Finite Group Databasebeta

← C2×C61×#2371 #29537579

Subgroups — C2×C61×#2371 #29537579

Order 15616.

2422 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#125×C61C211yesmaximal, characteristic
2C61×#2371C2116yesmaximal, direct product factor
3C2×C61×#387C214yesmaximal
4C2×C61×#387C214yesmaximal
5C2×C4×C4⋊C4×C61C214yesmaximal
6C2×C61×#376C212yesmaximal
7C2×#125×C61C4116yescomplemented
8C2×#125×C61C2218yes—
9C22×C4⋊C4×C61C2214yes—
10C22×C4⋊C4×C61C2214yes—
11C22×C4⋊C4×C61C2212yes—
12C22×C42×C61C2214yes—
13C22×C42×C61C2214yes—
14C22×C42×C61C2211yescharacteristic
15C61×#387C22132yes—
16C61×#387C22132yes—
17C4×C4⋊C4×C61C22132yes—
18C61×#376C22116yes—
19C2×#95×C61C2212yes—
20C2×#95×C61C2212yes—
21C2×#95×C61C2214yes—
22C2×#95×C61C2214yes—
23C2×#95×C61C2214yes—
24C23×C4×C61C2314yes—
25C23×C4×C61C2311yescharacteristic
26C23×C4×C61C2312yes—
27C23×C4×C61C2314yes—
28C23×C4×C61C2314yes—
29#125×C61C2×C4164yescomplemented
30C2×C4⋊C4×C61—216no—
31C2×C4⋊C4×C61C2×C4132yes—
32C2×C4⋊C4×C61C2×C4116yes—
33C2×C4⋊C4×C61C23116yes—
34C2×C4⋊C4×C61C23116yes—
35C2×C4⋊C4×C61C2318yes—
36C2×C42×C61—216no—
37C2×C42×C61—216no—
38C2×C42×C61C2×C418yes—
39C2×C42×C61C23116yes—
40C2×C42×C61C23116yes—
41C2×C42×C61C2314yes—
42#95×C61C2318yes—
43#95×C61C2318yes—
44#95×C61C23116yes—
45#95×C61C23116yes—
46#95×C61C23116yes—
47C24×C61C2411yescharacteristic
48C22×C4×C61—28no—
49C22×C4×C61—28no—
50C22×C4×C61—28no—
51C22×C4×C61—28no—
52C22×C4×C61—24no—
53C22×C4×C61—28no—
54C22×C4×C61—28no—
55C22×C4×C61—28no—
56C22×C4×C61—28no—
57C22×C4×C61C22×C418yes—
58C22×C4×C61C22×C414yes—
59C22×C4×C61C22×C4116yes—
60C22×C4×C61#41116yes—
61C22×C4×C61C2418yes—
62C22×C4×C61C2412yes—
63C22×C4×C61C2414yes—
64C22×C4×C61C2418yes—
65C22×C4×C61C2418yes—
66C4⋊C4×C61—232no—
67C4⋊C4×C61C22×C4164yes—
68C4⋊C4×C61C22×C4132yes—
69C42×C61—232no—
70C42×C61—232no—
71C42×C61C22×C4116yes—
72C23×C61C23×C412yes—
73C23×C61C2×#4118yes—
74C23×C612−1+414yes—
75C23×C61C2511yescharacteristic
76C2×C4×C61—216no—
77C2×C4×C61—232no—
78C2×C4×C61—28no—
79C2×C4×C61—232no—
80C2×C4×C61—28no—
81C2×C4×C61—28no—
82C2×C4×C61—28no—
83C2×C4×C61—28no—
84C2×C4×C61—24no—
85C2×C4×C61—28no—
86C2×C4×C61—28no—
87C2×C4×C61—28no—
88C2×C4×C61—28no—
89C2×C4×C61—416no—
90C2×C4×C61C23×C418yes—
91C2×C4×C61C23×C414yes—
92C2×C4×C61C23×C4116yes—
93C2×C4×C61C2×#41116yes—
94C2×#2371C6111yesmaximal, characteristic, direct product factor, Sylow
95C22×C61C4×#4118yes—
96C22×C61#51914yes—
97C22×C61#552116yes—
98C22×C61C24×C411yescharacteristic
99C22×C61C22×#4114yes—
100C22×C61C2×2−1+412yes—
101C4×C61—28no—
102C4×C61—216no—
103C4×C61—24no—
104C4×C61—216no—
105C4×C61—48no—
106C22×#125C2×C6111yescharacteristic
107#2371C2×C61116yesdirect product factor
108C2×#387C2×C6114yes—
109C2×#387C2×C6114yes—
110C2×C4×C4⋊C4C2×C6114yes—
111C2×#376C2×C6112yes—
112C2×C61#237118yesdirect product factor
113C2×C61C2×C4×#4112yes—
114C2×C61C2×#51911yescharacteristic
115C2×C61C2×#55214yes—
116C2×#125C4×C61116yescomplemented
117C2×#125C22×C6118yes—
118C22×C4⋊C4C22×C6114yes—
119C22×C4⋊C4C22×C6114yes—
120C22×C4⋊C4C22×C6112yes—
121C22×C42C22×C6114yes—
122C22×C42C22×C6114yes—
123C22×C42C22×C6111yescharacteristic
124#387C22×C61132yes—
125#387C22×C61132yes—
126C4×C4⋊C4C22×C61132yes—
127#376C22×C61116yes—
128C2×#95C22×C6112yes—
129C2×#95C22×C6112yes—
130C2×#95C22×C6114yes—
131C2×#95C22×C6114yes—
132C2×#95C22×C6114yes—
133C61C2×#237111yescharacteristic, direct product factor, Sylow
134C23×C4C23×C6114yes—
135C23×C4C23×C6114yes—
136C23×C4C23×C6111yescharacteristic
137C23×C4C23×C6112yes—
138C23×C4C23×C6114yes—
139#125C2×C4×C61164yescomplemented
140C2×C4⋊C4—216no—
141C2×C4⋊C4C2×C4×C61132yes—
142C2×C4⋊C4C2×C4×C61116yes—
143C2×C4⋊C4C23×C61116yes—
144C2×C4⋊C4C23×C61116yes—
145C2×C4⋊C4C23×C6118yes—
146C2×C42—216no—
147C2×C42—216no—
148C2×C42C2×C4×C6118yes—
149C2×C42C23×C61116yes—
150C2×C42C23×C61116yes—
151C2×C42C23×C6114yes—
152#95C23×C6118yes—
153#95C23×C6118yes—
154#95C23×C61116yes—
155#95C23×C61116yes—
156#95C23×C61116yes—
157C24C24×C6111yescharacteristic
158C22×C4—28no—
159C22×C4—28no—
160C22×C4—28no—
161C22×C4—28no—
162C22×C4—28no—
163C22×C4—24no—
164C22×C4—28no—
165C22×C4—28no—
166C22×C4—28no—
167C22×C4C22×C4×C6118yes—
168C22×C4C22×C4×C6114yes—
169C22×C4C22×C4×C61116yes—
170C22×C4#41×C61116yes—
171C22×C4C24×C6118yes—
172C22×C4C24×C6112yes—
173C22×C4C24×C6114yes—
174C22×C4C24×C6118yes—
175C22×C4C24×C6118yes—
176C4⋊C4—232no—
177C4⋊C4C22×C4×C61164yes—
178C4⋊C4C22×C4×C61132yes—
179C42—232no—
180C42—232no—
181C42C22×C4×C61116yes—
182C23C23×C4×C6112yes—
183C23C2×#41×C6118yes—
184C232−1+4×C6114yes—
185C23C25×C6111yescharacteristic
186C2×C4—232no—
187C2×C4—28no—
188C2×C4—28no—
189C2×C4—28no—
190C2×C4—28no—
191C2×C4—24no—
192C2×C4—28no—
193C2×C4—216no—
194C2×C4—232no—
195C2×C4—28no—
196C2×C4—28no—
197C2×C4—28no—
198C2×C4—28no—
199C2×C4—416no—
200C2×C4C23×C4×C6118yes—
201C2×C4C23×C4×C6114yes—
202C2×C4C23×C4×C61116yes—
203C2×C4C2×#41×C61116yes—
204C22C4×#41×C6118yes—
205C22C61×#51914yes—
206C22C61×#552116yes—
207C22C24×C4×C6111yescharacteristic
208C22C22×#41×C6114yes—
209C22C2×2−1+4×C6112yes—
210C4—28no—
211C4—216no—
212C4—24no—
213C4—216no—
214C4—48no—
215C2C61×#237118yesdirect product factor
216C2C2×C4×#41×C6112yes—
217C2C2×C61×#51911yescharacteristic
218C2C2×C61×#55214yes—
subgroupG/H
center, socle, prime radicalC24×C61C24
commutatorC22C24×C4×C61
Frattini, cosocleC23C25×C61
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24×C61 ◁ G

Complements

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

NH ≅ G/Ntype
2. C61×#2371C2direct (N × H)
7. C2×#125×C61C4semidirect
29. #125×C61C2×C4semidirect
94. C2×#2371C61direct (N × H)
107. #2371C2×C61direct (N × H)
112. C2×C61#2371direct (N × H)
116. C2×#125C4×C61semidirect
133. C61C2×#2371direct (N × H)
139. #125C2×C4×C61semidirect
215. C2C61×#2371direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#125×C612
C61×#23712
C2×C61×#3872
C2×C4×C4⋊C4×C612
C2×C61×#3762
C2×#237161