Finite Group Databasebeta

← #30625082

Subgroups — #30625082

Order 15616.

1800 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462365C213yesmaximal
2#21462353C2112yesmaximal
3#21462319C2112yesmaximal
4#125×#6923C213yesmaximal
5#564×D61C211yesmaximal, characteristic
6#16840713C2216yes—
7#16840713C22112yes—
8#16840713C22112yes—
9#16840711C22112yes—
10#16840711C22112yes—
11#16840711C22112yes—
12#16840711C22112yes—
13#16840711C22112yes—
14#16840707C2214yes—
15#16840706C2213yes—
16C4⋊C4×#6923C2216yes—
17C4⋊C4×#6923C22112yes—
18C4⋊C4×#6923C22112yes—
19C4×C4⋊#6923C22112yes—
20C42×#6923C2211yescharacteristic
21C61⋊#564C413yescomplemented
22C61⋊#564C4112yescomplemented
23#125×D61C2213yes—
24#125×D61C22112yes—
25C61×#564C411yescharacteristic, complemented
26C2×C4⋊#6923C23112yes—
27C2×C4⋊#6923C2316yes—
28C2×C4⋊#6923C23112yes—
29C2×C4×#6923C2313yes—
30C2×C4×#6923C23112yes—
31C2×C4×#6923C23112yes—
32C2×C4×#6923C2313yes—
33(C2×#6923)⋊C4C23112yes—
34(C2×#6923)⋊C4C23112yes—
35(C2×#6923)⋊C4C23112yes—
36(C2×#6923)⋊C4C23112yes—
37(C2×#6923)⋊C4C23112yes—
38#14519769C2×C413yes—
39#14519769C2×C4112yes—
40#14519768C2×C416yes—
41#14519768C2×C4112yes—
42#14519768C2×C4112yes—
43#14519767C2×C4112yes—
44#14519767C2×C4112yes—
45#14519767C2×C4112yes—
46#14519767C2×C4112yes—
47#14519767C2×C4112yes—
48C4⋊C4×D61C23112yes—
49C4⋊C4×D61C23112yes—
50C4⋊C4×D61C2316yes—
51C42×D61C2311yescharacteristic
52C42×D61C2314yes—
53#125×C61C2×C413yes—
54#125×C61C2×C4112yes—
55C22×#6923C2411yescharacteristic
56C22×#6923C24112yes—
57C22×#6923C2413yes—
58C2×C4×D61C2413yes—
59C2×C4×D61C24112yes—
60C4⋊#6923—224no—
61C4⋊#6923—412no—
62C4⋊#6923—46no—
63C4×#6923—26no—
64C4×#6923—412no—
65C4×#6923—412no—
66C4×#6923—43no—
67C4⋊C4×C61C22×C4112yes—
68C4⋊C4×C61C22×C416yes—
69C4⋊C4×C61C22×C4112yes—
70C42×C61C22×C411yescharacteristic
71C42×C61C22×C414yes—
72C4⋊Dic61C22×C4112yes—
73C4⋊Dic61C22×C4112yes—
74C4⋊Dic61C22×C416yes—
75#11758196C22×C4112yes—
76#11758196C22×C4112yes—
77#11758196C22×C4112yes—
78#11758196C22×C4112yes—
79#11758196C22×C4112yes—
80C4×Dic61C22×C413yes—
81C4×Dic61C22×C4112yes—
82C22×D61C2511yescharacteristic
83C2×#6923—26no—
84C2×#6923—23no—
85C2×#6923—212no—
86C2×#6923—212no—
87C2×#6923—212no—
88C2×#69232−1+416yes—
89C2×C4×C61C23×C413yes—
90C2×C4×C61C23×C4112yes—
91C2×Dic61C23×C413yes—
92C2×Dic61C23×C4112yes—
93C4×D61—23no—
94C4×D61—212no—
95C4×D61—212no—
96C4×D61—23no—
97#34526—611nomaximal, Sylow
98C22×C61C24×C411yescharacteristic
99C2×D61C2×2−1+413yes—
100C2×D61#58513yes—
101#6923—26no—
102#6923—412no—
103#6923—41no—
104C4×C61—23no—
105C4×C61—212no—
106Dic61—23no—
107Dic61—212no—
108C2×#564—611no—
109#2382—613no—
110#2370—6112no—
111#2336—6112no—
112C4×#125—613no—
113C2×C61C4×2−1+413yes—
114D61#355313yes—
115D61#355611yescharacteristic
116#564—6112no—
117#564—613no—
118#564#692311yescharacteristic, complemented
119C2×#125—613no—
120C2×#125—6112no—
121#389—616no—
122#389—6112no—
123#389—6112no—
124#387—6112no—
125#387—6112no—
126#387—6112no—
127#387—6112no—
128#387—6112no—
129#383—614no—
130#382—613no—
131C4×C4⋊C4—616no—
132C4×C4⋊C4—6112no—
133C4×C4⋊C4—6112no—
134C4×C4⋊C4—6112no—
135C43—611no—
136C61#3452611yescharacteristic, complemented, Sylow
137#125—6112no—
138#125—6112no—
139#125—6112no—
140#125—6112no—
141#125—6112no—
142#125—616no—
143#125—6112no—
144#125—6112no—
145#125—613no—
146#125—6112no—
147#125C2×#692313yes—
148#125C2×#6923112yes—
149C2×C4⋊C4—616no—
150C2×C4⋊C4—6112no—
151C2×C4⋊C4—6112no—
152C2×C4⋊C4—6112no—
153C2×C4⋊C4—616no—
154C2×C4⋊C4—6112no—
155C2×C42—611no—
156C2×C42—614no—
157C2×C42—613no—
158C2×C42—613no—
159C2×C42—6112no—
160C2×C42—6112no—
161#95—6112no—
162#95—6112no—
163#95—6112no—
164#95—6112no—
165#95—6112no—
166C22×C4—613no—
167C22×C4—613no—
168C22×C4—6112no—
169C22×C4—611no—
170C22×C4—6112no—
171C4⋊C4—6112no—
172C4⋊C4—6112no—
173C4⋊C4—6112no—
174C4⋊C4—6112no—
175C4⋊C4—6112no—
176C4⋊C4—616no—
177C4⋊C4—6112no—
178C4⋊C4—6112no—
179C4⋊C4—12224no—
180C4⋊C4—24412no—
181C4⋊C4—2446no—
182C4⋊C4C22×#692316yes—
183C4⋊C4C22×#6923112yes—
184C4⋊C4C22×#6923112yes—
185C42—613no—
186C42—6112no—
187C42—1226no—
188C42—24412no—
189C42—24412no—
190C42—2443no—
191C42C22×#692311yescharacteristic
192C42C22×#692314yes—
193C23—611no—
194C2×C4—616no—
195C2×C4—613no—
196C2×C4—6112no—
197C2×C4—1223no—
198C2×C4—12212no—
199C2×C4—1223no—
200C2×C4—12212no—
201C2×C4—1226no—
202C2×C4—1223no—
203C2×C4—12212no—
204C2×C4—12212no—
205C2×C4—12212no—
206C2×C4C23×#692313yes—
207C2×C4C23×#6923112yes—
208C22—613no—
209C22—613no—
210C22C24×#692311yescharacteristic
211C4—23no—
212C4—212no—
213C4—1226no—
214C4—1223no—
215C4—12212no—
216C4—24412no—
217C4—2441no—
218C2—613no—
219C2—611no—
220C22−1+4×#692313yes—
subgroupG/H
center, FrattiniC22C24×#6923
commutator, socleC22×C61C24×C4
FittingC61×#564C4
prime radical, cosocleC22×D61C25
hypercenter#564#6923
hypocenterC61#34526
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ #564

Complements

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

NH ≅ G/Ntype
21. C61⋊#564C4semidirect
22. C61⋊#564C4semidirect
25. C61×#564C4semidirect
118. #564#6923semidirect
136. C61#34526semidirect

Maximal subgroups

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

subgroupindex
#214623652
#214623532
#214623192
#125×#69232
#564×D612
#3452661