C23—224no—
173C23—232no—
174C23C2×C4⋊C4×C61124yes—
175C23C2×C4⋊C4×C6114yes—
176C23#135×C6118yes—
177C23C22×D4×C6114yes—
178C23C22×D4×C6113yes—
179C2×C4—216no—
180C2×C4—448no—
181C2×C4C2×C4⋊C4×C61148yes—
182C2×C4C2×C4⋊C4×C61112yes—
183C2×C4C2×C4⋊C4×C61124yes—
184C2×C4C23×C4×C6116yes—
185C2×C4C23×C4×C6118yes—
186C2×C4C22×D4×C6114yes—
187C2×C4C22×D4×C6113yes—
188C2×C4C22×Q8×C6118yes—
189C2×C4C22×Q8×C6116yes—
190C2×C4C25×C6111yescharacteristic
191C8C23×C4×C6118yes—
192C8C23×C4×C6118yes—
193C22—28no—
194C22—248no—
195C22C61×#427116yesdirect product factor
196C22C22×C4⋊C4×C6118yes—
197C22C22×C4⋊C4×C6116yes—
198C22C2×#135×C61112yes—
199C22C23×D4×C6111yescharacteristic
200C4—416no—
201C4C22×C4⋊C4×C6116yes—
202C4C22×C4⋊C4×C6118yes—
203C4C24×C4×C6111yescharacteristic
204C4C23×Q8×C6111yescharacteristic
205C2—28no—
206C2C2×C61×#427112yesdirect product factor
207C2C23×C4⋊C4×C6111yescharacteristic
208C2C22×#135×C6112yes—
subgroupG/H
centerC23×C4×C61D4
commutatorC4C24×C4×C61
Frattini, cosocleC2×C4C25×C61
socleC24×C61C2×D4
prime radicalC25×C61D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C4×C61 ◁ C24×C4×C61 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C61×#427C2direct (N × H)
4. C22×#107×C61C2semidirect
5. C22×#106×C61C2semidirect
7. C23×C8×C61C4semidirect
16. C61×#427C22direct (N × H)
17. C2×#107×C61C22semidirect
18. C2×#106×C61C22semidirect
29. C22×C8×C61C2×C4semidirect
39. #107×C61C23semidirect
40. #106×C61C23semidirect
60. C2×C8×C61C22×C4semidirect
88. C22×#427C61direct (N × H)
91. C22×C61#427direct (N × H)
103. C2×#427C2×C61direct (N × H)
104. C22×#107C2×C61semidirect
105. C22×#106C2×C61semidirect
107. C2×C61C2×#427direct (N × H)
111. C23×C8C4×C61semidirect
120. #427C22×C61direct (N × H)
121. C2×#107C22×C61semidirect
122. C2×#106C22×C61semidirect
123. C61C22×#427direct (N × H)
134. C22×C8C2×C4×C61semidirect
144. #107C23×C61semidirect
145. #106C23×C61semidirect
165. C2×C8C22×C4×C61semidirect
195. C22C61×#427direct (N × H)
206. C2C2×C61×#427direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C8×C612
C22×#117×C612
C2×C61×#4272
C22×#107×C612
C22×#106×C612
C22×#42761