Finite Group Databasebeta

← #34031053

Subgroups — #34031053

Order 18688.

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#22626477C213yesmaximal
2#22626465C2112yesmaximal
3#22626431C2112yesmaximal
4#125×#64408C213yesmaximal
5#564×D73C211yesmaximal, characteristic
6#17989538C2216yes—
7#17989538C22112yes—
8#17989538C22112yes—
9#17989536C22112yes—
10#17989536C22112yes—
11#17989536C22112yes—
12#17989536C22112yes—
13#17989536C22112yes—
14#17989532C2214yes—
15#17989531C2213yes—
16C4⋊C4×#64408C2216yes—
17C4⋊C4×#64408C22112yes—
18C4⋊C4×#64408C22112yes—
19C4×C4⋊#64408C22112yes—
20C42×#64408C2211yescharacteristic
21C73⋊#564C413yescomplemented
22C73⋊#564C4112yescomplemented
23#125×D73C2213yes—
24#125×D73C22112yes—
25#564×C73C411yescharacteristic, complemented
26C2×C4⋊#64408C23112yes—
27C2×C4⋊#64408C23112yes—
28C2×C4⋊#64408C2316yes—
29C2×C4×#64408C2313yes—
30C2×C4×#64408C2313yes—
31C2×C4×#64408C23112yes—
32C2×C4×#64408C23112yes—
33(C2×#64408)⋊C4C23112yes—
34(C2×#64408)⋊C4C23112yes—
35(C2×#64408)⋊C4C23112yes—
36(C2×#64408)⋊C4C23112yes—
37(C2×#64408)⋊C4C23112yes—
38#14556583C2×C413yes—
39#14556583C2×C4112yes—
40#14556582C2×C416yes—
41#14556582C2×C4112yes—
42#14556582C2×C4112yes—
43#14556581C2×C4112yes—
44#14556581C2×C4112yes—
45#14556581C2×C4112yes—
46#14556581C2×C4112yes—
47#14556581C2×C4112yes—
48C4⋊C4×D73C2316yes—
49C4⋊C4×D73C23112yes—
50C4⋊C4×D73C23112yes—
51C42×D73C2311yescharacteristic
52C42×D73C2314yes—
53#125×C73C2×C413yes—
54#125×C73C2×C4112yes—
55C22×#64408C2411yescharacteristic
56C22×#64408C24112yes—
57C22×#64408C2413yes—
58C4⋊#64408—224no—
59C4⋊#64408—412no—
60C4⋊#64408—46no—
61C4×#64408—26no—
62C4×#64408—412no—
63C4×#64408—412no—
64C4×#64408—43no—
65C2×C4×D73C2413yes—
66C2×C4×D73C24112yes—
67C4⋊Dic73C22×C416yes—
68C4⋊Dic73C22×C4112yes—
69C4⋊Dic73C22×C4112yes—
70#11924007C22×C4112yes—
71#11924007C22×C4112yes—
72#11924007C22×C4112yes—
73#11924007C22×C4112yes—
74#11924007C22×C4112yes—
75C4×Dic73C22×C413yes—
76C4×Dic73C22×C4112yes—
77C4⋊C4×C73C22×C416yes—
78C4⋊C4×C73C22×C4112yes—
79C4⋊C4×C73C22×C4112yes—
80C42×C73C22×C411yescharacteristic
81C42×C73C22×C414yes—
82C22×D73C2511yescharacteristic
83C2×#64408—26no—
84C2×#64408—23no—
85C2×#64408—212no—
86C2×#64408—212no—
87C2×#64408—212no—
88C2×#644082−1+416yes—
89C2×C4×C73C23×C413yes—
90C2×C4×C73C23×C4112yes—
91C2×Dic73C23×C413yes—
92C2×Dic73C23×C4112yes—
93C4×D73—23no—
94C4×D73—212no—
95C4×D73—212no—
96C4×D73—23no—
97C22×C73C24×C411yescharacteristic
98C2×D73C2×2−1+413yes—
99C2×D73#58513yes—
100#64408—26no—
101#64408—412no—
102#64408—41no—
103C4×C73—23no—
104C4×C73—212no—
105Dic73—23no—
106Dic73—212no—
107#34526—731nomaximal, Sylow
108C2×C73C4×2−1+413yes—
109D73#355313yes—
110D73#355611yescharacteristic
111C2×#564—731no—
112#2382—733no—
113#2370—7312no—
114#2336—7312no—
115C4×#125—733no—
116C73#3452611yescharacteristic, complemented, Sylow
117#564—7312no—
118#564—733no—
119#564#6440811yescharacteristic, complemented
120C2×#125—733no—
121C2×#125—7312no—
122#389—736no—
123#389—7312no—
124#389—7312no—
125#387—7312no—
126#387—7312no—
127#387—7312no—
128#387—7312no—
129#387—7312no—
130#383—734no—
131#382—733no—
132C4×C4⋊C4—736no—
133C4×C4⋊C4—7312no—
134C4×C4⋊C4—7312no—
135C4×C4⋊C4—7312no—
136C43—731no—
137#125—7312no—
138#125—7312no—
139#125—7312no—
140#125—7312no—
141#125—7312no—
142#125—736no—
143#125—7312no—
144#125—7312no—
145#125—733no—
146#125—7312no—
147#125C2×#6440813yes—
148#125C2×#64408112yes—
149C2×C4⋊C4—736no—
150C2×C4⋊C4—7312no—
151C2×C4⋊C4—736no—
152C2×C4⋊C4—7312no—
153C2×C4⋊C4—7312no—
154C2×C4⋊C4—7312no—
155C2×C42—733no—
156C2×C42—7312no—
157C2×C42—7312no—
158C2×C42—731no—
159C2×C42—734no—
160C2×C42—733no—
161#95—7312no—
162#95—7312no—
163#95—7312no—
164#95—7312no—
165#95—7312no—
166C22×C4—733no—
167C22×C4—7312no—
168C22×C4—733no—
169C22×C4—731no—
170C22×C4—7312no—
171C4⋊C4—29212no—
172C4⋊C4—2926no—
173C4⋊C4—7312no—
174C4⋊C4—7312no—
175C4⋊C4—7312no—
176C4⋊C4—7312no—
177C4⋊C4—7312no—
178C4⋊C4—736no—
179C4⋊C4—7312no—
180C4⋊C4—7312no—
181C4⋊C4—14624no—
182C4⋊C4C22×#6440816yes—
183C4⋊C4C22×#64408112yes—
184C4⋊C4C22×#64408112yes—
185C42—29212no—
186C42—29212no—
187C42—2923no—
188C42—733no—
189C42—7312no—
190C42—1466no—
191C42C22×#6440811yescharacteristic
192C42C22×#6440814yes—
193C23—731no—
194C2×C4—736no—
195C2×C4—7312no—
196C2×C4—733no—
197C2×C4—14612no—
198C2×C4—1463no—
199C2×C4—1463no—
200C2×C4—14612no—
201C2×C4—14612no—
202C2×C4—14612no—
203C2×C4—1463no—
204C2×C4—14612no—
205C2×C4—1466no—
206C2×C4C23×#64408112yes—
207C2×C4C23×#6440813yes—
208C22—733no—
209C22—733no—
210C22C24×#6440811yescharacteristic
211C4—212no—
212C4—23no—
213C4—29212no—
214C4—2921no—
215C4—1466no—
216C4—14612no—
217C4—1463no—
218C2—733no—
219C2—731no—
220C22−1+4×#6440813yes—
subgroupG/H
center, FrattiniC22C24×#64408
commutator, socleC22×C73C24×C4
Fitting#564×C73C4
prime radical, cosocleC22×D73C25
hypercenter#564#64408
hypocenterC73#34526
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ #564

Complements

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

NH ≅ G/Ntype
21. C73⋊#564C4semidirect
22. C73⋊#564C4semidirect
25. #564×C73C4semidirect
116. C73#34526semidirect
119. #564#64408semidirect

Maximal subgroups

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

subgroupindex
#226264772
#226264652
#226264312
#125×#644082
#564×D732
#3452673